In , we construct the shift with the additional entropy dissipation. Finally, in we prove the main theorem by using the additional entropy dissipation from the shift to translate in $x$ the piecewise-smooth solution artificially.

Hypotheses on the system {#hypotheses_on_system}
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We will consider the following structural hypotheses $(\mathcal{H})$, $(\mathcal{H})^*$ on the system ,

regarding the 1-shock and n-shock curves (they are closely related to hypotheses in [@Leger2011] and [@MR3519973]). For a fixed piecewise smooth solution $\bar{u}$ (as in the context of the main theorem ):

-   $(\mathcal{H}1)$: (Family of 1-shocks verifying the Liu condition) There exists $r_0>0$ such that for all $u_L\in\{\bar{u}(s(t)-,t) | t\in[0,T)\}\coloneqq I_{-}$, and for all $u\in B_{r_0}(u_L)$, there is a 1-shock curve (issuing from $u$) $S_u^1\colon[0,s_u)\to \mathcal{V}$ (possibly $s_u=\infty$) parameterized by arc length. Moreover, $S_u^1(0)=u$ and the Rankine-Hugoniot jump condition holds: $$\begin{aligned}
    f(S_u^1(s))-f(u)=\sigma^1_u(s)(S_u^1(s)-u),\end{aligned}$$ where $\sigma^1_u(s)$ is the velocity function. The map $u\mapsto s_u$ is Lipschitz on $\mathcal{V}$. Further, the maps $(s,u)\mapsto S_u^1(s)$ and $(s,u)\mapsto \sigma^1_u(s)$ 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) } \frac{\mbox{d}}{\mbox{d}s} \sigma^1_u(s) <0,\hspace{.2in} \sigma^1_u(0)=\lambda_1(u), 
    \\&\mbox{(b) (shock ``strengthens'' with $s$) }  \frac{\mbox{d}}{\mbox{d}s}\eta(u|S_u^1(s))>0, \hspace{.2in}\mbox{for all } s>0,
    \\&\mbox{(c) (the shock curve cannot wrap tightly 