around itself)}
    \\&\hspace{.5in}\mbox{For all $R>0$, there exists $\tilde{S}>0$ such that}  
    \\
    &\hspace{.8in}\Big\{S^1_{u}(s) \Big| s\in[0.s_u), \abs{u}\leq R \mbox{ and } \abs{S^1_u(s)}\leq R\Big\} \subseteq \Big\{S^1_u(s) \Big| \abs{u} \leq R \mbox{ and } s\leq \tilde{S}\Big\}.\end{aligned}$$

-   $(\mathcal{H}2)$: If $(u_L,u_R)$ is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$, then $\sigma> \lambda_1(u_R)$.

-   $(\mathcal{H}3)$: If $(u_L,u_R)$ (with $u_L\in B_{r_0}(\tilde{u}_L)$, for $\tilde{u}_L\in I_{-}$) is an entropic Rankine-Hugoniot discontinuity with shock speed $\sigma$ verifying $$\begin{aligned}
    \sigma\leq\lambda_1(u_L),\end{aligned}$$ then $u_R$ is in the image of $S_{u_L}^1$. In other words, there exists $s_{u_R}\in[0,s_{u_L})$ such that $S_{u_L}^1(s_{u_R})=u_R$ (and by implication, $\sigma=\sigma^1_{u_L}(s_{u_R})$).

Similarly, we will consider the following structural hypotheses $(\mathcal{H})^*$ on the system , regarding the n-shock curves:

-   $(\mathcal{H}1)^*$: (Family of n-shocks verifying the Liu condition) There exists $r_0>0$ such that for all $u_R\in\{\bar{u}(s(t)+,t) | t\in[0,T)\}\coloneqq I_{+}$, and for all $u\in B_{r_0}(u_R)$, there is an n-shock curve (issuing from $u$) $S_u^n\colon[0,s_u)\to \mathcal{V}$ (possibly $s_u=\infty$) parameterized by arc length. Moreover, $S_u^n(0)=u$ and the Rankine-Hugoniot jump condition holds: $$\begin{aligned}
    f(S_u^n(s))-f(u)=\sigma^n_u(s)(S_u^n(s)-u),\end{aligned}$$ where $\sigma^n_u(s)$ is the velocity function. The map $u\mapsto s_u$ is Lipschitz on $\mathcal{V}$. Further, the maps $(s,u)\mapsto S_u^n(s)$ and $(s,u)\mapsto \sigma^n_u(s)$ are both $C^1$ on $\{(s,u)|s\in [0,s_u), u\in\mathcal{V}\}$, and the following conditions are satisfied: $$\begin{aligned}
    &\mbox{(a) (Liu entropy condition) } 