Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 29C / i / Solution

For a first-order quasilinear partial differential equation, take $C^1$ coefficients $a_j,b$, $C^1$ prescribed data $u_0$ on a smooth $C^2$ hypersurface $S$, and a point $x_0\in S$. The data are non-characteristic there ifwhere $n$ is any nonzero normal to $S$. Then there is a neighborhood of $x_0$ on which a unique $C^1$ solution realizes the data. Uniqueness is local, after restricting to a neighborhood swept out by the nearby characteristics.

The method of characteristics solves $\dot{x}=a(x,u)$, $\dot{u}=b(x,u)$ from points of $S$. In local surface coordinates $\eta$, the map $(\eta ,s)\mapsto x(\eta ,s)$ has nonsingular derivative at $s=0$ exactly under the stated normal condition. The inverse function theorem makes these variables local coordinates, giving the asserted solution and uniqueness.