property if for each fixed $h$, the right and left limits $$\begin{aligned}
\lim_{y\to0^{+}}u(h(t)+y,t)\hspace{.7in}\mbox{and}\hspace{.7in}\lim_{y\to0^{-}}u(h(t)+y,t)\end{aligned}$$ exist for almost every $t$. In particular, a function $u\in L^\infty(\mathbb{R}\times[0,T))$ will have strong traces according to if $u$ has a representative which is in $BV_{\text{loc}}$. However, the strong trace property is weaker than $BV_{\text{loc}}$.

Technical Lemmas {#technical_lemmas}
================

Throughout this paper, we use the following definition for the relative flux $$\begin{aligned}
\label{Z_def}
f(a|b)\coloneqq f(a)-f(b)-\nabla f (b)(a-b),\end{aligned}$$ and the relative $\nabla\eta$: for $a,b\in\mathbb{M}^{n\times 1}$, $$\begin{aligned}
\nabla\eta(a|b)\coloneqq \nabla\eta(a)-\nabla\eta(b)-[a-b]^T\nabla^2\eta(b).\end{aligned}$$.

The following lemma from [@MR3537479] describes how the relative entropy obeys a sort of triangle inequality:

\[triangle\_inequality\_systems\_entropy\] For any $u,v,w\in\mathcal{V}$, we have $$\begin{aligned}
\eta(u|w)+\eta(w|v)=\eta(u|v)+(\nabla\eta(w)-\nabla\eta(v))\cdot(w-u),\end{aligned}$$ and $$\begin{aligned}
q(u;w)+q(w;v)=q(u;v)+(\nabla\eta(w)-\nabla\eta(v))\cdot(f(w)-f(u)).\end{aligned}$$

Thus, for any $\sigma\in\mathbb{R}$, $$\begin{aligned}\label{48_kang_vasseur_a_contraction}
q(u;v)-\sigma\eta(u|v)=&(q(u;w)-\sigma\eta(u;w))+(q(w;v)-\sigma\eta(w|v))
\\
&-(\nabla\eta(w)-\nabla\eta(v))\cdot(f(w)-f(u)-\sigma(w-u)).
\end{aligned}$$

The proof of follows immediately from the definition of $q(\hspace{.015in}\cdot\hspace{.015in};\hspace{.015in}\cdot\hspace{.015in})$ and $\eta(\hspace{.015in}\cdot\hspace{.015in}|\hspace{.015in}\cdot\hspace{.015in})$. In particular, see [@MR3519973 p. 360-1] for a simple proof.

\[a\_cond\_lemma\_itself\] Fix $B>0$. Then there exists a constant $C>0$ depending on $B$ such that the following holds:

If $u_L,u_R\in\mathcal{V}$ with $\abs{u_L},\abs{u_R}\leq B$, then whenever $\alpha,\theta\in(0,1)$ verify $$\begin{aligned}
\label{cond_a}
\alpha<\frac{\theta^2}{C},\end{aligned}$$ then $R_a\coloneqq\{u | \eta(u|u_L)\leq a\eta(u|u_R)\}\subset B_{\theta}(u_L)$ for all $0<a<\alpha$.

The set $R_a$ is compact.

The proof of is found in the proof of Lemma 4.3 in [@MR3519973]. We repeat the proof in for 