0;\;x(0) = \xi, $$ describes the position at time $t$ of a material point starting at $\xi$ at time 0; here ${D\over Dt}$ is the partial derivative in $t$ when $\xi$ is fixed, but we reserve ${\partial \over \partial t}$ to denote the partial derivative in $t$ when $x$ is fixed, i.e. in the Eulerian point of view; ${D\over Dt}$ is called the material derivative. It seems natural to ask for uniqueness of a solution, and the classical condition is the local version of the following global L{\eightrm IPSCHITZ} condition $$ |u(x,t)-u(y,t)|\le \lambda(t)|x-y|\hbox{ for all }x,y\hbox{ and }t\in(0,T), $$ with $\lambda\in L^{1}(0,T)$. There is a small improvement due to O{\eightrm SGOOD}, which gives uniqueness when one only assumes that $$ |u(x,t)-u(y,t)|\le \omega(|x-y|) \hbox{ for all }x,y\hbox{ and the modulus of uniform continuity }\omega\hbox{ satisfies }\int_{0}^{1} {ds\over \omega(s)} = +\infty. $$ This gives $x = \Phi(\xi,t)$, and with $u$ of class $C^{1}$ in $(x,t)$, it is not difficult to prove that $\xi\mapsto \Phi(\xi,t)$ is a local diffeomorphism and that the Jacobian matrix ${\partial\Phi\over \partial \xi}$ satisfies the linear differential equation $$ {D{\partial\Phi\over \partial \xi}\over Dt} = {\partial u\over \partial x}{\partial\Phi\over \partial \xi}\hbox{ on }(0,T);\; {\partial\Phi\over \partial \xi}(0) = I, $$ so that the Jacobian determinant $det{\partial\Phi\over \partial \xi}$ satisfies $$ {D(det{\partial\Phi\over \partial \xi})\over Dt} = Trace\Bigl( {\partial u\over \partial x} \Bigr)det{\partial\Phi\over \partial \xi} = div(u)\,det{\partial\Phi\over \partial \xi}\hbox{ on }(0,T);\; det{\partial\Phi\over \partial \xi}(0) = 1. $$ As $det{\partial\Phi\over \partial \xi}$ represents the increase in volume by the transformation $\xi\mapsto x(t)$, conservation of mass can then be written as $$ \rho(x(t),t)det{\partial\Phi\over \partial \xi} = \rho(\xi,0)\hbox{ almost everywhere}, $$ and the equation for the Jacobian determinant can therefore be written as $$ {D({\rho(\xi,0)\over \rho})\over Dt} = div(u)\,{\rho(\xi,0)\over \rho}, $$ or equivalently (using $\rho(\xi,0)>0$) $$ {D\rho\over Dt} + \rho\,div(u) = 0, $$ which is the desired equation as $$ {D\over Dt} = {\partial \over \partial t}+\sum_{j = 1}^{N} u_{j}{\partial \over \partial x_{j}}. $$ \medskip It seems then reasonable to admit the derived form of conservation of mass, but the regularity hypotheses invoked for proving it are a little too strong in some situations. For N{\eightrm AVIER}-S{\eightrm TOKES} equation, under the assumption that the fluid is incompressible and that the viscosity is independent of temperature (so that one just forgets about the equation of conservation of energy), one knows uniqueness of the solution in 2 dimensions, and the solution is smooth enough if the initial data are smooth enough. However, uniqueness is not known in 3 dimensions, and it is only for sufficiently small smooth data that one knows that the solution stays smooth; the dissipation of energy by viscosity gives directly that $u\in L^{2}\bigl( 0,T;H^{1}(\Omega;R^{3}) \bigr)$, and by improving an argument of Ciprian F{\eightrm OIAS}, I proved that $u\in L^{1}(0,T;Z)$, with $Z$ a little smaller that $W^{1,3}(\Omega;R^{3})$ (so that $Z\subset C^{0}(\overline{\Omega};R^{3})$, for example), but that is far from the $W^{1,\infty}$ regularity required for deriving the equation. \medskip For incompressible 2-dimensional E{\eightrm ULER} equation, there is a global existence (and maybe uniqueness result) due to T. K{\eightrm ATO}, I believe. The vorticity $\omega = {\partial u_{1}\over \partial x_{2}} - {\partial u_{2}\over \partial x_{1}}$ is transported by the flow, i.e. satisfies ${D\omega\over Dt} = 0$, and therefore if the initial vorticity is in $L^{\infty}(R^{2})$ it stays in this space. At a given time one has then $curl(u)\in L^{\infty}$ and $div(u) = 0$, but that does not implies $u\in W^{1,\infty}(R^{2},R^{2})$ ($L^{\infty}$ is not a good space for singular integrals). \par The singular integrals which often appear in linear systems of partial differential equations with constant coefficients in $R^{N}$ are convolution equations with a kernel which is homogeneous of degree $-N$ and whose integral on the sphere is 0; very often they are polynomials in the R{\eightrm IESZ} operators $R_{j}$, which are the natural generalization to $R^{N}$ of the H{\eightrm ILBERT} transform in $R$. Singular operators act on spaces like $C^{k,\alpha}$ (results proved in the 1920s/30s by G{\eightrm IRAUD}, I believe), and were extended in the 1950s to $L^{p}$ with $1

0$, then one quickly checks that ${dH\over dx} = \delta_{0}$.
A simple ``paradox'' will show that not all formulas extend to distributions: let $u$ be the sign function ($u = -1+2H$), so that ${du\over dx} = 2\delta_{0}$, and notice that $u^{2} = 1$ and $u^{3} = u$, but the formula ${du^{3}\over dx} = 3u^{2}{du\over dx}$ does not hold as the left side is $2\delta_{0}$ while the right side is $6\delta_{0}$; using S{\eightrm OBOLEV} imbedding theorem, the $N$-dimensional formula ${\partial u^{3}\over \partial x_{j}} = 3u^{2}{\partial u\over \partial x_{j}}$ is actually valid on $W^{1,p}(R^{N})$ for $p\ge {3N\over N+2}$.
\medskip
Let us consider now a finite number of point masses moving around, the particle \# i having mass $m_{i}$, position $M_{i}(t)$ and velocity $V_{i}(t) = {dM_{i}\over dt}$ at time $t$.
Conservation of mass is expressed by the fact that $m_{i}$ is independent of $t$; although two particles can go through the same point at some time, there is no exchange of mass between them during these ``collisions''.
The analog of a smooth density $\rho(x,t)$ is the measure $\mu$ defined by
$$
\langle \mu,\varphi \rangle = \sum_{i} \int_{0}^{T} m_{i}\varphi(M_{i}(t),t)\,dt,
$$
and the analog of the mass density at time $t$ is the measure $\mu_{t} = \sum_{i} m_{i}\delta_{M_{i}(t)}$.
We introduce then a {\it momentum} measure $\pi$ by
$$
\langle \pi,\varphi \rangle = \sum_{i} \int_{0}^{T} m_{i}V_{i}(t)\varphi(M_{i}(t),t)\,dt,
$$
and the analog of the momentum density at time $t$ is the measure $\pi_{t} = \sum_{i} m_{i}V_{i}(t)\delta_{M_{i}(t)}$.
Notice that $V_{i}$ are vectors, and therefore $\pi$ is vector valued measure, and its components will be written as $(\pi)_{j}$ for $j = 1,\ldots,N$.
Then conservation of mass implies that
$$
{\partial \mu\over \partial t} + \sum_{j = 1}^{N} {\partial (\pi)_{j}\over \partial x_{j}} = 0.
$$
Indeed for a test function $\varphi\in C^{\infty}_{c}\bigl( \Omega\times(0,T) \bigr)$, it means that $\bigl\langle \mu,{\partial \varphi\over \partial t} \bigr\rangle + \sum_{j = 1}^{N} \bigl\langle (\pi)_{j},{\partial \varphi\over \partial x_{j}} \bigr\rangle = 0$, and this means that $\sum_{i} m_{i}\int_{0}^{T} {\partial \varphi\over \partial t}(M_{i}(t),t)\,dt + \sum_{j = 1}^{N} \sum_{i} m_{i}\int_{0}^{t} (V_{i})_{j}(t)\,{\partial \varphi\over \partial x_{j}}(M_{i}(t),t)\,dt = 0$, which follows from the fact that the coefficient of $m_{i}$ is 0; this coefficient is
$\int_{0}^{T} \bigl[ {\partial \varphi\over \partial t}(M_{i}(t),t)\,dt + \sum_{j = 1}^{N} (V_{i})_{j}(t)\,{\partial \varphi\over \partial x_{j}}(M_{i}(t),t) \bigr]\,dt$, and as the bracket is the total derivative with respect to $t$ of $\varphi(M_{i}(t),t)$ the integral is indeed 0.
\par
If by a limiting process the measure $\mu$ converges vaguely to $\rho(x,t)\,dx\,dt$ and the measure $\pi$ converges vaguely to $p(x,t)\,dx\,dt$, then one obtains the conservation of mass ${\partial \rho \over \partial t}+div(p) = 0$, and $p$ represents the density of momentum, and the macroscopic velocity is defined by $u = {p\over \rho}$.
\par
Notice that the physical quantities, which are additive, are $\rho$ and $p$, and not $u$.
\par
Notice that a particle can leave the domain $\Omega$ without any difficulty in the preceding proof, as it stops being taken into account when it goes out of the support of $\varphi$ and $\varphi(M_{i}(t),t) = 0$ before the particle exits.
However, particles can also enter $\Omega$ without any problem, and the conservation of mass is only expressed inside $\Omega$: R{\eightrm ADON} measures or distributions in $\Omega$ do not see the boundary $\partial\Omega$, and in order to treat boundary conditions one will have to use various S{\eightrm OBOLEV} spaces and check what is the meaning of boundary conditions.
\par
As mentioned, there is no problem having different particles go through the same point with different velocities, and therefore we have not been following an Eulerian point of view, but we have discovered that the velocity $u$ is actually an average, and in cases where the velocity has oscillations, it will be important to understand which are the physical quantities and what equations they satisfy.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
6. Monday January 25.
\medskip
Let us now consider the equations describing the conservation of momentum and the conservation of angular momentum.
\par
E{\eightrm ULER} is credited for writing the equations for an ideal fluid (non viscous), which are
$$
\eqalign{
{\partial \rho\over \partial t}+\sum_{j}{\partial (\rho\,u_{j})\over \partial x_{j}} &= 0\cr
{\partial (\rho\,u_{k})\over \partial t}+\sum_{j}{\partial (\rho\,u_{k}\,u_{j})\over \partial x_{j}} +{\partial p\over \partial x_{k}} &= 0\hbox{ for all }k,}
$$
where $p$ is the pressure.
\par
The equation for the motion of a viscous fluid are attributed to N{\eightrm AVIER} and to S{\eightrm TOKES}, but S{\eightrm TOKES} only considered the linearized problem, and so one uses the term S{\eightrm TOKES} equation when inertial terms are neglected but one uses the term N{\eightrm AVIER}-S{\eightrm TOKES} equation when they are taken into account, although N{\eightrm AVIER} had discovered it alone.
It is unfortunate that so many results are not attributed correctly: the shock conditions expressing the conservation of mass and momentum in gas dynamics, now known after R{\eightrm ANKINE} and H{\eightrm UGONIOT}, were actually first derived in 1848 by S{\eightrm TOKES}, and then rediscovered in 1860 by R{\eightrm IEMANN} for an isentropic gas; S{\eightrm TOKES} is therefore credited for a discovery of N{\eightrm AVIER} but forgotten for some of his discoveries; it could be by his own fault, as when he edited his complete works around 1870 he did not reproduce his derivation of the jump conditions, and he apologized for his mistake, because he had been (wrongly) convinced by Lord K{\eightrm ELVIN} and Lord R{\eightrm AYLEIGH} that his discontinuous solutions were not physical, as they did not conserve energy.
It is a quite amazing fact that such great scientists as S{\eightrm TOKES}, K{\eightrm ELVIN} and R{\eightrm AYLEIGH} did not understand as late as 1870 that heat was a form of energy and that the missing energy had been transformed into heat (C{\eightrm ARNOT} and W{\eightrm ATT} did not need partial differential equations to understand that).
\par
The form of the S{\eightrm TOKES} equation is very similar to that of linearized Elasticity, which C{\eightrm AUCHY} had derived, and that involves something more general than pressure, as he had to introduce stress (what we call now the C{\eightrm AUCHY} stress tensor, which is symmetric, and appears in the Eulerian point of view, while in the Lagrangian point of view the P{\eightrm IOLA}-K{\eightrm IRCHHOFF} stress tensor appears, which is not usually symmetric).
\par
Pressure might be considered an easy concept, but I do not think that A{\eightrm RCHIMEDES} knew that the reason why a body receives an upward force from the water in which one tries to submerse it is that the body receives a stronger force from below than from above because the hydrostatic pressure is higher below.
Even in the beginning of this Century, after people had giggled at the idea of making flying machines that would be heavier than air, it was thought that the reason a plane could fly was that it was sustained from the air below it, while it is more because it is sucked upwards from the air above it, as an important depression is created above the wing by the flow (if the profile of the wing is well designed).
The difficulty, of course, was that static questions about pressure had been well understood for some time, while dynamic questions were quite new.
For the static question, and some dynamic effects, it is clear from some of his drawings that D{\eightrm A} V{\eightrm INCI} had well understood what the pressure is, and that should not be so surprising if one remembers that he was first of all an hydraulic engineer.
After T{\eightrm ORRICELLI} had invented the barometer, P{\eightrm ASCAL} was the first to study the laws governing hydrostatic pressure, and both were remembered when units were chosen, a Torr for a pressure of a millimeter of mercury, and a Pascal for the rather small pressure of 1 Newton per square meter.
\medskip
One B{\eightrm ERNOULLI} had studied the movement of a vibrating string by considering the approximation of many smalls masses connected by small springs; he apparently only derived the modes of vibration and it was D'A{\eightrm LEMBERT} who first wrote the 1-dimensional wave equation.
H{\eightrm UYGHENS} had some insight about the wave nature of Light, but it might have been L{\eightrm APLACE} or P{\eightrm OISSON} who first wrote down the 3-dimensional wave equation.
\par
C{\eightrm AUCHY} derived the linearized Elasticity equation using the same idea of masses with small springs, but he only found a one parameter family of isotropic materials, and it was L{\eightrm AM\'E} who introduced the two parameter family that we use now for the constitutive equation (strain-stress law) $\sigma_{ij} = 2\mu\,\varepsilon_{ij} +\lambda\delta_{ij}\sum_{k} \varepsilon_{kk}$, where $\varepsilon_{ij} = {1\over 2}\bigl( {\partial u_{i}\over \partial x_{j}}+{\partial u_{j}\over \partial x_{i}} \bigr)$.
If one lets $\lambda$ go to $\infty$ and $\mu$ go to 0, one finds that $\sum_{k} \varepsilon_{kk}$, which is $div(u)$, tends to 0, and that $\lambda\,div(u)$ tends to a limit, giving the law for an inviscid incompressible fluid $\sigma_{ij} = -p\,\delta_{ij}$ (but this ``pressure'' for an incompressible fluid is not so physical).
It is worth noticing that in his Physics course, F{\eightrm EYNMAN} qualified the E{\eightrm ULER} equation as the equation for dry water, and the N{\eightrm AVIER}-S{\eightrm TOKES} equation as the equation for wet water.
\par
C{\eightrm AUCHY} may have understood the force exerted by a part of an elastic body onto its complement as the resultant of all these tiny forces transmitted through these microscopic springs, but if that description might be found convenient for a solid, it does not look so realistic for a liquid or a gas.
\par
The first explanation of what creates the pressure in a gas might have appeared in the work on kinetic theory of B{\eightrm OLTZMANN} and M{\eightrm AXWELL} (whose name was actually C{\eightrm LERK} when he was born, and became C{\eightrm LERK} M{\eightrm AXWELL} after his father had inherited from an uncle).
\medskip
In kinetic theory, one considers a gas with so many particles inside that one can take a limit and describe a density $f(x,v,t)$ for particles near the point $x$, having their velocity near $v$ around the time $t$ (in order to simplify, I assume that all particles have the same mass).
If these particles were not interacting and were feeling no exterior forces, the evolution of the density would be given by the free transport equation
$$
{\partial f\over\partial t}+\sum_{j} v_{j}{\partial f\over\partial x_{j}} = 0,
$$
the density of mass $\rho$ and of momentum $p$, and the (macroscopic) velocity $u$ being defined by
$$
\eqalign{
\rho(x,t) &= \int_{R^{3}} f(x,v,t)\,dv\cr
p(x,t) &= \int_{R^{3}} v\,f(x,v,t)\,dv\cr
&= \rho(x,t)u(x,t),}
$$
so that if one integrates in $v$ the free transport equation, one obtains the equation of conservation of mass
$$
{\partial \rho\over \partial t}+\sum_{j}{\partial (\rho\,u_{j})\over \partial x_{j}} = 0.
$$
Actually, there are exterior forces, depending both on position and velocity.
For electrically charged particles, one must take into account the L{\eightrm ORENTZ} force $e(E+v\times B)$ for a particle with charge $e$; in Oceanography one must take into account Gravity and the C{\eightrm ORIOLIS} force created by the rotation of the Earth, and the form is similar.
If all particles have the same mass $m$ and the same charge $e$, and we still assume that particles do not interact, the evolution of the density of charged particles would be given by the transport equation
$$
{\partial f\over\partial t}+\sum_{j} v_{j}{\partial f\over\partial x_{j}} + \sum_{j} {e\over m}\Bigl( E_{j}(x,t)+\sum_{kl} \varepsilon_{jkl}v_{k}B_{l}(x,t) \Bigr){\partial f\over\partial v_{j}} = 0,
$$
and integration in $v$ (assuming that $f$ is 0 for large $v$ for example) would give the same form of the conservation of mass because $\sum_{k} \varepsilon_{jkl}\delta_{jk} = 0$.
I will describe another time the C{\eightrm ORIOLIS} force, and forget about these exterior forces now and concentrate on interior forces, due to ``collisions'' of particles.
\medskip
B{\eightrm OLTZMANN} equation, in the absence of exterior forces, has the form
$$
{\partial f\over\partial t}+\sum_{j} v_{j}{\partial f\over\partial x_{j}} +Q(f,f) = 0,
$$
where $Q(f,f)$ is a somewhat complicated term, but for our purpose, because collisions are supposed to conserve mass, momentum and kinetic energy, we will admit that it always satisfies
$$
\eqalign{
\int_{v\in R^{3}} Q(f,f)\,dv &= 0\cr
\int_{v\in R^{3}} v_{j}Q(f,f)\,dv &= 0\hbox{ for all }j\cr
\int_{v\in R^{3}} |v|^{2}|Q(f,f)\,dv &= 0.}
$$
Integrating the B{\eightrm OLTZMANN} equation in $v$ gives then again the same equation for conservation of mass, and integrating after multiplicating by $v_{i}$ will give us the form of the equationm of conservation of momentum, and integrating after multiplicating by $|v|^{2}$ will give us the form of the equation of conservation of energy.
The form is independent of what $Q$ is, as long as $Q$ satisfies the above constraints, but we will have to add constitutive relations.
Let us define the symmetric stress tensor $\sigma$ by
$$
\sigma_{ij}(x,t) = -\int_{v\in R^{3}} f(x,v,t)\Bigl( v_{i}-u_{i}(x,t) \Bigr)\Bigl( v_{j}-u_{j}(x,t) \Bigr)\,dv.
$$
Then as $v_{i}v_{j} = u_{i}u_{j} + u_{i}(v_{j}-u_{j}) + u_{j}(v_{i}-u_{i}) + (v_{i}-u_{i})(v_{j}-u_{j})$, and $\int_{v} f(x,v,t)\bigl( v_{i}-u_{i}(x,t) \bigr)\,dv = 0$ by definition of $u$, one deduces that
$$
\int_{v\in R^{3}} v_{i}v_{j}f(x,v,t)\,dv = \rho(x,t)u_{i}(x,t)u_{j}(x,t) -\sigma_{ij}(x,t),
$$
and the equation of conservation of momentum becomes
$$
{\partial (\rho\,u_{i})\over\partial t}+\sum_{j} {\partial (\rho\,u_{i}\,u_{j})\over\partial x_{j}} - \sum_{j} {\partial \sigma_{ij}\over\partial x_{j}} = 0\hbox{ for all }i.
$$
In the case where $\sigma_{ij} = -p\,\delta_{ij}$, $p$ is the pressure, which is nonnegative, as the definition of $\sigma$ shows that it is a negative definite tensor, because $f$ is a nonnegative function with positive total mass (assuming that all the mass does not move at the same velocity $u$).
This is acceptable in a gas, but not in a solid where extension is possible, and I will rederive the same equation using the point of view of B{\eightrm ERNOULLI} and C{\eightrm AUCHY} based on little springs.
\medskip
The pressure has a simple explanation if we look at what happens on the boundary.
If the normal to the boundary going inside the gas is $\nu$, the usual law of reflection, called specular reflection, is that a particle arriving with velocity $v$ with $(v.\nu)<0$ is reflected with velocity $w$ given by $w = -2\nu(v.\nu)+v$, so that $(w.\nu) = -(v.\nu)>0$.
Each particle bouncing on the wall receives then a momentum in the direction of $\nu$, and the pressure exerted by the gas is precisely the effect that all the particles transmit to the boundary a momentum in the direction $-\nu$ when they collide the boundary.
The specular reflection is not exactly true, because the boundary is also made of particles and if a particle from the gas has enough velocity it may enter slightly into the solid, interact with the particles in the solid, and get back in various direction, after a small delay.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
7. Wednesday January 27.
\medskip
Let us look now at the conservation of energy, by multiplying the B{\eightrm OLTZMANN} equation by ${|v|^{2}\over 2}$ and then integrating in $v$; as before, the term $Q(f,f)$ will disappear in this process.
\par
One defines the internal energy per unit of mass $e$ by the formula
$$
\rho(x,t)e(x,t) = \int_{R^{3}} {|v-u(x,t)|^{2}\over 2}f(x,v,t)\,dv,
$$
and there is then an automatic relation
$$
\rho\,e = -{1\over 2}trace(\sigma),\hbox{ i.e. } {3p\over 2}\hbox{ in the case where }\sigma_{ij} = -p\,\delta_{ij},
$$
and this is an obvious defect of the B{\eightrm OLTZMANN} equation, that it implies constitutive relations which are not exactly true for real gases.
Actually B{\eightrm OLTZMANN} equation should only be considered as a model for rarefied gases, in agreement with the way the equation was derived, by assuming that two nearby particles only see each other and none of the other particles in the gas.
\par
One also defines the heat flux $q$ by the formula
$$
q_{j}(x,t) = \int_{R^{3}} \Bigl( v_{j}-u_{j}(x,t) \Bigr){|v-u(x,t)|^{2}\over 2}f(x,v,t)\,dv\hbox{ for all }j.
$$
Apart from the relation already noticed between $\rho\,e$ and $\sigma$, there is no other automatic relation between the thermodynamic quantities $\rho,\sigma,e,q$, i.e. quantities pertaining to the gas and which therefore do not change in a Galilean transformation (consisting in adding a constant velocity to $u$).
As $\rho$ is the moment of order 0 of $f$, the moments of order 1 are 0, the moments of order 2 give $\sigma$ (and $\rho\,e$ is a particular combination of these moments), and a particular combination of moments of order 3 is $q$, one can show that the only relations between these moments are the nonnegative character of $\rho$ and $-\sigma$.
\medskip
One has to compute the term $\int_{v} {|v|^{2}\over 2}f\,dv$, and putting $v = u+\xi$, one finds that ${|v|^{2}\over 2} = {|u|^{2}\over 2}+(u.\xi)+{|\xi|^{2}\over 2}$, and therefore, as $\int_{v} \xi\,f\,dv = 0$, one finds $\int_{v} {|v|^{2}\over 2}f\,dv = {\rho\,u^{2}\over 2} +\rho\,e$.
Then, for each $j$, one has to compute the term $\int_{v} {|v|^{2}\over 2}v_{j}f\,dv$, and one finds that ${|v|^{2}\over 2}v_{j} = {|\xi|^{2}\over 2}\xi_{j} + u_{j}{|\xi|^{2}\over 2} +\xi_{j}(\xi.u) +\xi_{j}{|u|^{2}\over 2} +u_{j}(\xi.u)+u_{j}{|u|^{2}\over 2}$, and therefore, $\int_{v} {|v|^{2}\over 2}v_{j}f\,dv = q_{j} + u_{j}\rho\,e + \sum_{k} \sigma_{jk}u_{k} + \rho\,u_{j}{|u|^{2}\over 2}$.
The conservation of energy appears then as
$$
{\partial \Bigl( {\rho\,|u|^{2}\over 2}+\rho\,e \Bigr)\over \partial t}+\sum_{j} {\partial \Bigl[ \Bigl( {\rho\,|u|^{2}\over 2}+\rho\,e \Bigr)u_{j} + \sum_{k} (\sigma_{jk}u_{k}) + q_{j} \Bigr]\over \partial x_{j}} = 0.
$$
\medskip
One sees from the formula $\int_{v} {|v|^{2}\over 2}f\,dv = {\rho\,|u|^{2}\over 2}+\rho\,e$ that $\rho\,e$ is the part of the kinetic energy which is hidden at a microscopic level.
In B{\eightrm OLTZMANN} model all energy is kinetic, i.e. comes from translation effects and none of it comes from rotation effects (as it would if the particles in the gas were also rotating), and the internal energy is that part of the kinetic energy which cannot be explained by looking only at the macroscopic quantities like $u$.
The First Principle of Thermodynamics asserts that Energy is conserved, but one should count all the various forms of energy (in nuclear reactions even mass must be considered a form of energy, with the celebrated E{\eightrm INSTEIN} formula $e = m\,c^{2}$); for a gas made of molecules (i.e. all real gases apart from the rare gases), besides translation and rotation effects of a molecule considered as rigid, there are also vibration effects due to the internal degrees of freedom of the molecule.
\medskip
B{\eightrm OLTZMANN} had also noticed his famous H-theorem (I think that he may have chosen H as the capital letter for $\eta$, used for entropy); it follows from the relation
$$
\int_{v} Q(f,f)\log\,f\,dv\ge 0,
$$
which implies
$$
{\partial (\int_{v} f\log\,f\,dv)\over \partial t} +\sum_{j} {\partial (\int_{v} v_{j}f\log\,f\,dv)\over \partial x_{j}} \le 0.
$$
It is a consequence of the symmetric form of the collision operator (and the nonnegativity of the kernel), and equality only occurs for local Maxwellian distributions, i.e.
$$
f = \alpha\,exp(-\beta|v-u|^{2})\hbox{ with }\alpha,\beta,u\hbox{ depending only upon }x, t.
$$
One has $\beta = {1\over k\,T}$, where $k$ is the B{\eightrm OLTZMANN} constant and $T$ the absolute temperature, and then $\beta^{3/2}\rho = \alpha\pi^{3/2}$, so for locally Maxwellian distributions, one can check that $e$ is proportional to $T$, that $\sigma = -p\,\delta_{ij}$ and $q = 0$, with $p$ computed as shown before, etc.
\par
For a real gas, there is an equation of state which relates the various thermodynamic quantities, not necessarily the one that comes out of the (formal) computation for B{\eightrm OLTZMANN} equation.
\medskip
The qualitative form of the collision operator is obtained as follows.
Two particles with initial velocities $v$ and $w$ ``collide'' and give two particles of velocities $v^{\prime}$ and $w^{\prime}$ and, as the masses are equal, conservation of momentum and conservation of kinetic energy are equivalent to the relations
$$
\eqalign{
v+w &= v^{\prime}+w^{\prime}\cr
|v|^{2}+|w|^{2} &= |v^{\prime}|^{2}+|w^{\prime}|^{2},}
$$
which give $|v-w| = |v^{\prime}-w^{\prime}|$, and putting $v^{\prime} = v+z$ and $w^{\prime} = w-z$ give $(v-w.z)+|z|^{2} = 0$, so that by putting $\alpha = {z\over |z|}$ (if $z = 0$ one takes for $\alpha$ any unit vector orthogonal to $v-w$), one can parametrize all the solutions by using $w$ and a unit vector $\alpha$:
$$
v^{\prime} = v+(w-v.\alpha)\alpha;\;w^{\prime} = w-(w-v.\alpha)\alpha,
$$
and if one defines $\theta$ by $|v-w|\cos\theta = |(v-w.\alpha)|$, then the deflection is $2\theta$ or $\pi-2\theta$, i.e. in the Galilean frame of the center of mass (moving at velocity ${v+w\over 2}$) the final velocity direction makes and angle $2\theta$ or $\pi-2\theta$ with the initial direction.
The kernel only depends upon $|v-w|$ (twice the velocity of approach in the frame of the center of mass), and $\theta$, as in the center of mass there is a symmetry around the direction of the initial velocity.
The term $Q(f,f)$ has therefore the form
$$
Q(f,f) = \int_{w\in R^{3}} \int_{\alpha\in S^{2}} B(|v-w|,\theta)\Bigl( f(v)f(w)-f(v^{\prime})f(w^{\prime}) \Bigr)\,dw\,d\alpha,
$$
and the kernel $B$ is nonnegative.
Because $\theta = {\pi\over 2}$ corresponds to $v^{\prime} = v$ and $w^{\prime} = w$ (or $v^{\prime} = w$ and $w^{\prime} = v$, as particles are undiscernable), and particle collisions are avoided outside a small effective scattering cross section, $B(|v-w|,\theta)$ tends to $+\infty$ as $\theta$ tends to ${\pi\over 2}$.
That makes B{\eightrm OLTZMANN} equation quite difficult, and following G{\eightrm RAD} one usually considers an angular cut-off, i.e. one truncates $B$ near $\theta = {\pi\over 2}$.
\par
If one notices that the kernel $B$ does not change if one exchanges $v$ and $w$, or if one exchanges the roles of $(v,w)$ and $(v^{\prime},w^{\prime})$ (which is like reversing time so the collision of $v^{\prime}$ and $w^{\prime}$ may produce $v$ and $w$), but $f(v)f(w)-f(v^{\prime})f(w^{\prime})$ stays the same for the first transformation and changes sign for the second, then one deduces that
$$
\eqalign{
\int_{v\in R^{3}} Q(f,f)\log\,f\,dv = {1\over 4}\int_{v,w\in R^{3}} \int_{\alpha\in S^{2}} B(|v-w|,\theta)\Bigl( f(v)f(w)&-f(v^{\prime})f(w^{\prime}) \Bigr)\Bigl( \log\,f(v)+\log\,f(w)\cr
&-\log\,f(v^{\prime})-\log\,f(w^{\prime}) \Bigr)\,dv\,dw\,d\alpha \ge 0,}
$$
the last inequality coming from $\log\,f(v)+\log\,f(w) - \log\,f(v^{\prime})- \log\,f(w^{\prime}) = \log\,f(v)f(w)-\log\,f(v^{\prime})f(w^{\prime})$, and the fact
that the logarithm is increasing.
Equilibrium corresponds to $\int_{v} Q(f,f)\log\,f\,dv = 0$, and this is equivalent to $f(v)f(w)-f(v^{\prime})f(w^{\prime}) = 0$ for all collisions, or $\log\,f(v)+ \log\,f(w) = \log\,f(v^{\prime})+\log\,f(w^{\prime}) = 0$ for all collisions, and that is certainly true if $\log\,f(v) = a+(b.v)+c{|v|^{2}\over 2}$ for some $a, b, c$ independent of $v$, giving the locally Maxwellian functions.
That there are no other solutions requires a little care.
\medskip
In order to find more relations between $\rho$, $e$, $\sigma$ and $q$, one usually quotes a formal argument of H{\eightrm ILBERT}, or one of C{\eightrm HAPMAN} \& E{\eightrm NSKOG}, which start by considering a term in ${1\over \varepsilon}Q(f,f)$, relating $\varepsilon$ to the mean free path between collisions.
The formal argument of H{\eightrm ILBERT} consists in assuming that $u = u_{0}+\varepsilon\,u_{1}+\ldots$ and identifying the various terms, the term in ${1\over \varepsilon}$ imposing that $u_{0}$ is a local Maxwellian, and the next terms giving E{\eightrm ULER} equation for an inviscid perfect gas.
The argument of C{\eightrm HAPMAN} \& E{\eightrm NSKOG} produces N{\eightrm AVIER}-S{\eightrm TOKES} equation with a viscosity of order $\varepsilon$.
\par
As I mentioned before, letting the mean free path between collisions tend to 0 is in contradiction with the assumption that one deals with a rarefied gas in order to compute the kernel.
It does not seem reasonable to assume that B{\eightrm OLTZMANN} equation is valid for dense gases and liquids, one reason being that if two many ``particles'' get nearby, then the only way to deal with them is to consider that they are waves, and not classical particles.
Actually, as B{\eightrm OLTZMANN} equation (formally) predicts a perfect gas behaviour, and real gases are not perfect gases, either B{\eightrm OLTZMANN} equation is not satisfied by real gases, or the formal argument of H{\eightrm ILBERT} is not valid.
\par
From a philosophical point of view, it is rather curious to observe the efforts made to derive E{\eightrm ULER} or N{\eightrm AVIER}-S{\eightrm TOKES} equation out of B{\eightrm OLTZMANN} equation, as if starting with B{\eightrm OLTZMANN} equation was a flawless assumption.
On the contrary, B{\eightrm OLTZMANN} equation has already postulated some irreversibility, and this is seen by the fact that nonnegative initial data create a nonnegative solutions, a property that is lost after time reversal.
Formally this is due to the form of the equation:
$$
{\partial f\over \partial t}+\sum_{j} v_{j}{\partial f\over \partial x_{j}} +f\,A(f) = B(f),
$$
with $B(f)\ge 0$ a.e. when $f\ge 0$ a.e.; if $A$ and $B$ were locally L{\eightrm IPSCHITZ} continuous one could obtain the solution by the iterative process
$$
{\partial f^{(n+1)}\over \partial t}+\sum_{j} v_{j}{\partial f^{(n+1)}\over \partial x_{j}} +f^{(n+1)}\,A(f^{(n)}) = B(f^{(n)}),
$$
which gives $f^{(n+1)}$ nonnegative when the initial condition and $f^{(n)}$ are nonnegative.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
8. Friday January 29.
\medskip
Let us look at the way B{\eightrm ERNOULLI} and D'A{\eightrm LEMBERT} were led to discover the 1-dimensional wave equation, and later C{\eightrm AUCHY} was led in the same way to discover the equation for linearized Elasticity.
\par
The simplest case, for what concerns its analysis, is that of a 1-dimensional longitudinal wave.
The motion of a violin string is different, as it is a transversal wave: the waves propagate along the string but the displacement is mostly perpendicular to the string.
A 1-dimensional longitudinal wave corresponds to the experimental situation of a metallic bar which one hits on one end with a hammer.
In linearized Elasticity, in two or three dimensions and in an isotropic material, P-waves (pressure waves) are longitudinal waves, while S-waves (shear waves) are transversal waves (they travel at different speeds).
\par
Let us consider the motion of $N-1$ small masses connected with springs, with the purpose of letting $N$ tend to $\infty$; let $x_{0}(t) = 0$, and $x_{N}(t) = L$, corresponding to the fixed walls where the first and last masses are attached.
Let $x_{i}(t)$, $i = 1,\ldots,N-1$, be the positions at time $t$ of the mass $\#i$, let $m_{i}$ be its mass.
Let us assume that the springs are at equilibrium if the masses are at their rest point, corresponding to $\xi_{i}$ for the mass $\#i$ (one take $\xi_{0} = 0$ and $\xi_{N} = L$), and let $\kappa_{i,i+1}$ be the constant of the spring connecting mass $\#i$ and mass $\#(i+1)$ (with $\#0$ and $\#N$ designating the walls), i.e. an increase in length of $\Delta>0$ creates a restoring force $\kappa\Delta$ (and similarly for a compression); of course, this is only realistic if the displacements are small.
\par
The force acting on mass $\#i$ by the spring connecting it to mass $\#(i+1)$ is $\kappa_{i,i+1}(x_{i+1}-x_{i}-\xi_{i+1}+\xi_{i})$, and it is therefore natural to put $x_{i}(t) = \xi_{i}+y_{i}(t)$, and the equation of motion (N{\eightrm EWTON}'s law) for mass $\#i$ is then
$$
m_{i}{d^{2}y_{i}\over dt^{2}} = \kappa_{i,i+1}(y_{i+1}-y_{i})- \kappa_{i-1,i}(y_{i}-y_{i-1})\hbox{ for } i = 1,\ldots,N-1,
$$
and the initial position and velocity of each of the $N-1$ masses must also be given, and as it is a linear differential system there exists a unique solution (global in time).
However, we need precise estimates if we want to understand what happens when $N$ tends to $\infty$.
\par
Before doing that, it is useful to repeat that the reason that one can do the analysis is that one has chosen a linearized problem without saying it expressly: if a spring has size 1 at rest and is elongated of an amount $\Delta$, the restoring force may be of the order of $\kappa\Delta$ if $|\Delta|$ is small, but it makes no sense having $\Delta$ go to $-1$, where the spring is compressed to zero length, or $\Delta$ tend to $\infty$ as no known material can sustain such a deformation without going through permanent plastic deformation before breaking.
Of course, these springs are only an idealized classical version of what happens at a microscopic level: electric forces may be attracting of repulsing and both occur in an ionic crystal like salt (NaCl), but forces that bind a metallic crystal are all similar and it is more than the nearby neighbours which play a role in the stability of the crystalline arrangement (at least in liquids, one sometimes invoke L{\eightrm ENNARD}-J{\eightrm ONES} potentials, which have a long range attraction potential in $1/r^{6}$ and a short range repulsion potential in $1/r^{12}$).
Then crystals are not very good at Elasticity and cannot support much strain and they change their microstructure to polycrystals, so the idealized description of Elasticity with little springs could have seemed reasonable to C{\eightrm AUCHY} (and I am not even sure if that is the way he thought), but is known to contradict our actual knowledge.
\par
However, one may look at this description in another way, and consider it a Numerical Analysis point of view.
Indeed, if one uses finite difference schemes or finite elements (where finite is in opposition to infinitesimal and not to infinite), it is quite natural to replace the wave equation by the system that I have written, and interpret it as moving masses connected by springs, and replace the equation of linearized Elasticity by a system very similar in nature.
The important difference is that in the Numerical Analysis point of view, mathematicians start from the partial differential equations and want to show that the finite dimensional approximation chosen will indeed approach the solution as the mesh size tends to 0 (while engineers might not even write down the partial differential equations and may only play with the finite dimensional description), and it is not so good for detecting the effects of nonlinearity.
We will first follow this point of view, neglecting nonlinearities by pretending that they are small, and later we will try to take them into account.
\medskip
The system written has an invariant, which is the total energy: multiplying equation $\#i$ by ${dy_{i}\over dt}$ and summing in $i$, one obtains
$$
{d\over dt}\Bigl( \sum_{j = 1}^{N-1} {m_{j}\over 2}\Bigl| {dy_{j}\over dt} \Bigr|^{2} + \sum_{j = 0}^{N-1} {\kappa_{j,j+1}\over 2}|y_{j+1}-y_{j}|^{2} \Bigr) = 0,
$$
the first part being the kinetic energy, and the second part being the potential energy, i.e. the energy stored inside the springs (and there is one more spring than masses).
