On Quasilinear PDEs

Lee Smooth Manifolds

A first-order quasilinear Cauchy problem is of the form

$$a^1(x,u(x))\frac{\partial u}{\partial x^1}+\dots +a^n(x,u(x))\frac{\partial u}{\partial x^n}=f(x,u(x))$$

where $a^1,\dots ,a^n,f\in C^{\infty }(W)$ for some open $W\subset {\mathbb{R}}^{n+1}$. We are also given the initial condition

$$u{|}_S=\phi$$

where $S\subset {\mathbb{R}}^n$ is an embedded hypersurface and $\phi :S\to \mathbb{R}$ is a smooth function such that its graph is contained in $W$. We call a quasilinear Cauchy problem noncharacteristic if the vector field $A^{\phi }:S\to T{\mathbb{R}}^n$ defined by

$$A^{\phi }{|}_x=a^1(x,\phi (x))\frac{\partial }{\partial x^1}{|}_x+\dots +a^n(x,\phi (x))\frac{\partial }{\partial x^n}{|}_x$$

is nowhere tangent to $S$. We wish to prove that for any such noncharacteristic Cauchy problem, for any $p\in S$ there exists a neighborhood $U$ of $p$ in ${\mathbb{R}}^n$ on which there exists a unique solution $u$.

Proof) The crux of this problem is formulating it in terms of a stationary vector field. We define the characteristic vector field $\xi :W\to T{\mathbb{R}}^{n+1}$ as

$$\xi (x,z)=a^1(x,z)\frac{\partial }{\partial x^1}{|}_{(x,z)}+\dots +a^n(x,z)\frac{\partial }{\partial x^n}{|}_{(x,z)}+f(x,z)\frac{\partial }{\partial z}$$

Assume $u$ is some smooth function such that its graph $\Gamma (u)$ is in $W$. $u$ is then a solution to the PDE iff, for any $(x,z)\in \Gamma (u)$, ${\xi }_{(x,z)}(u(x)-z)=0$. As $u(x)-z$ is a defining function of the embedded submanifold $\Gamma (u)$ of $W$, this holds iff ${\xi }_{(x,z)}\in T\Gamma (U{)}_{(x,z)}$ for every $(x,z)\in \Gamma (u)$; i.e. $\xi$ is tangent to $\Gamma (u)$. Intuitively, this means we can obtain a solution to the Cauchy problem by taking the flowout of $\xi$ from the initial submanifold and representing it as a graph. Of course, we must first show a flowout exists.

Let $\Gamma (\phi )\subset W$ be the graph of $\phi :S\to \mathbb{R}$, which is an embedded $n-1$ dimensional submanifold. We wish to show that, at every point $p\in \Gamma (\phi )$, ${\xi }_p$ is independent of $T_p\Gamma (\phi )$. This follows from the fact that the projection $\pi :{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n$ that simply omits the last coordinate maps $T_p\Gamma (\phi )$ into $T_{p}S$and ${\xi }_p$ into $A^{\phi }{|}_p$, which is independent of $T_{p}S$ as the PDE is noncharacteristic.

Let $\Xi :{\mathcal{O}}_{\delta }\to {\mathbb{R}}^{n+1}$ be the flowout of $\xi$ from $\Gamma (\phi )$ where ${\mathcal{O}}_{\delta }$ is some neighborhood of $\{0\}\times \Gamma (\phi )$ in $\mathbb{R}\times \Gamma (\phi )$. Note that $\Xi ({\mathcal{O}}_{\delta })$ is an immersed $n$-manifold in ${\mathbb{R}}^{n+1}$, and $\Xi ({\mathcal{O}}_{\delta })\subset W$. Let $p\in \Gamma (\phi )$ be arbitrary; $T_p\Xi ({\mathcal{O}}_{\delta })=\text{Span}(T_p\Gamma (\phi ),{\xi }_p)$. Recall that the projection $\pi :{\mathbb{R}}^{n+1}\to {\mathbb{R}}^n$ maps $T_p\Xi ({\mathcal{O}}_{\delta })$ into a linear subspace of dimension $n$; this is an isomorphism and thus there is a neighborhood $U$of $p$ in $\Xi ({\mathcal{O}}_{\delta })$ and $V$ of $\pi (p)$ in ${\mathbb{R}}^n$ such that $\pi$ is a diffeomorphism from $U$ to $V$. This forces $U$ to be the graph of the smooth function $u:V\to \mathbb{R}$ defined by $u={\pi }^{\prime }\circ {\pi }^{-1}$ where ${\pi }^{\prime }:{\mathbb{R}}^{n+1}\to \mathbb{R}$ is the projection onto the last coordinate. We have thus proven existence.

For uniqueness, restrict $U$ such that it is of the form $\Xi ((-ϵ,ϵ)\times P)$ where $P$ is some neighborhood of $p$ in $\Gamma (\phi )$ and $V$ such that $V=\pi (U)$. Assume there is some other solution $\tilde{u}$ defined on $V$ to the Cauchy problem. Note that $\Xi ((-ϵ,ϵ)\times P)$ is the union of integral curves starting at points of $P\subset \Gamma (\phi )$. Each of these integral curves intersects the graph $\Gamma (\tilde{u})$, which is a properly embedded submanifold of $V\times \mathbb{R}$ such that $\xi$ is tangent to $\Gamma (\tilde{u})$. This forces each of the integral curves defined on $|t|<ϵ$ to be fully contained in $V\times \mathbb{R}$, and thus $\Xi ((-ϵ,ϵ),P)\subset \Gamma (\tilde{u})$. Obviously, this is only possible if $u=\tilde{u}$ on $V$.