Theorem 12.2.2: Local Existence and Uniqueness of $1^{st}$ order Quasilinear PDE

Theorem 12.2.2 (Local Existence and Uniqueness of $1^{st}$ order Quasilinear PDE).label For the Cauchy problem above such that $(a_{1},...,a_{n})$ is not tangent to $S$ at any $x\in S$. Then for any sufficiently small neighborhood $\OM$ of $S$ in $\R^{n}$, there exists a unique solution to the Cauchy problem.

Proof. Let us express non-tangency condition. Suppose $S$ is parametrized by

\begin{align*}U\ni s=(s_{1},...,s_{n-1})\mto (g_{1}(s),...,g_{n}(s))&&\det\begin{pmatrix}\pder{g_1}{s_1}&\cds&\pder{g_1}{s_{n-1}}&a_{1}(\OBC{g(s)}{x\in S},\OBC{\p(g(s))}{y(x)})\\ \vdots&&\vdots\\ \pder{g_n}{s_1}&\cds&\pder{g_n}{s_{n-1}}&a_{n}(g(s),\p((g(s))))\end{pmatrix}\neq 0\end{align*}

$\fall s\in U$. If we were in the linear case then this $\det\neq 0$ condition is equivalent to $(a_{1},...,a_{n})$ is non-characteristic.

We now turn this into ODE system

\begin{align*}\der{x_j}{t}=a_{j}(x,y)&&\der{y}{t}=b(x,y)\end{align*}

with initial conditions

\begin{align*}x_{j}(s,0)=g_{j}(s)&&y(s,0)=\p(g(s))\end{align*}

By $\exists!$ for ODE (Cauchy-Lipschitz Picard) $\exists!$ solution

\begin{align*}(s_{1},...,s_{n-1},t)\mto (x_{1}(s,t),...,x_{n}(s,t))\end{align*}

which is $C^{1}$ in both $s$ (parametrizes the initial data) and $t$ (flow parameter). By IFT, $\exists C^{1}$ inverse map

\begin{align*}s=s(x)&&t=t(x)\end{align*}

in a small neighborhood $\OM$ of any point in $S$. Then we may define $u(x)\define y(s(x),t(s))$. If $x\in S$ then we have $u(x)=\p(s)$ hence the Cauchy condition is satisfied. It remains to show that $u(x)$ solves the PDE. We have

\begin{align*}\sums{n }{j=1}a_{j}\pd_{j}u&=\sums{n }{j=1}a_{j} \parens{\sums{n-1 }{k=1}\pder{u}{s_k}\pder{s_k}{x_j}+\pder{u}{t}\pder{t}{x_j}}\\&=\sums{n-1}{k=1}\pder{u}{s_k}\sums{n }{j=1}\UBC{a_j}{\pder{x_j}{t}}\pder{s_k}{x_j}+\pder{u}{t}\sums{n}{j=1}\UBC{a_j}{\pder{x_j}{t}}\pder{t}{x_j}\\&=\sums{n-1}{k=1}\pder{u}{s_k}\sums{n }{j=1}\UBC{\pder{x_j}{t}\pder{s_k}{x_j}}{\pder{s_k}{t}=0}+\pder{u}{t}\sums{n }{j=1}\UBC{\pder{x_j}{t}\pder{t}{x_j}}{\pder{t}{t}=1}\\&=\pder{u}{t}=b\end{align*}

The integral curve of ODE system is called characteristic curve$\square$

The Eigenspace

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