There is an obvious Hamiltonian framework behind our equation, but one should be aware of the fact that for partial differential equations which are not linear, a Hamiltonian framework is not always so useful, the main reason being that things which are conserved like energy may suddenly start converting to a new form like heat, which may not be described by the same equation, and suddenly the ``conserved quantity'' starts to change!
\par
Then one wants to let $N$ tend to $\infty$, and all questions of scaling should be done with care, as there might be different regimes to consider, but here the matter is straightforward.
One way to guess the right scaling is to consider that the values $y_{j}$ are extended by interpolation, filling the intervals in the space variable $\xi$ and time $t$ (that is a Lagrangian point of view), defining a function $u$, and that the kinetic part should look like ${1\over 2}\int_{0}^{L} \rho(\xi)\bigl| {\partial u\over \partial t} \bigr|^{2}\,d\xi$ and the potential part like ${1\over 2}\int_{0}^{L} \kappa(\xi)\bigl| {\partial u\over \partial \xi} \bigr|^{2}\,d\xi$.
For example, taking $\xi_{j} = {j\,L\over N}$ and $m_{j} = {M\over N}$ so that $M$ is the total mass of the springs, and $\kappa_{j.j+1} = N\kappa$, corresponds to a uniform density of mass $\rho = {M\over L}$ and constant $\kappa$, and the equation becomes
$$
M{\partial^{2} u\over \partial t^{2}} - \kappa\,L^{2}{\partial^{2} u\over \partial \xi^{2}} = 0,
$$
corresponding to a propagation speed
$$
c = \sqrt{\kappa\over M}L,
$$
and its solutions are of the form $f(x-c\,t)+g(x+c\,t)$, as noticed by D'A{\eightrm LEMBERT}.
One should add the boundary conditions $u(0,t) = u(L,t) = 0$, and the initial conditions
$$
u(\xi,0) = v(\xi);\;{\partial u\over \partial t}(\xi,0) = w(\xi)\hbox{ a.e. in }(0,L).
$$
This can be proved using standard results of Functional Analysis (from any bounded sequence in $L^{2}$, one can extract a weakly converging subsequence) and a little use of distributions (for pushing the derivatives to the test functions), but one must be careful that the initial condition should be approached in the right way, i.e. $v\in H^{1}_{0}(0,L)$ and $w\in L^{2}(0,L)$ (I will use this type of method extensively later on, and I will then explain the details of the argument).
\par
As the unit of $\kappa_{j,j+1}$ is mass/time$^{2}$, and the mass scales naturally in $m_{j} = M/N$, the scaling $\kappa_{j,j+1} = N\kappa$ corresponds to a characteristic time in $1/N$, which is quite natural for a characteristic length $L/N$ and a finite propagation speed, but the argument is circular because the discrete system does not have finite propagation speed (a change of position of the first mass is immediately felt at the last one), and it is only the limiting equation that has the finite propagation speed property.
However, if the total energy of the initial data is kept fixed and if one takes $\kappa_{j,j+1} = h(N)$ with $h(N)/N$ tending to $\infty$, then the solution tends to 0 and all the energy goes into vibration, while if $h(N)/N$ tends to 0 there is only kinetic energy at the limit and no interaction between particles; therefore there is only one good scaling!
\medskip
It seems that B{\eightrm ERNOULLI} only considered the solutions of the form $y_{j}(t) = e^{i\,\omega\,t}z_{j}$, which he found to be $z_{j} = \gamma\sin\bigl( {j\over N}m\pi \bigr)$, corresponding to $\omega^{2} = {4N^{2}\kappa\over M}\sin^{2}\bigl( {m\pi\over 2N} \bigr)$, which tends to ${K\,m^{2}\pi^{2}\over M}$ as $N$ tends to $\infty$, and that is not as precise as deriving the wave equation.
Physicists often find information for special solutions oscillating at a unique frequency, and the result may show that no partial differential equation of a given type may create the same kind of relation, but even if one has to write down a pseudo-differential equation, it is better to understand what all solutions do; actually in a nonlinear setting one cannot expect to reconstruct the solution easily from the knowledge of special solutions, and even in linear situations it does not help much for understanding what the boundary conditions are (as every function in $L^{2}(0,1)$ can be written as an infinite sum of functions vanishing at 0, one must be careful).
\medskip
If one considers a 2-dimensional or 3-dimensional array of masses connected by springs, or even the transversal vibrations of a string, the first thing to realize if that without linearization the problem becomes terribly difficult.
With linearization, the idea if that if a spring connects points $A$ and $B$ and that these points move of $\delta A$ and $\delta B$, which are small compared to the length of $AB$, then the new length is $|B-A+\delta B-\delta A| = \sqrt{|B-A|^{2}+2(B-A,\delta B-\delta A)+|\delta B-\delta A|^{2}} = |B-A| +(B-A,\delta B-\delta A)/|B-A| +o(|\delta B-\delta A|)$, and therefore only the displacement perpendicular to the initial position of the spring is taken into account.
In a two dimensional setting, denoting by $x$ and $y$ the space variables, by $u$ and $v$ the displacement, one sees that a spring parallel to the $x$ axis corresponds to a potential energy involving $|u_{x}|^{2}$ (where subscript denotes differentiation), a spring parallel to the $y$ axis corresponds to a potential energy involving $|v_{y}|^{2}$, a spring along the first diagonal corresponds to a potential energy involving $|u_{x}+u_{y}+v_{x}+v_{y}|^{2}$, and a spring along the second diagonal corresponds to a potential energy involving $|u_{x}-u_{y}-v_{x}+v_{y}|^{2}$.
One understands then that the notation
$$
\varepsilon_{ij} = {1\over 2}\Bigl( {\partial u_{i}\over \partial x_{j}} + {\partial u_{j}\over \partial x_{i}} \Bigr),
$$
helps in writing the limiting equations as
$$
\rho(\xi){\partial^{2}u_{i}\over \partial t^{2}} -\sum_{j} {\partial\sigma_{ij}\over \partial x_{j}} = 0\hbox{ for all i},
$$
where the stress $\sigma$ has the form
$$
\sigma_{ij} = \sum_{k,l} C_{ijkl}(\xi)\varepsilon_{kl}.
$$
Linearization has the defect of mixing up the Eulerian and the Lagrangian point of views, and the C{\eightrm AUCHY} stress, which is symmetric, should appear in the Eulerian point of view while the P{\eightrm IOLA}-K{\eightrm IRCHHOFF} stress, which is not symmetric, should appear in the Lagrangian point of view.
In the isotropic case, C{\eightrm AUCHY} had found the relation $\sigma_{ij} = 2\mu\varepsilon_{ij}+\delta_{ij}\sum_{k} \varepsilon_{kk}$, but with a special relation between $\mu$ (the shear modulus) and $\lambda$ (the L{\eightrm AM\'E} parameter), as he had $\lambda = \mu$, and it was L{\eightrm AM\'E} who then pointed out that there was a two dimensional family of isotropic materials.
Because the tensors $\varepsilon$ and $\sigma$ are symmetric, there is no restriction in assuming that
$$
C_{ijkl} = C_{jikl}\hbox{ and }C_{ijkl} = C_{ijlk}\hbox{ for all }i,j,k,l,
$$
but there is another symmetry relation, for hyperelastic materials, i.e. those materials which have a stored energy function (and this symmetry is probably a necessary condition for the evolution problem to be well posed with the finite propagation speed property, assuming that some kind of ellipticity condition is satisfied),
$$
C_{ijkl} = C_{klij}\hbox{ for all }i,j,k,l.
$$
Under this last condition, the conservation of energy becomes
$$
\int \Bigl( {\rho(\xi)\over 2} \sum_{i} \Bigl| {\partial u_{i}\over \partial t} \Bigr|^{2} + {1\over 2}\sum_{i,j,k,l} C_{ijkl}(\xi)\varepsilon_{ij}\varepsilon_{kl} \Bigr)\,dx = constant.
$$
\medskip
The preceding discussion was to show the form of the equation, and under an hypothesis of ``very strong ellipticity'' one can show existence and uniqueness for the evolution problem (and the finite propagation speed property), and the convergence of some natural approximation processes, like the one involving little masses and springs.
\par
However, the C{\eightrm AUCHY} stress should be discussed in an Eulerian framework, and the argument of C{\eightrm AUCHY} that there must exists a stress tensor used the equilibrium of a small tetrahedron.
He assumed that for a domain $\omega$ (with L{\eightrm IPSCHITZ} boundary!), the force acting on a small set of the boundary of $\omega$ by the exterior of $\omega$ is a force proportional to the surface of the element and depending upon the position and the normal to $\partial\omega$ (the exterior of $\omega$ receiving an opposite force, so that conservation of momentum is satisfied); writing then the equilibrium of a tetrahedron small enough so that the dependence is only in the normal, the following argument was used to deduce the fact that the dependence with respect to the normal must be linear.
For $a_{1},a_{2},a_{3}>0$ and small, the faces of the tetrahedron are the planes $x_{i} = 0$ and the face $T$ of equation ${x_{1}\over a_{1}}+{x_{2}\over a_{2}}+{x_{3}\over a_{3}} = 1$ with $x_{j}\ge 0$.
Let $F_{i}$ be the force by unit area on the face $x_{i} = 0$ and let $G$ be the force per unit area on the face $T$, then the equilibrium of the tetrahedron is $a_{2}a_{3}F_{1}+a_{3}a_{1}F_{2}+a_{1}a_{2}F_{3}+S\,G = 0$, where $S$ is the area of the triangle $T$, but as the normal $\nu$ to $T$ has the form $\nu_{j} = {\lambda\over a_{j}}$ for some $\lambda>0$, and $S = {a_{1}a_{2}\over \nu_{3}} = {a_{1}a_{2}a_{3}\over \lambda}$, one finds that $G = -\nu_{1}F_{1}-\nu_{2}F_{2}-\nu_{3}F_{3}$, and therefore $G$ is linear with respect to $\nu$.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
9. Monday February 1.
\medskip
Linearization may often seem a reasonable step when some quantities are believed to be small, and a function that one may want to neglect may indeed be small, but the danger comes from the fact that its derivative might not be small.
For what concerns hyperbolic equations, which are more or less the partial differential equations for which information travels at finite speed, the difference between the linear and the nonlinear case (actually, the quasilinear case) is quite important.
\par
In the case of an elastic string, taking into account large deformations leads to an equation of the form
$$
{\partial^{2} w\over \partial t^{2}} -{\partial\over \partial x}\Bigl( f\Bigl( {\partial w\over \partial x} \Bigr) \Bigr) = 0,
$$
where $w$ denotes the vertical displacement, ${\partial w\over \partial t}$ the velocity, ${\partial w\over \partial x}$ the strain, and $f\bigl( {\partial w\over \partial x} \bigr)$ the stress, and the function $f$ is no longer affine, but it satisfies $f^{\prime}>0$, and $\sqrt{f^{\prime}}$ appears to be the local speed of propagation of perturbations.
\par
The first to study such an equation was P{\eightrm OISSON}, around 1807, but he was concerned with gas dynamics in a simplified form, i.e. the system
$$
\eqalign{
&{\partial \rho\over \partial t}+{\partial (\rho\,u)\over \partial x} = 0\cr
&{\partial (\rho\,u)\over \partial t}+{\partial (\rho\,u^{2}+p)\over \partial x} = 0,}
$$
with $p$ being a nonlinear function of $\rho$.
One of the reasons why P{\eightrm OISSON} was interested in compressible gases was to compute the exit velocity of a shell out of the barrel of a gun.
N{\eightrm EWTON} had apparently computed the velocity of sound in air, but his calculation had given a value almost 100 m/s short of the measured velocity (which is a little above 300 m/s under usual conditions).
He had certainly not written the wave equation, but he had indeed used what he knew about compressibility of air, i.e. he had used $p$ as a linear function of $\rho$, according to the law of perfect gases $P\,V = constant$ (as the relation $P\,V = R\,T$ appeared much later).
P{\eightrm OISSON} was using a relation $p = c\,\rho^{\gamma}$, which L{\eightrm APLACE} may have suggested, and the thermodynamic interpretation came much later: as the wave are fast phenomena, the mechanical energy has no time to be transformed into heat, and the process is therefore adiabatic ($\delta Q = 0$), or equivalently isentropic (as $\delta Q = T\,dS$).
Thermodynamics tells us that $\gamma$ is the ratio ${c_{p}\over c_{v}}$, where $c_{p}$ is the heat capacity per unit mass at fixed pressure, and $c_{v}$ the heat capacity per unit mass at fixed volume; it is about 5/3 for air.
P{\eightrm OISSON}'s solution was not analytical but had an implicit form, and in 1848 C{\eightrm HALLIS} found that his formula could not be true for all time, which prompted S{\eightrm TOKES} to explain that profiles were becoming steeper and steeper, until one had to introduce a discontinuity, for which he computed the velocity, by expressing the conservation of mass and the conservation of momentum.
\par
The basic ideas are more easily explained on the inviscid B{\eightrm URGERS} equation
$$
{\partial u\over \partial t}+u{\partial u\over \partial x} = 0,
$$
which will have to be written as
$$
{\partial u\over \partial t}+{\partial ({u^{2}\over 2})\over \partial x} = 0,
$$
as some solutions will not be smooth (but will not be general distributions, for which one cannot define $u^{2}$).
In order to be consistent, $u$ must have the dimension of a velocity.
In 1948, B{\eightrm URGER} had proposed the equation
$$
{\partial u\over \partial t}+u{\partial u\over \partial x} -\varepsilon {\partial^{2}u\over \partial x^{2}} = 0,
$$
as a 1-dimensional model of turbulence, and apart from pointing out that turbulence was something very different, Eberhard H{\eightrm OPF} had been able to study the limiting case $\varepsilon\rightarrow 0$ by using a nonlinear transformation which changes the equation into the linear heat equation (that transformation is now known as the H{\eightrm OPF}-C{\eightrm OLE} transformation, as Julian C{\eightrm OLE} had also found it independently).
The work of Peter L{\eightrm AX} and of Olga O{\eightrm LEINIK} opened then the way for more general cases.
\par
If $a(x,t)$ is L{\eightrm IPSCHITZ} continuous in $x$, the solution of ${\partial u\over \partial t}+a{\partial u\over \partial x} = 0$ and $u(x,0) = v(x)$ is obtained by the method of characteristic curves, going back to C{\eightrm AUCHY}: along the solution of ${dx(t)\over dt} = a(x(t),t)$ and $x(0) = \xi$, the solution $u$ satisfies ${d\over dt}\bigl( u(x(t),t) \bigr) = 0$ and so $u(x(t),t) = v(\xi)$.
Assuming that the solution $u$ of B{\eightrm URGERS} equation is L{\eightrm IPSCHITZ} continuous in $x$ for $0\le t p>F(b)$, and 0 otherwise; it will also extend the case of a holomorphic function $F$ for a domain in $C$ bounded by a smooth J{\eightrm ORDAN} curve $\Gamma$, where the number is ${1\over 2i\pi}\int_{\Gamma} {F^{\prime}(z)\over F(z)-p}\,dz$ (always a nonnegative integer).
\par
The degree $deg(F,\Omega,p)$ is defined for any continuous function $F$ from $\overline{\Omega}$ into $R^{N}$ which does not take the value $p$ on $\partial\Omega$, and it depends only upon the restriction of $F$ to $\partial\Omega$.
It is an integer and if $deg(F,\Omega,p)\ne 0$ then there exists $x\in\Omega$ with $F(x) = p$.
If the degree is defined for $\Omega_{1}$ and for $\Omega_{2}$, and $\Omega_{1}\cup\Omega_{2}\subset\Omega\subset\overline{\Omega_{1}}\cup\overline{\Omega_{2}}$, then $deg(F,\Omega,p) = deg(F,\Omega_{1},p)+deg(F,\Omega_{2},p)-deg(F,\Omega_{1}\cap\Omega_{2},p)$.
The degree is invariant by homotopy, i.e. $deg(F,\Omega,p) = deg(G,\Omega,p)$ if there exists a homotopy between $F$ and $G$, i.e. there exists a continuous function $H$ from $\overline{\Omega}\times[0,1]$ into $R^{N}$, such that $H$ does not take the value $p$ on $\partial\Omega\times[0,1]$, and such that $H(x,0) = F(x)$ and $H(x,1) = G(x)$ on $\overline{\Omega}$.
In the case where $F$ is continuous from $\overline{\Omega}$ into $R^{N}$, of class $C^{1}$ in $\Omega$, and only takes the value $p$ at a finite number of points $a_{j}, j = 1,\ldots,r$, of $\Omega$, and if the Jacobian determinant of $F$ is nonzero at each of these points, then $deg(F,\Omega,p) = \sum_{j = 1}^{r} sign\bigl( det(\nabla F)(a_{j}) \bigr)$.
For instance, if $F(x) = x$ on $\partial\Omega$ then $deg(F,\Omega,p) = 1$ if $p\in\Omega$, 0 if $p\notin\overline{\Omega}$ (and is not defined if $p\in\partial\Omega$); if $F(x) = -x$ on $\partial\Omega$ then $deg(F,\Omega,p) = (-1)^{N}$ if $-p\in\Omega$, 0 if $-p\notin\overline{\Omega}$ (and is not defined if $-p\in\partial\Omega$).
\par
A first application is that there exists no nonzero continuous vector field from $S^{2}$ into $R^{3}$ which is everywhere tangent; more generally every nonzero continuous vector field from $S^{2N}$ into $R^{2N+1}$ is normal at (at least) one point of $S^{2N}$.
Indeed suppose that $F$ is a nonzero continuous vector field from $S^{2N}$ into $R^{2N+1}$ which is nowhere normal; then the homotopy $H$ defined by $H(x,t) = (1-t)F(x)+t\,x$ is not 0 on $S^{2N}$ so $deg(F,B(0,1),0) = deg(id,B(0,1),0) = 1$, and similarly the homotopy $K$ defined by $K(x,t) = (1-t)F(x)-t\,x$ is not 0 on $S^{2N}$ so $deg(F,B(0,1),0) = deg(-id,B(0,1),0) = -1$, a contradiction.
\par
A second application is that there does not exist a continuous retraction from a bounded open set $\Omega\subset R^{N}$ onto its boundary $\partial\Omega$, i.e. a continuous function $F$ from $\overline{\Omega}$ into $\partial\Omega$ such that $F(x) = x$ on $\partial\Omega$.
Indeed for $p\in\Omega$ one has $deg(F,\Omega,p) = deg(id,\Omega,p) = 1$ and therefore there exists $x\in \Omega$ with $F(x) = p$, contradicting the fact that the range of $F$ is inside $\partial\Omega$.
A consequence is B{\eightrm ROUWER} fixed point theorem: every continuous mapping $\Phi$ from the closed unit ball of $R^{N}$ into itself has at least one fixed point.
Let $\Omega = B(0,1)$, and assume that $\Phi$ has no fixed point in $\overline{\Omega}$; then for every $x\in \overline{\Omega}$ the line joining $x$ to $\Phi(x)$ is well defined and intersects $\partial\Omega$ in two points, and if one takes $F(x)$ to be that point on the side of $x$ one sees that $F$ is a continuous retraction from $\Omega$ onto its boundary.
Analytically, $F(x) = (1-t)x+t\Phi(x)$ with $t\le 0$ and $|(1-t)x+t\,\Phi(x)| = 1$, i.e. $t^{2}|x-\Phi(x)|^{2}+2t(x-\Phi(x).x)+|x|^{2}-1 =0$.
The theorem extends to any nonempty compact convex set $C\subset R^{N}$: one first restricts attention to the affine subspace generated by $C$, so that $C$ has a nonempty interior in that subspace, and one notices that a compact convex set with nonempty interior in $R^{M}$ is homeomorphic to the closed unit ball in $R^{M}$ (taking 0 inside $C$, and using the M{\eightrm INKOWSKI} functional $p_{C}$, the mapping $c\mapsto {p_{C}(c)\over ||c||}c$ is a homeomorphism of $C$ onto the closed unit ball).
\par
The result does not extend in infinite dimension to the closed unit ball of the H{\eightrm ILBERT} space $l^{2}$: one defines $\Phi$ by $\Phi(x) = (\sqrt{1-|x|^{2}},x_{1},\ldots)$ for $x = (x_{1},\ldots,x_{n},\ldots)$; then $\Phi$ is L{\eightrm IPSCHITZ} continuous, maps the closed unit ball into its boundary and has no fixed point; one can deduce that there exists a continuous retraction of the unit ball onto its boundary.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
18. Monday February 22.
\medskip
I had read about topological degree in some lecture notes of the C{\eightrm OURANT} Institute by J. T. S{\eightrm CHWARTZ} ``Nonlinear functional Analysis''; I did not find these notes very clear, and in 1974 I had simplified the exposition of the essential results by the following approach, based on the study of functionals $J_{\varphi}(u) = \int_{\Omega} \varphi(u)det(\nabla\,u)\,dx$.
[In the Fall of 1975 I learned from John M. B{\eightrm ALL} about null Lagrangians and the property that Jacobian determinants are sequentially weakly continuous and I immediately linked this kind of robustness to that encountered in the study of topological degree.]
\par
Although topological degree will be defined for some continuous functions, the integrals that we start with assume that the functions are of class $C^{1}$, and even to have two derivatives in some proofs; a density argument is then necessary for extending to continuous functions the results obtained.
\medskip
Let $\Omega$ be a bounded regular open set of $R^{N}$, i.e. whose boundary is given locally by a L{\eightrm IPSCHITZ} function so that the exterior normal $n$ to $\partial \Omega$ is defined almost everywhere on $\partial \Omega$ and the formula of integration by parts is valid (and uses the measure $d\sigma$ on $\partial \Omega)$.
All our functions are assumed to be continuously extended to $\partial \Omega$, so that they will be defined on $\overline{\Omega}$.
\par
Let $u$ be a $C^{1}$ function from $\overline{\Omega}$ into $R^{N}$ and $\varphi$ a continuous scalar function on $R^{N}$; we define the functional $J_{\varphi}$ by the formula
$$
J_{\varphi}(u)=\int_{\Omega}\varphi(u)det(\nabla\,u)\,dx,
$$
where $\nabla\,u(x)$ is the Jacobian matrix at the point $x$, whose entries are the partial derivatives ${\partial u_{i}\over\partial x_{j}}$.
Remark that the definition makes sense for functions $u$ in the S{\eightrm OBOLEV} space $W^{1,N}(\Omega;R^{N})$ and $\varphi$ bounded; Louis N{\eightrm IRENBERG} and Ha\"{\i}m B{\eightrm REZIS} have recently extended topological degree to functions in $VMO$ (Vanishing Mean Oscillation, the closure of $C^{\infty}$ functions in $BMO$).
The crucial property of the functional $J_{\varphi}$ is the following
\medskip
\noindent
{\bf Main Lemma}: Assume that $u$ is $C^{2}$ from $\overline\Omega$ into $R^{N}$ and that $\varphi$ is $C^{1}$ from $R^{N}$ into $R$; let $v$ be a $C^{1}$ function from $\overline\Omega$ into $R^{N}$, then
$$
{d\Bigl( J_{\varphi}(u+\varepsilon\,v) \Bigr)\over d\varepsilon}\Bigl|_{\varepsilon = 0} = \int_{\partial\Omega} \varphi(u)\Bigl( \sum_{k = 1}^{N} \psi_{k}n_{k} \Bigr)\,d\sigma,
$$
where $\psi_{k}$ is defined by
$$
\psi_{k} = det\Bigl( {\partial u\over \partial x_{1}},\ldots, {\partial u\over \partial x_{k-1}},v, {\partial u\over \partial x_{k+1}},\ldots, {\partial u\over \partial x_{N}} \Bigr),
$$
i.e. $\psi_{k}(x)$ is obtained from the Jacobian matrix $\nabla\,u(x)$ by replacing the $k^{\rm th}$ column by the vector $v(x)$.
\medskip
A consequence is that if $u$ and $w$ are $C^{1}$ functions from $\overline\Omega$ into $R^{N}$ which are equal on the boundary $\partial \Omega$ and if $\varphi$ is continuous, then $J_{\varphi}(u) = J_{\varphi}(w)$.
Indeed, by an argument of density, it is enough to prove the corollary when $u$ and $w$ are of class $C^{2}$ and $\varphi$ is of class $C^{1}$.
The lemma is used for computing the derivative of $J_{\varphi}\bigl( (1-\theta)u+\theta\,w \bigr)$ with respect to $\theta$ and it says that the derivative is equal to an integral on $\partial\Omega$, and this integral is 0 as the functions $\psi_{k}$ vanish on $\partial \Omega$ because $v$ is $w-u$, which is assumed to be 0 on the boundary.
\par
More generally one can change the values of $u$ on the boundary without changing the value of $J_{\varphi}(u)$ if one avoids the support of $\varphi$ and this gives the following property called invariance by homotopy.
\medskip
\noindent
{\bf Lemma}: Assume that $u$ and $w$ are $C^{1}$ functions from $\overline\Omega$ into $R^{N}$ and $\varphi\in C_{c}(R^{N})$.
We assume that $u$ and $w$ can be joined by a homotopy having the property that on the boundary $\partial\Omega$ it avoids the support of $\varphi$, then $J_{\varphi}(u) = J_{\varphi}(w)$.
(The hypothesis means that there exists a continuous function $F$ defined on $\overline\Omega\times[0,1]$ with values in $R^{N}$ such that $F(\cdot,0) = u$, $F(\cdot,1) = w$ on $\overline\Omega$ and $F(x,\theta)\not\in support(\varphi)$ for $(x,\theta)\in \partial\Omega\times[0,1])$.
\par
\noindent
{\it Proof}: Indeed one can regularize $u,w,\varphi$ and $F$ and still satisfy the same conditions; then one considers $G(\theta) = J_{\varphi}\bigl( F(\cdot,\theta) \bigr)$ and the lemma applies with $u$ replaced by $F(\cdot,\theta)$ and $v$ by ${\partial F\over\partial \theta}$: it says that $G^{\prime}(\theta)$ is equal to an integral on the boundary and the integrand is 0 because it contains a term $\varphi \bigl( F(\cdot,\theta) \bigr)$ which is 0 on the boundary, and therefore $G(0) = G(1)$ which is our assertion.
\medskip
The invariance by homotopy enables us to defines $J_{\varphi}(u)$ when $u$ is only continuous: if $u$ is a continuous function from $\overline\Omega$ into $R^{N}$ satisfying the condition $u(x)\not\in support(\varphi)$ when $x\in\partial\Omega$, then one can define $J_{\varphi}(u)$ by taking any sequence $v_{n}$ of $C^{1}$ functions from $\overline\Omega$ into $R^{N}$ which converges uniformly to $u$, because for $n$ large $J_{\varphi}(v_{n})$ is constant and $J_{\varphi}(u)$ is defined as this limiting value.
Indeed let $\varepsilon>0$ be small enough so that for $x\in \partial \Omega$ the distance of $u(x)$ to the support of $\varphi$ is at least $2\varepsilon$; if 2 functions $v_{n}$ and $v_{m}$ of class $C^{1}$ are in the ball of center $u$ and radius $\varepsilon$ in the $C^{0}$ distance, then they can be joined by the homotopy $(1-\theta )v_{n}+\theta v_{m}$ and then $J_{\varphi}(v_{n}) = J_{\varphi}(v_{m})$, so $J_{\varphi}(v)$ is constant in a ball around $u$.
\par
With this extension of the definition to some continuous functions, we can see that the preceding results are true for continuous functions.
\medskip
\noindent
{\bf Lemma}: If $u$ is a continuous function from $\overline\Omega$ into $R^{N}$ such that $J_{\varphi}(u)\ne 0$ (the condition $u(x)\not\in support(\varphi)$ for $x\in \partial\Omega$ being satisfied in order to define $J_{\varphi}(u)$), then there exists $x\in \Omega$ such that $u(x)\in support(\varphi)$.
\par
\noindent
{\it Proof}: As $J_{\varphi}(v_{n})\ne 0$ for large $n$, $\varphi(v_{n})$ cannot vanish identically and so there exists $x_{n}\in \Omega$ such that $v_{n}(x_{n})\in support(\varphi)$; every limit point $x\in\overline\Omega$ of the sequence $x_{n}$ is then such that $u(x)\in support(\varphi)$ and $x\not\in \partial\Omega$ by hypothesis.
\medskip
\noindent
{\it Proof of Main Lemma}: The derivative in $\varepsilon$ that we are considering is
$$
{d\Bigl( J_{\varphi}(u+\varepsilon\,v) \Bigr)\over d\varepsilon} \Bigl|_{\varepsilon = 0} = \int_{\Omega} \Bigl[ \sum_{i = 1}^{N} {\partial \Bigl( \varphi(u) \Bigr)\over \partial u_{i}}v_{i}\,det(\nabla\,u) + \varphi(u)\sum_{k = 1}^{N} H_{k} \Bigr]\,dx,
$$
where the functions $H_{k}$ are
$$
H_{k} = det\Bigl( {\partial u\over \partial x_{1}},\ldots, {\partial u\over \partial x_{k-1}},{\partial v\over \partial x_{k}}, {\partial u\over \partial x_{k+1}} ,\ldots, {\partial u\over \partial x_{N}}\Bigr),
$$
expressing the multilinearity of the determinant.
The Main Lemma will be proved by integration by parts if we show that
$$
\sum_{i = 1}^{N} {\partial \Bigl( \varphi(u) \Bigr)\over \partial u_{i}}v_{i}\,det(\nabla\,u) + \varphi(u) \sum_{k = 1}^{N} H_{k} = \sum_{k = 1}^{N} {\partial \Bigl( \varphi(u)\psi_{k} \Bigr)\over \partial x_{k}},
$$
and this will follow from the following two identities
$$
\sum_{k = 1}^{N} {\partial \psi_{k}\over \partial x_{k}} = \sum_{k = 1}^{N} H_{k}.
$$
and
$$
\sum_{i = 1}^{N} {\partial u_{i}\over \partial x_{k}}\,\psi_{k} = v_{i}\,det(\nabla\,u),
$$
The first identity requires $u$ to be of class $C^{2}$; once again the multilinearity of the determinant is used and we must show that the sum of the terms containing second derivatives of $u$ is 0.
Here it is the antisymmetry of the determinant that is needed because there are two terms showing a given second derivative ${\partial^{2}u\over\partial x_{i}\partial x_{k}}$: one has it in column $i$ with $v$ in column $k$ and the other has it in column $k$ with $v$ in column $i$, all the other columns being similar.
The second identity is linear in $v$, so we check it in the case where only one component of $v$, say $v_{1}$, is 1 and the others are 0; then the left hand side consists in developping with respect to the first row a determinant obtained from $\nabla\,u$ by replacing the first row by $grad(u_{i})$ so it gives $det(\nabla\,u)$ if $i = 1$ and, again from antisymmetry, it gives 0 if $i\ne 1$ because two rows are identical.
\medskip
The usual definition of the topological degree will consist in computing an algebraic number of solutions of $u(x)=p$ for a point $p\in R^{N}$ and it is obtained by letting the function $\varphi$ approach the D{\eightrm IRAC} mass at the point $p$.
We are led to the following definition.
\medskip
\noindent
{\bf Definition}: If $u$ is a continuous function from $\overline\Omega$ into $R^{N}$ satisfying the condition $u(x)\ne p$ when $x\in \partial\Omega$ then one can define $deg(u,\Omega,p)$ as the limit of the values $J_{\varphi_{n}}(u)$ for a sequence of functions $\varphi_{n}$ whose supports converge to the point $p$ and whose integrals converge to 1.
\medskip
Obviously one has $u(x)\not\in support(\varphi_{n})$ for $x\in \partial\Omega$ for large $n$ so that $J_{\varphi_{n}}(u)$ has a meaning, but one difficulty is to show that this limit exists; the proof actually gives an important property, namely that the degree $deg(u,\Omega,p)$ is always an integer, for the values $p$ for which this degree is defined, i.e. $p\not\in u(\partial\Omega)$.
The first step is to notice that there is a discrete formula for computing the degree in the case where $u$ is of class $C^{1}$ under a slight restriction.
\medskip
\noindent
{\bf Lemma}: Let $u$ be of class $C^{1}$ from $\overline\Omega$ into $R^{N}$ such that $\nabla\,u(z)$ is invertible at every point $z$ solution of $u(z) = p$, none of these solutions being on the boundary, so that there is only a finite number of them; then $deg(u,\Omega,p)$ is an integer, the sum of the signs of the Jacobian at all these points
$$
deg(u,\Omega,p) = \sum_{u(z_{\alpha }) = p}sign\Bigl( det\Bigl( \nabla\,u(z_{\alpha}) \Bigr)\Bigr).
$$
\par
\noindent
{\it Proof}: For $n$ large enough $\varphi_{n}$ is 0 except in small disjoint neighborhoods of the $z_{\alpha}$ solutions of $u(z_{\alpha}) = p$; aroung each $z_{\alpha }$ one can use a change of variable in the integral by taking $y = u(x)$ as the new variable: one then obtains a contribution $sign\bigl( det\bigl( \nabla u(z_{\alpha}) \bigr) \bigr)\int \varphi_{n}(y)\,dy$ and this gives our formula in the limit $n\rightarrow\infty$.
\medskip
The second step is to notice that every $C^{1}$ function from $\Omega$ into $R^{N}$ (and thus every continuous function from $\Omega$ into $R^{N})$ can be approximated by such special functions $u$; this is done by adding a small constant vector to $u$ and using S{\eightrm ARD}'s lemma which states that the set of critical values $p$ such that $\nabla u(x)$ is not invertible at some solution of $u(x)=p$ has measure 0 and so has an empty interior.
\par
The third step is to notice that one can extend all the properties of the functionals $J_{\varphi}$ to the topological degree $deg(u,\Omega,p)$ and in particular the invariance by homotopy.
Another easy consequence is that the degree is continuous in $p$ and, because it is an integer, it is locally constant in each connected component of the complement of $u(\partial \Omega)$ (that one needs to avoid in order to give a meaning to the definition).
\medskip
It is useful to notice that in practical situations one computes the degree by using a homotopy to a simple $C^{1}$ function for which one can obtain the degree explicitly using the formula, and so S{\eightrm ARD}'s lemma is only a technical tool used to show that the degree is defined for every continuous function (the proof of S{\eightrm ARD}'s lemma in the case of spaces of same dimension, which is the one that interests us here, is relatively easy: one covers the initial open set with small cubes and notices that around one critical point the image of the corresponding cube is inside a flat cylinder of much smaller volume; using the uniform continuity of derivatives one concludes that the set of critical values is covered by sets of arbitrarily small volume, and so has measure 0).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
19. Wednesday February 24.
\medskip
Let us look now at the evolution equations, first without nonlinearity for the S{\eightrm TOKES} equation, then with the nonlinearity for the N{\eightrm AVIER}-S{\eightrm TOKES} equation.
\par
I switch now to more traditional notations and denote
$$
V = \{u\in H^{1}_{0}(\Omega;R^{N}), div(u) = 0\hbox{ in }\Omega\},
$$
previously denoted $W$, and
$$
H = \{u\in L^{2}(\Omega;R^{N}), div(u) = 0\hbox{ in }\Omega, u.n = 0\hbox{ on }\partial\Omega\},
$$
for which we will have to show that $u.n$ makes sense on the boundary; we will also have to prove that $V$ is dense in $H$.
Before doing so, we start with some abstract results on evolution equations, where the spaces denoted $V$ or $H$ do not necessarily mean those above (which are adapted to the treatment of S{\eightrm TOKES} equation).
\medskip
There is an abstract theory for linear evolution equations, the theory of semi-groups, which was developped independently in the 40s by K\^osaku Y{\eightrm OSIDA} in Japan, and by H{\eightrm ILLE} and then Ralph P{\eightrm HILIPPS} in United States, but the theory has proved difficult to generalize to nonlinear equations, apart from situations where the maximum principle plays a role, which is usually not the case for equations of Continuum Mechanics.
One advantage of the theory is that it puts into the same framework lots of linear evolution equations with coefficients independent of $t$, but that is also a defect as it does not take into account the particular properties that the equations may have: transport equations ${\partial u\over \partial t}+\sum_{j} a_{j}{\partial u\over \partial x_{j}} = 0$, diffusion equations like the heat equation ${\partial u\over \partial t} -\sum_{ij}{\partial \over \partial x_{i}}\bigl( a_{ij}{\partial u\over \partial x_{j}} \bigr) = 0$, S{\eightrm CHR\"ODINGER} equations $i{\partial u\over \partial t} -\Delta\,u +V\,u = 0$, wave equations $\rho{\partial^{2} u\over \partial t^{2}} -\sum_{ij}{\partial \over \partial x_{i}}\bigl( a_{ij}{\partial u\over \partial x_{j}} \bigr) = 0$, the systems of linearized Elasticity, M{\eightrm AXWELL}, or S{\eightrm TOKES}, can all be considered in such an abstract framework.
The framework uses one B{\eightrm ANACH} space, which of course changes from an equation to the other, and it is sometimes an important restriction, because a good understanding of some equations from Continuum Mechanics often requires the use of more than one functional space: for S{\eightrm TOKES} equation, the bound on the kinetic energy corresponds to a bound in $L^{\infty}(0,T;H)$, while the bound on the energy dissipated by viscosity corresponds to a bound in $L^{2}(0,T;V)$.
Despite these shortcomings, I quickly sketch the main ideas of the semi-group approach.
\par
For an abstract evolution equation ${du\over dt}+A\,u = 0$, where $A$ is a partial differential operator, one cannot define $e^{-t\,A}$ by the usual series $\sum_{k = 0}^{\infty} {(-t)^{k}\over k!}A^{k}$, as in the case where $A\in{\cal L}(E,E)$ for a B{\eightrm ANACH} space $E$, and write then the solution as $u(t) = e^{-t\,A}u(0)$.
Nevertheless if one finds a way to define the solution in a unique way, one may expect that the mapping $u(0)\mapsto u(t)$ defines an operator $S(t)\in{\cal L}(E,E)$ for $t\ge 0$, satisfying $S(0) = I$ and $S(t_{1})S(t_{2}) = S(t_{1}+t_{2})$ (the semi-group property), and some sort of continuity in $t$ for $S(t)e$ for each $e\in E$, for example $S(t)e\rightarrow e$ in $E$ strong as $t\rightarrow 0$.
Given such a (strongly continuous) semi-group, the uniform boundedness principle implies that $||S(t)||$ is bounded for $t\in[0,1]$, and then that $||S(t)||\le M\,e^{\omega\,t}$ for $t\ge 0$; putting $u(t) = v(t)\,e^{\omega\,t}$ creates a bounded semigroup $S_{1}(t) = S(t)\,e^{-\omega\,t}$ for $v$, satisfying $||S_{1}(t)||\le M$ for all $t\ge 0$; in the equivalent norm $||e||_{1} = \sup_{t\ge 0} ||S_{1}(t)e||$, $S_{1}$ becomes a semi-group of contractions.
