Noncharacteristic Cauchy surfaces for first-order transport

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be the domain of the transport equation, and let $\sigma :U\subseteq {\mathbb{R}}^n\to \mathrm{\Omega }$ be a $C^1$ parametrized hypersurface, written

$\sigma (\eta )=(\gamma (\eta ),\tau (\eta ))\mathrm{.}$

For the transport equation $u_t+a(x,t)\cdot {\mathrm{\nabla }}_{x}u+c(x,t)u=f(x,t)$, let the space-time transport vector be

$B(x,t):=(a(x,t),1)\in {\mathbb{R}}^{n+1}\mathrm{.}$

The surface $\sigma$ is noncharacteristic at ${\eta }_0$ when the $(n+1)\times (n+1)$ matrix whose columns are the $n$ tangent vectors ${\mathrm{\partial }}_{{\eta }_1}\sigma ({\eta }_0),\dots ,{\mathrm{\partial }}_{{\eta }_n}\sigma ({\eta }_0)$ and the vector $B(\sigma ({\eta }_0))$ has nonzero determinant. Equivalently, $B(\sigma ({\eta }_0))$ is not tangent to the data surface.

A prescribed datum for the transport equation is a function $g:U\to \mathbb{R}$, interpreted as the Cauchy condition $u(\gamma (\eta ),\tau (\eta ))=g(\eta )$.