Planar shocks and piecewise smooth traces

Let $I\subseteq \mathbb{R}$ be open, $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, $\mathcal{O}\subseteq {\mathbb{R}}^m$ be the state domain of a conservation system, $n\in {\mathbb{R}}^d$ satisfy $\mathrm{\mid }n\mathrm{\mid }=1$, and $s\in \mathbb{R}$. The planar interface and its two open sides are $\mathrm{\Sigma }=\{(t,x)\in I\times \mathrm{\Omega }:n\cdot x-st=0\},Q^{-}=\{(t,x)\in I\times \mathrm{\Omega }:n\cdot x-st<0\},Q^{+}=\{(t,x)\in I\times \mathrm{\Omega }:n\cdot x-st>0\}\mathrm{.}$ A state field is piecewise $C^1$ across $\mathrm{\Sigma }$ if on each side it is $C^1$ and, near every point of $\mathrm{\Sigma }$, each side field is the restriction of a $C^1$ map from a neighborhood in $I\times \mathrm{\Omega }$ into $\mathcal{O}$. The left and right traces $U_L,U_R:\mathrm{\Sigma }\to \mathcal{O}$ are the restrictions of those extensions to $\mathrm{\Sigma }$; they are independent of the chosen extensions because they agree with the same field on their respective open sides. For any state or flux quantity $G$ with these traces, define its jump by $[G]=G_R-G_L$. A planar shock profile is such a piecewise $C^1$ state field with its specified constant interface normal $n$ and speed $s$; the definition does not assert that it solves a conservation system.

conservation-system state and flux domains

This definition supplies actual one-sided traces only for fields with the stated piecewise smooth extensions. It does not assert a trace theorem for arbitrary locally bounded weak solutions. The value assigned on $\mathrm{\Sigma }$ is not part of either one-sided trace.