\par
The domain $D(A)$ of the infinitesimal generator $A$ of a (strongly continuous) semi-group $S$ is defined as the subspace of elements $e\in E$ for which $S(t)e$ has a derivative at $t = 0$, denoted $-A\,e$; one deduces that if $e\in D(A)$ then $S(t)e\in D(A)$ and its derivative is $-A\,S(t)e$, so that $S(t)$ does play the role of $e^{-t\,A}$.
One shows then that $D(A)$ is dense in $E$, and that $A$ is closed.
If $S(t)$ is a semi-group of contraction, one shows then that $I+\lambda\,A$ is invertible for $\lambda\ge 0$ with $||(I+\lambda\,A)^{-1}||\le 1$.
\par
Conversely, if a closed operator $A$ with dense domain is such that $I+\lambda\,A$ is invertible for $\lambda\ge 0$ with $||(I+\lambda\,A)^{-1}||\le 1$, then one can construct a semi-group $S$ of contractions, of which $A$ is the infinitesimal generator.
Without going into the details (what I am sketching is a simplified view of the H{\eightrm ILLE}-Y{\eightrm OSIDA} theorem), the idea is to consider the implicit approximation scheme ${u_{n+1}-u_{n}\over \Delta\,t}+A\,u_{n+1} = 0$, where $u_{n}$ serves as an approximation of $u(n\,\Delta\,t)$, and as $u_{n+1} = (I+\Delta\,t\,A)^{-1}u_{n}$, the way to use the bounds $||(I+\lambda\,A)^{-1}||\le 1$ for $\lambda\ge 0$ appears easily (the explicit scheme ${u_{n+1}-u_{n}\over \Delta\,t}+A\,u_{n} = 0$ requires $u_{n}\in D(A)$, and therefore one needs $u(0)\in D(A^{k})$ for all $k$ just for defining all the $u_{n}$, so this scheme is not of great use).
\medskip
I will present now a different framework, where two H{\eightrm ILBERT} spaces $V$ and $H$ are used (or three if one count $V^{\prime}$, $H^{\prime}$ being identified to its dual); this framework is adapted to solving diffusion equations, or S{\eightrm TOKES} equation, for example (in semi-group theory it is related to analytic
semi-groups, which can be extended for $t$ in a sector of the complex plane.
I have learned many of the results that I present from Jacques-Louis L{\eightrm IONS}, and I have only improved small technical details.
\par
Let $V$ and $H$ be two (real) H{\eightrm ILBERT} spaces, with norms $||\cdot||$ for $V$ and $|\cdot|$ for $H$, $V$ being continuously imbedded in $H$ and being dense in $H$; $H$ is identified to its dual $H^{\prime}$, which is continuously imbedded in $V^{\prime}$ and dense in $V^{\prime}$ (in some cases the identification of $H$ to its dual $H^{\prime}$ may create a few problems, and I will consider that question later).
Let $A\in {\cal L}(V,V^{\prime})$ be such that there exists $\alpha>0$ and $\beta\in R$ for which
$$
\langle A\,u,u \rangle\ge\alpha||u||^{2}-\beta|u|^{2}\hbox{ for all }u\in V,
$$
(for simplification, I assume that $A$ is independent of $t$; in practical situations one may have a bilinear continuous form $a(t,u,v)$ measurable in $t$).
\medskip
\noindent
{\bf Lemma}: Given $u_{0}\in H$, $f_{1}\in L^{1}(0,T;H)$ and $f_{2}\in L^{2}(0,T;V^{\prime})$, there exists a unique $u\in C([0,T];H)\cap L^{2}(0,T;V)$ with ${du\over dt}\in L^{1}(0,T;H) + L^{2}(0,T;V^{\prime})$, solution of
$$
{du\over dt}+A\,u = f_{1}+f_{2}\hbox{ in }(0,T);\;u(0) = u_{0},
$$
which in variational form means
$$
\eqalign{
&\int_{0}^{T} \Bigl( -{d\varphi\over dt}(u.v) + \varphi\langle A\,u,v \rangle \Bigr)\,dt = \varphi(0)(u_{0},v) +\int_{0}^{T} \varphi\Bigl( (f_{1}.v)+\langle f_{2},v \rangle \Bigr)dt\cr
&\hbox{for all }v\in V\hbox{ and all }\varphi\in C^{\infty}([0,T])\hbox{ satisfying }\varphi(T) = 0.}
$$
\medskip
Jacques-Louis L{\eightrm IONS} always considered $f\in L^{2}(0,T;V^{\prime})$, i.e. the case $f_{1} = 0$, which gives ${du\over dt}\in L^{2}(0,T;V^{\prime})$; because of the natural bounds $u\in L^{\infty}(0,T;H)\cap L^{2}(0,T;V)$, I find natural to take $f\in L^{1}(0,T;H) + L^{2}(0,T;V^{\prime})$.
\par
I will assume that $V$ is separable (and then $H$ is separable as $V$ is dense in $H$); this is not a restriction for applications, and it avoids some tecnical difficulties about measurability of functions with values in $V$, $H$ or $V^{\prime}$.
Let $e_{1},\ldots$ be a any (R{\eightrm ITZ}-) G{\eightrm ALERKIN} basis of $V$, and let $V_{m}$ be the subspace generated by $e_{1},\ldots,e_{m}$.
One looks for a function $u_{m}$ from $[0,T]$ into $V_{m}$, i.e. $u_{m}(t) = \sum_{i = 1}^{m} \xi_{mi}(t)e_{i}$, and the coefficients $\xi_{mi}$ will belong to $W^{1,1}(0,T)$, which is continuously imbedded in $C([0,T])$; one asks $u_{m}$ to satisfy
$$
\Bigl( {du_{m}\over dt}.e_{k} \Bigr) + \langle A\,u_{m},e_{k} \rangle = (f_{1}.e_{k})+\langle f_{2},e_{k} \rangle \hbox{ a.e. in }(0,T)\hbox{ and } (u_{m}(0).e_{k}) = (u_{0}.e_{k}) \hbox{ for }k = 1,\ldots,m.
$$
This is an ordinary linear differential equation in $R^{m}$, of the form $\xi^{\prime}+A_{m}\xi = \eta_{m}$ in $(0,T)$ and $\xi(0) = \xi_{0m}$, with $\xi_{0m}\in R^{m}$ and $\eta_{m}\in L^{1}(0,T;R^{m})$; it has a unique solution in $W^{1,1}(0,T;R^{m})$, which is given explicitly by $\xi(t) = e^{-t\,A_{m}}\xi_{0m} + \int_{0}^{t} e^{-(t-s)\,A_{m}}\eta_{m}(s)\,ds$ for $t\in[0,T]$; one may prefer to deal with classical $C^{1}$ solutions in $V_{m}$, and that consists in choosing $u_{0m}\in V_{m}$, $f_{1m}, f_{2m}\in C([0,T];V_{m})$ approaching in a strong or weak way $u_{0}$ in $H$, $f_{1}$ in $L^{1}(0,T;H)$ and $f_{2}$ in $L^{2}(0,T;V)$.
Because the equation is linear, we immediately know existence and uniqueness on the whole interval $[0,T]$, but when we will deal with a nonlinear equation like N{\eightrm AVIER}-S{\eightrm TOKES} equation, we will have to start with a local existence result and then we will show that the solution exists on $[0,T]$.
\par
We need now precise bounds (independent of $m$) in order to take the limit $m\rightarrow\infty$, and we will need some technical results.
\medskip
\noindent
{\bf Lemma}: i) $W^{1,1}(0,T)\subset C([0,T])$,
\par
ii) $u, v\in W^{1,1}(0,T)$ imply $u\,v\in W^{1,1}(0,T)$ and $(u\,v)^{\prime} = u\,v^{\prime} + u^{\prime}v$ a.e. in $(0,T)$,
\par
iii) G{\eightrm RONWALL} inequality: if $\varphi\in L^{\infty}(0,T)$ satisfies $\varphi(t)\ge 0$ a.e. on $(0,T)$ and $\varphi(t)\le\psi(t) = A+\int_{0}^{t} (\lambda_{1}\varphi+\lambda_{2})\,ds$ a.e. in $(0,T)$, where $\lambda_{1}, \lambda_{2}\in L^{1}(0,T)$, then $\psi(t)\le \bigl( A+\int_{0}^{t} |\lambda_{2}(s)|\,ds \bigr)exp\bigl( \int_{0}^{t} |\lambda_{1}(s)|\,ds \bigr)$ for $t\in[0,T]$.
\par
\noindent
{\it Proof}: For $f\in L^{1}(0,T)$, let $f_{n}\in C([0,T])$ converge to $f$ in $L^{1}(0,T)$, then $u_{n}\in C^{1}([0,T])$ defined by $u_{n}(t) = \int_{0}^{t} f_{n}(s)\,ds$ converges uniformly to $u$ defined by $u(t) = \int_{0}^{t} f(s)\,ds$, and as $u_{n}^{\prime}$ converges in the sense of distributions to $u^{\prime}$, one has $u^{\prime} = f$ a.e. in $(0,T)$; if $v\in W^{1,1}(0,T)$ has $v^{\prime} = f$, then $v-u$ has derivative 0 and is therefore a constant, showing that $v$ is continuous and that the constant is $v(0)$.
\par
The proof of i) has shown that $C^{1}([0,T])$ is dense in $W^{1,1}(0,T)$.
If $u_{n}\in C^{1}([0,T])$ with $u^{\prime}_{n}\rightarrow u^{\prime}$ in $L^{1}(0,T)$, and $v_{n}\in C^{1}([0,T])$ with $v^{\prime}_{n}\rightarrow v^{\prime}$ in $L^{1}(0,T)$, then $u_{n}v_{n}$ converges uniformly to $u\,v$ and $(u_{n}v_{n})^{\prime} = u_{n}^{\prime}v_{n}+u_{n}v_{n}^{\prime}$ converges in $L^{1}(0,T)$ to $u^{\prime}v+u\,v^{\prime}$, and to $(u\,v)^{\prime}$ in the sense of distributions, and therefore $(u\,v)^{\prime} = u^{\prime}v+u\,v^{\prime}$.
\par
As $\varphi\ge 0$, the inequality stays true if one replaces $\lambda_{1}$ by its absolute value, and one may then assume that $\lambda_{1}\ge 0$ a.e. on $(0,T)$.
Then $\psi^{\prime} = \lambda_{1}\varphi+\lambda_{2}\le\lambda_{1}\psi+\lambda_{2}$ a.e. on $(0,T)$.
As for the proof of i) and ii), if one defines $E$ by $E(t) = exp\bigl( -\int_{0}^{t} \lambda_{1}(s)\,ds \bigr)$ then $E\in W^{1,1}(0,T)$ and $E^{\prime} = -E\,\lambda_{1}$.
One deduces that $(E\,\psi)^{\prime}\le E\,\lambda_{2}$, so that $\psi(t)\le A\,exp\bigl( \int_{0}^{t} \lambda_{1}(s)\,ds \bigr) +\int_{0}^{t} \lambda_{2}(s) exp\bigl( \int_{s}^{t} \lambda_{1}(\sigma)\,d\sigma \bigr)\,ds$, from which the bound follows.
\medskip
For obtaining estimates on $u_{m}$, one replaces $e_{k}$ bu $u_{m}$, which is a linear combination of the $e_{k}$, and one obtains
$$
{1\over 2}{d|u_{m}|^{2}\over dt}+\alpha||u_{m}||^{2}-\beta|u_{m}|^{2}\le |f_{1m}|\,|u_{m}|+||f_{2m}||_{*}||u_{m}||\hbox{ a.e. in }(0,T).
$$
Using the inequalities $|f_{1m}|\,|u_{m}|\le {1\over 2}|f_{1m}|+{1\over 2}|f_{1m}|\,|u_{m}|^{2}$ and $||f_{2m}||_{*}||u_{m}||\le{\alpha\over 2}||u_{m}||^{2}+{1\over 2\alpha}||f_{2m}||_{*}^{2}$, gives then
$$
{d|u_{m}|^{2}\over dt}+\alpha||u_{m}||^{2}\le (2\beta+|f_{1m}|)|u_{m}|^{2}+{1\over \alpha}||f_{2m}||_{*}^{2}\hbox{ a.e. in }(0,T),
$$
and, forgetting for a while the term $\alpha||u_{m}||^{2}$, G{\eightrm RONWALL} inequality applies with $\varphi = |u_{m}|^{2}$ after integrating in $t$, and it gives the bound
$$
|u_{m}(t)|^{2}\le \Bigl( |u_{0m}|^{2}+{1\over \alpha}\int_{0}^{t} ||f_{2m}(s)||_{*}^{2}\,ds \Bigr)exp\Bigl( \int_{0}^{t} (2\beta+|f_{1m}(s)|)\,ds \Bigr)\hbox{ for }t\in[0,T],
$$
and then taking into account the term in $\alpha||u_{m}||^{2}$ gives
$$
\alpha\int_{0}^{T} ||u_{m}(t)||^{2}\,dt\le \Bigl( |u_{0m}|^{2}+{1\over \alpha}\int_{0}^{T} ||f_{2m}(t)||_{*}^{2}\,dt \Bigr)exp\Bigl( \int_{0}^{T} (2\beta+|f_{1m}(t)|)\,dt \Bigr)-|u_{0m}|^{2}.
$$
\medskip
These bounds are good enough for our purpose, but show a strange dependence with respect to the norm of $f_{1m}$, and one way to avoid it is to use linearity, i.e. to consider first the case $f_{1m} = 0$ for which the above bound is acceptable and then the case where $f_{2m} = 0$, where from the bound
$$
{1\over 2}{d|u_{m}|^{2}\over dt}+\alpha||u_{m}||^{2}-\beta|u_{m}|^{2}\le |f_{1m}|\,|u_{m}|\hbox{ a.e. in }(0,T),
$$
one forgets the term $\alpha||u_{m}||^{2}$ and one deduces
$$
{d|u_{m}|\over dt}\le\beta|u_{m}|+|f_{1m}|\hbox{ a.e. in }(0,T),
$$
giving
$$
|u_{m}(t)|\le |u_{0m}|e^{\beta\,t}+\int_{0}^{t} |f_{1m}(s)|e^{\beta(t-s)}\,ds,
$$
and giving the expected affine dependence in the norm of $f_{1m}$.
In our finite dimensional situation one does have $|u_{m}|\in W^{1,1}(0,T)$ and ${d|u_{m}|^{2}\over dt} = 2|u_{m}|{d|u_{m}|\over dt}$, but the argument would not work in infinite dimension, where it is better to consider $\sqrt{\varepsilon+|u_{m}|^{2}}$ for $\varepsilon>0$, and therefore
$$
{d\sqrt{\varepsilon+|u_{m}|^{2}}\over dt} = {1\over 2\sqrt{\varepsilon+|u_{m}|^{2}}}{d|u_{m}|^{2}\over dt}\le (\beta|u_{m}|+|f_{1m}|){|u_{m}|\over \sqrt{\varepsilon+|u_{m}|^{2}}}\le \beta\sqrt{\varepsilon+|u_{m}|^{2}}+|f_{1m}|\hbox{ a.e. in }(0,T),
$$
and in the bound obtained for $\sqrt{\varepsilon+|u_{m}|^{2}}$ one lets $\varepsilon$ tend to 0, or one gets the inequality for ${d|u_{m}|\over dt}$, which shows that $|u_{m}|\in BV(0,T)$.
\par
This way of getting bounds is not possible if one is dealing with a nonlinear equation, and another way to deal with the bounds is to use the (Y{\eightrm OUNG} inequality) $||f_{2m}||_{*}||u_{m}||\le{\alpha\over 2}||u_{m}||^{2}+{1\over 2\alpha}||f_{2m}||_{*}^{2}$, which gives
$$
{d|u_{m}|^{2}\over dt}+\alpha||u_{m}||^{2}\le 2\beta|u_{m}|^{2}+2|f_{1m}||u_{m}|+{1\over \alpha}||f_{2m}||_{*}^{2}\hbox{ a.e. in }(0,T),
$$
which, forgetting again the term $\alpha||u_{m}||^{2}$ for a while, gives
$$
|u_{m}(t)|^{2}\le |u_{0m}|^{2}+{1\over \alpha}\int_{0}^{t} ||f_{2m}(s)||_{*}^{2}\,ds + \int_{0}^{t} (2\beta|u_{m}(s)|^{2}+2|f_{1m}||u_{m}(s)|)\,ds,
$$
and then to use a variant of G{\eightrm RONWALL} inequality
$$
\varphi(t)\le \psi(t) = A+\int_{0}^{t} (2\mu_{1}\varphi+2\mu_{2}\sqrt{\varphi})\,ds
$$
which gives
$$
\psi^{\prime}\le 2|\mu_{1}|\psi+2|\mu_{2}|\sqrt{\psi},
$$
from which one gets
$$
\sqrt{\psi(t)}\le \Bigl( A+\int_{0}^{t} |\mu_{2}(s)|\,ds \Bigr)exp\Bigl( \int_{0}^{t} |\mu_{1}(s)|\,ds \Bigr),
$$
and therefore as $A = |u_{0m}|^{2}+{1\over \alpha}\int_{0}^{\tau} ||f_{2m}(t)||_{*}^{2}\,dt$ as long as $t\le\tau$, one deduces
$$
|u_{m}(t)|\le \Bigl( \sqrt{|u_{0m}|^{2}+{1\over \alpha}\int_{0}^{t} ||f_{2m}(s)||_{*}^{2}\,ds}+\int_{0}^{t} |f_{1m}(s)|)\,ds \Bigr)e^{\beta\,t}\hbox{ on }[0,T].
$$
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
20. Friday February 26.
\medskip
By taking $u_{0n}$ bounded in $H$, $f_{1n}$ bounded in $L^{1}(0,T;H)$ and $f_{2n}$ bounded in $L^{2}(0,T;V^{\prime})$, one has obtained a uniform bound for $u_{n}$ in $C([0,T];H)\cap L^{2}(0,T;V)$, and one can extract a subsequence $u_{p}$ converging to $u_{\infty}$ in $L^{\infty}(0,T;H)$ weak $\star$ and in $L^{2}(0,T;V)$ weak, i.e.
$$
\int_{0}^{T} (u_{p}.v)\,dt\rightarrow \int_{0}^{T} (u_{\infty}.v)\,dt\hbox{ for all }v\in L^{1}(0,T;H);\;\int_{0}^{T} \langle u_{p},v \rangle\,dt\rightarrow \int_{0}^{T} \langle u_{\infty},v \rangle\,dt\hbox{ for all }v\in L^{2}(0,T;V).
$$
If $u_{0n}$ converges weakly to $u_{0}$ in $H$, $f_{1n}$ converges weakly to $f_{1}$ in $L^{1}(0,T;H)$ and $f_{2n}$ converges weakly to $f_{2}$ in $L^{2}(0,T;V^{\prime})$, then one can take the limit as $p\rightarrow\infty$.
For $\varphi\in C^{\infty}([0,T])$ satisfying $\varphi(T) = 0$, one rewrites the term $\int_{0}^{T} \varphi\bigl( {du_{p}\over dt}.e_{k} \bigr)\,dt = -\int_{0}^{T} {d\varphi\over dt} (u_{p}.e_{k})\,dt -\varphi(0)(u_{0p}.e_{k})$, which enables us to obtain the limit equation
$$
-\int_{0}^{T} {d\varphi\over dt}(u_{\infty}.e_{k})\,dt +\int_{0}^{T} \varphi\langle A\,u_{\infty},e_{k} \rangle = \varphi(0)(u_{0}.e_{k}) +\int_{0}^{T} \varphi(f_{1}.e_{k})\,dt + \int_{0}^{T} \varphi\langle f_{2},e_{k} \rangle\,dt\hbox{ for all }k,
$$
and for all test functions $\varphi\in C^{\infty}([0,T])$ such that $\varphi(T) = 0$.
Using linearity one can replace $e_{k}$ by any linear combination of the elements of the basis, and then by density, by any element $v\in V$.
Putting $g_{2} = f_{2}-A\,u_{\infty}\in L^{2}(0,T;V^{\prime})$, one has
$$
-\int_{0}^{T} {d\varphi\over dt}(u_{\infty}.v)\,dt = \varphi(0)(u_{0}.v) +\int_{0}^{T} \varphi(f_{1}.v)\,dt + \int_{0}^{T} \varphi\langle g_{2},v \rangle\,dt\hbox{ for all }v\in V,
$$
and for all $\varphi\in C^{\infty}([0,T])$ such that $\varphi(T) = 0$.
This means that $(u_{\infty}.v)\in W^{1,1}(0,T)$, that its value at 0 is $(u_{0}.v)$ and that its derivative is $(f_{1}.v)+\langle g_{2},v \rangle$, but if one defines $u_{*}\in C([0,T];V^{\prime})$ by $u_{*}(t) = u_{0}+\int_{0}^{T} \bigl( f_{1}(s)+g_{2}(s) \bigr)\,ds$, then $\langle u_{*},v \rangle$ has the same properties than $(u_{\infty}.v)$ and therefore they are equal.
This shows that $u_{\infty}\in W^{1,1}(0,T;V^{\prime})$ with ${du_{\infty}\over dt} = f_{1}+g_{2}$, i.e. $u_{\infty}$ solves the equation ${du_{\infty}\over dt}+A\,u_{\infty} = f_{1}+f_{2}$ in $(0,T)$ and $u(0) = u_{0}$.
As we will show that this equation has a unique solution, all the sequence does converge weakly to $u_{\infty}$.
\medskip
Uniqueness, which could have been proved before proving existence, follows from the formula
$$
\Bigl\langle {du\over dt},u \Bigr\rangle = {1\over 2}{d|u|^{2}\over dt}\hbox{ in }(0,T),
$$
as ${du\over dt}+A\,u = 0$ implies then ${1\over 2}{d|u|^{2}\over dt}+\alpha||u||^{2}-\beta|u|^{2}\le 0$, and therefore $|u(t)|\le|u(0)|e^{\beta\,t}$, proving that $u = 0$ if $u(0) = 0$.
The formula is valid if $u\in W_{1}(0,T) = \{u\in L^{2}(0,T;V),{du\over dt}\in L^{1}(0,T;H)+L^{2}(0,T;V^{\prime})\}$, or in the smaller space $W(0,T) = \{u\in L^{2}(0,T;V),{du\over dt}\in L^{2}(0,T;V^{\prime})\}$, used by Jacques-Louis L{\eightrm IONS} for the case where $f_{1} = 0$.
The formula to be proved is true pointwise if $u\in C^{1}([0,T];V)$, and in weak formulation it can be written as $\int_{0}^{T} \varphi\bigl\langle {du\over dt},u \bigr\rangle\,dt = -{1\over 2}\int_{0}^{T} {d\varphi\over dt}|u|^{2}\,dt$ for all $\varphi\in C^{\infty}_{c}(0,T)$.
\par
We will first prove that $C^{\infty}([0,T];V)$ is dense in $W_{1}(0,T)$; then we will use the density in order to prove that $W_{1}(0,T)\subset C([0,T];H)$; then we will deduce that the formula is true in $W_{1}(0,T)$, because both sides of the variational formulation are continuous bilinear forms on $W_{1}(0,T)$ (the formula implies that $|u|^{2}\in W^{1,1}(0,T)$ for $u\in W_{1}(0,T)$).
First one notices that the space $W_{1}(0,T)$ is local, i.e. if $\psi\in C^{\infty}([0,T])$ and $u\in W_{1}(0,T)$ then $\psi\,u\in W_{1}(0,T)$, because $u$ being in $L^{2}(0,T;V)$ is automatically in $L^{1}(0,T;H)$ or in $L^{2}(0,T;V^{\prime})$.
Choosing $\theta\in C^{\infty}([0,T])$, equal to 1 on $[0,T/3]$ and 0 on $[2T/3,T]$, one can consider $\theta\,u$ as being 0 on $[T,\infty)$ and $(1-\theta)u$ as being 0 on $(-\infty,0]$.
One regularizes then $\theta\,u$ by convolution with a regularizing sequence with support on $(-\infty,0)$, and similarly one regularizes $(1-\theta)u$ by convolution with a regularizing sequence with support on $(0,\infty)$, and the usual properties of regularization show that $C^{\infty}([0,T],V)$ is dense in $W_{1}(0,T)$.
\par
We want to prove now that $W_{1}(0,T)$ is continuously imbedded in $C([0,T];H)$, and for that one only needs to show that there exists C such that $||u||_{C([0,T];H)}\le C||u||_{L^{2}(0,T;V)}+ C\bigl| \bigl| {du\over dt} \bigr| \bigr|_{L^{1}(0,T;H)+L^{2}(0,T;V^{\prime})}$ for all functions $u\in C^{\infty}([0,T];V)$.
The norm in $L^{1}(0,T;H)+L^{2}(0,T;V^{\prime})$ is the infimum of $||h_{1}||_{L^{1}(0,T;H)}+||h_{2}||_{L^{2}(0,T;V^{\prime})}$ over all the decompositions ${du\over dt} = h_{1}+h_{2}$ with $h_{1}\in L^{1}(0,T;H)$ and $h_{2}\in L^{2}(0,T;V^{\prime})$.
By reasoning on $\theta\,u$ and then $(1-\theta)u$, one may assume that $u$ is 0 at one end of the interval; for example, assuming that $u(0) = 0$ one has $|u(t)|^{2} = 2\int_{0}^{t} \bigl( {du\over dt}.u \bigr)\,ds = 2\int_{0}^{t} \bigl\langle {du\over dt},u \bigr\rangle\,ds = 2\int_{0}^{t} \bigl( (h_{1}.u) + \langle h_{2},u \rangle \bigr)\,ds$, from which one deduces
$$
\eqalign{
||u||^{2}_{C([0,T];H)}&\le 2||h_{1}||_{L^{1}(0,T;H)}||u||_{C([0,T];H)} + 2||h_{2}||_{L^{2}(0,T;V^{\prime})}||u||_{L^{2}(0,T;V)}\cr
&\le 2||h_{1}||^{2}_{L^{1}(0,T;H)} + {1\over 2}||u||^{2}_{C([0,T];H)} + 2||h_{2}||_{L^{2}(0,T;V^{\prime})}||u||_{L^{2}(0,T;V)},}
$$
and therefore
$$
||u||^{2}_{C([0,T];H)}\le 4||h_{1}||^{2}_{L^{1}(0,T;H)} + 4||h_{2}||_{L^{2}(0,T;V^{\prime})}||u||_{L^{2}(0,T;V)},
$$
and by taking the infimum on the decompositions ${du\over dt} = h_{1}+h_{2}$ with $h_{1}\in L^{1}(0,T;H)$ and $h_{2}\in L^{2}(0,T;V^{\prime})$, it proves the continuous imbedding of $W_{1}(0,T)$ into $C([0,T];H)$.
\par
This being done, all the terms of the weak formulation are now seen to be continuous on $W_{1}(0,T)$, and the formula is true by density.
\medskip
It must be noticed that one does not have in general ${du_{n}\over dt}$ bounded in $L^{1}(0,T;H)+L^{2}(0,T;V^{\prime})$.
In order to obtain bounds for ${du_{n}\over dt}$, one can either make time regularity hypotheses on $f_{1}$ and $f_{2}$ and a regularity hypothesis on $u_{0}$ (which corresponds to regularity in space variables in applications to partial differential equations), or use a special (R{\eightrm ITZ}-) G{\eightrm ALERKIN} basis.
\par
The first idea consists in noticing that formally $u^{\prime} = {du\over dt}$ satisfies ${du^{\prime}\over dt}+A\,u^{\prime} = f_{1}^{\prime}+f_{2}^{\prime}$ and $u^{\prime}(0) = f_{1}(0)+f_{2}(0)-A\,u_{0}$, which suggests that if $f_{1}\in W^{1,1}(0,T;H)$, $f_{2}\in W^{1,1}(0,T;V^{\prime})$ and $u_{0}\in V$ with $A\,u_{0}-f_{2}(0)\in H$, then $u^{\prime}\in W_{1}(0,T)$ and one can expect a bound on ${du_{n}\over dt}$.
Indeed, one can easily choose $f_{1n}\in C^{\infty}([0,T];H)$ and $f_{2n}\in C^{\infty}(0,T;V^{\prime})$ converging respectively to $f_{1}$ and $f_{2}$, so that $u_{n}\in C^{\infty}(0,T;V_{n})$, but one must be a little careful for the bound on $u^{\prime}_{n}(0)$; Jacques-Louis L{\eightrm IONS} taught the trick of taking $u_{0}$ as the first element of the basis (if it is not 0), so that one can take $u_{0n} = u_{0}$ and then one asks that $f_{2n}(0)-f_{2}(0)$ converges weakly to 0 in $H$.
\par
It is useful to notice that if $f_{1}\in W^{1,1}(0,T;H)$, $f_{2}\in W^{1,1}(0,T;V^{\prime})$, but $u_{0}\in H$ only, then one does not obtain $u^{\prime}\in W_{1}(0,T)$ by lack of the needed regularity on $u_{0}$, but one has $t\,u^{\prime}\in W_{1}(0,T)$, as $v = t\,u^{\prime}$ satisfies ${dv\over dt}+A\,v = t{df_{1}\over dt}+t{df_{2}\over dt}+u^{\prime}$ and $v(0) = 0$.
This is a form of regularization effect for the solutions of the equation, already apparent from the fact that one does not need $u_{0}\in V$ in order to have the solution taking its values in $V$.
For obtaining the corresponding estimate for $t\,u^{\prime}_{n}$, it is better to use $w = t\,u^{\prime}-u$, which satisfies ${dw\over dt}+A\,w = t{df_{1}\over dt}+t{df_{2}\over dt}-A\,u$ and $v(0) = 0$, and the corresponding bounds for $t{du_{n}\over dt}-u_{n}$ are obtained easily.
\par
The choice of a special (R{\eightrm ITZ}-) G{\eightrm ALERKIN} basis is a different trick, and we will use it for N{\eightrm AVIER}-S{\eightrm TOKES} equation (at least in dimension 3, as the case of dimension 2 can be handled more easily), but it requires the symmetry of $A$, or simply $A^{T}-A\in{\cal L}(V,H)$, and the compact injection of $V$ into $H$.
One assumes that $A = A_{0}+B$ with $A_{0}$ symmetric $V$-elliptic and $B\in{\cal L}(V,H)$.
As $A_{0}$ is an isomorphism from $V$ onto $V^{\prime}$, its inverse $A_{0}^{-1}$ maps $V^{\prime}$ and therefore $H$ into $V$, so $A_{0}^{-1}$ is a compact operator on $H$, and as it is symmetric, R{\eightrm IESZ} theory asserts that $H$ has an orthonormal basis made of eigenvectors of $A_{0}^{-1}$, $e_{n}, n\ge 1$, with real positive eigenvalues $\mu_{n}$ converging to 0.
Therefore $A_{0}\,e_{n} = \lambda_{n}e_{n}$ with $\lambda_{n} = {1\over \mu_{n}}$ tending to $+\infty$, and if one replaces the norm $||u||$ on $V$ by the equivalent norm $\sqrt{\langle A_{0}u,u \rangle}$, then the basis is also orthogonal in $V$, and therefore it is also orthogonal in $V^{\prime}$.
The estimate for ${du_{n}\over dt}$ comes easily once one has observed that for a finite linear combination $v = \sum_{i} v_{i}e_{i}$, one has $||v||^{2} = \sum_{i} \lambda_{i}|v_{i}|^{2}$, $|v|^{2} = \sum_{i} |v_{i}|^{2}$, and $||v||_{*}^{2} = \sum_{i} {1\over \lambda_{i}}|v_{i}|^{2}$.
\par
It is not necessary to take this special basis of eigenvectors in order to deduce estimates on ${du_{n}\over dt}$, and a more general condition is obtained in the following way.
Let $P_{n}$ be the orthogonal projection of $H$ onto the (closed finite dimensional) subspace $V_{n}$, where orthogonality is understood in the scalar product of $H$, so that $P_{n}$ is a contraction if one uses the norm of $H$ for $V_{n}$; let $C_{n}$ be the norm of $P_{n}$ considered as a mapping from $V$ onto $V_{n}$ equipped with the norm of $V$; then the basis is special enough in order to obtain a bound for ${du_{n}\over dt}$ if $C_{n}$ is bounded (the choice of eigenvectors of $A_{0}$ gives $C_{n} = 1$ if one uses $\sqrt{\langle A_{0}\cdot,\cdot \rangle}$ for the norm on $V$).
Indeed, if $k\in V^{\prime}$ and $k_{n}\in V_{n}$ is defined by $(k_{n}.v) = \langle k,v \rangle$ for all $v\in V_{n}$, then for $w\in V$, one has $\langle k_{n},w \rangle = (k_{n},w) = (k_{n}.P_{n}w) = \langle k,P_{n}w \rangle = \langle P_{n}^{T}k,w \rangle$, so that $||k_{n}||_{*}\le C_{n}||k||_{*}$.
\medskip
There are interesting results of uniqueness which are not based on the ellipticity of $A$ but on its symmetry, and they can be used as well for $A$ or $-A$ with data at 0, or for $A$ with either initial data at 0 or final data at $T$ (one talks of backward uniqueness then).
One approach, due to Shmuel A{\eightrm GMON} and Louis N{\eightrm IRENBERG} consists in proving that if $u$ is a non vanishing solution of $u^{\prime}+A\,u = 0$, then $|u|$ is log-convex, i.e. $t\mapsto\log|u(t)|$ is convex.
Indeed, the derivative of $\log|u|$ is ${(u^{\prime}.u)\over |u|^{2}}$, whose derivative is ${(u^{\prime\prime}.u)+|u^{\prime}|^{2}\over |u|^{2}} - 2{(u^{\prime}.u)^{2}\over |u|^{4}}$, and as $(u^{\prime\prime}.u) = (A^{2}u.u) = |A\,u|^{2} = |u^{\prime}|^{2}$, one concludes by using C{\eightrm AUCHY}-S{\eightrm CHWARTZ} inequality.
In the early 70s I extended this method with Claude B{\eightrm ARDOS}, and we could apply it to N{\eightrm AVIER}-S{\eightrm TOKES} equation in two dimensions.
I have noticed that the log-convexity property is true if $A$ is normal (i.e. $A$ commutes with $A^{T}$, or $|A^{T}\,v| = |A\,v|$ for all $v$), and it is equivalent in finite dimension, while Shmuel F{\eightrm RIEDLAND} has noticed that a sufficient condition is $|A^{T}\,v|\le|A\,v|$ for all $v$, which may happen without equality in infinite dimension.
There is a second approach, by Jacques-Louis L{\eightrm IONS} and Bernard M{\eightrm ALGRANGE}, based on C{\eightrm ARLEMAN} estimates.
I have also introduced another approach, which is useful for improving the localization of the trajectory which results from the log-convexity property: if the solution exists on $[0,T]$, then for $0\le\tau_{1}<\tau_{2}\le T$, the trajectory for $t\in (\tau_{1},\tau_{2})$ lies inside the closed ball with diameter the segment $[u(\tau_{1}),u(\tau_{2})]$; this could certainly be more useful if one knew how to prove similar results for nonlinear equations.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
21. Wednesday March 3.
\medskip
For applying the preceding abstract framework to S{\eightrm TOKES} equation, there are questions about the functional spaces.
We are already familiar with $V$ (which was denoted $W$ in the stationary case), but we have to identify its closure in $L^{2}(\Omega;R^{N})$, which is the space $H$ of the abstract theory.
As we will see later, $H = \{u\in L^{2}(\Omega;R^{N}), div(u) = 0$, and $u.n = 0$ on $\partial\Omega\}$, and we will have to explain the meaning of $u.n$ on the boundary, the normal trace of $u$ (physically, $u.n = 0$ means that the flow is tangent, the so called slip condition).
\par
There is however another question which is about the ``pressure'': have we really solved the S{\eightrm TOKES} equation
$$
\eqalign{
&{\partial u_{i}\over \partial t}-\nu\Delta\,u_{i}+{\partial p\over \partial x_{i}} = f_{i}, i = 1,\ldots,N,\hbox{ in }\Omega\cr
&div(u)\hbox{ in }\Omega\cr
&u(\cdot,0) = u_{0}\hbox{ in }\Omega?}
$$
Certainly, if $f\in L^{2}\bigl( 0,T;H^{-1}(\Omega;R^{N}) \bigr)$ and if the solution satisfies $u\in L^{2}(0,T;V)\cap C^{0}([0,T];H)$, ${\partial u\over \partial t}\in L^{2}\bigl( 0,T;H^{-1}(\Omega;R^{N}) \bigr)$ and $p\in L^{2}\bigl( 0,T;L^{2}(\Omega) \bigr)$, then using a test function $v\in V$, one deduces that
$$
{d(u.v)\over dt} + \nu\,a(u,v) = \langle f,v \rangle\hbox{ in }(0,T),
$$
and with the data $u_{0}\in H$, one has a solution of the abstract problem, solution which we know to be unique.
The question is: can we deduce from the abstract formulation that ${\partial u\over \partial t}\in L^{2}\bigl( 0,T;H^{-1}(\Omega;R^{N}) \bigr)$ and $p\in L^{2}\bigl( 0,T;L^{2}(\Omega) \bigr)$?
Of course, as one can add to $p$ an arbitrary function of $t$ without changing the equation (which is quite unphysical, but is the price to pay for the unrealistic hypothesis of incompressibility), one must normalize $p$ by asking for example that $\int_{\Omega} p(x,t)\,dx = 0$ in $(0,T)$.
\par
If $f_{1} = 0$, the abstract formulation has given ${du\over dt} = g\in L^{2}(0,T;V^{\prime})$, and the problem comes from the fact that $V^{\prime}$ is not a space of distributions in $\Omega$, as $C^{\infty}_{c}(\Omega;R^{N})$ is certainly not dense in $V$, as it is not even included in $V$ because of the constraint $div(u) = 0$.
I have mentioned that for a bounded open set $\Omega$ with L{\eightrm IPSCHITZ} boundary the elements of $H^{-1}(\Omega;R^{N})$ orthogonal to $V$ have the form $grad(q)$ with $q\in L^{2}(\Omega)$ (I have deduced it in the case where $meas(\Omega)<\infty$ from $X(\Omega) = L^{2}(\Omega)$, but I have not proved that last assertion).
For any $L\in V^{\prime}$, one can solve for $w_{L}\in V$, unique solution of $a(w_{L},v) = L(v)$ for every $v\in V$, and even without the interpretation of this equation using the gradient of a pressure, one sees that $-\Delta\,w_{L}\in H^{-1}(\Omega;R^{N})$, that it defines the same linear form than $L$ on $V$, and that $||w_{L}|| = ||L||_{*}$.
