A macroscopic traffic model with phase transitions and local point constraints on the flow

Figure 1.  Geometrical meaning of the notations used through the paper. In particular, $\Omega_{\rm f} = \Omega_{\rm f}^- \cup \Omega_{\rm f}^+$ and $\Omega_{\rm c}$ are the free-flow and congested domains, respectively; $V_{\rm f}^+$ and $V_{\rm f}^-$ are the maximal and minimal speeds in the free-flow phase, respectively, and $V_{\rm c}$ is the maximal speed in the congested phase

Figure 2.  Geometrical meaning of the cases (T₁¹a) and (T₁¹b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 3.  Geometrical meaning of the cases (T₁²a) and (T₁²b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 4.  Geometrical meaning of the cases (T₂¹a), (T₂¹b) and (T₂²a). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 5.  $(\rho_1, v_1) \doteq \mathcal{R}_1^{\rm c}[u_\ell,u_r]$ and $(\rho_2, v_2) \doteq \mathcal{R}_2^{\rm c}[u_\ell,u_r]$ in the case considered in Example 1.

Figure 6.  The invariant domains described in Proposition 6 and Proposition 9

Figure 7.  $u_1 \doteq \mathcal{R}_1^{\rm c}[u_0,u_0]$ and $u_2 \doteq \mathcal{R}_2^{\rm c}[u_0,u_0]$ in the case considered in Example 2. Above $\hat{u}_1,\check{u}_1$ are given by (T₁¹b) and $\hat{u}_2,\check{u}_2$ by (T₂¹b); we let $w_0 = W(u_0)$, $\check{v}_i = V(\check{u}_i)$, $\hat{w}_2 = W(\hat{u}_2)$, $\check{w}_i = W(\check{u}_i)$

Figure 8.  Notations used in Section 5

Figure 9.  The solutions constructed in Subsection 5.1 on the left and in Subsection 5.2 on the right represented in the $(x,t)$-plane. The red thick curves are phase transitions. In particular, those along $x=0$ are stationary undercompressive phase transitions

Figure 10.  Quantitative representation of density, on the left, and velocity, on the right, corresponding to the solutions constructed in Subsection 5.1 and Subsection 5.2. Recall that the two solutions coincide up to the interaction $i_5$

Figure 11.  Quantitative representation of density, on the left, and velocity, on the right, corresponding to the solution constructed in Subsection 5.2