n.\end{aligned}$$ By convention, the relation is rewritten as $$\begin{aligned}
\nabla q = \nabla \eta \nabla f,\end{aligned}$$ where $\nabla f$ denotes the matrix $(\partial_j f_i)_{i,j}$.

For $u\in\mathcal{V}$ where $\eta$ exists , the system is hyperbolic, and the matrix $\nabla f(u)$ is diagonalizable, with eigenvalues $$\begin{aligned}
\lambda_1(u)\leq \ldots \leq \lambda_n(u),\end{aligned}$$ called *characteristic speeds*.

We consider both bounded *classical* and bounded *weak* solutions to . A weak solution $u$ is bounded and measurable and satisfies in the sense of distributions. I.e., for every Lipschitz continuous test function $\Phi:\mathbb{R}\times[0,T)\to \mathbb{M}^{1\times n}$ with compact support, $$\begin{aligned}\label{u_solves_equation_integral_formulation_chitchat}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty} \Bigg[\partial_t\Phi u + \partial_x\Phi f(u) \Bigg]\,dxdt +\int\limits_{-\infty}^{\infty} \Phi(x,0)u^0(x)\,dx
\\
=-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Phi G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$

We only consider solutions $u$ which are entropic for the entropy $\eta$. That is, they satisfy the following entropy condition: $$\begin{aligned}
\label{entropy_condition_distributional_system_chitchat}
\partial_t \eta(u)+\partial_x q(u) \leq \nabla\eta(u)G(u(\cdot,t))(x),\end{aligned}$$ in the sense of distributions. I.e., for all positive, Lipschitz continuous test functions $\phi:\mathbb{R}\times[0,T)\to\mathbb{R}$ with compact support: $$\begin{aligned}\label{u_entropy_integral_formulation_chitchat}
\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\Bigg[\partial_t\phi\big(\eta(u(x,t))\big)+&\partial_x \phi \big(q(u(x,t))\big)\Bigg]\,dxdt+ \int\limits_{-\infty}^{\infty}\phi(x,0)\eta(u^0(x))\,dx\geq
\\
&-\int\limits_{0}^{T} \int\limits_{-\infty}^{\infty}\phi\nabla\eta(u(x,t))G(u(\cdot,t))(x)\,dxdt.
\end{aligned}$$

For $u_L,u_R\in\mathbb{R}^n$, the function $u:\mathbb{R}\times[0,\infty)\to\mathbb{R}^n$ defined by $$\begin{aligned}
\label{shock_solution_system}
u(x,t)\coloneqq
\begin{cases}
u_L &\mbox{ if } x<\sigma t ,\\
u_R &\mbox{ if } x>\sigma t
\end{cases}\end{aligned}$$ is a weak solution to if and only if $u_L,u_R$, and $\sigma$ satisfy the Rankine-Hugoniot 