The element $g\in L^{2}(0,T;V^{\prime})$ can be transformed in this way into a $w_{g}\in L^{2}(0,T:V)$ and for every $v\in V$ and every $\varphi\in C^{\infty}_{c}(0,T)$, one has
$$
-\int_{0}^{T} (u(t).v){d\varphi\over dt}\,dt = \int_{0}^{T} \Bigl\langle {du(t)\over dt}.v \Bigr\rangle\varphi\,dt = \int_{0}^{T} a(w_{g},v)\varphi\,dt.
$$
One defines then $W_{g}\in H^{1}(0,T;V)$ by
$$
W_{g}(t) = \int_{0}^{t} w_{g}(s)\,ds,
$$
and as $\int_{0}^{T} a(w_{g},v)\varphi\,dt = -\int_{0}^{T} a(W_{g},v){d\varphi\over dt}\,dt$ for $\varphi\in C^{\infty}_{c}(0,T)$, one deduces that $(u(t).v)-a(W_{g},v)$ is a constant, and therefore taking $t = 0$ one has
$$
(u(t).v) = (u_{0}.v)+a(W_{g},v)\hbox{ in }(0,T),\hbox{ for every }v\in V.
$$
As $u-u_{0}+\Delta\,W_{g}\in C^{0}\bigl( 0,T;H^{-1}(\Omega;R^{N}) \bigr)$, it shows that
$$
u-u_{0}+\Delta\,W_{g} = grad(q)\hbox{ for some }q\in C^{0}\Bigl( 0,T;L^{2}(\Omega) \Bigr),
$$
Taking the derivative in $t$ (in the sense of distributions), one finds that
$$
{\partial u\over \partial t} +\Delta\,w_{g} = grad(p),\hbox{ but }p = {\partial q\over \partial t}.
$$
In order to avoid having the pressure in a space of distributions, one can use a regularity theorem.
\medskip
\noindent
{\bf Lemma} Assume that $A^{T} = A$.
If $u_{0}\in V$ and $f\in L^{2}(0,T;H)$, then ${\partial u\over \partial t}, A\,u\in L^{2}(0,T;H)$ and $u\in C^{0}([0,T];V)$.
If $u_{0}\in H$ and $\sqrt{t}\,f\in L^{2}(0,T;H)$, then $\sqrt{t}\,{\partial u\over \partial t}, \sqrt{t}\,A\,u\in L^{2}(0,T;H)$ and $\sqrt{t}\,u\in C^{0}([0,T];V)$.
\par
\noindent
{\it Proof}: Formally, one can multiply the equation by ${\partial u\over \partial t}$ or by $A\,u$, and one gets either $|u^{\prime}|^{2}+{1\over 2}a(u,u)^{\prime} = (f.u^{\prime})$ or ${1\over 2}a(u,u)^{\prime} +|A\,u|^{2} = (f.A\,u)$, and each implies $a(u,u)^{\prime}\le {1\over 2}|f|^{2}$, giving the bound of $u$ in $L^{\infty}(0,T;V)$; then one gets either a bound for $u^{\prime}$ or a bound for $A\,u$ in $L^{2}(0,T;H)$, the other bound being given by the equation.
Multiplying by $u^{\prime}$ can be done on the (R{\eightrm ITZ}-) G{\eightrm ALERKIN} approach, but not multiplying by $A\,u$ unless one uses a special basis; the same estimates are obtained in the finite dimensional case, and the limit inherits of these bounds, but there is a little work necessary in order to improve $u\in L^{\infty}(0,T;V)$ into $u\in C^{0}([0,T];V)$.
For example, if $f\in H^{1}(0,T;H)$ and $u_{0}\in D(A)$, then the hypotheses for time regularity are satisfied and $u\in H^{1}(0,T;V)\subset C^{0}([0,T];V)$, one concludes by a density argument.
\par
The regularizing effect in the case where one only has $u_{0}\in H$, is obtained by multiplying by $t\,u^{\prime}$ or $t\,A\,u$, the first one being more adapted to the (R{\eightrm ITZ}-) G{\eightrm ALERKIN} approach.
\medskip
In the case where $A^{T}\ne A$, one has the same result by replacing $u_{0}\in V$ by $u_{0}\in[D(A),H]_{1/2} = (D(A),H)_{1/2,2}$; however, Jacques-Louis L{\eightrm IONS} has shown that if $D(A^{T}) = D(A)$, then the interpolation space mentioned is actually $V$.
\medskip
For the application to S{\eightrm TOKES} equation (or to N{\eightrm AVIER}-S{\eightrm TOKES} equation), one must be careful about the strange consequences of having identified $H$ and its dual, as this identification is not compatible with the usual basic identification of $L^{2}(\Omega)$ with its dual.
The hypothesis $f\in L^{2}(0,T;H)$ actually means $f\in L^{2}(0,T;H^{\prime})$, so that $f\in L^{2}\bigl( 0,T;L^{2}(\Omega;R^{N}) \bigr)$ is actually possible, without imposing $div(f) = 0$, which is one condition for taking values in $H$.
One way to think about that question is to remember that gradients have no effect on $V$ or $H$ (if $p\in H^{1}(\Omega)$) and therefore any element of the form $h+grad(p)$ with $h\in H$ and $p\in H^{1}(\Omega)$ belongs to $H^{\prime}$; when we will study the space $H$, we will actually prove that the orthogonal of $H$ in $L^{2}(\Omega;R^{N})$ is $\{grad(p),p\in H^{1}(\Omega)\}$.
When we interpret $A\,u\in L^{2}(0,T;H)$, it means $L^{2}(0,T;H^{\prime})$, because $A\,u$ is only defined through the bilinear form $a(u,v)$ with $v\in V$ (although with a constant viscosity, the divergence of $\Delta\,u$ is 0, the normal trace is not 0 in general).
However, when we interpret $u^{\prime}\in L^{2}(0,T;H)$, it does mean $H$ and not $H^{\prime}$, as $u$ takes values in $V\subset H$, and $u^{\prime}$ is a limit of ${u(t+h)-u(t)\over h}$, which take values in $V$.
\par
If $f\in L^{2}\bigl( 0,T;L^{2}(\Omega;R^{N}) \bigr)$ and $u_{0}\in V$, then one has $u^{\prime}\in L^{2}(0,T;H)\subset L^{2}\bigl( 0,T;L^{2}(\Omega;R^{N}) \bigr)$, and therefore $S = {\partial u\over \partial t}-\nu\Delta\,u-f\in L^{2}\bigl( 0,T;H^{-1}(\Omega;R^{N}) \bigr)$.
As $\int_{0}^{T} \varphi\langle S(t),v \rangle\,dt = 0$ for all $v\in V$ and all $\varphi\in C^{\infty}_{c}(0,T)$, one deduces that for almost every $t\in(0,T)$, $S(t)$ is orthogonal to $V$ (using the separability of $V$), and therefore $S(t) = -grad\bigl( p(t) \bigr)$ with $p(t)\in L^{2}(\Omega)$; if one normalizes $p(t)$ by adding a constant so that its integral in $\Omega$ is 0, one has $||p(t)||_{L^{2}(\Omega)}\le C||S(t)||_{H^{-1}(\Omega;R^{N})}$ and therefore $S = -grad(p)$ with $p\in L^{2}\bigl( 0,T;L^{2}(\Omega) \bigr)$.
\medskip
Another case where the pressure can be estimated easily is the case $\Omega = R^{N}$, where one can use F{\eightrm OURIER} transform (in $x$ alone): the S{\eightrm TOKES} equation becomes
$$
\eqalign{
&{\partial{\cal F}u\over \partial t}+4\nu\pi^{2}|\xi|^{2}{\cal F}u+2i\pi\xi{\cal F}p = {\cal F}f\hbox{ in }R^{N}\times(0,T)\cr
&({\cal F}u.\xi) = 0\hbox{ in }R^{N}\times(0,T)\cr
&{\cal F}u(\cdot,0) = {\cal F}u_{0},}
$$
and taking the scalar product with $\xi$ gives
$$
{\cal F}p = {1\over 2i\pi}{({\cal F}f.\xi)\over |\xi|^{2}}\hbox{ in }R^{N}\times(0,T).
$$
Of course, as P{\eightrm OINCAR\'E} inequality does not hold for $R^{N}$, one must be careful: if $f\in L^{2}\bigl( 0,T;L^{2}(R^{N};R^{N}) \bigr)$, then one finds a bound for $grad(p)$, but not for $p$, and therefore one does not find that $p$ takes values in $H^{1}(R^{N})$, but in a different space (which has been studied first by Jacques D{\eightrm ENY} and Jacques-Louis L{\eightrm IONS}, and present a particular difficulty for $N = 2$); however if
$$
f_{i} = \sum_{j = 1}^{N} {\partial g_{ij}\over \partial x_{j}}\hbox{ with }g_{i,j}\in L^{2}\bigl( 0,T;L^{2}(R^{N}) \bigr), i, j = 1,\ldots,N,\hbox{ in }R^{N}\times(0,T),
$$
then
$$
p\in L^{2}\Bigl( 0,T;L^{2}(R^{N} \Bigr)\hbox{ with } ||p(\cdot,t)||_{L^{2}(R^{N})}\le C\sum_{i,j = 1}^{N} ||g_{ij}(\cdot,t)||_{L^{2}(R^{N})}\hbox{ a.e. }t\in(0,T).
$$
\bigskip
For N{\eightrm AVIER}-S{\eightrm TOKES} equation, the dimension $N$ will play a very important role, more important than for the stationary case.
For $N = 2$, we will be able to prove an existence and uniqueness result.
For $N\ge 3$, we will prove existence of weak solutions defined on $(0,T)$, but the uniqueness of weak solutions is an open question; for smooth data, we will also prove that strong solutions exist locally, and that they are unique, but it is an open question to show that they can be extended up to $T$; however, for small smooth data the strong solution will exist globally.
\par
It should be noticed, however, that the approach for proving existence goes absolutely against physical intuition: there is a transport operator
$$
{D\over Dt} = {\partial \over \partial t}+\sum_{j} u_{j}{\partial \over \partial x_{j}},
$$
and there are various physical quantities transported along the flow, like mass, momentum, energy (or vorticity, helicity, thermodynamic entropy); each component of the velocity satisfies an equation
$$
\Bigl( {D\over DT}-\nu\Delta \Bigr)u_{i} = f_{i}-{\partial p\over \partial x_{i}}\hbox{ in }\Omega\times(0,T),
$$
and the operator ${D\over DT}-\nu\Delta$ which is applied to each $u_{i}$ has good properties, some of the bound using the maximum principle and requiring little smoothness of the coefficients $u_{j}$, but as ${\partial p\over \partial x_{i}}$ is needed and the equations are coupled via $div(u) = 0$, it would be useful to have an equation for $p$; taking the divergence of the equation gives
$$
-\Delta\,p = -div(f)+\sum_{i,j = 1}^{N} {\partial u_{i}\over \partial x_{j}}{\partial u_{j}\over \partial x_{i}},
$$
where one has used $div(u) = 0$ for simplifying the divergence of the nonlinear term.
The difficulty comes from the fact that one does not have adequate boundary conditions for $p$.
The nonlinearity appearing in the equation for $p$ is actually a little special, with slightly better bounds than expected.
\par
The usual approach, however, does not work with the operator ${D\over DT}-\nu\Delta$, but cuts the operator ${D\over DT}$ into two parts: sending the nonlinear term to play with $f$, one considers N{\eightrm AVIER}-S{\eightrm TOKES} equation as a perturbation of S{\eightrm TOKES} equation, and this is obviously not a good idea, but no one has really found how to do better yet.
\medskip
We have seen in studying the stationary N{\eightrm AVIER}-S{\eightrm TOKES} equations that the nonlinear operator $B$ defined by
$$
\langle B(u,v),w \rangle = \int_{\Omega} u_{j}{\partial v_{i}\over \partial x_{j}}w_{i}\,dx,
$$
is continuous from $V\times V$ into $V^{\prime}$ for $N\le 4$, and satisfies $\langle B(u,v),w \rangle+\langle B(u,w),v \rangle = 0$ (in particular $\langle B(u,v),v \rangle = 0$, but for the evolution problem we will need more precise bounds, and as $||B(u,v)||_{V^{\prime}}\le C\sum_{ij} ||u_{j}v_{i}||_{L^{4}(\Omega)}$, one deduces
$$
\eqalign{
&||B(u,u)||_{*}\le C ||u||^{2}\hbox{ in dimension }N = 4,\cr
&||B(u,u)||_{*}\le C ||u||^{3/2}|u|^{1/2}\hbox{ in dimension }N = 3,\cr
&||B(u,u)||_{*}\le C ||u||\,|u|\hbox{ in dimension }N = 2.}
$$
The first two inequalities follow from S{\eightrm OBOLEV} imbedding theorem $H^{1}_{0}(\Omega)\subset L^{4}(\Omega)$ in dimension $N = 4$, $H^{1}_{0}(\Omega)\subset L^{6}(\Omega)$ in dimension $N = 3$, the second using also H{\eightrm \"OLDER} inequality $||v||_{L^{4}}\le ||v||_{L^{6}}^{3/4}||u||_{L^{2}}^{1/4}$.
The third inequality, attributed to Olga L{\eightrm ADYZHENSKAYA}, uses the same method with which Emilio G{\eightrm AGLIARDO} and Louis N{\eightrm IRENBERG} proved S{\eightrm OBOLEV} imbedding theorem: $|u|^{2}\le F(x_{2}) = \int_{R} |u|\,\bigl| {\partial u\over \partial x_{1}} \bigr|\,dx_{1}$ and $|u|^{2}\le G(x_{1}) = \int_{R} |u|\,\bigl| {\partial u\over \partial x_{2}} \bigr|\,dx_{2}$, and therefore $\int_{R^{2}} |u|^{4}\,dx\le C\int_{R^{2}} |u|\,\bigl| {\partial u\over \partial x_{1}} \bigr|\,dx\int_{R^{2}} |u|\,\bigl| {\partial u\over \partial x_{1}} \bigr|\,dx$, which gives the desired result by using C{\eightrm AUCHY}-S{\eightrm CHWARZ} inequality.
\par
The natural bounds for a solution are $u\in L^{\infty}(0,T;H)$, which corresponds to the fact that the kinetic energy is bounded, and $u\in L^{2}(0,T;V)$, which corresponds to the fact that the energy dissipated by viscosity between time 0 and $T$ is bounded.
The dependence with $N$ becomes then
$$
u\in L^{2}(0,T;V)\cap L^{\infty}(0,T;H)\hbox{ imply } B(u,u)\in
\cases{
L^{1}(0,T;V^{\prime})\hbox{ in dimension }N = 4,\cr
L^{4/3}(0,T;V^{\prime})\hbox{ in dimension }N = 3,\cr
L^{2}(0,T;V^{\prime})\hbox{ in dimension }N = 2,}
$$
and therefore it is only for $N = 2$ that $B(u,u)$ falls into a space which is allowed for the (abstract) S{\eightrm TOKES} equation; for $N\ge 3$, the nonlinearity is then a much too strong nonlinear operator, and N{\eightrm AVIER}-S{\eightrm TOKES} equation is then not a mere perturbation of S{\eightrm TOKES} equation.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
22. Friday March 5.
\medskip
For $\Omega\subset R^{N}$, $V =\{u\in H^{1}_{0}(\Omega;R^{N}), div(u) = 0$ in $\Omega\}$ and $H = \{u\in L^{2}(\Omega;R^{N}), div(u) = 0$ in $\Omega$ and $(u.n) = 0$ on $\partial\Omega\}$.
We will see a little later the meaning of $(u.n)$, where $n$ is the exterior normal to $\partial\Omega$.
If $V$ is dense in $H$ (which we will show if $\Omega$ is bounded with $\partial\Omega$ smooth enough), then one can consider the (incompressible) N{\eightrm AVIER}-S{\eightrm TOKES} equation as an abstract evolution equation $u^{\prime} +B(u,u)+A\,u = f$ in $(0,T)$, where $A\in{\cal L}(V,V^{\prime})$ is given by $\langle A\,u,v \rangle = \nu\int_{\Omega}\bigl( \sum_{ij}{\partial u_{i}\over \partial x_{j}} {\partial v_{i}\over \partial x_{j}} \bigr)\,dx$ for all $u, v\in V$, and $B$ is the bilinear continuous mapping from $V\times V$ into $V^{\prime}$ (for $N\le 4$), given by $\langle B(u,v),w \rangle = \int_{\Omega} \bigl( \sum_{ij} u_{j}{\partial v_{i}\over \partial x_{j}}w_{i} \bigr)\,dx$ for all $u, v, w\in V$, and it satisfies $\langle B(u,v),w \rangle + \langle B(u,w),v \rangle = 0$, and in particular $\langle B(u,v),v \rangle = 0$.
As mentioned before, $u^{\prime} + B(u,u)$ is ${Du\over Dt}$ where ${D\over Dt} = {\partial\over \partial t}+\sum_{j}u_{j}{\partial \over \partial x_{j}}$ is the operator of transport along the flow, and it is not a good idea to handle this operator by cutting it in two parts (geometers use a formalism involving affine connections, but up to now it has not helped understand more on that question of transport, and it is therefore not clear yet what is the right way to handle this operator).
However this bad way of treating the nonlinearity does not hurt for $N = 2$ and one can prove uniqueness, or more precisely the following continuous dependence with respect to the data.
\medskip
\noindent
{\bf Proposition}: For $N = 2$, $\Omega$ being any open set in $R^{2}$, if
$$
f_{j} = \sum_{k = 1}^{2} {\partial g_{jk}\over \partial x_{k}}, j = 1,2,\hbox{ with }g_{jk}\in L^{2}\Bigl( 0,T;L^{2}(\Omega;R^{2}) \Bigr),
$$
and if $u_{j}\in L^{2}(0,T;V)\cap C^{0}([0,T];H)$ solve
$$
u_{j}^{\prime} + B(u_{j},u_{j}) + A\,u_{j} = f_{j}, j = 1, 2, \hbox{ in }(0,T); u_{j}(0) = u_{0j}\in H, j = 1, 2,
$$
then one has
$$
\eqalign{
||u_{2}-u_{1}||^{2}_{C^{0}([0,T];H)} + \nu||u_{2}-u_{1}||^{2}_{L^{2}(0,T;V)}&\le C\Bigl( |u_{02}-u_{01}|^{2}_{H} + {1\over \nu}\sum_{k = 1}^{2} ||g_{2k}-g_{1k}||^{2}_{L^{2}(0,T;L^{2}(\Omega;R^{2}))} \Bigr)K\cr
K(u_{1},u_{2}) &= exp\Bigl( {C\over \nu}\int_{0}^{T} \min\{|grad(u_{1})|_{L^{2}}^{2},|grad(u_{2})|_{L^{2}}^{2}\}\,dt \Bigr),}
$$
where $C$ is a universal constant.
\par
\noindent
{\it Proof}: As $u_{j}\in L^{2}(0,T;V)\cap C^{0}([0,T];H)$ implies $B(u_{j},u_{j})\in L^{2}(0,T;V^{\prime})$ in dimension 2, one has $u_{j}\in W(0,T)$.
Subtracting the two equations and multiplying by $u_{2}-u_{1}$ gives
$$
{1\over 2}{d|u_{2}-u_{1}|_{H}^{2} \over dt}+\langle B(u_{2},u_{2})-B(u_{1},u_{1}),u_{2}-u_{1} \rangle +\nu|grad(u_{2}-u_{1})|_{L^{2}}^{2} = -\sum_{k = 1}^{2} \Bigl( g_{2k}-g_{1k}.grad(u_{2}-u_{1}) \Bigr).
$$
One has $-\sum_{k = 1}^{2} \Bigl( g_{2k}-g_{1k}.grad(u_{2}-u_{1}) \Bigr)\le {\nu\over 3}|grad(u_{2}-u_{1})|_{L^{2}}^{2} +{3\over 4\nu}\sum_{k = 1}^{2} |g_{2k}-g_{1k}|_{L^{2}}^{2}$ and $\langle B(u_{2},u_{2})-B(u_{1},u_{1}),u_{2}-u_{1} \rangle = \langle B(u_{2},u_{2}-u_{1})+B(u_{2}-u_{1},u_{1}),u_{2}-u_{1} \rangle =
\langle B(u_{2}-u_{1},u_{1}),u_{2}-u_{1} \rangle$, or $\langle B(u_{2},u_{2})-B(u_{1},u_{1}),u_{2}-u_{1} \rangle = \langle B(u_{2}-u_{1},u_{2})+B(u_{1},u_{2}-u_{1}),u_{2}-u_{1} \rangle = \langle B(u_{2}-u_{1},u_{2}),u_{2}-u_{1} \rangle$, and therefore
$$
|\langle B(u_{2},u_{2})-B(u_{1},u_{1}),u_{2}-u_{1} \rangle|\le C\min\{|grad(u_{1})|_{L^{2}},|grad(u_{2})|_{L^{2}}\}|u_{2}-u_{1}|_{H}\,|grad(u_{2}-u_{1})|_{L^{2}},
$$
where $C$ is a universal constant, independent of $\Omega$ (one does not assume here that P{\eightrm OINCAR\'E} inequality holds, which is the reason for the restriction on $f_{j}$).
Using the bound $|\langle B(u_{2},u_{2})-B(u_{1},u_{1}),u_{2}-u_{1} \rangle|\le {\nu\over 3}|grad(u_{2}-u_{1})|_{L^{2}}^{2}+{3C^{2}\over 4\nu}\min\{|grad(u_{1})|_{L^{2}}^{2},|grad(u_{2})|_{L^{2}}^{2}\}|u_{2}-u_{1}|_{H}^{2}$, one obtains
$$
{1\over 2}{d|u_{2}-u_{1}|_{H}^{2} \over dt}+{\nu\over 3}|grad(u_{2}-u_{1})|_{L^{2}}^{2} \le {3\over 4\nu}\sum_{k = 1}^{2} |g_{2k}-g_{1k}|_{L^{2}}^{2} + {3C^{2}\over 4\nu}\min\{|grad(u_{1})|_{L^{2}}^{2},|grad(u_{2})|_{L^{2}}^{2}\}|u_{2}-u_{1}|_{H}^{2},
$$
and one concludes by G{\eightrm RONWALL} inequality.
\medskip
The same type of proof applies in dimension $N = 3$ or higher, if the solutions are regular enough.
\par
It would be better if one did not cut the operator ${D\over Dt}$ into two pieces, and if one could use the fact that the maximum principle applies to ${D\over Dt}-\nu\Delta$, but a difficulty appears because of the pressure.
As a way to handle a similar situation, I want to show a uniqueness result of Michel A{\eightrm RTOLA} for an equation
$$
{\partial u\over \partial t} -div\bigl( A(x,u)grad(u) \bigr) = f,
$$
with D{\eightrm IRICHLET} conditions (there is an extension to the case $f(x,u)$ that we worked out together, with the type of proof that I show, which is a little more general than Michel's original proof, which extended a result of Neil T{\eightrm RUDINGER}).
Of course $A$ is a C{\eightrm ARATH\'EODORY} function; one assumes that $|A(x,u)|\le\beta$ and $(A(x,u).u)\ge\alpha|u|^{2}$ for all $u\in R^{N}$ with $\alpha>0$, and $|A(x,u)-A(x,v)|\le\omega(|v-u|)$ with $\omega$ nondecreasing and satisfying $\int_{0}^{1} {ds\over \omega^{2}(s)} = +\infty$.
Under these conditions, if $f = div(g)$ with $g\in L^{2}\bigl( 0,T;L^{2}(\Omega;R^{N}) \bigr)$ and $u_{0}\in L^{2}(\Omega)$ there is a unique solution $u\in C^{0}\bigl( [0,T];L^{2}(\Omega) \bigr)\cap L^{2}\bigl( 0,T;H^{1}_{0}(\Omega) \bigr)$ (there is actully a contraction property in $L^{1}(\Omega)$ also).
If $u_{1}$ and $u_{2}$ are two such solutions, one subtract the two equations and one multiplies by $\varphi^{\prime}(u_{2}-u_{1})$ where $\varphi$ is convex, $\varphi^{\prime}(0) = 0$ and $\varphi^{\prime}$ is bounded.
One obtains
$$
{d(\int_{\Omega} \varphi(u_{2}-u_{1})\,dx)\over dt} + \int_{\Omega} \varphi^{\prime\prime}(u_{2}-u_{1})\bigl( A(x,u_{2})grad(u_{2})-A(x,u_{1})grad(u_{1}).grad(u_{2}-u_{1}) \bigr)\,dx = 0,
$$
and using $A(x,u_{2})grad(u_{2})-A(x,u_{1})grad(u_{1}) = A(x,u_{2})grad(u_{2}-u_{1})+\bigl( A(x,u_{2})-A(x,u_{1}) \bigr)grad(u_{1})$, one deduces
$$
\eqalign{
{d(\int_{\Omega} \varphi(u_{2}-u_{1})\,dx)\over dt} &+ \alpha\int_{\Omega} \varphi^{\prime\prime}(u_{2}-u_{1})|grad(u_{2}-u_{1})|^{2}\,dx \le\cr
&\int_{\Omega} \varphi^{\prime\prime}(u_{2}-u_{1})\omega(|u_{2}-u_{1}|)|grad(u_{1})|\,|grad(u_{2}-u_{1})|\,dx,}
$$
from which one deduces
$$
\eqalign{
{d(\int_{\Omega} \varphi(u_{2}-u_{1})\,dx)\over dt}&\le C\int_{\Omega} \varphi^{\prime\prime}(u_{2}-u_{1})\omega^{2}(|u_{2}-u_{1}|)|grad(u_{1})|^{2}\,dx\cr
&\le C\max_{s\in R} \{\varphi^{\prime\prime}(s)\omega^{2}(|s|)\}\int_{\Omega} |grad(u_{1})|^{2}\,dx.}
$$
One chooses then $0<\varepsilon<\eta$, and $\varphi_{\varepsilon\eta}$ even and defined by $\varphi_{\varepsilon\eta}(0) = \varphi_{\varepsilon\eta}^{\prime}(0) = 0$ and $\varphi_{\varepsilon\eta}^{\prime\prime}(s) = {1\over \omega^{2}(s)}$ for $\varepsilon 0$ and a constant $M$ such that a sequence $u_{n}$ is bounded in $L^{p}(0,T;E)$ with $\bigl( \int_{0}^{T-h} ||u_{n}(t+h)-u_{n}(t)||^{p}\,dt \bigr)^{1/p}\le M\,|h|^{\eta}$ for all $h\in(0,T/2)$.
Then $u_{n}$ is bounded in $L^{q}(0,T;E)$ with $q 0$, one wants an estimate of the measure of $\omega_{2\lambda} = \{t\in(0,T/2):|u(t)|\ge 2\lambda\}$, and one chooses $\varepsilon>0$ such that $||u||_{L^{a}}\bigl( {K\over \varepsilon} \bigr)^{1/a} = \lambda$, so that $u = v+w$ with $v = \rho_{\varepsilon}\star u$ bounded by $\lambda$; therefore $\omega_{2\lambda}$ is included in the set where $|w| = |u - (\rho_{\varepsilon}\star u)|\ge \lambda$, which gives the estimate $\lambda^{p}meas(\omega_{2\lambda})\le M^{p}\varepsilon^{\eta\,p}$, and using the choice of $\varepsilon$, i.e.
$K\lambda^{-a}||u||_{L^{a}}^{a}$, one obtains the estimate $\lambda^{p+a\eta\,p}meas(\omega_{2\lambda})\le M^{p}K^{\eta\,p}||u||_{L^{a}}^{a\eta\,p}$.
\par
This estimate shows that $u\in L^{b}$ for $b 0$ and a constant $M$ such that $\bigl( \int_{0}^{T-h} ||u_{n}(t+h)-u_{n}(t)||^{p}\,dt \bigr)^{1/p}\le M\,|h|^{\theta}$ for all $h\in(0,T/2)$, then $u_{n}$ belong to a compact set of $L^{p}(0,T;E_{3})$.
\par
\noindent
{\it Proof}: After localization so that the support of all $u_{n}$ is in $(0,T/2)$, one chooses $h>0$ small and one defines $v_{n}(t) = {1\over h}\int_{t}^{t+h} u_{n}(s)\,ds$ and $w_{n} = u_{n}-v_{n}$.
As before, $w_{n}(t) = {1\over h}\int_{t}^{t+h} \bigl( u_{n}(t)-u_{n}(s) \bigr)\,ds$, giving the bound $||w_{n}||_{L^{p}(0,T;E_{3})}\le C\,|h|^{\theta}$.
For a fixed $h>0$, $v_{n}$ takes its values in a bounded set of $E_{1}$, and therefore in a compact set of $E_{3}$, but as from the previous lemma $u_{n}$ is bounded in some $L^{q}(0,T;E_{3})$ with $q>1$, one sees that $v_{n}$ has its derivative bounded in $L^{q}(0,T;E_{3})$ and is thereforeuniformly H{\eightrm \"OLDER} continuous with values in $E_{3}$; by A{\eightrm SCOLI}'s theorem a subsequence $v_{m}$ converges uniformly.
One deduces that $\limsup_{m,m^{\prime}\rightarrow\infty} ||u_{m}-u_{m^{\prime}}||_{L^{p}(0,T;E_{3})}\le \limsup_{m,m^{\prime}\rightarrow\infty} ||w_{m}-w_{m^{\prime}}||_{L^{p}(0,T;E_{3})}\le 2C\,|h|^{\theta}$, and letting $h$ tend to 0 shows that $u_{m}$ is a C{\eightrm AUCHY} sequence in $L^{p}(0,T;E_{3})$.
\bigskip
In our application to N{\eightrm AVIER}-S{\eightrm TOKES} equation using a special (R{\eightrm ITZ}-) G{\eightrm ALERKIN} basis, one has $u_{n}$ bounded in $L^{2}(0,T;V)$ and in $L^{\infty}(0,T;H)$, and ${du_{n}\over dt}$ is bounded in $L^{p}(0,T;V^{\prime})$, with $p = 2$ for $N = 2$, $p = 4/3$ for $N = 3$, and $p = 1$ for $N = 4$; moreover $V$ is continuously and compactly imbedded into $H$ (and therefore into $V^{\prime}$).
One can take $\theta = 1$ and one first deduces that $u_{n}$ belongs to a compact of $L^{p}(0,T;V^{\prime})$; but as it is bounded in $L^{\infty}(0,T;V^{\prime})$, it belongs to a compact of $L^{q}(0,T;V^{\prime})$ for all $q<\infty$; then it belongs to a compact of $L^{2}(0,T;H)$, the limitation by 2 being due to the estimate of $u_{n}$ in $L^{2}(0,T;V)$, and one can extract subsequences which converge almost everywhere in $\Omega\times(0,T)$.
In dimension $N = 2$, using an interpolation inequality, each component of $u_{n}$ is bounded in $L^{4}\bigl( \Omega\times(0,T) \bigr)$ and therefore the term $(u_{m})_{j}(u_{m})_{i}$ for which one needs the limit (in order to compute the limit of the term $\langle B(u_{m},u_{m}),e_{k} \rangle$) is bounded in $L^{2}\bigl( \Omega\times(0,T) \bigr)$ and converges almost everywhere to $(u_{\infty})_{j}(u_{\infty})_{i}$; in dimension $N = 3$, each component of $u_{n}$ is bounded in $L^{8/3}\bigl( 0,T;L^{4}(\Omega) \bigr)$ and therefore the term $(u_{m})_{j}(u_{m})_{i}$ is bounded in $L^{4/3}\bigl( 0,T;L^{2}(\Omega) \bigr)$ and converges then almost everywhere to $(u_{\infty})_{j}(u_{\infty})_{i}$.
The case $N\ge 4$ uses interior regularity for the $e_{k}$.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
25. Friday March 12.
\medskip
We have obtained the existence of a weak solution for an abstract formulation of N{\eightrm AVIER}-S{\eightrm TOKES} equation; our solution is defined on $(0,T)$, and one may take $T = +\infty$ without much change in the proof.
Even in dimension $N = 2$, where we know the solution to be unique, it is useful to know whether or not it is regular when the data are more regular.
For $N = 3$ (or $N\ge 4$ for purely mathematical reasons), one may wonder if one can find a strong solution, i.e. a solution having a better regularity so that the solution would be unique, for example, or if the pressure would be found in a space of locally integrable functions in $(x,t)$.
Some of the regularity results depend upon the smoothness of the boundary $\partial\Omega$, for example in order to use the fact that $D(A)\subset H^{2}(\Omega;R^{N})$, which is not always true for L{\eightrm IPSCHITZ} domains.
\medskip
\noindent
{\bf Lemma}: If $N = 2$, if $f\in L^{2}\bigl( 0,T;L^{2}(\Omega;R^{2}) \bigr)$ and $u_{0}\in V$, if $\Omega$ is smooth enough so that P{\eightrm OINCAR\'E} inequality holds and $D(A)\subset H^{2}(\Omega;R^{2})$, then the solution of $u^{\prime}+B(u,u)+A\,u = f$ and $u(0) = u_{0}$ satisfies $u\in C^{0}([0,T];V)$, $u\in L^{2}\bigl( 0,T;H^{2}(\Omega;R^{2}) \bigr)$, and $u^{\prime}\in L^{2}(0,T;H)$.
\par
\noindent
{\it Proof}: One proves the estimates for the approximation with the special basis, multiplying by $A\,u_{n}$, and one needs to bound terms like $(u_{n})_{j}{\partial (u_{n})_{i}\over \partial x_{j}}$ in $L^{2}(\Omega)$, for example from a bound of $(u_{n})_{j}$ in $L^{\infty}(\Omega)$; one may use a bound $||v||_{L^{\infty}(\Omega)}\le C||v||_{L^{2}(\Omega)}^{1/2}||v||_{H^{2}(\Omega)}^{1/2}\le C|v|^{1/2}|A\,v|^{1/2}$ (valid in dimension 2), and one obtains
$$
\eqalign{
{1\over 2}{d(||u_{n}||^{2})\over dt}+|A\,u_{n}|^{2}&\le |f|\,|A\,u_{n}|+ C|u_{n}|^{1/2}||u_{n}||\,|A\,u_{n}|^{3/2}\cr
&\le \varepsilon|A\,u_{n}|^{2}+C(\varepsilon)|f|^{2} +C(\varepsilon) |u_{n}|^{2}||u_{n}||^{4},}
$$
where one has used Y{\eightrm OUNG}'s inequality $ab\le |\varepsilon\,a|^{p}/p + |b/\varepsilon|^{p^{\prime}}/p^{\prime}$, with $p = 4/3, p^{\prime} = 4$, and one deduces
$$
||u_{n}(t)||^{2}\le ||u_{0}||^{2}+C\int_{0}^{t} |f(s)|^{2}\,ds + C\int_{0}^{t} \lambda_{n}(s)||u_{n}(s)||^{2}\,ds,\hbox{ with }\lambda_{n} = C|u_{n}|^{2}||u_{n}||^{2},
$$
from which a uniform bound for $||u_{n}||$ is deduced by applying G{\eightrm RONWALL} inequality, as $\lambda_{n}$ is bounded in $L^{1}(0,T)$, and this implies that $|A\,u_{n}|$ is bounded in $L^{2}(0,T)$.
\medskip
In the preceding proof, one can use different estimates for the bound in $L^{\infty}$; for example, trying to get the power of $|A\,u_{n}|$ as low as possible, one can use $||v||_{L^{\infty}(\Omega)}\le C(\eta)||v||_{L^{2}(\Omega)}^{1/(1+\eta)}||v||_{H^{1+\eta}(\Omega)}^{\eta/(1+\eta)}$ (valid in dimension 2), and if one uses $\theta = \eta/(1+\eta)\in(0,1/2)$, it gives a bound for $B(u_{n},u_{n})$ in $L^{2}(\Omega;R^{2})$ of the form $C(\theta)|u_{n}|^{\theta}||u_{n}||^{2-2\theta}|A\,u_{n}|^{\theta}$.
The application of Y{\eightrm OUNG}'s inequality, with $p = 2/(1+\theta), p^{\prime} = 2/(1-\theta)$, gives a term in $\varepsilon|A\,u_{n}|^{2}+C(\varepsilon,\theta)|u_{n}|^{2\theta/(1-\theta)}||u_{n}||^{4}$, and therefore one gains on the power of $|u_{n}|$ but not on the power of $||u_{n}||$.
\medskip
\noindent
{\bf Lemma}: If $N = 3$, if $f\in L^{2}\bigl( 0,T;L^{2}(\Omega;R^{3}) \bigr)$ and $u_{0}\in V$, if $\Omega$ is smooth enough so that P{\eightrm OINCAR\'E} inequality holds and $D(A)\subset H^{2}(\Omega;R^{3})$, then there exists $T_{c}\in(0,T]$ depending upon the norms of the data such that there exists a unique solution of $u^{\prime}+B(u,u)+A\,u = f$ and $u(0) = u_{0}$ on $[0,T_{c}]$ which satisfies $u\in C^{0}([0,T_{c}];V)$, $u\in L^{2}\bigl( 0,T_{c};H^{2}(\Omega;R^{3}) \bigr)$, and $u^{\prime}\in L^{2}(0,T_{c};H)$.
\par
\noindent
{\it Proof}: Same type of proof than before, but now one has $||v||_{L^{\infty}(\Omega)}\le C||v||^{1/2}|A\,v|^{1/2}$ (valid in dimension 3), giving a bound for $B(u_{n},u_{n})$ in $L^{2}(\Omega;R^{3})$ of the form $||u_{n}||^{3/2}|A\,u_{n}|^{1/2}$.
The application of Y{\eightrm OUNG}'s inequality, with $p = 4/3, p^{\prime} = 4$, gives a term in $\varepsilon|A\,u_{n}|^{2}+C(\varepsilon)||u_{n}||^{6}$, and the exponent is too large for obtaining a bound by G{\eightrm RONWALL}'s inequality, and one can only obtain a local bound from the inequality
$$
{d(||u_{n}||^{2})\over dt}+\alpha|A\,u_{n}|^{2}\le C\,|f|^{2} + C\,||u_{n}||^{6}.
$$
After integration, and omission of the term in $|A\,u_{n}|^{2}$, one obtains
$$
||u_{n}(t)||^{2}\le ||u_{0}||^{2}+C\int_{0}^{t} |f(s)|^{2}\,ds + C\int_{0}^{t} ||u_{n}(s)||^{6}\,ds\le K+ C\int_{0}^{t} ||u_{n}(s)||^{6}\,ds,\hbox{ in }(0,T),
$$
where $K = ||u_{0}||^{2}+C\int_{0}^{T} |f(s)|^{2}\,ds$, and if one defines $\varphi$ by $\varphi(t) = \int_{0}^{t} ||u_{n}(s)||^{6}\,ds$, then one has $\varphi^{\prime}\le (K+C\,\varphi)^{3}$, which implies $\bigl( (K+C\,\varphi)^{-2} \bigr)^{\prime} = -2C(K+C\,\varphi)^{-3}\varphi^{\prime}\ge -2C$ and therefore $\bigl( K+C\,\varphi(t) \bigr)^{-2}\ge K^{-2}-2C\,t$, which is only useful on $[0,T_{c}]$ if $K^{-2}-2C\,T_{c}>0$, in which case it gives the desired bounds.
\par
For proving uniqueness one assumes that one has two solutions $u^{1}, u^{2}$, one subtracts the equations and one multiplies by $A(u^{2}-u^{1})$, and one has to estimate the norm in $L^{2}(\Omega;R^{3})$ of $B(u^{2},u^{2})-B(u^{1},u^{1})$, which one writes as $B(u^{2},u^{2}-u^{1})+B(u^{2}-u^{1},u^{1})$; the first term can be bounded as $||u^{2}||_{L^{6}(\Omega;R^{3})}||u^{2}-u^{1}||_{W^{1,3}(\Omega;R^{3})}$, bounded by $C||u^{2}||\,||u^{2}-u^{1}||^{1/2}|A\,u^{2}-A\,u^{1}|^{1/2}$; the second term can be bounded as $||u^{2}-u^{1}||_{L^{\infty}(\Omega;R^{3})}||u^{1}||$, bounded by $C||u^{1}||\,||u^{2}-u^{1}||^{1/2}|A\,u^{2}-A\,u^{1}|^{1/2}$.
