Piecewise smooth shocks and one-sided traces

Let $n\ge 1$, $T>0$, and $f\in C^1(\mathbb{R};{\mathbb{R}}^n)$ (Scalar conservation laws, fluxes and Cauchy data). Let $\mathrm{\Gamma }$ be a $C^1$ hypersurface in ${\mathrm{\Pi }}_T={\mathbb{R}}^n\times (0,T)$ with a two-sided open neighbourhood $U\subset {\mathrm{\Pi }}_T$, so that for disjoint open sides $U^{\pm }$ ($C^k$ Euclidean maps and diffeomorphisms, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder). Fix the unit normal $\nu =({\nu }_t,{\nu }_x)$ on $\mathrm{\Gamma }$ oriented from $U^{-}$ toward $U^{+}$, with ${\nu }_t^2+\mathrm{\mid }{\nu }_x{\mathrm{\mid }}^2=1$.

A piecewise $C^1$ weak solution near $\mathrm{\Gamma }$ is a distributional weak solution $u$ of $u_t+{\mathrm{div}}_{x}f(u)=0$ on ${\mathrm{\Pi }}_T$ (Distributional weak solutions of the Cauchy problem) such that at each $\zeta \in \mathrm{\Gamma }$ there is a neighbourhood $V_{\zeta }\subset U$ and functions ${\tilde{u}}_{\zeta }^{\pm }\in C^1(V_{\zeta })$ representing $u$ on $V_{\zeta }\cap U^{\pm }$, respectively. The one-sided traces at $\zeta$ are $u^{\pm }(\zeta ):={\tilde{u}}_{\zeta }^{\pm }(\zeta )\mathrm{.}$ They are independent of the chosen local extensions, since continuous extensions agreeing almost everywhere on an open side agree throughout that side and at its interface points. Thus genuine traces are specified at every point of $\mathrm{\Gamma }\subset U$; arbitrary representative values on $\mathrm{\Gamma }$ do not affect the weak-solution class or these traces. Write $[u](\zeta )=u^{+}(\zeta )-u^{-}(\zeta ),[f](\zeta )=f(u^{+}(\zeta ))-f(u^{-}(\zeta ))\mathrm{.}$ A point is a shock point when $[u](\zeta )\mathrm{\ne }0$.

In one space dimension, this includes a $C^1$ graph $x=s(t)$, with sides $x<s(t)$ and $x>s(t)$; the definition is not restricted to that case, and the existence of the strong one-sided traces is part of the piecewise-smooth hypothesis, not a conclusion for arbitrary $L^{\mathrm{\infty }}$ weak solutions.