Method of Characteristics

The method of characteristics applies to some first order linear equations Suppose $u$ solves $$ a \cdot \grad u = 0 \,, $$ for some given function $a$. Let $x$ solve $\dot x = a(x)$, and note that $u(x(t))$ is constant. So if the value of $u$ is known at some point on this curve (called a characteristic), then one can determine $u$ everywhere along this curve. A generalization of this is the method of characteristics.

The characteristic equation

Consider a general first order PDE of the form $$\label{e:F}\tag{F} F(Du, u, x) = 0 \quad\text{in } U \,, $$ where $F$ is smooth, and $U \subseteq \R^d$ is some domain. Our aim is to solve \eqref{e:F}, with boundary conditions $$ u = g \quad\text{on } \Gamma\,, $$ where $\Gamma \subseteq U$, and $g \colon \Gamma \to \R$ is given.

We want to reduce \eqref{e:F} to a system of ODEs. Suppose we find a curve in $U$ parametrized by $x(t)$. Define $$ z(t) \defeq u(x(t)) \quad\text{and}\quad p(t) = \grad u(x(t)) \,. $$

Proposition 1. Suppose $u \in C^2(U)$ solves \eqref{e:F}. If $x$ solves $$\label{e:x}\tag{x} \dot x = \grad_p F(p, z, x)\,, $$ then $p, z$ solve \begin{align} \label{e:p}\tag{p} \dot p &= -\grad_x F(p, z, x) - \partial_z F(p, z, x) p \\ \label{e:z}\tag{z} \dot z &= \grad_p F(p, z, x) \cdot p \,. \end{align}

Remark 2. Getting a closed system of ODEs for a PDE can only be done in very special cases.

Remark 3. The solutions to the system \eqref{e:x}, \eqref{e:p}, \eqref{e:z} are called the characteristics of \eqref{e:F}. The trajectories of only $x$ are called the projected characteristics (or sometimes just called characteristics).

Problem 4. The PDE \eqref{e:F} is called quasi-linear if $F$ is affine linear in $Du$; that is \eqref{e:F} is of the form $$ b(u, x) \cdot \grad u + c(u, x) = 0 $$

  1. Show that you can get a closed system of ODEs for $x$ and $z$, without involving $p$.
  2. Find the projected characteristics for the transport equation, and use this to derive the general solution of the transport equation as stated in this remark.
  3. Find the general solutions of the PDEs
    1. $(1 + x^2) \partial_x u + \partial_y u = y u^2$.
    2. $(1-xy) \partial_x u + \partial_y u + \partial_z u = (y-z) u$.

The boundary value problem

The above suggests that the PDE \eqref{e:F} may be (uniquely) solvable if $\Gamma$ is a $d-1$ dimensional manifold. This is true, under certain compatibility conditions.

Lemma 5. Fix $x_0 \in \Gamma$, an open $V \ni x_0$ and suppose $\Phi\colon V \to V’ \subseteq \R^d$ is an invertible $C^1$ function such that $\Phi(V \cap \Gamma) \subseteq R^{d - 1}$. Let $v = u \circ \Phi^{-1}$. Then $v$ satisfies an equation of the form \begin{align} G(Dv, v, x) &= 0 && \text{in } V’ \\ v &= h &&\text{on } \R^{d-1} \cap V’ \,. \end{align}

By transforming coordinates (as above) we now assume $u$ solves \eqref{e:F}, with $\Gamma \subseteq R^{d-1}$. We aim to use \eqref{e:x}, \eqref{e:p} and \eqref{e:z} to construct a solution of \eqref{e:F} near $x_0$. Suppose we impose initial conditions $$ p(0) = p_0 \,,\quad z(0) = z_0 \,,\quad x(0) = x_0 \,. $$ Clearly we should have the compatibility conditions $$\label{e:C}\tag{C} \begin{aligned} z_0 &= g(x_0) \\ p_{0, i} &= p_0 \cdot e_i = \partial_i g(x_0) &&i \in \set{1, \dots, d-1} \,, \\ F(p_0, z_0, x_0) &= 0 \,. \end{aligned} $$

Definition 6. We say $(p_0, z_0, x_0)$ is admissible if it satisfies \eqref{e:C}.

