\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}$$

For a fixed $T>0$ (including possibly $T=\infty$), the *unknown* is $u\colon\mathbb{R}\times[0,T)\to \mathbb{M}^{n\times 1}$. The function $u^0\colon\mathbb{R}\to\mathbb{M}^{n\times 1}$ is in $L^\infty(\mathbb{R})$ and is the *initial data*. The function $f\colon\mathbb{M}^{n\times 1}\to\mathbb{M}^{n\times 1}$ is the *flux function* for the system. The *source term* $G\colon (L^2(\mathbb{R}))^n\to (L^2(\mathbb{R}))^n$ is translation invariant. We also ask that $G$ be Lipschitz continuous from $(L^2(I))^n\to (L^2(I))^n$ for every interval $I\subseteq\mathbb{R}$, with a Lipschitz constant uniform in $I$. In other words, there exists $C_G>0$ such that $$\begin{aligned}
\label{G_acts_like}
\norm{G(g_1)-G(g_2)}_{L^2(I)}\leq C_G \norm{g_1-g_2}_{L^2(I)},\end{aligned}$$ for every $g_1,g_2\in(L^2(\mathbb{R}))^n$ and for every interval $I\subseteq\mathbb{R}$. Furthermore, we require that $G$ is bounded on $(L^\infty(\mathbb{R}))^n$: $$\begin{aligned}
\label{G_acts_like_2}
\norm{G(g)}_{L^\infty(\mathbb{R})}\leq C_G \norm{g}_{L^\infty(\mathbb{R})},\end{aligned}$$ for every $g\in(L^\infty(\mathbb{R}))^n$.

We assume the system is endowed with a strictly convex entropy $\eta$ and associated entropy flux $q$. Note the system will be hyperbolic on the state space where $\eta$ exists. We assume the functions $f, \eta$, and $q$ are defined on an open convex state space $\mathcal{V}\subset\mathbb{R}^n$. We assume $f,q\in C^2(\mathcal{V})$ and $\eta \in C^3(\mathcal{V})$. By assumption, the entropy $\eta$ and its associated entropy flux $q$ verify the following compatibility relation: $$\begin{aligned}
\label{compatibility_relation_eta_system}
\partial_j q =\sum_{i=1}^n \partial_i\eta\partial_j f_i,\hspace{.25in} 1\leq j \leq 