= Solution
For the first-order quasilinear equation $a(x,u)\cdot\nabla u=b(x,u)$, suppose the coefficients and initial data $u=g$ on a smooth hypersurface $\Sigma$ are $C^1$. At an initial point $x_0$, require $a(x_0,g(x_0))\cdot n(x_0)\ne0$ for a normal $n$ to $\Sigma$; this is precisely the condition that $\Sigma$ be a <non-characteristic hypersurface>. Then in a neighborhood of $x_0$ there is a unique $C^1$ solution taking the given data. Smooth coefficients and data give corresponding higher regularity.
The <method of characteristics> solves $\dot x=a(x,u)$, $\dot u=b(x,u)$ from each initial point. The non-characteristic condition makes the map from the initial coordinates and characteristic time to $x$ locally invertible, so these characteristic solutions define the asserted function.
Back to article page
