characteristic velocity $\lambda^{(2)}$ equals the vehicle velocity and its associated characteristic field is linearly degenerate (i.e., its associated waves are contact discontinuities that are transported with the flow).

\[lem:lambdas\_decreasing\] Both characteristic velocities are strictly decreasing w.r.t. $\rho$, i.e., ${\frac{\partial\lambda^{(1)}}{\partial\rho}} < 0$ and ${\frac{\partial\lambda^{(2)}}{\partial\rho}} < 0$.

We have that $\lambda^{(1)} = V+\rho{\frac{\partialV}{\partial\rho}} = {\frac{\partialQ}{\partial\rho}}$. Since $Q$ is assumed concave w.r.t. $\rho$, it follows that ${\frac{\partial\lambda^{(1)}}{\partial\rho}} = \frac{\partial^2 Q}{\partial \rho^2} < 0$. Moreover, ${\frac{\partial\lambda^{(2)}}{\partial\rho}} = {\frac{\partialV}{\partial\rho}} < 0$ by Lemma \[lem:concave\_Q\_decreasing\_U\].

We continue with the discussion of the characteristic fields. The scalar function $I^{(1)} = q/\rho = w$ satisfies $\nabla I^{(1)}\cdot {\mathbf{\gamma}}^{(1)} = 0$, and it is a Riemann invariant to $\lambda^{(1)}$. Hence, across waves of the first family, the empty road velocity $w$ is constant. The field associated with the second eigenvalue $\lambda^{(2)}$ is linearly degenerate. It is given by $I^{(1)} = \lambda^{(2)} = u$ and across waves of the second family, the velocity $u$ is constant. The solution to a Riemann problem, i.e., the Cauchy problem to system on the real line with discontinuous piecewise constant initial data ${\mathbf{U}}(x,0) = (1-H(x)){\mathbf{U}}_\text{L}+H(x){\mathbf{U}}_\text{R}$, where $H$ is the Heaviside function, generalizes naturally from the ARZ model as well. In general, the solution is 