Proposition 7. If $(p_0, z_0, x_0)$ is an admissible triple, and $F$ satisfies the non-characteristic condition $$\label{e:N}\tag{N} \partial_{p_d} F(p_0, z_0, x_0) \neq 0\,, $$ then for every $y \in \Gamma$, sufficiently close to $x_0$, there exists a unique solution to the compatibility conditions at $y$. That is, there exists $q = q(y)$ such that $$\label{e:Cy}\tag{C$_y$} \begin{aligned} q_{i}(y) &= \partial_i g(y) &&i \in \set{1, \dots, d-1} \,, \\ F(q(y), g(y), y) &= 0 \,. \end{aligned} $$

Remark 8. If $\Gamma$ is not flat, then \eqref{e:N} should be replaced with $$ \grad_p F(p_0, z_0, x_0) \cdot \hat n \neq 0\,, $$ where $\hat n$ is a unit normal to $\Gamma$ at $x_0$.

Let $\phi$ be the flow map of the characteristic equations.

Lemma 9. Suppose $(p_0, z_0, x_0)$ is an admissible triple where the non-characteristic condition \eqref{e:N} holds. There exists an open $W \subseteq R^{d-1}$ containing $x_0$, $T > 0$, and an open $V \subseteq \R^d$, $V \ni x_0$, such that for every $x’ \in V$, there exists a unique $t \in (-T, T)$, $y \in W$ such that the projected characteristic $x(y, t)$ (which starts at $x(y, 0) = y$) satisfies $$ x’ = x(y, t) $$

Theorem 10. Given $x’ \in V$, define $$ u(x’) = z(y, t)\,, $$ where $y = y(x’) \in \Gamma$ and $t = t(x’) \in (-T, T)$ are the points such that $x’ = x(y, t)$. Then $u$ satisfies \eqref{e:F} in $V$.

Proof sketch.

  1. Characteristic equation implies $F(p(y, t), u(x’), x’) = F(p(y, t), z(y, t), x(y, t)) = 0$.
  2. We claim $p(y, t) = \grad u(x’)$:
    1. Set $r_i(t) \defeq \partial_{y_i} z - \sum_j p_j(y, t) \partial_{y_i} x_j$.
    2. Check $r_i(0) = 0$, and $\partial_t r_i = - \partial_z F r_i$, and conclude $r \equiv 0$.
    3. Write $\partial_{x_j} u = \partial_t z \partial_{x_j} t + \sum_{i=1}^{d - 1} \partial_{y_i} z \partial_{x_j} y_i$, use the above and simplify to obtain $\partial_{x_j} u = p_j$ as desired.

Remark 11. Consider a first order linear (inhomogeneous) equation with $\Gamma = \partial U$. By uniqueness of solutions to the characteristic ODE, projected characteristics from different points can not cross.

  1. Suppose the (projected) characteristics have an attracting point inside $U$. The solution to \eqref{e:F} can be defined locally near each point in $\partial U$, and can likely be continued to some neighborhood of $\partial U$, but can’t extend to all of $U$ (unless it’s identically constant).

  2. Suppose the (projected) characteristics flow through the domain (and hit the boundary at multiple points). Then again the solution can be defined in some neighborhood of $\partial U$ but can’t extend, unless the data is compatible.

Remark 12. For a first order quasi-linear equation, projected characteristics emanating from different points wont cross in $V$ (the neighborhood where we have local existence) but may cross outside $V$. This usually means that the local solutions may not be extendable.

General existence theorems

The method of characteristics gives local solutions to the boundary value problem for certain first order equations. You may wonder if there’s a more general method that gives solutions to a more general class of equations.

  1. The Cauchy–Kovalevskaya theorem guarantees existence for analytic (quasi-linear) equations, with analytic data on a non-characteristic analytic surface.

  2. The Malgrange-Ehrenpreis theorem says that linear PDEs with constant coefficients have at least one solution.

  3. Lewy found an example of a simple equation which almost always has no solutions. A simpler example (by Mizohata) is: $$ \partial_x u + i x \partial_y u = F(x, y) \,. $$ which has no solutions for certain $F$.

Existence theorems in PDE are usually equation specific.