One obtains then $(||u^{2}-u^{1}||^{2})^{\prime}+2|A\,u^{2}-A\,u^{1}|^{2}\le C(||u^{1}||+||u^{2}||)||u^{2}-u^{1}||^{1/2}|A\,u^{2}-A\,u^{1}|^{3/2}\le |A\,u^{2}-A\,u^{1}|^{2}+C(||u^{1}||+||u^{2}||)^{4}\,||u^{2}-u^{1}||^{2}$, and one concludes by using G{\eightrm RONWALL} inequality.
\medskip
If the data are small enough, one can take $T_{c} = T$, but one can even obtain global existence on $[0,\infty)$ if the data are small enough; for simplicity, I consider first the case $f = 0$.
\medskip
\noindent
{\bf Lemma}: If $N = 3$, if $\Omega$ is smooth enough so that P{\eightrm OINCAR\'E} inequality holds and $D(A)\subset H^{2}(\Omega;R^{3})$, then if $u_{0}\in V$, and if $|u_{0}|\,||u_{0}||$ is small enough, then the solution of $u^{\prime}+B(u,u)+A\,u = 0$ with $u(0) = u_{0}$ exists for all $t\in[0,\infty)$ and satisfies $u\in C^{0}([0,\infty);V)$, $u\in L^{2}\bigl( 0,\infty;H^{2}(\Omega;R^{3}) \bigr)$, and $u^{\prime}\in L^{2}(0,\infty;H)$.
\par
\noindent
{\it Proof}: One bounds the norm of $v_{j}{\partial v_{i}\over \partial x_{j}}$ in $L^{2}(\Omega)$ by $||v_{j}||_{L^{3}(\Omega)}\bigl| \bigl| {\partial v_{i}\over \partial x_{j}} \bigr| \bigr|_{L^{6}(\Omega)}$, $\bigl| \bigl| {\partial v_{i}\over \partial x_{j}} \bigr| \bigr|_{L^{6}(\Omega)}$ by $C||v||_{H^{2}(\Omega)}$ and $||v_{j}||_{L^{3}(\Omega)}$ by $C|v|^{1/2}||v||^{1/2}$, so that
$$
|\langle B(u_{n},u_{n}),A\,u_{n} \rangle|\le C_{0}|u_{n}|^{1/2}\,||u_{n}||^{1/2}\,|A\,u_{n}|^{2}\hbox{ for every }u_{n}\in D(A).
$$
Because $f = 0$, one obtains
$$
\eqalign{
&{1\over 2}{d(|u_{n}|^{2})\over dt}+||u_{n}||^{2}\le 0\cr
&{1\over 2}{d(||u_{n}||^{2})\over dt}+|A\,u_{n}|^{2}\le C_{0}|u_{n}|^{1/2}\,||u_{n}||^{1/2}\,|A\,u_{n}|^{2}\hbox{ on }(0,T),}
$$
and therefore as long as $C_{0}|u_{n}|^{1/2}\,||u_{n}||^{1/2}<1$, both the quantities $|u_{n}|$ and $||u_{n}||$ are nonincreasing and their product is less than its value at time 0; consequently, if the initial data $u_{0}\in V$ is chosen so that
$$
C_{0}|u_{0}|^{1/2}\,||u_{0}||^{1/2}\le 1,
$$
then one has $|u_{n}(t)|\le |u_{0}|$ and $||u_{n}(t)||\le||u_{0}||$ on $(0,\infty)$, and a global solution exists on $(0,\infty)$.
\medskip
In the case $f\ne 0$, one has $|u_{n}(t)|\le|u_{0}|+\int_{0}^{t} |f(s)|\,ds$, but if one wants to avoid assuming $f\in L^{1}\bigl( 0,\infty;L^{2}(\Omega;R^{3}) \bigr)$, one may use P{\eightrm OINCAR\'E} inequality $||v||^{2}\ge \lambda_{1}|v|^{2}$ for all $v\in H^{1}_{0}(\Omega)$, and the inequality $(|u_{n}|^{2})^{\prime}+ 2\lambda_{1}|u_{n}|^{2}\le 2|f|\,|u_{n}|\le 2\lambda_{1}|u_{n}|^{2} +|f|^{2}/2\lambda_{1}$ gives $|u_{n}(t)|^{2}\le |u_{0}|^{2}+ {1\over 2\lambda_{1}}\int_{0}^{t} |f(s)|^{2}\,ds$.
If one can enforce the condition $C_{0}|u_{n}|^{1/2}||u_{n}||^{1/2}\le 1/2$, then one has $(||u_{n}||^{2})^{\prime}+|A\,u_{n}|^{2}\le 2|f|\,|A\,u_{n}|\le |A\,u_{n}|^{2} +|f|^{2}$ and therefore $||u_{n}||^{2}\le ||u_{0}||^{2}+\int_{0}^{t} |f|^{2}\,dt$, and the condition to enforce is satisfied if one asks that
$$
\Bigl( |u_{0}|^{2}+{1\over 2\lambda_{1}}\int_{0}^{\infty} |f(s)|^{2}\,ds \Bigr)\Bigl( ||u_{0}||^{2}+ \int_{0}^{\infty} |f(s)|^{2}\,ds \Bigr)\le {1\over 16C_{0}^{4}}.
$$
\par
In the case where $f = 0$, instead of putting conditions on $|u_{0}|\,||u_{0}||$ one can impose a more natural condition that $u_{0}$ be small in the domain of $A^{1/4}$; this is done by multiplying by $A^{1/2}u_{n}$, and using the estimate $||v||_{L^{3}(\Omega;R^{3})}\le C|A^{1/4}v|$, which implies $|(B(u_{n},u_{n}),A^{1/2}u_{n})|\le C_{1}|A^{1/4}u_{n}|\,|A^{3/4}u_{n}|^{2}$, and therefore if $C_{1}|A^{1/4}u_{0}|\le 1$, then the norm of $|A^{1/4}u_{n}|$ is nonincreasing and stays then $\le{1\over C_{1}}$; one easily extends this idea to the case $f\ne 0$.
\medskip
All these types of inequalities are quite standard, and although I may have improved on details, I had learned most of these techniques in lectures of Jacques-Louis L{\eightrm IONS} in the late 70s; I had taught these tecgniques in my 1974/75 course in Madison (the lecture notes were written by graduate students).
After that I advocated following a little more the Physics of the fluid flows in order to get better results, and I still insist that one should not cut the transport term into two pieces, and I thought that everything important had been found out of these differential inequalities.
I was wrong; in January 1980, Colette G{\eightrm UILLOP\'E} showed me some handwritten pages by Ciprian F{\eightrm OIAS}, and I took a copy which I looked at during the following month, which I spent at the T{\eightrm ATA} Institute in Bangalore; I made an improvement on Ciprian F{\eightrm OIAS}'s original computation, but the idea is his.
\par
In the case $N = 3$, taking $f = 0$ in order to simplify, one starts from the already mentioned differential inequality $(||u_{n}||^{2})^{\prime}+|A\,u_{n}|^{2}\le C||u_{n}||^{6}$, together with $(|u_{n}|^{2})^{\prime}+2||u_{n}||^{2} = 0$, which gives the existence on $(0,T)$ of the approximate solution $u_{n}$; the idea of Ciprian F{\eightrm OIAS} was to divide by $1+||u_{n}||^{6}$, while my improvement is to divide only by $1+||u_{n}||^{4}$!
One obtains
$$
{d\Bigl( \arctan(||u_{n}||^{2}) \Bigr)\over dt}+{|A\,u_{n}|^{2}\over 1+||u_{n}||^{4}} = {1\over 1+||u_{n}||^{4}}\Bigl( (||u_{n}||^{2})^{\prime}+|A\,u_{n}|^{2} \Bigr)\le {C||u_{n}||^{6}\over 1+||u_{n}||^{4}}\le C||u_{n}||^{2}.
$$
Integrating from 0 to $T$ (which can be $+\infty$), one obtains
$$
\int_{0}^{T} {|A\,u_{n}|^{2}\over 1+||u_{n}||^{4}}\,dt\le \arctan(||u_{n}(0)||^{2})+C\int_{0}^{T} ||u_{n}||^{2}\,dt\le {\pi\over 2} + {C|u_{0}|^{2}\over 2}.
$$
One has ${|A\,u_{n}|^{1/2}\over 1+||u_{n}||}$ bounded in $L^{4}(0,T)$, but as $||u_{n}||^{1/2}(1+||u_{n}||)$ is bounded in $L^{4/3}(0,T)$, one has
$$
||u_{n}||^{1/2}|A\,u_{n}|^{1/2}\hbox{ bounded } in L^{1}(0,T),
$$
from which one obtains
$$
u_{n}\hbox{ is bounded in }L^{1}\Bigl( 0,T;L^{\infty}(\Omega;R^{3}) \Bigr).
$$
One deduces the same properties for the limit.
\par
Notice how far this estimate is from that which would give well defined curves followed by particles along the flow.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
26. Monday March 15.
\medskip
If $U_{0}$ is a characteristic velocity and $L_{0}$ is a characteristic length of a flow, then the corresponding R{\eightrm EYNOLDS} number ${U_{0}\,L_{0}\over \nu}$ is adimensional; large $\nu$ or small $R$ correspond to laminar flows, while small $\nu$ or large $R$ correspond to turbulent flows.
In the ocean, the characteristic lengths $L_{0}$ are large.
\par
As $u$ denotes a velocity, it has dimension $L\,T^{-1}$, where $L$ denotes length and $T$ denotes time, and the kinematic viscosity $\nu = \mu/\rho$ has the dimension $L^{2}T^{-1}$ (while $\mu$ has dimension $M\,L^{-1}T^{-1}$, where $M$ denotes mass).
In dimension $N = 3$, the norm $|u|$ has the dimension $L^{5/2}T^{-1}$ (and therefore as $\rho$ has dimension $M\,L^{-3}$, $\rho_{0}|u|^{2}$ has dimension $M\,L^{2}T^{-2}$, i.e. energy), the norm $||u||$ has the dimension $L^{3/2}T^{-1}$ (and $\mu||u||^{2}$ has the dimension $M\,L^{2}T^{-3}$, as energy dissipated per unit of time), and $|u|^{1/2}||u||^{1/2}$ has the dimension $L^{2}T^{-1}$, as the does $||u||_{L^{3}}$.
\par
The term $\langle B(u,u),A\,u \rangle$, as an integral $\nu\int u\,\partial u\,\partial^{2}u\,dx$ has the dimension $L^{5}T^{-4}$, while $|u|^{1/2}||u||^{1/2}|A\,u|^{2}$ has the dimension $L^{7}T^{-5}$, and therefore $C_{1}$ has the dimension $L^{-2}T$, i.e. $1/C_{1}$ has the same dimension $L^{2}T^{-1}$ as $\nu$.
As the only parameter in the equation is $\nu$ (the term $1/\rho$ in front of $grad(p)$ is hidden, and as $\rho$ is assumed constant, it is $p/\rho_{0}$ which we have called pressure!), it is natural to compare norms to that number, but that only makes sense for norms whose dimension is a power of $L^{2}T^{-1}$.
\medskip
The limitations of the estimates shown before are due in part to the fact that one uses norms which give a global information on the solution and not a local information; the total kinetic energy at time $t$ is seen by $\rho_{0}|u(t)|^{2}$ and the energy dissipated by viscosity between time 0 and $T$ is seen by $\mu\int_{0}^{T} ||u(t)||^{2}\,dt$, but these norms do not tell if some regions corresponds to large velocities or to a large dissipation of energy (as we have assumed that $\rho_{0}$ and $\mu$ are independent of temperature, the energy dissipated by viscosity appears in the equation of balance of energy, which is decoupled from the equation of motion that we have been dealing with up to now).
\bigskip
A different approach, which I initiated in 1979 for a different class of equations, consists in avoiding the semi-group approach where one deals with functional spaces which are functions in $x$ alone and one defines the domain of a nonlinear operator, and instead deals with functional spaces in $(x,t)$ adapted to the equation.
In the class of discrete velocity models in kinetic theory, which are supposed to simplify the B{\eightrm OLTZMANN} equation, there is a particular model attributed to B{\eightrm ROADWELL} (Ren\'ee G{\eightrm ATIGNOL} attributes this kind of model to M{\eightrm AXWELL}); in two dimensions $(x,y)$ it is
$$
\eqalign{
&{\partial u_{1}\over \partial t}+{\partial u_{1}\over \partial x}+(\alpha\,u_{1}u_{2}-\beta\,u_{3}u_{4}) = 0\hbox{ in }R^{2}\times(0,T), u_{1}(x,y,0) = u_{01}(x,y)\hbox{ in }R^{2}\cr
&{\partial u_{2}\over \partial t}-{\partial u_{2}\over \partial x}+(\alpha\,u_{1}u_{2}-\beta\,u_{3}u_{4}) = 0\hbox{ in }R^{2}\times(0,T), u_{2}(x,y,0) = u_{02}(x,y)\hbox{ in }R^{2}\cr
&{\partial u_{3}\over \partial t}+{\partial u_{3}\over \partial y}-(\alpha\,u_{1}u_{2}-\beta\,u_{3}u_{4}) = 0\hbox{ in }R^{2}\times(0,T), u_{3}(x,y,0) = u_{03}(x,y)\hbox{ in }R^{2}\cr
&{\partial u_{4}\over \partial t}-{\partial u_{4}\over \partial y}-(\alpha\,u_{1}u_{2}-\beta\,u_{3}u_{4}) = 0\hbox{ in }R^{2}\times(0,T), u_{4}(x,y,0) = u_{04}(x,y)\hbox{ in }R^{2},}
$$
where $u_{1}, u_{2}, u_{3}, u_{4}$ denote the density of particles at $(x,y,t)$; these particles have all the same mass but their velocities are respectively $(+1,0), (-1,0), (0,+1), (0,-1)$ (Jim G{\eightrm REENBERG} denotes the unknowns $l,r,u,d$, for left, right, up, down); $\alpha, \beta$ are positive parameters related to probability of collisions, which are usually taken equal to $1/\varepsilon$, where $\varepsilon$ is related to a mean free path between collisions (B{\eightrm ROADWELL} was actually interested in the formal fluid limit $\varepsilon\rightarrow 0$, and there are plenty of open questions in that direction).
Local existence for data in $L^{\infty}(R^{2})$ is standard (locally L{\eightrm IPSCHITZ} perturbation of a linear semigroup), and if the data are nonnegative the solution is nonnegative.
The model conserves mass and momentum (density of mass is $u_{1}+u_{2}+u_{3}+u_{4}$, density of momentum is $(u_{1}-u_{2},u_{3}-u_{4})$), and also kinetic energy as it is proportional to mass (so there is no temperature for this model).
An analog of the H-theorem of B{\eightrm OLTZMANN} holds, and there is an entropy which decreases (density of entropy is $u_{1}\log(u_{1})+u_{2}\log(u_{2})+u_{3}\log(u_{3})+u_{4}\log(u_{4})$).
Takaaki N{\eightrm ISHIDA} and I (independently) have noticed that there is a global existence of a solution for small nonnegative data in $L^{2}(R^{2})$, and it is useful to notice that the $L^{2}(R^{2})$ norm in invariant by scaling: if $U = (u_{1},u_{2},u_{3},u_{4})$ is a solution, then $V$ defined by $V(x,y,t) = \lambda\,U(\lambda\,x,\lambda\,y,\lambda\,t)$ is also a solution for any $\lambda>0$, and the norm of the initial data in $L^{2}(R^{2};R^{4})$ is the same for $U$ or $V$; my argument generalized what I had done for the one dimensional case, which I describe now.
\par
If $u_{3}$ and $u_{4}$ are equal at time 0 and independent of $y$ then they stay equal and independent of $y$ for $t>0$ (also for $t<0$ as long as the solution exists, but nonnegativity is only conserved when $t$ increases); one considers then the simplified model where $\alpha = \beta = 1$
$$
\eqalign{
&u_{t}+u_{x}+u\,v-w^{2} = 0\hbox{ in }R\times(0,\infty), u(x,0) = u_{0}(x)\hbox{ in }R\cr
&v_{t}-v_{x}+u\,v-w^{2} = 0\hbox{ in }R\times(0,\infty), v(x,0) = v_{0}(x)\hbox{ in }R\cr
&w_{t}-u\,v+w^{2} = 0\hbox{ in }R\times(0,\infty), w(x,0) = w_{0}(x)\hbox{ in }R.}
$$
In 1975, in collaboration with Michael C{\eightrm RANDALL}, we had proved global existence for bounded nonnegative data by using finite propagation speed, the entropy estimate as a compactness argument in $L^{1}$, and a crucial result that M{\eightrm IMURA} and Takaaki N{\eightrm ISHIDA} had just published, where they had shown that for small nonnegative data in $L^{1}$ and arbitrary bound in $L^{\infty}$, the $L^{\infty}$ estimate was controlled for all $t$.
As our argument could only be used in one dimension (as the generalization of the estimate of M{\eightrm IMURA} and N{\eightrm ISHIDA} to more than one dimension was unlikely), I thought of using more physical spaces than $L^{\infty}$, and I thought that BMO was a good substitute, as it controls the portion of the mass which is out of equilibrium, but I could not get my colleague Yves M{\eightrm EYER} to help me, and I never went forward with this idea.
In 1979, I was wondering which discrete velocity models of kinetic theory were stable by weak convergence (as I had noticed that the C{\eightrm ARLEMAN} model was not, although it is not really a model of kinetic theory as it does not conserve momentum), and I found that (apart from the affine case) it only happened in one space dimension for models of the form
$$
{\partial u_{i}\over \partial t}+C_{i}{\partial u_{i}\over \partial x}+\sum_{j,k} A_{ijk}u_{j}u_{k} + affine(u) = 0\hbox{ in }R\times(0,\infty), u_{i}(x,0) = u_{0i}(x)\hbox{ in }R,
$$
where the interaction coefficients (with $A_{ijk} = A_{ikj}$ for all $i, j, k$) satisfy the condition
$$
C_{j} = C_{k}\hbox{ implies }A_{ijk} = 0\hbox{ for all }i.
$$
I looked into the existence of solutions for these models, as I had found them in connection with a question of Compensated Compactness, for which better bounds were known or conjectured (a topic which I call now Compensated Integrability in order to point out the difference with Compensated Compactness, because I had noticed that in an interesting article of Ronald C{\eightrm OIFMAN}, Pierre-Louis L{\eightrm IONS}, Yves M{\eightrm EYER} and Stephen S{\eightrm EMMES} using H{\eightrm ARDY} spaces, they had wrongly claimed to improve the Compensated Compactness theory, while they were actually improving one of my argument of Compensated Integrability based on using L{\eightrm ORENTZ} spaces); I noticed a simple trick, which gave global existence (from $-\infty$ to $+\infty$) for small data in $L^{1}(R)$.
\par
Let $V_{c} = \{u:u_{t}+c\,u_{x}\in L^{1}(R^{2}),u(\cdot,0)\in L^{1}(R)\}$ and $W_{c} = \{u:|u(x,t)|\le U(x-c\,t)$ a.e., $U\in L^{1}(R)\}$, then $V_{c}\subset W_{c}$ and if $u\in V_{c}, v\in V_{c^{\prime}}$ with $c\ne c^{\prime}$, then $u\,v$ belongs to $L^{1}(R^{2})$, and $|c-c^{\prime}|\,||u\,v||_{L^{1}(R^{2})}\le ||u||_{W_{c}}||v||_{W_{c^{\prime}}}$.
Then using an iterative scheme in $V = \prod_{i} V_{c_{i}}$ one finds a strict contraction in a small ball centered at 0, and this gives the global existence for small data in $L^{1}(R)$.
Therefore one does not try to define the domain of the nonlinear operator, one finds that all products $u_{j}u_{k}$ appearing in the equation belong to $L^{1}(R^{2})$ and therefore by F{\eightrm UBINI}'s theorem, for almost every $t$ the product $u_{j}u_{k}$ belong to $L^{1}(R)$.
\par
For the two dimensional case, it is $u_{i}^{2}$ which belongs to a space like $V_{c}$, and for example the analog of the space $W_{c}$ are $|u_{1}(x,y,t)|\le U_{1}(x-t,y),|u_{2}(x,y,t)|\le U_{2}(x+t,y), |u_{3}(x,y,t)|\le U_{3}(x,y-t), |u_{4}(x,y,t)|\le U_{4}(x,y+t)$, with $U_{1}, U_{2}, U_{3}, U_{4}\in L^{2}(R^{2})$; one has then to show that $u_{1}u_{3}u_{4}\in L^{1}(R^{3})$, and this is analogous to the trick used in the proof of S{\eightrm OBOLEV} imbedding theorem in the methods of Emilio G{\eightrm AGLIARDO} and of Louis N{\eightrm IRENBERG}.
\par
I have not found how to use this idea for N{\eightrm AVIER}-S{\eightrm TOKES} equation, but there has been some application to B{\eightrm OLTZMANN} equation or F{\eightrm OKKER}-P{\eightrm LANCK} equation by my student Kamel H{\eightrm AMDACHE}, and if the idea to use $|f(x,v,t)|\le F(x-v\,t,v)$ was clear, it was not obvious how to choose $F$, and Kamel H{\eightrm AMDACHE} extended an initial result of Reinhard I{\eightrm LLNER} and S{\eightrm HINBROT}, who had taken $F(\xi,v) = M\,e^{-\alpha|\xi|^{2}}$.
\bigskip
A second idea, is to use pointwise estimates with maximal functions, and the possibility of using that idea for S{\eightrm TOKES} equation only occurred to me two years ago, but I have not found a way to handle the question of transport and extend it to N{\eightrm AVIER}-S{\eightrm TOKES} equation.
I had learned the trick in an argument of Lars H{\eightrm EDBERG}, reproduced by Ha\"{\i}m B{\eightrm REZIS} and Felix B{\eightrm ROWDER} for a question of truncation (which some justly call the H{\eightrm EDBERG} truncation method); after finding how to use the trick for the heat equation or S{\eightrm TOKES} equation two years ago (during a meeting dedicated to Jindrich N{\eightrm E\v{C}AS} in Lisbon), I exchanged e-mail with Lars H{\eightrm EDBERG} in order to learn about the origin of the idea, which is his but he pointed out an earlier result of Lennart C{\eightrm ARLESON} in the late 60s and a result of Elias S{\eightrm TEIN} in his book from the early 70s.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
27. Wednesday March 17.
\medskip
At the moment, uniqueness of weak solutions of 3-dimensional incompressible N{\eightrm AVIER}-S{\eightrm TOKES} equation is an open problem.
Jean L{\eightrm ERAY} had conjectured that there could be point singularities of the equation and he had imagined that they could be self similar solutions, of the form $u(x,t) = {1\over\sqrt{2a(T-t)}}U\bigl( {x\over\sqrt{2a(T-t)}} \bigr)$; he also thought that it was related to turbulence, but the prevailing ideas on turbulence now are those imagined much later by K{\eightrm OLMOGOROV} (I do not think that K{\eightrm OLMOGOROV} was absolutely right, but L{\eightrm ERAY}'s idea does not fit well with what I understand about effective properties of microstructures).
Recently Jind\v{r}ich N{\eightrm E\v{C}AS}, M. R{\eightrm U\v{Z}ICKA} and Vladimir \v{S}{\eightrm VER\'AK} have shown that the self similar solutions imagined by L{\eightrm ERAY} cannot have $U\in L^{3}(R^{3};R^{3})$.
\par
Measuring the H{\eightrm AUSDORFF} dimension of the singular set of a solution has been a way to determine how far it is from being smooth, and at the moment the best result has been obtained by Luis C{\eightrm AFFARELLI}, Robert K{\eightrm OHN} and Louis N{\eightrm IRENBERG}; they showed that the 1-dimensional H{\eightrm AUSDORFF} measure of the singular set is 0, and therefore it cannot be a point singularity moving along a nice curve; Michael S{\eightrm TRUWE} has obtained a similar result for the stationary case in 5 dimensions.
\par
Jean L{\eightrm ERAY} had already obtained results bounding the H{\eightrm AUSDORFF} dimension of the singular set in $t$ alone; I have not read his argument, but I think that it is based on the already mentioned differential inequality $(||u||^{2})^{\prime}\le C||u||^{6}$ as follows.
Let $\varphi = ||u||^{2}$, so we start with the information $\varphi\in L^{1}(0,T)$ and $\varphi^{\prime}\le a\varphi^{3}$; the differential inequality implies $(\varphi^{-2})^{\prime}\le -2a$ and therefore $\varphi(t)\le \varphi(0)\bigl( 1-2a\,t\,\varphi(0)^{2} \bigr)^{-1/2}$ as long as $1-2a\,t\,\varphi(0)^{2}>0$, and therefore the blow up time satisfies $T_{c}\ge {1\over 2a\,\varphi(0)^{2}}$.
One divides $(0,T)$ into $N$ equal intervals $I_{1},\ldots,I_{N}$, of length $\tau = T/N$; if $j \delta$, and this term is bounded by $C\,\delta(M|\Delta\,u|)$ by using the argument on convolution with radial functions, and the second is bounded by $(C/\delta)E\star |\Delta\,u| = C\,v/\delta$; then the best $\delta$ is chosen (depending upon $x$).
\par
The estimate of L{\eightrm IU} is about $|u(x)-u(y)|\le C|x-y|\bigl( M|grad(u)|(x)+M|grad(u)|(y) \bigr)$ for a.e. $x,y\in R^{N}$; one starts from $u(x)-u(y) = \int_{0}^{1} \bigl( grad(u)\bigl( x+t(y-x) \bigr).y-x \bigr)\,dt$, from which one deduces
$$
\int_{B(x,\rho)} {|u(y)-u(x)|\over |x-y|}\,dy\le \int_{0}^{1} \int_{B(x,\rho)} \Bigl| grad(u)\Bigl( x+t(y-x) \Bigr) \Bigr|\,dt\,dy,
$$
and one uses the change of variable $z = -t(y-x)$; the variable $t$ varies from $|z|/\rho$ to 1, and the last integral is $(N-1)\int_{B(x,\rho)} |grad(u)(x-z)|(|z|^{1-N}-1)\,dz$, which is a convolution of $|grad(u)|$ by a radial decreasing function and is therefore bounded by $M|grad(u)|(x)$ multiplied by the $L^{1}$ norm of the radial function, which is the value obtained when one replaces $|grad(u)|$ by 1, i.e. the volume of $B(0,\rho)$.
One integrates then ${|u(y)-u(x_{1})|\over |y-x_{1}|}+{|u(y)-u(x_{2})|\over |y-x_{2}|}$ on $A = B(x_{1},\rho)\bigcap B(x_{2}\rho)$, and it is bounded by the integral on $B = B(x_{1},\rho)\bigcup B(x_{2}\rho)$, i.e. by $M|grad(u)|(x_{1})+M|grad(u)|(x_{2})$ multiplied by twice the volume of $B(0,\rho)$; one choose $\rho = |x_{1}-x_{2}|$ for example and one finds a $y\in A$ such that ${|u(y)-u(x_{1})|\over |y-x_{1}|}+{|u(y)-u(x_{2})|\over |y-x_{2}|}\le C\bigl( M|grad(u)|(x_{1})+M|grad(u)|(x_{2}) \bigr)$ and as $|y-x_{1}|, |y-x_{2}|\le |x_{1}-x_{2}|$ it implies the desired inequality.
\medskip
The inequality that Lars H{\eightrm EDBERG} used in his truncation method is reminiscent of G{\eightrm AGLIARDO}-N{\eightrm IRENBERG} inequality, and indeed one can find a pointwise version of G{\eightrm AGLIARDO}-N{\eightrm IRENBERG} inequality, as I checked last December, only to discover a week or two after that Patrick G{\eightrm \'ERARD} had made the same observation; however his proof is different from mine, relying on a dyadic decomposition in the style of L{\eightrm ITTLEWOOD}-P{\eightrm ALEY}, while mine is more elementary and uses a parametrix.
Let $E$ be the usual elementary solution of $-\Delta$, i.e. $E(x) = C_{N}/|x|^{N-2}$ if $N>2$, or $E(x) = C\log(|x|)$ for $N = 2$; let $\varphi\in C^{\infty}_{c}(R^{N})$ be such that $\varphi(x) = 1$ for $|x|\le 1$ and $\varphi(x) = 0$ for $|x|\ge 2$, and for $\alpha>0$ let us consider the parametrix $P_{\alpha}$ defined by $P_{\alpha}(x) = E(x)\varphi(x/\alpha)$, so that $-\Delta\,P_{\alpha} = \delta_{0}+g$ with $g(x) = \psi(x/\alpha)/|x|^{N}$, with $\psi\in C^{\infty}_{c}(R^{N})$.
Taking the convolution by $\partial_{j}u$ gives $\partial_{j}P_{\alpha}\star(-\Delta\,u) = \partial_{j}u+ \partial_{j}g\star u$, and as $||r\,grad(\partial_{j}P_{\alpha})||_{L^{1}(R^{N})} = C\,\alpha$ and $||r\,grad(\partial_{j}g)||_{L^{1}(R^{N})} = C/\alpha$, one deduces $|\partial_{j}u|\le C\,M(\Delta\,u)\alpha\,+C\,M(u)/\alpha$, and taking then the best $\alpha$ (depending on $x$) gives $|grad(u)|^{2}\le C\,Mu\,M(\Delta\,u)$.
\bigskip
I want to finish with some remarks on compensated integrability.
\par
In the Summer 1982, at a meeting in Oxford, I heard about a result of W{\eightrm ENTE}, and back in Paris I derived a proof by interpolation which I mentioned around.
If $u, v\in H^{1}(R^{2})$ then one cannot assert that $u_{x}v_{y}-u_{y}v_{x}$ belongs to $H^{-1}(R^{2})$ because $L^{1}(R^{2})$ is not imbedded in $H^{-1}(R^{2})$ as $H^{1}(R^{2})$ is not imbedded into $L^{\infty}(R^{2})$; it does not follow either from writing that quantity as $(u\,v_{y})_{x}-(u\,v_{x})_{y}$ or $(u_{x}v)_{y})-(u_{y}v){y}$, which is the key to the sequential weak lower semicontinuity observed by M{\eightrm ORREY} and which I learned from by John M.
B{\eightrm ALL} before extending this kind of property into the Compensated Compactness method with Fran\c{c}ois M{\eightrm URAT}.
However, it is indeed true that $u_{x}v_{y}-u_{y}v_{x}\in H^{-1}(R^{2})$, and W{\eightrm ENTE} also proved that if one solves the equation $-\Delta\,w = u_{x}v_{y}-u_{y}v_{x}$, then one also has $w\in H^{1}(R^{2})\bigcap C_{0}(R^{N})$.
\par
I do not know what the original proof of W{\eightrm ENTE} was, but my first proof used interpolation and L{\eightrm ORENTZ} spaces.
It would take us too far if I explained what interpolation of B{\eightrm ANACH} spaces is according to the theory developed by Jacques-Louis L{\eightrm IONS} and Jaak P{\eightrm EETRE}, and how the theory applied to $L^{1}$ and $L^{\infty}$ creates the family of L{\eightrm ORENTZ} spaces; therefore I will just state the ideas for those readers who know these tools.
First one uses the fact that $H^{1/2}(R^{2})\subset L^{4,2}(R^{2})$, as noticed by Jaak P{\eightrm EETRE} (a result used was $||u||_{L^{4}(R^{2})}\le C\,||u||_{L^{2}(R^{2})}^{1/2}||grad(u)||_{L^{2}(R^{2})}^{1/2}$, and it is not as precise because a theorem of L{\eightrm IONS} and P{\eightrm EETRE} asserts that this statement is equivalent to the fact that the interpolation space $\bigl( H^{1}(R^{2}),L^{2}(R^{2}) \bigr)_{1/2,1}$, which is smaller than $H^{1/2}(R^{2})$, is included in $L^{4}(R^{2})$, which is bigger than $L^{4,2}(R^{2})$).
Then one uses the fact that the product of two functions in $L^{4,2}(R^{2})$ is in $L^{2,1}(R^{2})$.
This shows that $B(u,v) = u_{x}v_{y}-u_{y}v_{x}$ is a sum of derivatives of functions in $L^{2,1}(R^{2})$, in the case where $u\in H^{1/2}(R^{2})$ and $v\in H^{3/2}(R^{2})$ by using the formula $(u\,v_{y})_{x}-(u\,v_{x})_{y}$, or in the case where $u\in H^{3/2}(R^{2})$ and $v\in H^{1/2}(R^{2})$ by using the formula $(u_{x}v)_{y})-(u_{y}v){y}$; by another theorem of L{\eightrm IONS} and P{\eightrm EETRE} on bilinear mappings the same property is then true for $u, v\in H^{1}(R^{2})$.
Then by C{\eightrm ALDER\'ON}-Z{\eightrm YGMUND} theorem and interpolation, one finds that $w$ has its two partial derivatives in $L^{2,1}(R^{2})$ (a smaller space than $L^{2}(R^{2})$), and this implies that $w\in C_{0}(R^{2})$.
\par
In 1984, I described a second method which extends immediately to more general situations similar to those found in Compensated Compactness theory for the quadratic forms which are sequentially weakly continuous; the method uses F{\eightrm OURIER} transform and interpolation, but not C{\eightrm ALDER\'ON}-Z{\eightrm YGMUND} theorem, and the results are slightly different.
The example that I had chosen was the equation for the pressure in N{\eightrm AVIER}-S{\eightrm TOKES} equation in 2 dimensions, but that is similar to the previous example.
Using $x_{1}, x_{2}$, instead of $x, y$, one has $|\xi|^{2}{\cal F}w(\xi) = \int_{R^{2}} B(\xi-\eta,\eta){\cal F}u(\xi-\eta){\cal F}v(\eta)\,d\eta$, where $B(\zeta,\eta) = \zeta_{1}\,\eta_{2}-\zeta_{2}\,\eta_{1}$, but using the fact that $B(\xi,\xi) = 0$ for all $\xi$, one has $B(\xi-\eta,\eta) = B(\xi,\eta) = B(\xi,\eta-\xi)$, so that one has the two bounds $|B(\xi-\eta,\eta)|\le C\,|\xi|\,|\eta|$ and $|B(\xi-\eta,\eta)|\le C\,|\xi|\,|\xi-\eta|$ and therefore $|B(\xi-\eta,\eta)|\le C\,|\xi|\,|\eta|^{1/2}|\xi-\eta|^{1/2}$.
One deduces that $|\xi|\,|{\cal F}w|\le C\,|\xi|^{1/2}|{\cal F}u|\star |\xi|^{1/2}|{\cal F}v|$, but as $|\xi|^{-1/2}\in L^{4,\infty}(R^{2})$, one deduces that $\xi\,{\cal F}u\in L^{2}(R^{2})$ implies $|\xi|^{1/2}|{\cal F}u|\in L^{4/3,2}(R^{2})$ and the convolution product of two functions in $L^{4/3,2}(R^{2})$ is in $L^{2,1}(R^{2})$.
In particular $|\xi|\,|{\cal F}w|\in L^{2}(R^{2})$ but I do not know how to compare the informations $grad(w)\in L^{2,1}(R^{2};R^{2})$ and $|\xi|{\cal F}w\in L^{2,1}(R^{2};R^{2})$.
As $|\xi|^{-1}\in L^{2,\infty}(R^{2})$, one deduces that ${\cal F}w\in L^{1}(R^{2})$, and then $w\in {\cal F}L^{1}(R^{2})\subset C_{0}(R^{2})$.
\par
My approach has been slightly improved by Ronald C{\eightrm OIFMAN}, Pierre-Louis L{\eightrm IONS}, Yves M{\eightrm EYER} and Stephen S{\eightrm EMMES}, using the H{\eightrm ARDY} spaces ${\cal H}^{1}$; their result has the advantage of showing that the second derivatives of $w$ belong to ${\cal H}^{1}(R^{2})$, and therefore $w\in W^{2,1}(R^{2})$; however, contrary to what they have claimed, many applications do not require their improvement and can be obtained by using my second method.
\bigskip
So much for technical details around N{\eightrm AVIER}-S{\eightrm TOKES} equation.
Let us go back to Oceanography!
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
29. Monday March 29.
\medskip
Up to now, I have not mentioned the effect of C{\eightrm ORIOLIS} force due to the rotation of the Earth; it is small but it does have some effect.
\par
Assume that we have a first frame, called fixed, in which N{\eightrm EWTON}'s law of Classical Mechanics, $force = mass\times acceleration$ applies, and let us see what it implies for the equation in a moving frame.
Let $x(t)$ be the position of a material point in the fixed frame, and let $\xi(t)$ be the position of the same point in the moving frame; let $a(t)$ be the position of the origin of the moving frame and let $P(t)$ be the rotation which maps the basis of the initial frame into the basis of the moving frame, so that one has
$$
x(t) = a(t) + P(t)\xi(t).
$$
As $P(t)^{T}P(t) = I$, if $\prime$ denotes the derivative with respect to $t$, one has $P^{\prime}(t)^{T}P(t)+P(t)^{T}P^{\prime}(t) = 0$, and so if one defines $B(t)$ by $P^{\prime}(t) = P(t)B(t)$, one obtains $B(t)^{T}+B(t) = 0$, and therefore, as we work in $R^{3}$, there exists a vector $\Omega(t)$ such that $B(t)x = \Omega(t)\times x$ for every $x\in R^{3}$; one deduces
$$
x^{\prime}(t) = a^{\prime}(t) + P(t)\Bigl( \xi^{\prime}(t)+\Omega(t)\times \xi(t)
\Bigr),
$$
and
$$
x^{\prime\prime}(t) = a^{\prime\prime}(t)+P(t)\Bigl( \Omega^{\prime}(t)\times\xi(t) \Bigr)+ P(t)\Bigl[ \xi^{\prime\prime}(t)+2\Omega(t)\times \xi^{\prime}(t) + \Omega(t)\times\Bigl( \Omega(t)\times\xi(t) \Bigr) \Bigr].
$$
In the case of the rotation of the Earth, one considers that $\Omega^{\prime}(t) = 0$.
The term $2\Omega(t)\times \xi^{\prime}(t)$ is the C{\eightrm ORIOLIS} acceleration (although L{\eightrm AGRANGE} had introduced it in 1778-79 in his studies of tides, while C{\eightrm ORIOLIS}'s work dates from 1835).
If one uses the formula $a\times(b\times c) = (a.c)b-(a.b)c$, one deduces that $\Omega\times(\Omega\times\xi) = (\Omega.\xi)\Omega - |\Omega|^{2}\xi$, and therefore the term $\Omega\times(\Omega\times\xi)$, which is related to the centrifugal acceleration with $a^{\prime\prime}(t)$, derives from a potential, which changes slightly the gravitation potential, creating the {\it geopotential}.
