\frac{\mbox{d}}{\mbox{d}s} \sigma^n_u(s) >0,\hspace{.2in} \sigma^n_u(0)=\lambda_n(u), 
    \\&\mbox{(b) (shock ``strengthens'' with $s$) }  \frac{\mbox{d}}{\mbox{d}s}\eta(u|S_u^n(s))>0, \hspace{.2in}\mbox{for all } s>0,
    \\&\mbox{(c) (the shock curve cannot wrap tightly around itself)}
    \\&\hspace{.5in}\mbox{For all $R>0$, there exists $\tilde{S}>0$ such that}  
    \\
    &\hspace{.8in}\Big\{S^n_{u}(s) \Big| s\in[0.s_u), \abs{u}\leq R \mbox{ and } \abs{S^n_u(s)}\leq R\Big\} \subseteq \Big\{S^n_u(s) \Big| \abs{u} \leq R \mbox{ and } s\leq \tilde{S}\Big\}.\end{aligned}$$

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

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

See [@Leger2011; @MR3519973] for remarks on these hypotheses. We include them here for completeness. In particular,

-   Note that the system verifies the hypotheses $(\mathcal{H}1)$-$(\mathcal{H}3)$ on the 1-shock family if and only if the system $$\begin{aligned}
    \begin{cases}
    \partial_t u - \partial_x f(u)=G(u(\cdot,t))(x),\mbox{ for } x\in\mathbb{R},\mbox{ } t>0,\\
    u(x,0)=u^0(x) \mbox{ for } x\in\mathbb{R}.
    \end{cases}\end{aligned}$$ verifies the properties $(\mathcal{H}1)^*$-$(\mathcal{H}3)^*$ for the n-shock family. It is in this way that $(\mathcal{H}1)$-$(\mathcal{H}3)$ are dual to $(\mathcal{H}1)^*$-$(\mathcal{H}3)^*$.

-   On top of the Liu entropy condition (Property (a) in $(\mathcal{H}1)$), we also assume 