to $(\mathcal{H}1)$.

    Recall also that by hypothesis $(\mathcal{H}1)$, $S_u^1$ is parameterized by arc length. Thus, $\abs{S_u^1(s)-u}\leq B$ for all $s\in[0,B]$. We can then use and to get, $$\begin{aligned}
    \label{shock_strength_comparable_s_systems1}
    (s-\rho)d_1\leq\abs{u-S_u^1(s)}^2\leq (s-\rho) d_2\end{aligned}$$ for all $u\in\mathcal{V}$ with $\abs{u}\leq B$ and for all $s\in[\rho,B]$. The constants $d_1,d_2>0$ depend only on $B$ and $\rho$. This says that $s-\rho$ is comparable to the shock strength $\abs{u-S_u^1(s)}^2$.

-   On the state space $\mathcal{V}$ where the strictly convex entropy $\eta$ is defined, the system is hyperbolic. Further, by virtue of $f\in C^2(\mathcal{V})$, the eigenvalues of $\nabla f (u)$ vary continuously on the state space $\mathcal{V}$. Further, if the eigenvalue $\lambda_1(u)$ ($\lambda_n(u)$) is simple for $u\in\mathcal{V}$ (such as when the system is strictly hyperbolic), the map $u\mapsto \lambda_1(u)$ ($u\mapsto \lambda_n(u)$) will be in $C^1(\mathcal{V})$ due to the implicit function theorem.

We study solutions $u$ to among the class of functions verifying a strong trace property (first introduced in [@Leger2011]):

\[strong\_trace\_definition\] Fix $T>0$. Let $u\colon\mathbb{R}\times[0,T)\to\mathbb{R}^n$ verify $u\in L^\infty(\mathbb{R}\times[0,T))$. We say $u$ has the *strong trace property* if for every fixed Lipschitz continuous map $h\colon [0,T)\to\mathbb{R}$, there exists $u_+,u_-\colon[0,T)\to\mathbb{R}^n$ such that $$\begin{aligned}
\lim_{n\to\infty}\int\limits_0^{t_0}\operatorname*{ess\,sup}_{y\in(0,\frac{1}{n})}\abs{u(h(t)+y,t)-u_+(t)}\,dt=\lim_{n\to\infty}\int\limits_0^{t_0}\operatorname*{ess\,sup}_{y\in(-\frac{1}{n},0)}\abs{u(h(t)+y,t)-u_-(t)}\,dt=0\end{aligned}$$ for all $t_0\in(0,T)$.

Note that for example a function $u\in L^\infty(\mathbb{R}\times[0,T))$ will satisfy the strong trace 