For the rotation of the Earth, $|\Omega| = {2\pi\over 86400}\approx 3.6\,10^{-5}$, so that at the equator the centrifugal acceleration is about $8.3\,10^{-3}$, less than one thousandth of the acceleration of gravity.
\medskip
Because the term $\Omega\times(\Omega\times\xi)$ is a gradient, it changes only what $p$ is, and therefore adding the C{\eightrm ORIOLIS} term $\Omega\times u$ in N{\eightrm AVIER}-S{\eightrm TOKES} equation does not change much in the proofs that we have seen, because this term is orthogonal to $u$ and therefore does not work, and the basic estimates are the same as before.
\medskip
The C{\eightrm ORIOLIS} force depends upon the velocity, in a way that reminds of Electromagnetism, where the L{\eightrm ORENTZ} force acting on a charge $\rho$ moving with velocity $v$ in an electric field $E$ and magnetic induction field $B$ is $\rho\bigl( E+v\times B \bigr)$.
The analogy goes further and it has been used in connection with MHD (Magnetohydrodynamics), at least by M{\eightrm OFFATT}: in MHD the fluid is a plasma, which has electrical charges moving around, but the forces acting on a neutral fluid are very similar, as we will see by computing $u\times curl(u)$ in a domain of $R^{3}$.
\medskip
Let $\varepsilon_{ijk}$ be the totally antisymmetric tensor, which is 0 if two of the indices $i, j, k$, are equal, and equal to the signature of the permutation $123\mapsto ijk$ in other cases, i.e. $\varepsilon_{123} = \varepsilon_{231} = \varepsilon_{312} = 1$ and $\varepsilon_{321} = \varepsilon_{213} = \varepsilon_{132} = -1$.
Then the definition of the exterior product of two vectors in $R^{3}$ is
$$
c = a\times b\hbox{ means }c_{i} = \sum_{j,k} \varepsilon_{ijk}a_{j}b_{k},
$$
and the $curl$ of a function $u$, sometimes denoted $\nabla\times u$, is defined by
$$
\Bigl( curl(u) \Bigr)_{i} = \sum_{j,k} \varepsilon_{ijk}{\partial u_{k}\over \partial x_{j}}.
$$
One has
$$
\eqalign{
\Bigl( u\times curl(u) \Bigr)_{i} &= \sum_{j,k} \varepsilon_{ijk}u_{j} \Bigl( \sum_{l,m} \varepsilon_{klm}{\partial u_{m}\over \partial x_{l}} \Bigr) = \sum_{j,k} \varepsilon_{ijk}u_{j} \Bigl( \varepsilon_{kij}{\partial u_{j}\over \partial x_{i}} + \varepsilon_{kji}{\partial u_{i}\over \partial x_{j}} \Bigr)\cr
&= \sum_{j,k} \varepsilon_{ijk}^{2}u_{j} \Bigl( {\partial u_{j}\over \partial x_{i}} -{\partial u_{i}\over \partial x_{j}} \Bigr) = \sum_{j\ne i} u_{j} \Bigl( {\partial u_{j}\over \partial x_{i}} -{\partial u_{i}\over \partial x_{j}} \Bigr) = \sum_{j} u_{j}{\partial u_{j}\over \partial x_{i}} -\sum_{j} u_{j}{\partial u_{i}\over \partial x_{j}},}
$$
and therefore
$$
\sum_{j} u_{j}{\partial u_{i}\over \partial x_{j}} = \Bigl( u\times curl(-u) \Bigr)_{i} +{1\over 2}{\partial |u|^{2}\over \partial x_{i}},
$$
so that N{\eightrm AVIER}-S{\eightrm TOKES} equation becomes
$$
{\partial u\over \partial t} -\nu\Delta\,u+u\times curl(-u)+grad\Bigl( {p\over \rho_{0}}+{|u|^{2}\over 2} \Bigr) = 0,\;div(u) = 0,
$$
and C{\eightrm ORIOLIS} acceleration justs adds $2\Omega$ to $curl(-u)$.
\par
In the case $\nu = 0$, corresponding to E{\eightrm ULER} equation, one sees that a stationary irrotational flow (i.e. satisfying $curl(u) = 0$), corresponds to ${p\over \rho_{0}}+{|u|^{2}\over 2} = constant$ (B{\eightrm ERNOULLI}'s law); one also sees that in the all space $R^{3}$ the helicity $\bigl( u.curl(u) \bigr)$ is conserved ($curl$ is a symmetric operator); this was first observed by Jean-Jacques M{\eightrm OREAU} (and also by someone else, whose name I do not remember), and M{\eightrm OFFATT} has given an interpretation of this quantity in terms of linking of vorticity lines (in order to avoid boundary conditions, the result is considered in the whole space, as I do not care much for unrealistic periodic conditions).
As the quantity integrated is not positive, the conservation of helicity has not helped for questions of global existence or smoothness of solutions of N{\eightrm AVIER}-S{\eightrm TOKES} equation in 3 dimensions.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
30. Wednesday March 31.
\medskip
I want to derive now the equation describing the evolution of the vorticity in two dimensions, and then in three dimensions, as a consequence of N{\eightrm AVIER}-S{\eightrm TOKES} equation.
\medskip
In two dimensions the N{\eightrm AVIER}-S{\eightrm TOKES} equation, with zero exterior forces (the gravitational force being included in the pressure term) is
$$
\eqalign{
&{\partial u_{1}\over \partial t}+u_{1}{\partial u_{1}\over \partial x_{1}}+u_{2}{\partial u_{1}\over \partial x_{2}}-\nu\Delta\,u_{1}+{1\over \rho_{0}}{\partial p\over \partial x_{1}} = 0\cr
&{\partial u_{2}\over \partial t}+u_{1}{\partial u_{2}\over \partial x_{1}}+u_{2}{\partial u_{2}\over \partial x_{2}}-\nu\Delta\,u_{2}+{1\over \rho_{0}}{\partial p\over \partial x_{2}} = 0\cr
&div(u) = 0,}
$$
and the vorticity is the scalar quantity
$$
\omega = {\partial u_{2}\over \partial x_{1}}-{\partial u_{1}\over \partial x_{2}}.
$$
Applying $-{\partial \over \partial x_{2}}$ to the first equation, ${\partial \over \partial x_{1}}$ to the second equation and adding, one finds that the vorticity $\omega$ satisfies the equation
$$
{\partial \omega\over \partial t}+u_{1}{\partial \omega\over \partial x_{1}}+u_{2}{\partial \omega\over \partial x_{2}} - \nu\Delta\omega = 0,
$$
because the pressure disappears and the supplementary terms coming from the first equation are $-{\partial u_{1}\over \partial x_{2}}{\partial u_{1}\over \partial x_{1}}-{\partial u_{2}\over \partial x_{2}}{\partial u_{1}\over \partial x_{2}} = -{\partial u_{1}\over \partial x_{2}}div(u)$, while the supplementary terms coming from the second equation are ${\partial u_{1}\over \partial x_{1}}{\partial u_{2}\over \partial x_{1}}+{\partial u_{2}\over \partial x_{1}}{\partial u_{2}\over \partial x_{2}} = {\partial u_{2}\over \partial x_{1}}div(u)$.
\medskip
In three dimensions the computation is a little more involved; the N{\eightrm AVIER}-S{\eightrm TOKES} equation, with zero exterior forces, is
$$
\eqalign{
&{\partial u_{i}\over \partial t}+\sum_{j = 1}^{3} u_{j}{\partial u_{i}\over \partial x_{j}}-\nu\Delta\,u_{i}+{1\over \rho_{0}}{\partial p\over \partial x_{i}} = 0\hbox{ for } i = 1, 2, 3,\cr
&div(u) = 0,}
$$
and the vorticity is the vector valued quantity
$$
\omega = curl(u),\hbox{ i.e. }\omega_{i} = \sum_{j,k = 1}^{3} \varepsilon_{ijk}{\partial u_{k}\over \partial x_{j}}\hbox{ for } i = 1, 2, 3.
$$
The equation for $\omega$ is
$$
{\partial \omega_{i}\over \partial t}+\sum_{j = 1}^{3} u_{j}{\partial \omega_{i}\over \partial x_{j}}-\sum_{j = 1}^{3} \omega_{j}{\partial u_{i}\over \partial x_{j}}-\nu\Delta\,\omega_{i} = 0\hbox{ for } i = 1, 2, 3.
$$
Indeed, the pressure disappears and the supplementary term in the equation for $\omega_{i}$ is $\sum_{j,k,l = 1}^{3} \varepsilon_{ijk}{\partial u_{l}\over \partial x_{j}}{\partial u_{k}\over \partial x_{l}}$.
As only the terms where $\varepsilon_{ijk}\ne 0$ are useful, $l$ takes the values $i, j$ and $k$, and the sum is $\sum_{j,k = 1}^{3} \varepsilon_{ijk}\bigl( {\partial u_{i}\over \partial x_{j}}{\partial u_{k}\over \partial x_{i}}+{\partial u_{j}\over \partial x_{j}}{\partial u_{k}\over \partial x_{j}}+{\partial u_{k}\over \partial x_{j}}{\partial u_{k}\over \partial x_{k}} \bigr)$, and using $div(u) = 0$ it is $\sum_{j,k = 1}^{3} \varepsilon_{ijk}\bigl( {\partial u_{i}\over \partial x_{j}}{\partial u_{k}\over \partial x_{i}}-{\partial u_{i}\over \partial x_{i}}{\partial u_{k}\over \partial x_{j}} \bigr)$, which one writes $-\omega_{i}{\partial u_{i}\over \partial x_{i}}+\sum_{j,k = 1}^{3} \varepsilon_{ijk}{\partial u_{i}\over \partial x_{j}}\bigl( {\partial u_{k}\over \partial x_{i}}-{\partial u_{i}\over \partial x_{k}} \bigr)$, and for $j\ne i$ and $k$ being the third index, the term ${\partial u_{k}\over \partial x_{i}}-{\partial u_{i}\over \partial x_{k}}$ is indeed $-\omega_{j}$.
\medskip
Except in the whole space or the unrealistic peridoc case, there are no clear boundary conditions for the vorticity (vorticity is created at the boundary).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
31. Friday April 2.
\medskip
In a talk by Roger L{\eightrm EWANDOWSKI} we have seen a model used in Oceanography: an horizontal fixed boundary is used to model the interface between Ocean and Atmosphere and a boundary condition is chosen there, which is supposed to take into account the turbulent kinetic energy (TKE) arising in a neighbourhood of the interface.
Before looking at questions of averaging, I want to discuss the question of which types of boundary conditions are natural.
\medskip
In studying the stationary S{\eightrm TOKES} equation, I have mentioned the approach of considering Linearized Elasticity and letting the L{\eightrm AM\'E} coefficient $\lambda$ tend to $\infty$, which forces the constraint $div(u) = 0$ at the limit, and the limit $p$ of $-\lambda\,div(u)$ plays the role of a pressure.
This similitude disappears as soon as one considers the evolution problems, because in Linearized Elasticity $u$ denotes a displacement (whose gradient is supposed to be small), while for S{\eightrm TOKES} equation $u$ denotes a velocity; the acceleration involves then a term in ${\partial^{2}u\over \partial t^{2}}$ in the first case and a term in ${\partial u\over \partial t}$ in the second case.
\par
I have initially discussed the homogeneous D{\eightrm IRICHLET} condition $u = 0$ on $\partial\Omega$, and one may also consider the case of a nonhomogeneous D{\eightrm IRICHLET} condition, $u = g$ on $\partial\Omega$: one first chooses a function equal to $g$ on the boundary, and the difference satisfies the homogeneous D{\eightrm IRICHLET} condition; one must then have characterized the space of traces of functions in $H^{1}(\Omega)$ (which is $H^{1/2}(\partial\Omega)$ in the good cases), but in the limiting case $\lambda\rightarrow\infty$, one needs to add a constraint.
If $u\in H^{1}(\Omega;R^{N})$ satisfies $div(u) = 0$, and $u = g$ on $\partial\Omega$, then integrating $div(u)$ in $\Omega$ gives $\int_{\partial\Omega} (g.n)\,dx = 0$, where $n$ denotes the exterior normal to $\Omega$; conversely if $g\in H^{1/2}(\partial\Omega;R^{N})$ satisfies $\int_{\partial\Omega} (g.n)\,dx = 0$, then one first chooses $v\in H^{1}(\Omega;R^{N})$ equal to $g$ on the boundary and it remains to add a function $u\in H^{1}_{0}(\Omega;R^{N})$ with $div(u) = -div(v)$, but as $\int_{\Omega} div(v)\,dx = 0$ because of the condition on $g$, a function $u$ exists if $\Omega$ is smooth enough (bounded with $X(\Omega) = L^{2}(\Omega)$ for example).
\par
The case of N{\eightrm EUMANN} condition over all the boundary of $\Omega$ is of the form
$$
\eqalign{
&-\sum_{j = 1}^{N} {\partial \sigma_{ij}\over \partial x_{j}} = f_{i}\hbox{ in }\Omega\cr
&\sum_{j = 1}^{N} \sigma_{ij}n_{j} = g_{i}\hbox{ on }\partial\Omega,}
$$
and it requires the compatibility conditions
$$
\eqalign{
&\int_{\Omega} f_{i}\,dx+\int_{\partial\Omega} g_{i}\,d\sigma = 0\hbox{ for all }i\cr
&\int_{\Omega} \Bigl( \sum_{j,k} \varepsilon_{ijk}x_{j}f_{k} \Bigr)\,dx+\int_{\partial\Omega} \Bigl( \sum_{j,k} \varepsilon_{ijk}x_{j}g_{k} \Bigr)\,d\sigma = 0\hbox{ for all }i,}
$$
which express the fact that the total force and the total torque acting on $\overline{\Omega}$ are 0.
It is important to notice that this follows from the equilibrium equation and the symmetry of the stress tensor, so that it is true for Linearized Elasticity as well as for the general (nonlinear) Elasticity in the deformed configuration, where the symmetric C{\eightrm AUCHY} stress tensor appears.
Indeed, the variational formulation is
$$
\int_{\Omega} \sum_{ij} \sigma_{ij}{\partial v_{i}\over \partial x_{j}}\,dx = \int_{\Omega} \sum_{i} f_{i}v_{i}\,dx+\int_{\partial\Omega} g_{i}v_{i}d\sigma\hbox{ for all }v\in H^{1}(\Omega;R^{3}),
$$
and by the symmetry of the stress tensor one has $\sum_{ij} \sigma_{ij}{\partial v_{i}\over \partial x_{j}} = \sum_{ij} \sigma_{ij}\varepsilon_{ij}(v)$, where as usual $\varepsilon_{ij}(v) = {1\over 2}\bigl( {\partial v_{i}\over \partial x_{j}}+{\partial v_{j}\over \partial x_{i}} \bigr)$, and therefore the left side is 0 if $v$ is such that $\varepsilon_{ij}(v) = 0$ for all $i,j$; this is the case if $v_{i} = a_{i}+\sum_{j} M_{ij}x_{j}$ for all $i$ with $M$ antisymmetric, and in three dimensions it means $M\,x = m\times x$ for some $m\in R^{3}$, and writing that the right side is 0 for all these $v$ gives the necessary conditions on $f$ and $g$, corresponding to the physical interpretation of total force and total torque.
In Linearized Elasticity, i.e. $\sigma_{ij} = \sum_{k,l} C_{ijkl}\varepsilon_{kl}(u)$ for all $i,j$, with $C_{ijkl} = C_{jikl} = C_{ijlk}$ for all $i,j,k,l$, and under the hypothesis of Very Strong Ellipticity (i.e. there exists $\alpha>0$ such that $\sum_{ijkl} C_{ijkl}A_{ij}A_{kl}\ge\alpha\sum_{ij} |A_{ij}|^{2}$ for all symmetric $A$), then the necessary conditions are sufficient if the injection of $H^{1}(\Omega)$ into $L^{2}(\Omega)$ is compact and if K{\eightrm ORN}'s inequality holds, as a consequence of the Equivalence Lemma.
This requires that one identifies all the $v\in H^{1}(\Omega;R^{3})$ satisfying $\varepsilon_{ij}(v) = 0$ for all $i,j$, and it follows from the identity
$$
2{\partial^{2}u_{i}\over \partial x_{j}\partial x_{k}} = {\partial\over \partial x_{j}}\Bigl( {\partial u_{i}\over \partial x_{k}}+{\partial u_{k}\over \partial x_{i}} \Bigr)-{\partial\over \partial x_{i}}\Bigl( {\partial u_{k}\over \partial x_{j}}+{\partial u_{j}\over \partial x_{k}} \Bigr)+
{\partial\over \partial x_{k}}\Bigl( {\partial u_{j}\over \partial x_{i}}+{\partial u_{i}\over \partial x_{j}} \Bigr)\hbox{ for all }i,j,k,
$$
so that $\varepsilon_{ij}(v) = 0$ for all $i,j$ implies that all second derivatives are 0, so that $v = a+M\,x$ and $M$ must be antisymmetric.
The solution $u$ exists then and is defined up to the addition of $a+m\times x$ for $a,m\in R^{3}$, and it must be pointed out that these are not rigid displacements but linearized rigid displacements (the antisymmetric matrices appear as the tangent space at I for the manifold of all rotations $SO(3)$, which is compact).
If the necessary conditions are not satisfied, the evolution equation will still have a solution and the body will move away in the direction of those linearized rigid displacements.
\medskip
Let us imagine now, in the approximation of Linearized Elasticity, an elastic body with a flat part of its boundary put on an horizontal table, and assume that the system of forces applied to it does not take it away from the table (or consider the purely mathematical problem that the displacement satisfies $u_{3} = 0$ on this flat part of the boundary); the body is allowed to slide horizontally on the table, and one expects to have less stringent compatibility conditions, corresponding to the horizontal part of the total force being 0 (there is no friction on the table and so the table will give a vertical reaction which will cancel the vertical component of the total force), and the torque along the $x_{3}$ axis must be 0 (the reactions of the table being able to compensate for the rest of the total torque).
Mathematically, the condition $u_{3} = 0$ on a piece of the boundary sitting in the plane $x_{3} = H$, is imposed in the definition of the functional space, and $v$ is constrained to be in this space, so only the elements $a+m\times x$ satisfying this constraint are allowed, i.e. one must choose $a_{3} = m_{1} = m_{2} = 0$, and the necessary conditions corresponding to $a_{1},a_{2}$ imply that the horizontal part of the total force is 0, while the necessary condition corresponding to $m_{3}$ implies then that the total torque around any vertical axis is 0.
\medskip
Mathematically one can study nonhomogeneous conditions, like imposing $u_{3}$ on a piece of the boundary which is not necessarily flat, and the natural boundary conditions implied by the variational formulation will involve the traction $T$ defined by $T_{i} = \sum_{j} \sigma_{ij}n_{j}$ (as for normal traces in $H(div;\Omega)$), and $T_{1}$ and $T_{2}$ can be imposed, with natural compatibility conditions
\par
Mathematically, one could also impose the displacements $u_{1}$ and $u_{2}$ on a piece of the boundary, and the natural boundary condition implied by the variational formulation will involve $T_{3}$.
\medskip
For (Newtonian) fluids, one has
$$
\sigma_{ij} = 2\mu\varepsilon_{ij}-p\,\delta_{ij} = \mu\Bigl( {\partial u_{i}\over \partial x_{j}}+{\partial u_{j}\over \partial x_{i}} \Bigr)-p\,\delta_{ij}\hbox{ for all }i,j,
$$
so that
$$
\sum_{j} \sigma_{ij}n_{j} = \mu{\partial u_{i}\over \partial n} + \mu\sum_{j} {\partial u_{j}\over \partial x_{i}}n_{j} -p\,n_{i}\hbox{ for all }i.
$$
For an horizontal boundary, like the fixed interface separating Ocean from Atmosphere in the model considered by Roger L{\eightrm EWANDOWSKI}, one has $n_{1} = n_{2} = 0$, and $n_{3} = 1$ for Ocean and $n_{3} = -1$ for Atmosphere, so that in the Ocean one has
$T^{O}_{1} = \mu{\partial u^{O}_{1}\over \partial x_{3}}+\mu{\partial u^{O}_{3}\over \partial x_{1}}, T^{O}_{2} = \mu{\partial u^{O}_{2}\over \partial x_{3}}+\mu{\partial u^{O}_{3}\over \partial x_{2}}, T^{O}_{3} = 2\mu{\partial u^{O}_{3}\over \partial x_{3}}-p^{O}$, and similarly for Atmosphere
$T^{A}_{1} = -\mu{\partial u^{A}_{1}\over \partial x_{3}}-\mu{\partial u^{A}_{3}\over \partial x_{1}}, T^{A}_{2} = -\mu{\partial u^{A}_{2}\over \partial x_{3}}-\mu{\partial u^{A}_{3}\over \partial x_{2}}, T^{A}_{3} = -2\mu{\partial u^{A}_{3}\over \partial x_{3}}+p^{A}$, and it is usually the jump of these quantities which appears in the variational formulations.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
32. Monday April 5.
\medskip
Modelization of turbulent flows is an important scientific and technological question, and although engineers may say that they are able to control turbulent flows, it is mainly because adaptive control ideas seem to work even in situations where no one knows what the right equations are for describing the phenomena which one wants to control.
From a scientific point of view, not so much is understood about turbulence.
For what concerns Oceanography, some modelization of turbulent flows is necessary in order to describe correctly what goes on at ``small scales'', remembering that the scales used for the Ocean, or the Atmosphere, are quite large.
\medskip
It is quite common to experience the presence of microstructures in some fluid flows, but it is a very arduous task to propose a model that would describe accurately the important effects occuring in these flows.
My first experimental evidence concerns the structure of the interface in front of a rainstorm, as I had observed many times long before becoming a mathematician, after a hot Summer day in the French countryside: one knows that a storm is coming, although the air is still, perhaps because the pressure is higher than usual, and then one starts to hear the leaves of the trees moving while the branches stay still; soon after the small branches start to move too, followed by the large branches a little after and the whole trees are in motion when the rain arrives.
It clearly suggests that the classical idea of a sharp interface with some partial differential equations being satisfied on each side and with some boundary conditions being imposed on the ``interface'' might not be so efficient for describing the effects occuring in that living layer, with small vortices on the dry side and large vortices on the wet side.
My second experimental evidence concerns the structure of the ``wind'', as I had observed twenty years ago, on a week end where I had expected to sail between La Rochelle and Ile de R\'e, but the morning had provided us with what one calls ``calme plat'' in French: there was no wind, and the surface of the sea was extremely smooth and only showing a long swell (``houle'' in French), which combined with the steady movement sustained by the small engine of the boat to produce a beginning of seasickness; fortunately, it did not last two long, because after a while we saw what one calls ``une ris\'ee'' in French (light squall in English): the wind waiting for us; it is an amazing fact to come from the windless side with a smooth sea surface to the place where the wind is, with the surface of the sea all wrinkled with wavelengths of the order of 5 to 10 centimeters, and when one crosses the transition line (which seemed stationary, but it might have been moving at a much slower pace that the boat, which was carried by its small engine), the sails inflated, and sailing started.
\par
It was around twenty years ago too that I had heard Joe K{\eightrm ELLER} mention that at one time there had been a lot of articles about the statistical distribution of wavelengths of the waves at the surface of the sea, until one had been able to measure this distribution and it had appeared that all the theories had been wrong, as one had observed much more energy that any theorist had expected in the small capillary waves, those which I had observed as the signature of the wind waiting for us.
\par
In other words, many like to imagine that natural phenomena obey the probabilistic processes or the statistical laws that are already known, and these people usually do not care that the phenomena that they are trying to study are described by complicated systems of partial differential equations for which their standard processes are obviously not adapted.
In another meeting, Joe K{\eightrm ELLER} had mentioned the evolution from ideas about three dimensional turbulence by K{\eightrm OLMOGOROV}, the two dimensional turbulence ideas used in meteorology (where the stratification by gravity simplifies the full three dimensional aspects), some one dimensional ideas that were not so good, and the zero dimensional ideas of iterating maps, followed by continually improving numerical simulations in dimensions one, two and even three, but he emphasized that something important had been lost in the way: in the 40s, turbulence specialists talked about velocities, pressure, kinetic energy, temperature, heat flux, while now they talk about statistics without reference to any important physical quantity related to fluids.
\medskip
The only thing about turbulence that everybody agrees with is that it is created by oscillations in the velocity field, and R{\eightrm EYNOLDS} might have been the first to notice that if the ``average'' of $u_{i}$ is denoted $\overline{u_{i}}$, then the average of $u_{i}u_{j}$ is $\overline{u_{i}}\,\overline{u_{j}} + R_{ij}$, where the symmetric R{\eightrm EYNOLDS} tensor $R$ with entries $R_{ij}$ is not necessarily 0.
\par
Probabilists like to imagine that all functions in the fluid depend upon a parameter $\omega$ belonging to a space endowed with a probability measure, and integration with respect to this probability measure, the expectation, plays the role of the intuitive averaging technique.
\par
Some specialists of asymptotic expansions like to plug functions like $u_{0}(x)+\varepsilon_{n}u_{1}\bigl( x,{x\over \varepsilon_{n}} \bigr)+\ldots$ into the system of equations governing fluids, where the functions $u_{j}(x,y)$ are periodic in $y$, the vague idea of averaging becoming the precise technique of averaging in $y$, and this deterministic approach is sometime useful, although it is not able to explain some multiple scale effects that turbulent fluids are believed to show.
\par
For about twenty five years, I have been developping a mathematical approach to the study of ``oscillations'' in solutions of partial differential equations, partly in collaboration with Fran\c{c}ois M{\eightrm URAT}, and various notions of weak convergence appear in this approach, which definitely has an advantage on all the others, that it does not postulate anything about oscillations but tries to determine what kind of oscillations are compatible with linear differential balance laws and nonlinear constitutive relations.
First, I should point out that I use the term ``oscillations'' to englobe also ``concentration effects'', i.e. the meaning used is to consider weakly convergent sequences which are not strongly convergent, but convergences of a weak type but different from the usual weak convergence are also used.
Second, I should point out that the use of sequences is a purely mathematical trick whose object is to identify the correct topology (usually related to some kind of weak convergence) that one should use for various physical or nonphysical quantities (it is similar to the description of $R$ by starting from C{\eightrm AUCHY} sequences in $Q$, and once $R$ is understood a real number is not related to a sequence of rationals any more!).
\par
The classical weak convergence appears to be natural for some quantities and not for others, and the notion of differential forms will clarify this question.
In the equation expressing conservation of mass, ${\partial \rho\over \partial t}+div(\rho\,u) = 0$, the quantities $\rho$ and $q_{i} = \rho\,u_{i}$ are coefficients of differential forms, but $u_{i}$ only appears as a quotient of two quantities for which the adapted topology is weak convergence; therefore density and momentum are more easy to handle than velocity.
It will be useful then to describe some properties of H-convergence (introduced with Fran\c{c}ois M{\eightrm URAT}, and generalizing the notion of G-convergence introduced by Sergio S{\eightrm PAGNOLO}, with some ideas from Ennio {\eightrm DE} G{\eightrm IORGI}), and I will describe some properties of weakly converging sequences of solutions of equations like $div\bigl( A_{n}\,grad(u_{n}) \bigr) = f$.
It will then be natural to consider sequences of operators of the form ${\partial \over \partial t}+\sum_{i} u_{i}^{n}{\partial \over \partial x_{i}}$, and as nothing general is known in the case when the coefficients only converge weakly, I will describe in detail some special cases.
\par
It is worth mentioning that geometers like to think that they know how to write equations for fluid flows in intrinsic forms, but as long as one does not know how to pass to the limit in weakly convergent sequences of solutions of these equations, one cannot assert that geometers have or have not introduced the correct framework (my guess is that they have not!).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
33. Wednesday April 7.
\medskip
In questions of asymptotic expansions, one considers sequences of functions like $v^{n}(x) = u_{0}(x)+\varepsilon_{n}u_{1}\bigl( x,{x\over \varepsilon_{n}} \bigr)+\ldots$, where the functions $u_{j}(x,y)$ are periodic in $y$ (and smooth enough in $(x,y)$), and $\varepsilon_{n}$ tends to 0.
If $\varepsilon_{n}$ is a small characteristic length, and if the solution of a physical problem has this form, then if one measured the value of $v^{n}$ at a few points, quite far apart compared to the characteristic length $\varepsilon_{n}$, then one would find $u_{0}(x)+O(\varepsilon_{n})$ (plus some eventual errors due to the measuring process), and one might well believe that the measured solution is $u_{0}$, considering the little discrepancies as systematic errors.
\par
It is usual in Physics courses to be told that one term is small and that it will be neglected (it is not always clear if these terms are indeed small, as they may be small in the real world, but if the equation used is not a good model of the physical world the corresponding term might not be so small); having neglected some terms one performs some formal computations with the simplified equation, like taking derivatives, and the first remark, which seems to infuriate Physics teachers, is that the derivative of a small term might not be small; actually, in our example one has ${\partial v^{n}\over \partial x_{i}} = {\partial u_{0}\over \partial x_{i}}+{\partial u_{1}\over \partial y_{i}} +O(\varepsilon_{n})$, and ${\partial u_{1}\over \partial y_{i}}$ is not small when $u_{1}$ does depend upon $y$.
\par
Fortunately, in some cases like linear partial differential equations with smooth coefficients, the procedure can be shown to work, because of the generalized framework of the theory of distributions of Laurent S{\eightrm CHWARTZ} for example: if a sequence $v^{n}$ converges to $v^{\infty}$, then ${\partial v^{n}\over \partial x_{i}}$ converges to ${\partial v^{\infty}\over \partial x_{i}}$ for every $i$, but in this statement it must be realized that the meaning of convergence is not that the differences are uniformly small; therefore we do have $v^{n} = u_{0}(x)+O(\varepsilon_{n})$, while ${\partial v^{n}\over \partial x_{i}} \ne {\partial u_{0}\over \partial x_{i}}+O(\varepsilon_{n})$, but there is no contradiction as long as one only considers linear questions.
\medskip
If one wants to avoid the too general framework of distributions, one can instead mention classical results of Functional Analysis concerning weak topologies; weak convergence appears then natural for quantities which are integrated against test functions, or integrated on certain sets; in Continuum Mechanics it is often the case that such quantities are coefficients of differential forms (and it is probably only for those that the weak convergence should be used).
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For example, in the equation of conservation of mass ${\partial \rho\over \partial t}+\sum_{i}{\partial(\rho\,u_{i})\over \partial x_{i}} = 0$, $\rho$ and $\rho\,u_{i}$, $i = 1,2,3$, are the coefficients of a 3-differential form in space-time, namely
$$
\omega = \rho\,dx_{1}\wedge dx_{2}\wedge dx_{3}-\rho\,u_{1}dt\wedge dx_{2}\wedge dx_{3} +\rho\,u_{2}dt\wedge dx_{1}\wedge dx_{3} -\rho\,u_{3}dt\wedge dx_{1}\wedge dx_{2},
$$
and as
$$
d\,\omega = \Bigl( {\partial \rho\over \partial t} +\sum_{i = 1}^{3} {\partial (\rho\,u_{i})\over \partial x_{i}} \Bigr)dt\wedge dx_{1}\wedge dx_{2}\wedge dx_{3},
$$
the equation of conservation of mass is $d\,\omega = 0$.
One must notice that the components $u_{i}$ of the velocity field are not themselves coefficients of differential forms (and the weak convergence is not adapted for them), and it is the momentum which is the correct physical quantity, which has an additive character.
The velocity is not always mentioned when one deals with conservation of electric charge, and it is written as ${\partial \rho\over \partial t}+div(j) = 0$, and one does not even bother to define a velocity as ${j\over \rho}$, because it would usually be meaningless: indeed the electric charge is transported by light electrons and by heavy ions, and an average velocity would be of little use (it is better to think of two interacting populations, one of electrons and one of ions, eventually having their own temperature).
\par
I will show later an example where a quantity which is not a coefficient of a differential form necessitates a different type of weak topology (the H-convergence, which I have introduced with Fran\c{c}ois M{\eightrm URAT}; it generalizes to nonsymmetric operators the G-convergence introduced by Sergio S{\eightrm PAGNOLO}, but it also introduces a quite different point of view).
\medskip
Although I am using the framework of differential forms, my motivation is quite different from that of geometers, and it is worth describing the differences of points of view.
\par
The exterior calculus is purely algebraic: one considers the $p$-linear alternated forms on a finite dimensional vector space $E$, i.e. $f$ is multilinear and satisfies $f(e_{s(1)},\ldots,e_{s(p)}) = \varepsilon(s)f(e_{1},\ldots,e_{p})$ for all $e_{1},\ldots,e_{p}\in E$ and all permutations $s$ of $p$ elements, where $\varepsilon(s)$ is the signature of the permutation $s$.
One defines then the exterior product $\wedge$: if $f$ is $p$-linear alternated and $g$ is $q$-linear alternated then $f\wedge g$ is the $(p+q)$-linear alternated defined by $(f\wedge g)(e_{1},\ldots,e_{p+q}) = {1\over p!q!}\sum_{s} \varepsilon(s) f(e_{s(1)},\ldots,e_{s(p)})g(e_{s(p+1)},\ldots,e_{s(p+q)})$, where $s$ runs through the permutations of $p+q$ elements; one checks easily that $g\wedge f = (-1)^{pq}f\wedge g$.
The exterior product is associative.
\par
A differential form of order $p$, or a $p$-form, on an open set $\Omega$ of $E$, is a (smooth enough) mapping from $\Omega$ into the space of $p$-linear alternated forms; a 0-form is a function, and the derivative of a function is a 1-form.
Then one defines the exterior derivative $d$, which maps $p$-forms into $(p+1)$-forms, with the rules that $d(f\wedge g) = (df)\wedge g +(-1)^{p}f\wedge (dg)$ if $f$ is a $p$-form, and $df = \sum_{i} {\partial f\over \partial x_{i}}dx_{i}$ if $f$ is a function.
One shows that $d\circ d = 0$, and P{\eightrm OINCAR\'E}'s lemma asserts that if $df = 0$ then locally $f = dh$ for a $(p-1)$-form $h$ (asking for global results leads to questions of Algebraic Topology).
\par
One can restrict a differential form to a submanifold by considering its action only on vectors tangent to the submanifold, and actually one can develop all the theory of differential forms on abstract manifolds (not necessarily orientable), with or without boundary, and one proves the S{\eightrm TOKES} formula $\int_{\Omega} d\omega = \int_{\partial\Omega} \omega$.
As a student, before learning this framework, I was taught about formulas by G{\eightrm REEN}, S{\eightrm TOKES}, and O{\eightrm STROGRADSKI}, where $curl$ and $div$ appear: a vector field $V$ in $R^{3}$ can be attached to a 1-form (because vectors and covectors are identified) $\omega(V) = V_{1}dx_{1}+V_{2}dx_{2}+V_{3}dx_{3}$ but also to a 2-form $\pi(V) = V_{1}dx_{2}\wedge dx_{3}+V_{2}dx_{3}\wedge dx_{1}+V_{3}dx_{1}\wedge dx_{2}$ (again because one uses the Euclidean structure), and $d\omega(V) = \pi\bigl( curl(V) \bigr)$ and $d\pi(V) = div(V)dx_{1}\wedge dx_{2}\wedge dx_{3}$ (one usually suppresses the $\wedge$ and one replaces $dx_{1}\wedge dx_{2}\wedge dx_{3}$ by $dx$).
From a practical point of view, $curl$ appears for 1-forms and $div$ appears for $(N-1)$-forms in dimension $N$.
\par
I suppose that all this beautiful theory was developped by Henri P{\eightrm OINCAR\'E} and Elie C{\eightrm ARTAN}, but I have also heard the name of P{\eightrm FAFF} being mentioned.
\medskip
If differential forms are natural for geometers, as they are the right objects which transform well under change of variables, the reason why I am using them is different: they are adapted to weak convergence.
In the early 70s, I worked with Fran\c{c}ois M{\eightrm URAT} on questions that were not yet called Homogenization, and we had understood from reading some work of Henri S{\eightrm ANCHEZ}-P{\eightrm ALENCIA} (who was using asymptotic expansions for problems with periodic microstructures), that what we had done was related to effective properties of mixtures (we also discovered that Sergio S{\eightrm PAGNOLO} had solved earlier the first step of our program): we were considering a sequence of elliptic problems $-div\bigl( A_{n}\,grad(u_{n}) \bigr) = f$ in $\Omega$, together with some natural boundary conditions, and when $A_{n}$ converged weakly we could extract a subsequence $u_{m}$ converging weakly in $H^{1}(\Omega)$ to $u_{\infty}$, but the limit of $A_{m}\,grad(u_{m})$ could not be defined easily, and we had introduced an adapted notion (later called H-convergence, and generalizing the G-convergence introduced by Sergio S{\eightrm PAGNOLO}).
Using notations from Electrostatics, with $E_{n} = -grad(u_{n})$ and $D_{n} = A_{n}E_{n}$, I was considering the weak convergence natural for the electric field $E_{n}$, interpreting its weak limit $E_{\infty}$ as a macroscopic field, and similarly for the polarization field $D_{n}$, but the right limit for $A_{n}$ was to relate $D_{\infty}$ to $E_{\infty}$ by a different physical process where there was no averaging of $A_{n}$: one created a macroscopic field $E_{\infty}$ by choosing correctly $f$ (which is $\rho$ in Electrostatics) and one measured the limit $D_{\infty}$ and that gave a partial information of a tensor $A^{eff}$ such that $D_{\infty} = A^{eff}E_{\infty}$; therefore one does not ``measure'' $A^{eff}$ by computing averages, but one ``identifies''$A^{eff}$ from averages of the electric and polarization fields.
We had also discovered the Div-Curl lemma, which I will describe below, and during the year 1974/75 which I spent in Madison, Joel R{\eightrm OBBIN} had explained to me that our results became quite clear when expressed in the framework of differential forms (using H{\eightrm ODGE} decomposition; he had also taught me how to write M{\eightrm AXWELL} equation using differential forms).
In the Fall of 1975, I met John B{\eightrm ALL} and learned about the sequential weak continuity of Jacobians (which I thought he had proved, but understood later that M{\eightrm ORREY} had done that in the 50s), and in dimension 2 or 3 I could derive easily these results from the Div-Curl lemma; the general framework of Compensated Compactness appeared the year after, again with participation of Fran\c{c}ois M{\eightrm URAT}.
\par
The Div-Curl lemma states that if $\Omega\subset R^{N}$, if $E_{n}\rightharpoonup E_{\infty}$ in $L^{2}_{loc}(\Omega;R^{N})$ weak, $D_{n}\rightharpoonup D_{\infty}$ in $L^{2}_{loc}(\Omega;R^{N})$ weak, $div(D_{n})\rightarrow div(D_{\infty}$ in $H^{-1}_{loc}(\Omega)$ strong, and $curl(E_{n})\rightarrow curl(E_{\infty}$ in $H^{-1}_{loc}(\Omega;X)$ strong (where $X$ has the right dimension), then $(E_{n}.D_{n})$ converges to $(E_{\infty}.D_{\infty})$ in the sense of measures (i.e. integrated against test functions in $C_{c}(\Omega)$).
