is strictly concave. This condition implies (see Lemma \[lem:concave\_Q\_decreasing\_U\]) in particular that ${\frac{\partialV}{\partial\rho}}(\rho,w) < 0$, i.e., each velocity curve $u_w(\rho) = V(\rho,w)$ is strictly decreasing w.r.t. the density.

-   ${\frac{\partialV}{\partialw}}(\rho,w) > 0$, i.e., a faster empty road velocity results in a faster velocity for all possible densities.

-   $V(\rho,0) = 0$, i.e., for $w = 0$, the concavity of $Q$ and the slope of $V$ hold with an equality sign.

\[lem:concave\_Q\_decreasing\_U\] Consider a $C^2$ function $U(\rho)$, and let $Q(\rho) = \rho U(\rho)$. If $Q''(\rho)<0$ everywhere, then $U'(\rho)<0$ everywhere.

The function $a(\rho) = Q(\rho)-Q'(\rho)\rho$ satisfies: (i) $a(0) = 0$, and (ii) $a'(\rho) = -Q''(\rho)\rho > 0$ everywhere. Hence, $U'(\rho) = \frac{-a(\rho)}{\rho^2} < 0$ everywhere.

In order to define an inhomogeneous GARZ model, an equilibrium velocity curve must be specified. We assume that it is a member of the family of velocity curves defined by $V$, i.e. $$U_\text{eq}(\rho) = V(\rho,w_\text{eq})\;,$$ for some equilibrium empty road velocity $w_\text{eq}$. We choose to generalize to the GARZ case as follows: $$\label{eq:GARZ_model_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= \tfrac{1}{\tau}{\left(U_\text{eq}(\rho)-u\right)}\;, \\
\text{where~}u &= V(\rho,w)\;.
\end{split}$$ Note that it would alternatively be conceivable to propose a relaxation in of the form $$\label{eq:alternative_relaxation}
w_t+uw_x = \tfrac{1}{\tau}{\left(w_\text{eq}-w\right)}\;.$$ While 