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{\it Pattern Formation and Partial Differential Equations}

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\begin{center} 
{\bf Felix Otto}\\
Universit\"at Bonn\\
Institute for Applied Mathematics\\
{\tt otto@iam.uni-bonn.de}\\
\end{center}

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\noindent {\bf Abstract}:  In this course, I will discuss three partial
differential 
equations (PDE) that model pattern formation. Numerical simulations reveal
that solutions
of these deterministic equations have indeed stationary or self-similar
statistics,
which are independent of the system size and of the details of the initial
data.
We show how PDE methods can be used to understand some aspects of this
universal behavior.


The first PDE has the structure of a gradient flow
(a feature on which the analysis relies), the
second PDE has the structure of a {\it driven} gradient flow,
whereas the third PDE is half-way between a conservative
and a dissipative system.


{\bf 1. Bounds on the coarsening rate in spinodal decomposition}


The PDE --- the Cahn-Hilliard equation --- is given by
%
$$
\partial_t u+\triangle(u(1-u^2)+\triangle u)\;=\;0
$$
%
with periodic boundary conditions in the spatial domain
$(0,L)^d$. Here, $u$ denotes the (renormalized) volume fraction
of a binary mixture, which is quenched (slightly) below the critical 
temperature and thus wants to segregate. 



Numerical simulations reveal that for generic initial
data (e.\ g.\ small amplitude white noise) after an initial layer, $(0,L)^d$
divides into
a convoluted domain where $u\approx1$ and its complement where $u\approx-1$,
separated by a characteristic interfacial layer of width $O(1)$. This
domain configuration coarsens over time. More precisely, the average
length scale of the domains behaves as $O(t^{1/3})$.
This is reflected by the fact that the average energy per volume, i.\ e.\
$E=L^{-d}\int\frac{1}{2}|\nabla u|^2+\frac{1}{4}(1-u^2)^2\,dx$,
behaves as $O(t^{-1/3})$.



We shall prove that, in a time-averaged sense,
$E\ge O(t^{-1/3})$.
This is joint work with R.\ V.\ Kohn.

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{\bf 2. Bounds on the Nusselt number in Rayleigh-B\'enard convection}

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The system of PDEs is given by an advection-diffusion equation for the
temperature $T$, 
and the Stokes equations with buoyancy for the fluid velocity $u$, i.\ e.\
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\begin{eqnarray*}
\partial_tT+\nabla\cdot(Tu)-\triangle T&=&0,\\
-\triangle u+\nabla p&=&T\,(0,0,1),\\
\nabla\cdot u&=&0
\end{eqnarray*}
%
in the 3-d spatial domain $(0,L)^2\times(0,H)$ with
periodic boundary conditions in the two horizontal dimensions.
The PDE is complemented by inhomogeneous (and thus driving) Dirichlet boundary
conditions
at the top and bottom boundaries 

$$
T\;=\;1\;\;\mbox{for}\;z=0,\quad
T\;=\;0\;\;\mbox{for}\;z=H,\quad
u\;=\;0\;\;\mbox{for}\;z=0,H.
$$


Experiments and numerical simulations for $H,L\gg 1$ show a 
chaotic velocity field $u$, with regions of high temperature $T\approx 1$ in
form of mushrooms (plumes). This leads to a high upwards
heat transport --- much higher than the one mediated by diffusion allone.
This upwards heat flux is given by the Nusselt number 
$Nu:=L^{-2}H^{-1}\int T\,u\cdot(1,0,0)dx$. 
Experiments and asymptotic analysis suggest that $Nu=O(H)$.


With the help of the background field method, we will prove
that indeed $Nu\le O(H)$ in $H\gg 1$ (up to a logarithm).
This is joint work with C.\ Doering and M.\ Reznikoff-Westdickenberg.

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{\bf 3. Bounds on the average dissipation in the Kuramoto-Sivashinsky
  equation}

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The PDE --- the Kuramoto-Sivashinsky equation --- is given by
%
$$
\partial_tu+\partial_x(\frac{1}{2}u^2)+\partial_x^2u+\partial_x^4u\;=\;0
$$
%
with periodic boundary conditions on $(0,L)$. In one
particular application, $u$
denotes the slope $\partial_xh$ of a (one-dimensional)
crystal surface. The Kuramoto-Sivashinsky
equation describes the evolution of the crystal surface
in the presence of slope selection, curvature regularization
and strong deposition --- in a regime where there is
no coarsening of facets. It can also be seen as a
toy model for the energy transfer from large wave lengths
to small wave lengths in the Navier Stokes equations.


For $L\gg 1$,
numerical simulations reveal that the solutions have
an average length scale of $O(1)$, and an average
amplitude of $O(1)$ and display spatio-temporal chaos.

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We will argue that the average dissipation rate,
i.\ e.\
$\lim_{T\uparrow\infty}T^{-1}L^{-1}\int_0^T\int_0^L(\partial_x^2u)^2dxdt$, 
is $O(1)$ in $L\gg 1$ (up to a logarithm). The argument relies on a new
observation
on the inhomogeneous inviscid Burger's equation
%
$$
\partial_tu+\partial_x(\frac{1}{2}u^2)\;=\;\partial_xf.
$$

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