In the case where $E_{n} = grad(u_{n})$, this is integration by parts and uses the compactness of the injection of $H^{1}_{loc}(\Omega)$ into $L^{2}_{loc}(\Omega)$ by writing $(E_{n}.D_{n}) = -(grad(u_{n}).D_{n}) = -div(u_{n}D_{n}) +u_{n}\,div(D_{n})$, which one integrates against $\varphi\in C^{\infty}_{c}(\Omega)$.
\par
The Compensated Compactness quadratic theorem considers a general framework of linear differential equations with constant coefficients, $U^{n}\rightharpoonup U^{\infty}$ in $L^{2}_{loc}(\Omega;R^{p})$ weak and $\sum_{jk} A_{ijk} {\partial U^{n}_{j}\over \partial x_{k}} \rightarrow f_{i}$ in $H^{-1}_{loc}(\Omega)$ strong for $i = 1,\ldots,q$, and identifies the possible limits (in the sense of measures) of all quadratic functions in $U^{n}$: if $Q$ is quadratic and $Q(U^{n})$ converges to $Q(U^{\infty})+\nu$ in the sense of measures then $\nu\ge 0$ if $Q(\lambda)\ge 0$ for all $\lambda\in \Lambda = \{\lambda\in R^{p}:$ there exists $\xi\in R^{N}\setminus 0, \sum_{jk} A_{ijk} \lambda_{j}\xi_{k} = 0$ for $i = 1,\ldots,q\}$; this result is optimal.
In particular $Q(U^{n})\rightharpoonup Q(U^{\infty}$ in the sense of measures if $Q(\lambda) = 0$ for all $\lambda\in \Lambda$.
\par
The Compensated Compactness method adds the use of ``entropies'' (which geometers call C{\eightrm ASIMIR}s) in order to deduce which Y{\eightrm OUNG} measures could be associated to the sequence $U^{n}$, assumed also to satisfy the constraints $U^{n}(x)\in K$ a.e. $x\in\Omega$ (which corresponds to constitutive relations, while the differential equations corresponds to balance equations for problems in Continuuum Mechanics).
\par
If one did not know about differential forms, one would discover them by looking at sequences $U^{n}$ which converge strongly to $U^{\infty}$ in $L^{2}_{loc}(\Omega;R^{p})$ but only weakly in $H^{1}_{loc}(\Omega;R^{p})$, and wonder if one could compute the limit of some functions of $grad(U^{n})$; indeed the quadratic theorem of Compensated Compactness would show that $dU^{n}_{i}\wedge dU^{n}_{j}$ converges to $dU^{\infty}_{i}\wedge dU^{\infty}_{j}$ in the sense of measures; by reiteration, using the entropy conditions following from the formula for $d(f\wedge g)$, one could recover M{\eightrm ORREY}'s result about Jacobians.
The Compensated Compactness framework is of course more general than M{\eightrm ORREY}s result, and can be used for any system that one encounters in Continuum Mechanics, but the Compensated Compactness method still needs to be improved.
For example, when applied to M{\eightrm AXWELL} equation, one find three independent quadratic quantities which are sequentially weakly continuous, and if one knows the framework with differential forms, they come from exterior products of forms which have a good exterior derivarive.
More generally, let $f^{n}$ be a sequence of $p$-forms converging to $f^{\infty}$ in $L^{2}_{loc}(\Omega)$ (for its coefficients) and such that $df^{n}$ has its coefficients staying in a compact of $H^{-1}_{loc}(\Omega)$ strong, let $g^{n}$ be a sequence of $q$-forms converging to $g^{\infty}$ in $L^{2}_{loc}(\Omega)$ (for its coefficients) and such that $dg^{n}$ has its coefficients staying in a compact of $H^{-1}_{loc}(\Omega)$ strong, then $df^{n}\wedge dg^{n}$ converges to $df^{\infty}\wedge dg^{\infty}$ in the sense of measures (of course one has better convergences if one improves the hypotheses).
Of course, the Compensated Compactness method must sometimes be used in conjunction with the ideas of H-convergence that I developed with Fran\c{c}ois M{\eightrm URAT} for Homogenization.
\bigskip
I always wonder why there is still a group of people who pretend to be interested in Elasticity and wants to ignore this framework which I have taught more than 20 years ago, where one can naturally introduce the equilibrium equation; I have heard so many talks by mathematicians who pretend to be interested in Elasticity and never mention the word stress, that I wonder if it is really so hard for them to learn about Continuum Mechanics (maybe it is hard for them to quote my results when they need them, and they often prefer to quote some others who have used my methods and have forgotten to mention where they had learned about them).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
34. Friday April 9.
\medskip
In one dimension, one can solve explicitly all Homogenization problems by computing various weak limits, and the same is true in more than one dimension when the oscillating coefficients only depend upon one variable; the general case of a diffusion equation was solved by Fran\c{c}ois M{\eightrm URAT} in the early 70s, then I learned in 1975 about the computation by M{\eightrm C}C{\eightrm ONNELL} of the general case for Linearized Elasticity, and I derived the general approach shown below a few years after; in 1979, having been asked how to compute the effective properties of a material layering steel and rubber, I also explained how to carry out the computations in a nonlinear setting, although there is no general theory of Homogenization for (nonlinear) Elasticity (despite the claim of those who have fallen into the trap of the $\Gamma$-convergence approach, this is still the situation today).
\par
The basic idea is an application of the Div-Curl lemma: if $\Omega\subset R^{N}$ and $D^{n}\rightharpoonup D^{\infty}$ in $L^{2}_{loc}(\Omega;R^{N})$ weak with $div(D^{n})$ staying in a compact of $H^{-1}_{loc}(\Omega)$ strong, then ``$D^{n}_{1}$ does not oscillate in $x_{1}$'', i.e. whenever $f_{n}$ only depends upon $x_{1}$ and $f_{n}\rightharpoonup f_{\infty}$ in $L^{2}_{loc}(\Omega)$ weak, one has $D^{n}_{1}f_{n}\rightharpoonup D^{\infty}_{1}f_{\infty}$ in the sense of measures (the precise definition that a sequence is not oscillating in $x_{1}$ says that the corresponding H-measures do not charge the point $e^{1}$ of the unit sphere); of course this follows from the fact that $E^{n} = f_{n}e^{1}$ is a gradient.
\par
For a diffusion equation, if $E^{n} = grad(u_{n})$ and $D^{n} = A^{n}E^{n}$ satisfies $div(D^{n})\rightarrow f$ in $H^{-1}_{loc}(\Omega)$ strong, with $A^{n}$ only depending upon $x_{1}$, one remarks that $D^{n}_{1}$ does not oscillate in $x_{1}$ as well as $E^{n}_{2},\ldots,E^{n}_{N}$, because of the equation $curl(E^{n}) = 0$; from the components of $E^{n}$ and $D^{n}$, one creates a good vector $G^{n}$ whose components are $D^{n}_{1},E^{n}_{2},\ldots,E^{n}_{N}$, and a bad vector $B^{n}$ whose components are $E^{n}_{1},D^{n}_{2},\ldots,D^{n}_{N}$, and one has $B^{n} = \Phi(A^{n}) G^{n}$, where $\Phi(A^{n})$ is obtained by algebraic computations from $A^{n}$, and these computations (which start by eliminating $E^{n}_{1}$ in the equation giving $D^{n}_{1}$) only require that $A^{n}_{11}$ stay away from 0; as $A^{n}$ only depends upon $x_{1}$, so does $\Phi(A^{n})$ and one can pass to the limit in $\Phi(A^{n}) G^{n}$, so that $B^{\infty} =$ [weak limit $\Phi(A^{n})$]$G^{\infty}$, i.e. $\Phi(A^{eff})$ is the weak limit of $\Phi(A^{n})$.
For Linearized Elasticity, the good vector uses the components $\sigma_{i1}$ (and $\sigma_{1i}$ which is equal to $\sigma_{i1}$ because the C{\eightrm AUCHY} stress tensor is used), and the $\varepsilon_{ij}$ for $i,j\ge 2$, while the bad vector uses the other components; starting from $\sigma_{ij} = \sum_{kl} C_{ijkl}\varepsilon_{kl}$, the algebraic computations only require that the acoustic tensor $A(e^{1})$ be invertible (one defines $A_{ik}(\xi) = \sum_{jl} C_{ijkl}\xi_{j}\xi_{l}$).
For (nonlinear) Elasticity, the good vector uses the components $\sigma_{i1}$ (but not $\sigma_{1i}$ which is different from $\sigma_{i1}$ because the P{\eightrm IOLA}-K{\eightrm IRCHHOFF} stress tensor is used), and the ${\partial u_{i}\over \partial x_{j}}$ for $j\ge 2$, while the bad vector uses the other components (in the case of HyperElasticity where there is a stored energy function $W$, the computations only require a uniform rank-one convexity for $W$).
\medskip
If one considers now to the general problem of Homogenization, and I recall that I do not imply any restriction to periodic structures like so many do when using this term (probably because they have not understood the general framework that I had developed with Fran\c{c}ois M{\eightrm URAT}), one imposes a uniform ellipticity condition, which for the diffusion case is that there exists $0<\alpha\le\beta<\infty$ such that $(A^{n}(x)\xi.\xi)\ge\alpha|\xi|^{2}$ and $(A^{n}(x)\xi.\xi)\ge{1\over\beta}|A^{n}(x)\xi|^{2}$ for all $\xi\in R^{N}$ and a.e. $x\in\Omega$ (if $A^{n}$ is symmetric, it means that $\alpha\,I\le A^{n}\le \beta\,I$ almost everywhere).
\par
In the G-convergence approach, developed in the late 60s by Sergio S{\eightrm PAGNOLO} (helped by the insight of Ennio {\eightrm DE} G{\eightrm IORGI}), one only considers symmetric $A^{n}$ and one extracts a subsequence such that for every $f\in H^{-1}(\Omega)$ the solution $u_{m}\in H^{1}_{0}(\Omega)$ of the equation $-div\bigl( A^{m}grad(u_{m}) \bigr) = f$ converges weakly to $u_{\infty}$, and one shows that there exists $A^{eff}$ (symmetric with $\alpha\,I\le A^{eff}\le \beta\,I$ almost everywhere) such that $-div\bigl( A^{eff}grad(u_{\infty}) \bigr) = f$ (this is the convergence of the G{\eightrm REEN} kernels, and explains the choice of the prefix G).
\par
In the H-convergence approach, which I developed in the early 70s with Fran\c{c}ois M{\eightrm URAT} without knowing at the time what Sergio S{\eightrm PAGNOLO} had already done, one can consider nonsymmetric $A^{n}$ and one extracts a subsequence such that for every $f\in H^{-1}(\Omega)$ the solution $u_{m}\in H^{1}_{0}(\Omega)$ of the equation $-div\bigl( A^{m}grad(u_{m}) \bigr) = f$ converges weakly to $u_{\infty}$, but also $A^{m}grad(u_{m})$ converges weakly to $D^{\infty}$, and one shows that there exists $A^{eff}$ (with $(A^{eff}(x)\xi.\xi)\ge\alpha|\xi|^{2}$ and $(A^{eff}(x)\xi.\xi)\ge{1\over\beta}|A^{eff}(x)\xi|^{2}$ for all $\xi\in R^{N}$ and a.e. $x\in\Omega$) such that $D^{\infty} = A^{eff}grad(u_{\infty})$ and therefore $-div\bigl( A^{eff}grad(u_{\infty}) \bigr) = f$ (it is equivalent to G-convergence in the symmetric case, and the choice of the prefix H, chosen in the late 60s, reminds of the term Homogenization introduced by Ivo B{\eightrm ABU\v{S}KA}).
\par
It is important to realize that $A^{eff}$ cannot be computed using the Y{\eightrm OUNG} measures associated with the sequence $A^{n}$ in dimension at least 2; using Y{\eightrm OUNG} measures is the mathematical way of dealing with one-point statistics which physicists use in their probabilistic framework, and therefore the preceding statement says that one cannot deduce the effective properties of a mixture by using only the proportions of the different constituents used (something we had known since the early 70s).
\par
The preceding statement contradicts a few other claims for the following reasons.
Some mathematicians claim that Y{\eightrm OUNG} measures are the right objects to study microstructures in crystals for example, and it is because they have not understood what I had already taught in my 1978 lectures at H{\eightrm ERIOT}-W{\eightrm ATT} University, where I had used (Y{\eightrm OUNG}) paraletrized measures for describing the limits of sequences constrained (in a pointwise way) by constitutive relations, and the Compensated Compactness method for describing the constraints due to the (linear differential) balance relations; I had shown the importance of characterizing which Y{\eightrm OUNG} measures are compatible with a given set of linear differential equations and a nonlinear constitutive relation, but those who have used much later terms like ``gradient Y{\eightrm OUNG} measures'' usually forget to mention that I had introduced that notion for a general system because the laws of Continuum Mechanics cannot be expressed using only gradients (cf. the widespread disease of pretending to work on Elasticity without ever mentioning stress); even for questions like twinning, which are akin to the method for computing effective properties of layers which I have described before, the mistake consists in not understanding that one uses in a crucial way the directions of the twins and therefore the statement that Y{\eightrm OUNG} measures are the right objects should be restricted to one dimensional geometries.
Physicists do write formulas for effective properties of mixtures, but they are only approximations, or bounds, and I have initially developped the technique of H-measures to explain why some formulas guessed by physicists are good in situations where the properties of the constituents are very similar.
There are other situations where phycisists might be right, because they only observe the result of an evolution, like for mixtures of gases or liquids, and it might be that the evolution dissipates energy and ends up at a stable equilibrium, which they can compute (there are no good mathematical methods yet for studying the evolution of mixtures).
\par
Y{\eightrm OUNG} measures have been introduced in the 30s by Laurence C. Y{\eightrm OUNG} (son of William and Grace Y{\eightrm OUNG}, who were both mathematicians and collaborated extensively so that it is not clear if some of the famous results attributed to Y{\eightrm OUNG} are due to his father or his mother); I learned about these measures as parametrized measures in seminars on control theory in the late 60s (without attribution to Y{\eightrm OUNG}) and I first used them under that name; for a sequence of measurable functions $U^{n}$ on $\Omega\subset R^{N}$, taking values in a closed bounded set $K\subset R^{p}$, there is a subsequence and a measurable family $\nu_{x}$ of probability measures on $K$ such that for every continuous function $\varphi$ on $K$ the subsequence $\varphi(u_{m})$ converges in $L^{\infty}(\Omega)$ weak $\star$ to a limit $l_{\varphi}$ such that $l_{\varphi}(x) = \langle \nu_{x},\varphi \rangle$ for a.e. $x\in\Omega$.
If $K$ is unbounded, one may lose information at infinity (one may use a compactification of $K$), and this corresponds to concentration effects; notice that the Compensated Compactness quadratic theorem, or the theory of H-measures which generalizes it, can deal with oscillations and concentration effects simultaneously; losing mass at infinity was a classical question in problems of theoretical Physics, and observing concentration effects in minimizing sequences was a well known fact for geometers, and when Pierre-Louis L{\eightrm IONS} studied these questions he might not have been aware of all the earlier results, but he told me that had choosen to call his approach the Concentration-Compactness method with the goal of inducing people in error because of the similarity in name with the Compensated Compactness method (which he had obviously not understood well himself even a few years after telling me that).
\par
If one mixes two isotropic materials of conductivity (or permittivity) $\alpha,\beta$, it means that $A^{n} = (\chi_{m}\alpha + (1-\chi_{n})\beta)I$, and if $\chi_{n}\rightharpoonup\theta$ in $L^{\infty}(\Omega)$ weak $\star$, then $\theta(x)$ is the local proportion of the first material near $x$; the Y{\eightrm OUNG} measure in this case is $\nu_{x} = \theta(x)\delta_{\alpha\,I}+\bigl( 1-\theta(x) \bigr)\delta_{\beta\,I}$.
One can construct layers in $x_{1}$ or layers in $x_{2}$ for two sequences having the same Y{\eightrm OUNG} measure by taking $\theta$ constant, but the effective properties are different: if $a_{+}$ is the arithmetic average $\theta\alpha+(1-\theta)\beta$, and $a_{-}$ is the harmonic average $\bigl( {\theta\over \alpha}+{1-\theta\over \beta} \bigr)^{-1}$, then layering in the direction $x_{j}$ corresponds to $A^{eff}$ being diagonal with $A^{eff}_{ii} = a_{-}\delta_{ij}+a_{+}(1-\delta_{ij})$; it is a little more technical to construct sequences for which $A^{eff}$ is of the form $\gamma\,I$ and show that the value $\gamma$ can be different for two sequences using the same proportions.
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\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
35. Monday April 12.
\medskip
For questions of Oceanography, we have to face the extremely difficult problem of passing to the limit in N{\eightrm AVIER}-S{\eightrm TOKES} in situations where the R{\eightrm EYNOLDS} number gets large; if one maintains the size of the domain and the size of the velocities it corresponds to letting the kinematic viscosity $\nu$ tend to 0.
We have seen that for $\rho$ and $\rho\,u$, the density of mass and the density of momentum, the weak convergence is well adapted as they are coefficients of differential forms, but as turbulence is related to situations where the velocity $u$ fluctuates, we must understand how to average $u$, in the sense of finding the right topology adapted to that quantity.
The velocity $u$ appears inside the differential operator
$$
{D\over Dt} = {\partial \over \partial t}+\sum_{i = 1}^{3} u_{i}{\partial \over \partial x_{i}},
$$
which is a transport operator, and what is transported is mass, momentum, angular momentum, temperature, salinity, pollutants, etc...
It is natural to ask the same question that was solved for equations of the form $div\bigl( A^{n}\,grad(u_{n}) \bigr) = f$ for equations of the form ${\partial v^{n}\over \partial t}+\sum_{i = 1}^{3} u^{n}_{i}{\partial v_{n}\over \partial x_{i}} = f$, where $u^{n}$ is now a given oscillating field and the solution $v^{n}$ may represent any one of the transported quantities.
If $v^{n}$ is a scalar quantity, we are considering a first order operator, which is hyperbolic, but it can also be considered as a degenerate elliptic operator and this could give some hope that the results for elliptic operators could extend to degenerate cases, but this extension is not straightforward, as we will see on much simpler examples, because nonlocal effects appear.
\medskip
That nonlocal effects may appear by Homogenization had first been noticed by Henri S{\eightrm ANCHEZ}-P{\eightrm ALENCIA} (using asymptotic expansions in a periodic setting), for questions like ViscoElasticity or for some memory effects in Electricity corresponding to the fact that some coefficients depend upon frequency, and Jacques-Louis L{\eightrm IONS} had invented examples where one needed to introduce pseudo-differential operators (with an interpretation as memory effects).
I started thinking about this question in 1980 because I guessed that these effects were the main reason behind the strange rules of absorption and reemission used by physicists, and I looked at the very simplified following model
$$
{\partial u_{n}(x,t)\over \partial t}+a_{n}(x)u_{n}(x,t) = f(x,t)\hbox{ in }\Omega\times(0,T);\;u_{n}(x,0) = v(x),
$$
where $a_{n}$ takes values between $\alpha$ and $\beta$ and converges to $a_{\infty}$ in $L^{\infty}(\Omega)$ weak $\star$ (the Y{\eightrm OUNG} measure of $a_{n}$ contains all the information that we will need).
I guessed that the limiting equation would have a convolution term
$$
{\partial u_{\infty}(x,t)\over \partial t}+a_{\infty}(x)u_{\infty}(x,t) -\int_{0}^{t} K(x,t-s)u_{\infty}(x,s)\,ds = f(x,t)\hbox{ in }\Omega\times(0,T);\;u_{\infty}(x,0) = v(x),
$$
and I expected $K\ge 0$ for reasons related to the maximum principle.
If one defines $B_{n}(x,t) = e^{-t\,a_{n}(x)}$, then one has $u_{n}(x,t) = B_{n}(x,t)v(x)+\int_{0}^{t} B_{n}(x,t-s)f(x,s)\,ds$, and if $B_{n}\rightharpoonup B_{\infty}$ in $L^{\infty}\bigl( \Omega\times(0,T) \bigr)$ weak $\star$, then $u_{\infty}$ satisfies the same equation with $B_{n}$ replaced by $B_{\infty}$; as $B_{\infty}(x,t)\ne e^{-t\,a_{\infty}(x)}$ except if $a_{n}$ converges strongly to $a_{\infty}$ (in $L^{1}_{loc}(\Omega)$ for example), one cannot have $K = 0$, and this gives the simplest example of a sequence of semi-groups whose limit is not a semi-group.
Of course, the kernel $K$ must satisfy the equation ${\partial B_{\infty}(x,t)\over \partial t}+a_{\infty}(x)B_{\infty}(x,t) = \int_{0}^{t} K(x,t-s)B_{\infty}(x,s)\,ds$ for a.e. $x\in\Omega$; this was the first approach I had taken after trying the approach by L{\eightrm APLACE} transform which I will follow now.
\par
One defines the L{\eightrm APLACE} transform of a function $g$ defined on $(0,\infty)$ by ${\cal L}g(p) = \int_{0}^{\infty} g(t)e^{-p\,t}\,dt$, and usually the L{\eightrm APLACE} transform is holomorphic in some half space $\Re{p}>\gamma$, and the theory has been extended by Laurent S{\eightrm CHWARTZ} to some distributions; the important fact is that ${\cal L}(g\star h) = {\cal L}g\,{\cal L}h$, and ${\cal L}{dg\over dt} = p{\cal L}g+g(0)$.
One has
$$
\Bigl( p+a_{n}(x) \Bigr){\cal L}u_{n}(x,p) = {\cal L}f(x,p)+v(x),
$$
and
$$
\Bigl( p+a_{\infty}(x) -{\cal K}(x,p) \Bigr){\cal L}u_{\infty}(x,p) = {\cal L}f(x,p)+v(x),
$$
so that $K$ is characterized by
$$
p+a_{\infty} -{\cal K}(\cdot,p) = \Bigl( \hbox{weak limit }{1\over p+a_{n}} \Bigr)^{-1}.
$$
Of course, using the Y{\eightrm OUNG} measures associated with the sequence $a_{n}$, the weak limit of ${1\over p+a_{n}}$ is $\int_{[\alpha,\beta]} {d\nu_{x}(a)\over p+a}$, and the key to the formula for $K$ is a property about functions $F(z)$ which satisfy $\Im{F(z)}>0$ when $\Im{z}>0$ (I had first heard of it in talks by David B{\eightrm ERGMAN}, and the idea is attributed to various authors such as H{\eightrm ELMHOLTZ}, P{\eightrm ICK}, N{\eightrm EVANLINA} or S{\eightrm TIELTJES}): suppose as a simplification that $F$ is defined in the complex plane except for a bounded closed interval $I$ on the real axis, and satisfies $\Im{F(z)}\Im{z}>0$ for $\Im{z}\ne 0$, then there exists $A\ge 0$, $B\in R$ and a nonnegative R{\eightrm ADON} measure $\mu$ with support in $I$ such that
$$
F(z) = A\,z+B+\int_{I} {d\mu(\lambda)\over \lambda-z}\hbox{ for all }z\notin I.
$$
One deduces that
$$
{1\over \int_{I} {d\nu_{x}(a)\over p+a} } = p+a_{\infty}(x) + \int_{[-\beta,-\alpha]} {d\mu_{x}(\lambda)\over \lambda-p}\hbox{ for all }p\notin [-\beta,-\alpha], \hbox{ a.e. }x\in\Omega,
$$
as a T{\eightrm AYLOR} expansion near $p = \infty$ gives $A = 1$ and $B = a_{\infty}(x) = \int_{I} a\,d\nu_{x}(a)$; the inverse L{\eightrm APLACE} transform is then easily performed and gives
$$
K(x,t) = \int_{-I} e^{\lambda\,t}\,d\mu_{x}(\lambda).
$$
If $a_{n}$ takes only $k$ different values, then $\nu_{x}$ is a combination of at most $k$ D{\eightrm IRAC} masses, and $\mu_{x}$ is a combination of at most $(k-1)$ D{\eightrm IRAC} masses which are the roots of a polynomial for which there is no simple formula in general.
\par
In the preceding example, we found a solution $u_{\infty}$ and we looked then for an equation that it satisfies, and the reason that the one obtained is natural is that the operator ${d\over dt}+a_{n}$ is linear and commutes with translation in $t$, and a theorem of Laurent S{\eightrm CHWARTZ} says that every linear operator which commutes with translation is a convolution operator (with a distribution kernel), and the only kernel that works here is ${d\delta_{0}\over dt}+a_{\infty}\delta_{0}-K$; we will use again this argument below, but in nonlinear settings the situation is not as clear.
\medskip
Although the preceding example is not of great interest from a physical point of view (one could use it for a mixture of materials decaying at different rates for example), it shows something important from a philosophical point of view: the memory effect term is not related to any probabilistic argument!
It is actually possible to invent a probabilistic game, with particles absorbed and particles reemitted, which will create the equation that we have found; there is absolutely no reason other than ideology to give a better status to the probabilistic approach than to any other way of considering the preceding equation (Probability is a part of Analysis, but from the point of view of Analysis without Probability integral equations with smooth kernels are treated as mere perturbations, in semi-group theory for example).
\medskip
The model explains qualitatively something about irreversibility.
One may start from an equation for which one can reverse time and a limiting process may make an irreversible equation appear: diffusion equations arrive naturally in certain situations by letting the velocity of Light $c$ tend to $\infty$, but the equation that one starts from has already incorporated a modelisation of scattering which is not reversible.
A more puzzling question is asked by people who start from a finite dimensional Hamiltonian system and let the number of degrees of freedom tend to $\infty$, as numerical simulations show that something like entropy increases, but the system is reversible and the same occurs for the reversed equation.
The answer provided by the example is that one might have to consider memory effects in order to describe well what is going on, and an observer using time in a backward way will then do the same analysis and get an integral term from $t$ to $\infty$ instead in his equations; it is when one wants to get rid of the nonlocal effects and only use partial differential equations that the problems occur.
In the previous example, one can approach $d\mu$ by a finite combination of D{\eightrm IRAC} masses and transform the equation obtained into a system of differential equations.
\medskip
The method which I have shown above was applied by my former student Kamel H{\eightrm AMDACHE} (with A{\eightrm MIRAT} and AbdelHamid Z{\eightrm IANI}) to a question which is more relevant to the questions of fluids that we are interested in (but from a pedagogical point of view I prefer to start with the simpler problem which was done first); their motivation was in flows in porous media, and they considered
$$
{\partial u_{n}\over \partial t}+a_{n}(y){\partial u_{n}\over \partial x} = f(x,y,t)\hbox{ in }R\times\Omega\times(0,T);\;u_{n}(x,y,0) = v(x,y)\hbox{ in }R\times\Omega.
$$
The method is essentially the same, using L{\eightrm APLACE} transform in $t$, but also F{\eightrm OURIER} transform in $x$, due to the fact that the partial differential operator that we are dealing with commutes with translations in $t$ but also in $x$; this gives
$$
\Bigl( p+2i\pi\xi\,a_{n}(y) \Bigr){\cal LF}(\xi,y,p) = {\cal LF}f(\xi,y,p)+{\cal F}v(\xi,y),
$$
and if one uses the Y{\eightrm OUNG} measures $\nu_{y}$ associated with a subsequence, one needs the weak $\star$ limit of ${1\over p+2i\pi\xi\,a_{n}(y)}$, which is $\int_{I} {d\nu_{y}(a)\over p+2i\pi\xi\,a} = {1\over 2i\pi\xi} \int_{I} {d\nu_{y}(a)\over q+a}$ where $q = {p\over 2i\pi\xi}$, and the same formula that we used before appears, so one can perform the inverse L{\eightrm APLACE}-F{\eightrm OURIER} transform easily and one obtains the only convolution equation is $(x,t)$ (independent of $f$ and $v$) that the limit solution may satisfy
$$
{\partial u_{\infty}\over \partial t}+a_{\infty}(y){\partial u_{\infty}\over \partial x} -\int_{0}^{t} \int_{[-\beta,-\alpha]} {\partial^{2}u_{\infty}(x+\lambda(t-s),y,t-s)\over \partial x^{2}}\,d\mu_{y}(\lambda)\,ds = f(x,y,t)\hbox{ in }R\times\Omega\times(0,T),
$$
with $u_{\infty}(x,y,0) = v(x,y)\hbox{ in }R\times\Omega$.
Notice that the second derivatives are not computed at the point $(x,y,t)$ but on lines approaching the point with a velocity $-\lambda$, with a weight depending upon $\lambda$; the equation obtained has of course the finite propagation speed property and A{\eightrm MIRAT}, H{\eightrm AMDACHE} and Z{\eightrm IANI} checked that this is true for any nonnegative measure $d\mu_{y}$ with bounded support (the ones coming from the formula have a constraint on their mass for example); they also proposed a way to look at this equation as a possibly infinite hyperbolic system, by using the auxilliary functions
$$
\varphi(x,y,t;V) = \int_{0}^{t} {\partial u_{\infty}(x-V(t-s),y,t-s)\over \partial x}\,ds,
$$
for $V\in[\alpha,\beta]$, so that
$$
{\partial \varphi(x,y,t;V)\over \partial t}+V{\partial\varphi(x,y,t;V)\over \partial x} = {\partial u_{\infty}(x,y,t)\over \partial x}\hbox{ in }R\times\Omega\times(0,T),
$$
and the equation becomes
$$
{\partial u_{\infty}\over \partial t}+a_{\infty}(y){\partial u_{\infty}\over \partial x} -{\partial \over \partial x}\Bigl( \int_{[\alpha,\beta]} \varphi(x,y,t;V)\,d\mu_{y}(-V) \Bigr) = f(x,y,t)\hbox{ in }R\times\Omega\times(0,T),
$$
with the initial conditions $u(x,y,0) = v(x,y)$ and $\varphi(x,y,0;V) = 0$ in $R\times\Omega$ for $V\in[\alpha,\beta]$.
\medskip
This example suggests that if a general transport operator with oscillating coefficients is used, one may expect nonlocal effects, but as we lose the commutations properties, one has to find other methods of proofs.
In the example, the coefficients are divergence free, so that one could write the equation in conservation form, and the transport operator applied to the coefficients give 0; for fluids the coefficients are the components of $u$, which is divergence free, but the transport operator applied to $u$ does not give 0, as the gradient of the pressure and the viscous term appear.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
36. Wednesday April 14.
\medskip
Physicists often describe some properties of matter at a given frequency, and in the relations that they obtain then the frequency often occurs explicitly; they usually do not bother to explain what could be a general equation valid for all solutions.
In linear cases, one can usually give a meaning to these computations, and something like pseudo-differential operators, or nonlocal effects do appear, but not much is understood for nonlinear equations.
For example, if one considers M{\eightrm AXWELL} equation
$$
div(D) = \rho;\;-{\partial D\over \partial t}+curl(H) = j;\;div(B) = 0;\;{\partial B\over \partial t}+curl(E) = 0,
$$
one usually assumes that there are relations $D = \varepsilon(x)E$ between the polarization field $D$ and the electric field $E$ ($\varepsilon$ is the electric permittivity), and $B = \mu(x)H$ between the induction field $B$ and the magnetic field $H$ ($\mu$ is the magnetic susceptibility), and one often adds the relation $j = \sigma(x)E$ ($\sigma$ is the conductivity), and in that case one forgets about the equation $div(D) = \rho$, which is automatically satisfied if it is true at time 0 because of the relation ${\partial \rho\over \partial t}+div(j) = 0$.
The physicists' point of view is to look at solutions of the form $B(x,t) = e^{i\omega\,t}b(x), D(x,t) = e^{i\omega\,t}b(x),\ldots$, so that for example one has $(\sigma+i\omega\,\varepsilon)e+curl(h) = 0$ and one sees a complex conductivity $\sigma+i\omega\,\varepsilon$ appear.
The mathematians' point of view is to use L{\eightrm APLACE} transform, and the same equation becomes $(\sigma+p\varepsilon){\cal L}E+curl({\cal L}H) = \varepsilon\,E(\cdot,0)$.
Whatever the point of view, if one considers a mixture of such materials, an Homogenization process usually creates coefficients which depend upon $\omega$ or $p$ in a non polynomial way, but in the second case one can look for a convolution equation for linking $D$ and $j$ to $E$ and its history (of course, one imposes the principle of causality, i.e. nonlocal effects must only use the past), and this was done using asymptotic expansions in a periodic framework by Henri S{\eightrm ANCHEZ}-P{\eightrm ALENCIA}.
So the physicists say that $\varepsilon$ depends upon the frequency $\omega$, but the mathematicians go further and try to identify a memory kernel for a convolution equation valid for all solutions and not only for those of the form $e^{i\omega\,t}f(x)$, and this is what we have done on the model examples.
However, physicists do use the same approach for nonlinear problems, but mathematicians do not have a general theory for these cases, but one can start following the approach shown below, but I have not solved the bookkeeping problem and the convergence problem.
\par
One could in principle use pseudo-differential operators, which Joseph K{\eightrm OHN} and Louis N{\eightrm IRENBERG} had introduced for developping a calculus that one can use for expressing the solutions of elliptic equations, or the theory of F{\eightrm OURIER} integral operators, which Lars H{\eightrm \"ORMANDER} developped for similar questions for hyperbolic equations, but these theories have unfortunately been developped only with smooth coefficients, and this is a serious handicap even for linear problems originating in Continuum Mechanics or Physics.
\medskip
It is not known how to extend to more realistic questions of fluid dynamics the results obtained for the models that I have shown, but it is useful to derive the same results with different methods for which there is more hope for an extension; one of these methods, which physicists often use, is a perturbation method.
I consider now a time dependent model problem
$$
{\partial u_{n}\over \partial t} +a_{n}(x,t)u_{n} = f(x,t)\hbox{ in }\Omega\times(0,T);\;u_{n}(x,0) = v(x)\hbox{ in }\Omega.
$$
Under the assumption that $a_{n}$ is globally L{\eightrm IPSCHITZ} in $t$, this was first considered by my former student Luisa M{\eightrm ASCARENHAS}; she had used a time discretization, but the following method is more easy to apply, and although I assumed equicontinuity in $t$, the result seems valid without such an assumption (it simplifies that having extracted a subsequence such that $a_{n}(x,t) a_{n}(x,s)$ converges in $L^{\infty}(\Omega)$ weak $\star$ for $s,t$ belonging to a countable dense set of $\Omega$, it is then true for all $s,t\in(0,T)$, but only some integrals of the limits is really needed).
Assuming that $a_{n}\rightharpoonup a_{\infty}$ in $L^{\infty}(\Omega)$ weak $\star$, one defines $b_{n} = a_{n}-a_{\infty}$ and one considers for a parameter $\gamma$ the equation
$$
{\partial U^{n}(x,t;\gamma)\over \partial t} +\bigl( a_{\infty}(x,t) + \gamma\,b_{n}(x,t) \bigr)U^{n}(x,t;\gamma) = f(x,t)\hbox{ in }\Omega\times(0,T); \;U^{n}(x,0;\gamma) = v(x)\hbox{ in }\Omega,
$$
so that the preceding problem corresponds to $\gamma = 1$.
Obviously $U^{n}$ is analytic in $\gamma$ and we can consider the T{\eightrm AYLOR} expansion at $\gamma = 0$,
$$
U^{n}(x,t;\gamma) = \sum_{k = 0}^{\infty} \gamma^{k}V^{n}_{k}(x,t)\hbox{ in }\Omega\times(0,T),
$$
and one finds immediately that $V^{n}_{0}$ is independent of $n$, and solution of
$$
{\partial V_{0}(x,t)\over \partial t} +a_{\infty}(x,t)V_{0}(x,t) = f(x,t)\hbox{ in }\Omega\times(0,T); \;V_{0}(x,0) = v(x)\hbox{ in }\Omega,
$$
and that for $k\ge 1$, $V^{n}_{k}$ is solution of
$$
{\partial V^{n}_{k}(x,t)\over \partial t} +a_{\infty}(x,t)V^{n}_{k}(x,t) + b_{n}(x,t)V^{n}_{k-1}(x,t) = 0\hbox{ in }\Omega\times(0,T); \;V^{n}_{k}(x,0) = 0\hbox{
in }\Omega.
$$
As $b_{n}\rightharpoonup 0$ in $L^{\infty}(\Omega)$ weak $\star$, one sees that $V^{n}_{1}$ converges weakly to 0, but one needs the limit of $b_{n}V^{n}_{1}$ in order to compute the limit of $V^{n}_{2}$, and more generally one needs the explicit form of each $V^{n}_{k}$ for $k\ge 1$,
$$
V^{n}_{k}(x,t) = -\int_{0}^{t} exp\Bigl( -\int_{s}^{t} a_{\infty}(x,\sigma)\,d\sigma \Bigr)b_{n}(x,s)V^{n}_{k-1}(x,s)\,ds\hbox{ for }(x,t)\in\Omega\times(0,T),
$$
so that if one defines $R(x,s,t) = exp\bigl( -\int_{s}^{t} a_{\infty}(x,\sigma)\,d\sigma \bigr)$, one has $V^{n}_{1}(x,t) = -\int_{0}^{t} R(x,s,t)b_{n}(x,s)V_{0}(x,s)\,ds$ and therefore the limit of $b_{n}V^{n}_{1}$ involves limits of $b_{n}(x,t)b_{n}(x,s)$ and has the form $\int_{0}^{t} C(x,t,s)V_{0}(x,s)\,ds$; one can deal similarly with the following terms and integral terms having appeared naturally, one may look for a kernel having the analytic form
$$
K(x,t,s,\gamma) = \sum_{k = 2}^{\infty} \gamma^{k} K_{k}(x,t,s),
$$
and it is not difficult to obtains bounds for the functions $V^{n}_{k}$ and $K_{k}$, and these bounds show that the T{\eightrm AYLOR} expansions written have an infinite radius of convergence, and one can take $\gamma = 1$ safely.
The formula obtained for the kernel is quite different from the one which was obtained by using the representation formula for P{\eightrm ICK} functions.
\par
In principle one could do the same type of expansions for some nonlinear problems, but the bookkeeping is quite arduous (and F{\eightrm EYNMANN} seems to have introduced his famous diagrams for a similar purpose), and the convergence questions are not so clear (and it is for similar reasons that physicists like to use P{\eightrm AD\'E} approximants, or other ways to sum divergent series).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
37. Monday April 19.
\medskip
In the preceding analysis, I was studying questions of Homogenization for first order differential equations with oscillating coefficients, but the reality of fluid dynamics is a little different, in particular because of viscosty and pressure.
Of course, the questions we would like to understand are related to small viscosities, and as this problem is far from being understood now, it is useful to derive simpler models retaining as much as possible of the qualitative properties that we are interested in.
\par
I started in this direction in 1976, and my analysis was based on the fact that the nonlinear term in N{\eightrm AVIER}-S{\eightrm TOKES} equation could be written as $u\times curl(-u)+grad(|u|^{2}/2)$: I knew from Electromagnetism that force terms in $u\times b$ have the effect of making particles turn, and as I had heard turbulence to be associated with vorticity, I decided to replace $curl(-u)$ with a given oscillating function in order to study its effect.
Not knowing what to expect, I decided to start with the stationary case, and I first used the formal method of asymptotic expansions in a periodic setting, so that my problem was
$$
-\nu\Delta\,u_{\varepsilon} +u_{\varepsilon}\times {1\over \varepsilon}b\Bigl( {x\over \varepsilon} \Bigr)+grad(p_{\varepsilon}) = f,\;div(u_{\varepsilon}) = 0\hbox{ in }\Omega;\;u\in H^{1}_{0}(\Omega;R^{3}),
$$
for a periodic vector field $b$.
I did the formal computations with Michel F{\eightrm ORTIN} who was visiting Orsay that year and sharing my office, and the first thing that we noticed was that the average of $b$ must be 0 or we were looking at a different question; in that case we derived an equation satisfied by the first term of the formal expansion.
Using then my method of oscillating test functions in Homogenization, I did not have difficulties proving the formal result; it had the interesting feature that although the force is perpendicular to the velocity $u_{\varepsilon}$ and therefore does not work, it induces oscillations in $grad(u_{\varepsilon})$ and therefore more energy is dissipated by viscosity (per unit of time, as we are looking at a stationary problem), but the added dissipation which appeared in our equation was not quadratic in $grad(u)$, but quadratic in $u$, contrary to the usual belief about turbulent viscosity.
There was not much difference avoiding the periodicity hypothesis and considering terms of the form $u_{\varepsilon}\times curl(v_{\varepsilon})$ with $v_{\varepsilon}$ converging weakly, but when I wrote it down for a meeting in 1984 I noticed something else, a quadratic effect in a strength parameter $\lambda$; the new problem was
$$
-\nu\Delta\,u_{n} +u_{n}\times curl (v_{0}+\lambda\,v_{n})+grad(p_{n}) = f,\;div(u_{n}) = 0\hbox{ in }\Omega,
$$
with $v_{0}\in L^{3}(\Omega;R^{3})$ and $v_{n}\rightharpoonup 0$ in $L^{3}(\Omega;R^{3})$ weak, and I did not impose boundary conditions but I assumed that $u_{n}\rightharpoonup u_{\infty}$ in $H^{1}(\Omega;R^{3})$ weak (it is a classical requirement in Homogenization that if one wants to speak about the effective properties of a mixture one should obtain a result which is independent of the boundary conditions; if one does not do this, one can only mention the global properties of the mixture and the container).
I showed that there exists a symmetric nonnegative matrix $M$, depending only upon a subsequence of $v_{n}$ that one may have to extract, such that $u_{\infty}$ satisfies the equation
$$
-\nu\Delta\,u_{\infty} +u_{\infty}\times curl (v_{0})+ \lambda^{2}M\,u_{\infty} +grad(p_{\infty}) = f,\;div(u_{\infty}) = 0\hbox{ in }\Omega,
$$
and of course a more precise convergence result is
$$
u_{n}\times curl (v_{n})\rightharpoonup \lambda\,M\,u_{\infty}\hbox{ in } H^{-1}_{loc}(\Omega;R^{3})\hbox{ weak},
$$
and
$$
\nu|grad(u_{n})|^{2}\rightharpoonup \nu|grad(u_{\infty})|^{2} + \lambda^{2} (Mu_{\infty}.u_{\infty})\hbox{ in the sense of measures.}
$$
The way $M$ is defined follows my approach to Homogenization, but the quadratic dependence in $\lambda$ and a particular formula in the case where $div(v_{n}) = 0$ was my first hint about the possibility of defining H-measures, and after I was led to introduce H-measures for another purpose I checked that $M$ could indeed be computed from the H-measures associated to the sequence $v_{n}$.
\par
One extracts a subsequence from $v_{n}$ and one constructs $M$ in the following way: for $k\in R^{3}$ one solves
$$
-\nu\Delta\,w_{n} + k\times curl(v_{n})+grad(q_{n}) = 0,\;div(w_{n}) = 0\hbox{ in }\Omega,
$$
adding boundary conditions which imply that $w_{n}\rightharpoonup 0$ in $H^{1}(\Omega;R^{3})$ weak (D{\eightrm IRICHLET} conditions, or periodic conditions in the case where $v_{n}$ is defined in a periodic way, for example); one can indeed extract a subsequence such that this occurs for three independent vectors $k$ and one defines $M$ by
$$
w_{n}\times curl(v_{n})\rightharpoonup M\,k\hbox{ in }H^{-1}_{loc}(\Omega;R^{3})\hbox{ weak}.
$$
\par
The first remark is that $u\mapsto u\times curl(v)$ maps continuously $H^{1}(\Omega;R^{3})$ into $H^{-1}(\Omega;R^{3})$ if $v\in L^{3}(\Omega;R^{3})$, if the boundary of $\Omega$ is smooth, using S{\eightrm OBOLEV} imbedding theorem $H^{1}(\Omega)\subset L^{6}(\Omega)$; indeed if $u, \varphi\in H^{1}(\Omega)$, then $u\,\varphi\in W^{1,3/2}(\Omega)$; if $f\in H^{-1}(\Omega;R^{3})$ one finds that, after eventually adding a constant, $p_{n}$ is bounded in $L^{2}_{loc}(\Omega)$, and one can assume that $p_{n}\rightharpoonup p_{\infty}$ in $L^{2}_{loc}(\Omega)$ weak.
Similarly, in the problem for $w_{n}$ one can assume that $q_{n}\rightharpoonup 0$ in $L^{2}_{loc}(\Omega)$ weak.
Using elliptic regularity theory (and C{\eightrm ALDER\'ON}-Z{\eightrm YGMUND} theorem), $grad(w_{n})$ is bounded in $L^{3}_{loc}(\Omega;R^{3})$ and therefore $w_{n}\rightarrow 0$ in $L^{p}_{loc}(\Omega;R^{3})$ strong for every $p<\infty$; one has then a better convergence for $w_{n}\times curl(v_{n})$, which converges to $M\,k$ in $H^{-1}_{loc}(\Omega;R^{3})$ strong, because of writing products of the form $(w_{n})_{i}{\partial (v_{n})_{j}\over \partial x_{k}}$ as ${\partial [(w_{n})_{i}(v_{n})_{j}]\over \partial x_{k}}-{\partial (w_{n})_{i}\over \partial x_{k}}(v_{n})_{j}$ and terms like $(w_{n})_{i}(v_{n})_{j}$ converge strongly to 0 in $L^{2}_{loc}(\Omega)$ and terms like ${\partial (w_{n})_{i}\over \partial x_{k}}(v_{n})_{j}$ are bounded in $L^{3/2}_{loc}(\Omega)$ and converge strongly to 0 in $L^{q}_{loc}(\Omega)$ for every $q>3/2$ and therefore in $H^{-1}_{loc}(\Omega)$ strong.
One applies the method of oscillating test functions, multiplying the equation for $u_{n}$ by $\varphi\,w_{n}$ and the equation for $w_{n}$ by $\varphi\,u_{n}$, with $\varphi\in C^{1}_{c}(\Omega)$, and noticing that $div(\varphi\,w_{n}) = (grad(\varphi).w_{n})$ and $div(\varphi\,u_{n}) = (grad(\varphi).u_{n})$ and therefore the estimates on the pressures $p_{n}$ and $q_{n}$ are needed.
One assumes that $u_{n}\times curl(v_{n})\rightharpoonup g$ in $H^{-1}(\Omega;R^{3})$ weak and one wants to identify $g$; one finds
$$
\lim_{n\rightarrow\infty} \int_{\Omega} \nu\varphi\Bigl( grad(u_{n}).grad(w_{n}) \Bigr)\,dx + \lambda\langle \varphi\,u_{n}\times curl(v_{n}),w_{n} \rangle = 0,
$$
and
$$
\lim_{n\rightarrow\infty} \int_{\Omega} \nu\varphi\Bigl( grad(u_{n}).grad(w_{n}) \Bigr)\,dx =\langle \varphi\,g,k \rangle,
$$
but as
$$
\langle \varphi\,u_{n}\times curl(v_{n}),w_{n} \rangle = -\langle \varphi\,w_{n}\times curl(v_{n}),u_{n} \rangle \rightarrow -\langle \varphi\,M\,k,u_{\infty} \rangle,
$$
one has shown that
$$
g = \lambda\,M^{T}\,u_{\infty}.
$$
The fact that $M$ is symmetric follows easily by the same method, $w_{n}^{\prime}$ being the solution for $k^{\prime}$, multiplying the equation for $w_{n}^{\prime}$ by $\varphi\,w_{n}$, the equation for $w_{n}$ by $\varphi\,w_{n}^{\prime}$, and comparing.
The limit of $\nu|grad(u_{n})|^{2}$ is obtained by multiplying the equation for $u_{n}$ by $\varphi\,u_{n}$.
\par
In the case where $div(v_{n}) = 0$, which is the case for fluid dynamics, one can take
$$
w_{n} = \Bigl( k.grad(z_{n}) \Bigr),
$$
where $z_{n}$ solves
$$
-\nu\Delta\,z_{n} = v_{n},
$$
and therefore
$$
\nu\sum_{l,m} {\partial^{2}(z_{n})_{l}\over \partial x_{i}\partial x_{m}}{\partial^{2}(z_{n})_{l}\over \partial x_{j}\partial x_{m}}\rightharpoonup M_{ij}\hbox{ in }L^{3/2}(\Omega)\hbox{ weak}.
$$
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
38. Wednesday April 21.
\medskip
The same analysis can be done for the evolution problem if one can obtain a bound for the pressure, so I initially did it for the whole space; {\eightrm VON} W{\eightrm AHL} later told me that he had shown by semi-group methods that one can obtain estimates on the pressure for any smooth bounded open set.
In my original proof for the evolution case, I had not seen how to prove that the matrix $M$ corresponding to the added dissipation is symmetric; two years ago, I worked with Chun L{\eightrm IU} and Konstantina T{\eightrm RIVISA} about extending the formula using H-measures to the evolution case, and we first checked the symmetry, but then we noticed that one needed a new variant of H-measures, with a parabolic scaling, i.e. instead of identifying rays $s\xi$ through a nonzero $\xi$ with $s>0$, one had to identify curves $(s\xi,s^{2}\tau)$ through a nonzero $(\xi,\tau)$.
\medskip
Of course, we should not lose sight of the reasons why the preceding models were chosen and the previous computations were done: the initial purpose was to understand what was the adapted weak type topology for the velocity, appearing as coefficients of a transport equation, with or without viscosity.
Starting from a model without viscosity, we saw that various nonlocal terms could appear, and this analysis could also be useful for correcting the defects of N{\eightrm AVIER}-S{\eightrm TOKES} equations, but the class to consider should then at least contain some equations with memory effects.
Starting with a model with viscosity but magnifying the ocillations possible for the velocity field, we have seen some lower order terms appear, and it could have some analogy with the framework using affine connections which geometers have advocated.
Obviously one should improve the model, but it is worth mentioning that the formula using H-measures which gives an explicit form for $M$ has a ${1\over \nu}$ in front of an integral on the unit sphere, and although it is tempting to rescale the equation, one should remember that turbulence is supposed to show an infinite number of length scales, and that an object like H-measures which mixes different frequencies cannot reasonable describe that.
\medskip
Before being able to describe the tool of H-measures and the various formulas that one can deduce from it, it is worth starting with the previous theory, which I had developped with Fran\c{c}ois M{\eightrm URAT} in the late 70s, Compensated Compactness.
I make a distinction between the basic Compensated Compactness theorem, described below, and the Compensated Compactness Method, which I developpeed after, and which is more general.
The basic theorem, which I call the quadratic theorem, is the following.
\medskip
\noindent
{\bf Theorem}: Let $U^{n}\rightharpoonup U^{\infty}$ in $L^{2}_{loc}(\Omega;R^{p})$ weak, and assume that
$$
\sum_{jk} A_{ijk} {\partial U^{n}_{j}\over \partial x_{k}}\hbox{ stays in a compact of } H^{-1}_{loc}(\Omega)\hbox{ for }i = 1,\ldots,q.
$$
Define the two characteristic sets ${\cal V}$ and $\Lambda$
$$
{\cal V} = \Bigl{\{} (\lambda,\xi)\in R^{p}\times(R^{N}\setminus 0): \sum_{jk} A_{ijk} \lambda_{j}\xi_{k} = 0\hbox{ for }i = 1,\ldots,q \Bigr{\}},
$$
$$
\Lambda = \Bigl{\{} \lambda\in R^{p}:\hbox{ there exists }\xi\in R^{N}\setminus 0,(\lambda,\xi)\in {\cal V} \Bigr{\}}.
$$
Let $Q$ be a quadratic form on $R^{p}$ satisfying
$$
Q(\lambda)\ge 0\hbox{ for all }\lambda\in \Lambda,
$$
then
$$
Q(U^{n})\rightharpoonup Q(U^{\infty})+\nu\hbox{ in the sense of measures implies }\nu\ge 0.
$$
\medskip
I have already mentioned the Div-Curl lemma, which I had found with Fran\c{c}ois M{\eightrm URAT} in 1974 in connection with Homogenization: it is the particular case where $U = (E,D)$ with the list of differential information corresponding to $div(D)$ and the components of $curl(E)$; then ${\cal V} = \{(E,D,\xi)$ with $\xi$ parallel to $E$ and orthogonal to $D\}$ and $\Lambda = \{(E,D)$ with $(E.D) = 0\}$; then the quadratic form $Q_{0}$ defined by $Q_{0}(E,D) = (E.D)$ is 0 on $\Lambda$, and therefore by applying the theorem to $\pm Q_{0}$ one deduces that $(E^{n}.D^{n})$ converges to $(E^{\infty}.D^{\infty})$ in the sense of measures.
The proof of the quadratic theorem mimicks the one we had found for the Div-Curl lemma, using F{\eightrm OURIER} transform and P{\eightrm LANCHEREL} formula.
\par
I spent the year 1974/75 in Madison, and Joel R{\eightrm OBBIN} taught me how to translate some of my results in the language of differential forms, and he showed me a new proof of the Div-Curl lemma using H{\eightrm ODGE} theory.
We did not notice that the same method was giving the properties of sequential weak continuity of Jacobian determinants, which M{\eightrm ORREY} had actually proved in the 50s, and I only learned these results from John M. B{\eightrm ALL} in the Fall of 1975, mistakenly believing that he had proved them.
\par
Later, Jacques-Louis L{\eightrm IONS} asked Fran\c{c}ois M{\eightrm URAT} to extend the Div-Curl lemma (and he gave him an article by S{\eightrm CHULENBERGER} and W{\eightrm ILCOX} which he thought related), and Fran\c{c}ois extended it first to a bilinear setting, i.e. $U = (V,W)$ with some differential list for $V$ and a differential list for $W$, and then he looked at the bilinear forms $B(V,W)$ which are sequentially weakly continuous.
I pointed out that the splitting of $U$ and the restriction to bilinear forms was not natural, and therefore Fran\c{c}ois M{\eightrm URAT} proved the above theorem in the case where $Q(\lambda) = 0$ for all $\lambda\in \Lambda$ (deducing that $\nu = 0$ in this case); however, his proof was a little different than the one we had followed for the Div-Curl lemma, probably because he had also extended the Div-Curl lemma itself to a $(L^{p}, L^{q})$ setting (for which he used the M{\eightrm IKHLIN}-H{\eightrm \"ORMANDER} theorem on ${\cal F}L^{p}$ multipliers, and one needs to check the smoothness of the multiplier), and a similar approach forced him to impose an hypothesis of constant rank: if for each $\xi\in R^{N}\setminus 0$ one denotes $\Lambda_{\xi} = \Bigl{\{} \lambda\in R^{p}: \sum_{jk} A_{ijk} \lambda_{j}\xi_{k} = 0\hbox{ for }i = 1,\ldots,q \Bigr{\}}$, so that $\Lambda$ is the union of the subspaces $\Lambda_{\xi}$, the constant rank hypothesis imposes that the dimension of $\Lambda_{\xi}$ is independent of $\xi$.
It was Jacques-Louis L{\eightrm IONS} who then coined the term Compensated Compactness for the type of result that Fran\c{c}ois had obtained, as it looked like a compactness argument because one could deduce the weak limit of a nonlinear quantity, but it was the result of a compensation effect.
\par
I extended then the result as shown above, and it has the following consequence: if $U^{n}_{i}U^{n}_{j}\rightharpoonup U^{\infty}_{i}U^{\infty}_{j}+R_{ij}$ in the sense of measures, then this defines a symmetric matrix $R$ whose entries may be R{\eightrm ADON} measures: if all $R_{ij}$ are integrable functions, one has
$$
R(x)\hbox{ belongs to the closed convex hull of }\{\lambda\otimes\lambda, \lambda\in \Lambda\}\hbox{ a.e. }x\in\Omega,
$$
and the general case is similar once one uses R{\eightrm ADON}-N{\eightrm IKODYM} theorem: if $\tau = \sum_{k} R_{kk}$, then $R_{ij} = \rho_{ij}\tau$ with the functions $\rho_{ij}$ being $\tau$-integrable, and it is $\rho(x)$ which belongs to the convex hull of the elements of the form $\lambda\otimes\lambda$ for $\lambda\in \Lambda$, and this property holds $\tau$-almost everywhere.
A point in the convex hull of a set $K$ is the center of mass of a probability measure with support on $K$, and the theory of H-measures, which I developed in the late 80s, extends the Compensated Compactness theorem and gives explicitly a way to describe these probability measures; the interest is that the H-measures are measures in $(x,\xi)$ and that they permit to extend the Compensated Compactness theorem to the case of differential equations with variable coefficients, and in some situations the H-measures satisfy partial differential equations in $(x,\xi)$.
There are however parts of the Compensated Compactness Method which still require improvements, as there is not yet a characterization of which pairs (Y{\eightrm OUNG} measures/H-measures) can be created, although I have obtained some information in this direction with Fran\c{c}ois M{\eightrm URAT}).
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
39. Friday April 23.
\medskip
My analysis of problems of Continuum Mechanics in the 70s was that there was a dichotomy between the constitutive relations which are possibly nonlinear pointwise constraints of the form
$$
U(x)\in K\hbox{ a.e. }x\in \Omega,
$$
with $K$ eventually depending upon $x$ (with possible oscillations requiring techniques from Homogenization), and the balance equations which are linear differential constraints of the form
$$
\sum_{jk} A_{ijk}{\partial U_{j}\over \partial x_{k}} = f_{i}\hbox{ in }\Omega, \hbox{ for }i = 1,\ldots,q.
$$
Perhaps because I had learned some Continuum Mechanics as a student, I knew that Elasticity meant
$$
\rho(x){\partial^{2}u_{i}\over \partial t^{2}}-\sum_{j} {\partial \sigma_{ij}\over \partial x_{j}} = f_{i}\hbox{ in }\Omega,
$$
and in that case $U$ would contain the components of the momentum $\rho{\partial u\over \partial t}$, the strain $\nabla\,u$, the stress $\sigma$ (as I do not remember hearing the term P{\eightrm IOLA}-K{\eightrm IRCHHOFF} stress from my studies, it might be that I was told mostly the Eulerian point of view, where the symmetric C{\eightrm AUCHY} stress appears), and the constitutive relations relate $\sigma$ and $\nabla\,u$, while the list of balance equations contain the equilibrium equation above and the compatibility conditions related to using gradients (after I developped the Compensated Compactness Method in 1977, I added ``entropies'' to that description; it may be useful to point out that ``entropies'' have nothing to do with the fact that one considers an evolution equation, or that one is interested in hyperbolic problems, as it is just a name for designing supplementary differential equations which are consequences of those already written for smooth solutions; certainly Peter L{\eightrm AX} could have chosen a better name, and geometers call them Casimirs).
\par
It is of course a handicap that the Compensated Compactness theorem cannot handle variable coefficients, but H-measures do not suffer from this defect: each first order partial differential equation written in conservative form $\sum_{jk} {\partial (A_{jk}(x)U^{n}_{j})\over \partial x_{k}}\rightarrow f$ in $H^{-1}_{loc}(\Omega)$ strong, with the coefficients $A_{jk}$ being continuous, can be seen by H-measures (and the Localization Principle implies $\sum_{jk} \xi_{k}A_{jk}(x)\mu^{jl} = 0$ for every $l$).
\par
Y{\eightrm OUNG} measures cannot take into account partial differential equations, but it might be because I used them in order to describe some constraints that they must satisfy as a consequence of the quadratic Compensated Compactness theorem, that some may have misunderstood their role.
Suppose for example that a sequence $U^{n}$ is bounded in $L^{\infty}$, corresponds to a Y{\eightrm OUNG} measure $\nu_{x}, x\in\Omega$, and satisfies a partial differential equation with constant coefficient $\sum_{jk} A_{jk}{\partial U^{n}_{j}\over \partial x_{k}} = 0$; decompose $R^{N}$ as a union of cubes of size $1/n$ and for each of these cubes chose a rigid displacement mapping the cube onto itself and transport the values of $U^{n}$ accordingly, and let $V^{n}$ be any of the new functions obtained this way; it is not difficult to check that $V^{n}$ corresponds to the same Y{\eightrm OUNG} measure $\nu_{x}, x\in\Omega$, than $U^{n}$; however $V^{n}$ is unlikely to solve the same partial differential equation than $U^{n}$, and therefore Y{\eightrm OUNG} measures cannot feel if the sequence that they analyze satisfies or not a given partial differential equation.
A different way to express the same idea is to notice that in defining Y{\eightrm OUNG} measures the only important property of $\Omega$ is to be endowed with a nonnegative measure without atoms, like the L{\eightrm EBESGUE} measure, and therefore the structure of differentiable manifold is not seen by Y{\eightrm OUNG} measure.
However what the Compensated Compactness theorem says can be expressed in terms of Y{\eightrm OUNG} measures, as it says that if $Q$ is quadratic and satisfies $Q(\lambda)\ge 0$ for all $\lambda\in\Lambda$, then the limit of $Q(U^{n})$ is $\langle \nu_{x},Q \rangle$, while the limit of $U^{n}$ is $\langle \nu_{x},id \rangle$ (where $id$ is the identity mapping), and therefore one has the following inequality (reminiscent of J{\eightrm ENSEN}'s inequality, which says that is one replaces $Q$ by any convex function the following inequality is true)
$$
\langle \nu_{x},Q \rangle \ge Q(\langle \nu_{x},id \rangle)\hbox{ a.e. }x\in\Omega.
$$
\par
If $\Lambda = \{0\}$, one is looking at an elliptic system or some overdetermined system; the ellipticity of the system corresponds to saying that for every $\xi\ne 0$, the linear mapping $U\mapsto V$ with $V_{i} = \sum_{jk} A_{ijk}U_{j}\xi_{k}$ for $i = 1,\ldots,q$, is invertible (so that $q = p$); if it is not the case then one necessarily has $q>p$, but that by itself is not enough to imply that $\Lambda = \{0\}$.
In the case $\Lambda = \{0\}$ one has $U^{n}\rightarrow U^{\infty}$ in $L^{2}_{loc}(\Omega)$ strong, because one can take $Q$ positive definite, and as $Q$ is 0 on $\Lambda$ one finds that $Q(u^{n})\rightharpoonup Q(U^{\infty})$ in the sense of measures; if $B$ is the symmetric bilinear form associated to $Q$ one has then $Q(U^{n}-U^{\infty}) = Q(U^{n})-2B(U^{n},U^{\infty})+Q(U^{\infty})$, and therefore $Q(U^{n}-U^{\infty})\rightharpoonup 0$ in the sense of measures.
The case $\Lambda = \{0\}$ corresponds then to using a compactness argument.
\par
The case $\Lambda = R^{p}$ corresponds to using a convexity argument, as $Q\ge 0$ is the same as $Q$ convex for quadratic forms; this happens if there is no differential equation ($q = 0$), but also for some list of differential equations that are not constraining enough: if $U^{n}$ consists of the list or $k$ vector fields whose divergence is controlled, then one has $\Lambda = R^{p}$ if $k~~0$ one let $\varepsilon$ tend to 0, and as $\varphi_{\varepsilon\eta}\rightarrow+\infty$ on $(\eta,+\infty)$, one deduces that $|u_{2}(x,t)-u_{1}(x,t)|\le \eta$ for almost every $x\in \Omega$, and letting then $\eta$ tend to 0, one deduces that $u_{2} = u_{1}$.
\bigskip
Our next step is to study the functional space $H$, and show that there is a notion of normal trace $(u.n)$ on the boundary, so that the definition makes sense.
Then we will check that $V$ is dense in $H$.
\bigskip
\vfill
\eject
\noindent
{\bf 21-820. PDE Models in Oceanography}
\par
\noindent
Luc T{\eightrm ARTAR}, W{\eightrm EAN} Hall 6212, 268-5734, tartar@andrew.cmu.edu
\bigskip
\noindent
23. Monday March 8.
\medskip
I think that it was Jacques-Louis L{\eightrm IONS} who introduced the space $H(div;\Omega) = \{u\in L^{2}(\Omega;R^{N}),div(u)\in L^{2}(\Omega)\}$ and proved that one can give a meaning to $(u.n)$ on the boundary $\partial\Omega$ if $\Omega$ has a L{\eightrm IPSCHITZ} boundary.
First, one notices that $H(div;\Omega)$ is a local space, i.e. $\theta\,u\in H(div,\Omega)$ for all $u\in H(div;\Omega)$ and $\theta\in C^{\infty}(R^{N})$ as $div(\theta\,u) = \theta\,div(u)+(grad(\theta).u)$ (notice that we plan to use the results for the case $div(u) = 0$, but that property is lost by multiplication by smooth functions).
Then one shows that $C^{\infty}(\overline{\Omega};R^{N})$ is dense in $H(div,\Omega)$ if the boundary is smooth enough: after using a partition of unity for localizing the problem, one regularizes each $\theta_{i}\,u$ by convolution with a suitable regularizing sequence adapted to the support of $\theta_{i}$, and this is possible if $\Omega$ is an open set with compact boundary and if near each point of the boundary $\Omega$ is only on one side of the boundary and the boundary has an equation $x_{N} = F(x^{\prime})$ with $F$ continuous (one can have unbounded boundaries if one asks for a global equation of the corresponding piece with $F$ uniformly continuous).
Then, assuming that $\Omega$ has a L{\eightrm IPSCHITZ} (compact) boundary, so that one can define the normal to the boundary and traces on the boundary for functions in $H^{1}(\Omega)$, one has the formula
$$
\int_{\Omega} \Bigl[ \Bigl( u.grad(\varphi) \Bigr) +div(u)\varphi \Bigr]\,dx = \int_{\partial\Omega} (u.n)\varphi\,d\sigma,
$$
for $u\in C^{\infty}(\overline{\Omega};R^{N})$ and $\varphi\in H^{1}(\Omega)$.
The left side of the equation is a bilinear continuous form on $H(div;\Omega)\times H^{1}(\Omega)$ and therefore the right side is also continuous for that topology, but the right side is 0 for $\varphi\in H^{1}_{0}(\Omega)$ and therefore it is actually defined on the quotient $H^{1}(\Omega)/H^{1}_{0}(\Omega)$; here a natural choice is to use $T(\Omega)$, the space of traces of functions of $H^{1}(\Omega)$, equipped with the quotient norm
$$
||v||_{T(\Omega)} = \inf\{||u||_{H^{1}(\Omega)}, trace(u) = v\},
$$
and then the right side is continuous for the norm of $H(div;\Omega)\times T(\Omega)$, and therefore by density $(u.n)$ is defined on $H(div;\Omega)$ as a linear continuous form on $T(\Omega)$.
\par
If $\Omega$ has a compact L{\eightrm IPSCHITZ} boundary, then $T(\Omega) = H^{1/2}(\partial\Omega)$, and the proof (and definition of the space) is easily derived from the property for $R^{N}_{+}$, which I review below using F{\eightrm OURIER} transform.
The interest of the preceding result is that it applies even if the boundary is not so smooth and the trace space $T(\Omega)$ has not been characterized.
\par
It is important to notice that one cannot define each of the terms $u_{j}n_{j}$ on the boundary, but only their sum.
There is a framework using differential forms which is also useful to know (Jacques-Louis L{\eightrm IONS} was not aware of this aspect when he worked on the preceding question).
For smooth functions, one considers the $(N-1)$-form $\omega = \sum_{i} (-1)^{i-1}u_{i}dx_{1}\wedge\ldots\wedge dx_{i-1}\wedge dx_{i+1}\wedge\ldots\wedge dx_{N}$ (also written $\sum_{i} (-1)^{i-1}u_{i}\widehat{dx_{i}}$), whose exterior derivative is $d\omega = div(u)dx$; in the case of smooth (coefficients and) boundary, one can restrict a $p$-form to a manifold, as it is a $p$-linear alternating form and therefore it needs $p$ vectors to act upon and its restriction on the manifold uses only vectors from the tangent space to the manifold; if one restricts the $(N-1)$-form $\omega$ to the (smooth) boundary one obtains a form which has only one coefficient (as the dimension of the boundary is $N-1$) and that coefficient is $(u.n)$; it is natural, but not straightforward, that one can relax the hypotheses of regularity and still be able to define the intrinsic quantity $(u.n)$.
\par
There is another space which is important in applications (to Electromagnetism, but also to fluids once one considers the vorticity), i.e. $H(curl;\Omega) = \{u\in L^{2}(\Omega;R^{3}),curl(u)\in L^{2}(\Omega;R^{3})\}$; here one should consider the 1-form $\omega = \sum_{i} u_{i}dx_{i}$ and $d\omega = \sum_{i} \bigl( curl(u) \bigr)_{i}\widehat{dx_{i}}$ and of course $div\bigl( curl(u) \bigr) = 0$ expresses the fact that $d\,d = 0$; in the smooth case the exterior derivative commutes with the restriction and therefore the restriction is a 1-form on the boundary has its exterior derivative well defined.
It is the tangential component of $u$ which is well defined on $H(curl;\Omega)$, with a differential restriction corresponding to writing the exterior derivative, and this has been extended to the smooth case by L. P{\eightrm AQUET} (I did and taught the L{\eightrm IPSCHITZ} case a few years ago).
\medskip
For $u\in H^{1}(R^{N}_{+})$ its trace on $x_{N} = 0$ has been shown to belong to $L^{2}(R^{N-1})$ (after extending $u$ to a function in $H^{1}(R^{N})$); we want to show that it actually belongs to $H^{1/2}(R^{N-1})$, and that all elements of $H^{1/2}(R^{N-1})$ can be traces.
Of course for $s\ge 0$, the space $H^{s}(R^{m})$ is defined by F{\eightrm OURIER} transform as
$$
H^{s}(R^{m}) = \{u\in L^{2}(R^{m}), |\xi|^{s}|{\cal F}u|\in L^{2}(R^{N})\},
$$
but for $s<0$ it is $\{u\in{\cal S}^{\prime}(R^{N}):(1+|\xi|^{2})^{s/2}{\cal F}u\in L^{2}(R^{N})\}$.
\par
For $u\in C^{\infty}_{c}(R^{N})$, let $v\in C^{\infty}_{c}(R^{N-1})$ be the restriction of $u$ to $x_{N} = 0$; one defines the F{\eightrm OURIER} transforms of $u$ and $v$
$$
\eqalign{
{\cal F}u(\xi^{\prime},\xi_{N}) &= \int_{R^{N}} u(x^{\prime},x_{N})e^{-2i\pi(x^{\prime}.\xi^{\prime})-2i\pi\,x_{N}\xi_{N}}\,dx^{\prime}\,dx_{N}\cr
{\cal F}v(\xi^{\prime}) &= \int_{R^{N-1}} u(x^{\prime},0)e^{-2i\pi(x^{\prime}.\xi^{\prime})}\,dx^{\prime},}
$$
and the critical relation is
$$
{\cal F}v(\xi^{\prime}) = \int_{R} {\cal F}u(\xi^{\prime},\xi_{N})\,d\xi_{N}.
$$
Indeed for a given $\xi^{\prime}$, if one defines $w$ by $w(x_{N}) = \int_{R^{N-1}} u(x^{\prime},x_{N})e^{-2i\pi(x^{\prime}.\xi^{\prime})}\,dx^{\prime}$, then $w\in{\cal S}(R)$ and therefore $w(0) = \int_{R} {\cal F}w(\xi_{N})d\xi_{N}$, because $w = \overline{\cal F}{\cal F}w$; after putting back explicitly the dependence in $\xi^{\prime}$, it is exactly our relation.
Using C{\eightrm AUCHY}-S{\eightrm CHWARTZ} inequality, one deduces
$$
\eqalign{
|{\cal F}v(\xi^{\prime})|&\le \int_{R} {1\over \sqrt{1+|\xi^{\prime}|^{2}+|\xi_{N}^{2}}}\,\sqrt{1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2}}\,|{\cal F}u(\xi^{\prime},\xi_{N})|\,d\xi_{N}\cr
&\le \Bigl( \int_{R} {d\xi_{N}\over 1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2}} \Bigr)^{1/2}\Bigl( \int_{R} (1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2})|{\cal F}u(\xi^{\prime},\xi_{N})|^{2}\,d\xi_{N} \Bigr)^{1/2}\cr
&= \Bigl( {\pi\over \sqrt{1+|\xi^{\prime}|^{2}}} \Bigr)^{1/2}\Bigl( \int_{R} (1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2})|{\cal F}u(\xi^{\prime},\xi_{N})|^{2}\,d\xi_{N} \Bigr)^{1/2},}
$$
and therefore $(1+|\xi^{\prime}|^{2})^{1/4}{\cal F}v\in L^{2}(R^{N-1})$.
\par
Conversely, given $v\in H^{1/2}(R^{N-1})$, one must find $u\in H^{1}(R^{N})$ such that ${\cal F}v(\xi^{\prime}) = \int_{R} {\cal F}u(\xi^{\prime},\xi_{N})\,d\xi_{N}$, and one chooses
$$
{\cal F}u(\xi^{\prime},\xi_{N}) = {\cal F}v(\xi^{\prime})\varphi\Bigl( {\xi_{N}\over \sqrt{1+|\xi^{\prime}|^{2}}} \Bigr){1 \over \sqrt{1+|\xi^{\prime}|^{2}}},
$$
where $\varphi\in C^{\infty}_{c}(R)$ satisfies $\int_{R} \varphi(s)\,ds = 1$.
It remains to check that $u\in H^{1}(R^{N})$, and this follows from
$$
\eqalign{
\int_{R^{N}} (1+|\xi|^{2})|{\cal F}u(\xi)|^{2}\,d\xi &= \int_{R^{N}} (1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2})|{\cal F}v(\xi^{\prime})|^{2}\varphi^{2}\Bigl( {\xi_{N}\over \sqrt{1+|\xi^{\prime}|^{2}}} \Bigr){1 \over 1+|\xi^{\prime}|^{2}}\,d\xi^{\prime}\,d\xi_{N}\cr
&= \int_{R^{N-1}} |{\cal F}v(\xi^{\prime})|^{2} \Bigl( \int_{R} {1+|\xi^{\prime}|^{2}+|\xi_{N}|^{2}\over 1+|\xi^{\prime}|^{2}}\varphi^{2}\Bigl( {\xi_{N}\over \sqrt{1+|\xi^{\prime}|^{2}}} \Bigr)\,d\xi_{N} \Bigr)\,d\xi^{\prime}\cr
&= \Bigl( \int_{R} (1+s^{2})\varphi^{2}(s)\,ds \Bigr)\int_{R^{N-1}} \sqrt{1+|\xi^{\prime}|^{2}}|{\cal F}v(\xi^{\prime})|^{2}\,d\xi^{\prime}.}
$$
\medskip
One defines $H = \{u\in L^{2}(\Omega;R^{N}), div(u) = 0$ in $\Omega, (u.n) = 0$ on $\partial\Omega\}$; as one imposes $div(u) = 0$, one has $u\in H(div;\Omega)$ and therefore $(u.n)$ has a meaning; more precisely $(u.n) = 0$ means that for all $\varphi\in H^{1}(\Omega)$ one has $\int_{\Omega} \bigl( (grad(\varphi).u)+ \varphi\,div(u) \bigr)\,dx = 0$ for all $\varphi\in H^{1}(\Omega)$, and $u\in H$ implies then $\int_{\Omega} (grad(\varphi).u)\,dx = 0$ for all $\varphi\in H^{1}(\Omega)$, as $div(u) = 0$.
One sees then that $H$ is orthogonal to the subspace of gradients of functions in $H^{1}(\Omega)$.
\medskip
\noindent
{\bf Lemma}: If the injection of $H^{1}(\Omega)$ into $L^{2}(\Omega)$ is compact, then the orthogonal of $H$ in $L^{2}(\Omega;R^{N})$ is the (closed) subspace of $grad(\varphi)$ for $\varphi\in H^{1}(\Omega)$.
\par
\noindent
{\it Proof}: One can apply the equivalence lemma to the case where $E_{1} = H^{1}(\Omega)$, $A = grad$ with $E_{2} = L^{2}(\Omega;R^{N})$, and $B$ is the (compact) injection of $H^{1}(\Omega)$ into $E_{3} = L^{2}(\Omega)$; the equivalence lemma asserts that the range of $A$ is closed.
Let us check that the orthogonal of $R(A)$ is $H$, which proves that the orthogonal of $H$ is the closure of $R(A)$, i.e. $R(A)$ itself.
\par
Assume that $u\in L^{2}(\Omega;R^{N})$ is orthogonal to $R(A)$; taking $\varphi\in C^{\infty}_{c}(\Omega)$ and noticing that $\langle div(u),\varphi \rangle = -\langle u,grad(\varphi) \rangle = 0$ shows that $div(u) = 0$ in the sense of distributions.
This proves that $u\in H(div;\Omega)$, and using already the information that $div(u) = 0$ in $\Omega$, one deduces that $\int_{\Omega} (grad(\varphi).u)\,dx = \langle (u.n),trace(\varphi) \rangle$ for all $\varphi\in H^{1}(\Omega)$, and as the left side is 0 by definition of $u$, the right side is 0 and therefore $(u.n) = 0$ (as a linear continuous form on the space of traces of functions in $H^{1}(\Omega)$), i.e. $u\in H$.
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We can now look at the important question of density of $V$ into $H$, as this is basic to the framework used.
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{\bf Lemma}: If $meas(\Omega)<\infty$ and if $L^{2}(\Omega) = X(\Omega)$ (which is $\{u\in H^{-1}(\Omega), {\partial u\over \partial x_{j}}\in H^{1}(\Omega)$ for $j = 1,\ldots,N\}$), then $V$ is dense in $H$.
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{\it Proof}: Let $h\in L^{2}(\Omega;R^{N})$ belong to the orthogonal of $V$ in $L^{2}(\Omega;R^{N})$; then $h$ can be considered an element of $H^{-1}(\Omega;R^{N})$, orthogonal to $V$ for the duality product, and therefore of the form $grad(p)$ with $p\in L^{2}(\Omega)$, and as $grad(p)\in L^{2}(\Omega;R^{N})$ it means that $p\in H^{1}(\Omega)$.
Therefore $h = grad(p)$ with $p\in H^{1}(\Omega)$ and so $h$ is orthogonal to $H$, which proves that $V$ is dense in $H$.
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The fact that $X(\Omega) = L^{2}(\Omega)$ requires more regularity of the boundary that the simple compactness of $H^{1}(\Omega)$ into $L^{2}(\Omega)$: I had noticed in the Fall that it is not true in a plane domain of the form $\{(x,y): 0~~