We extend the Phase Transition model for traffic proposed in [
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Figure 1. The free phase $F$ and the congested phase $C$ resulting from (1) in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. In the $(\rho,\eta)$ plane, the curves $\eta= \check w \rho $, $\eta= \hat w \rho $ and the curve $\eta= \frac{V_{\max}}{\psi(\rho)}\rho $ that divides the two phases are represented. The densities $\sigma_-$ and $\sigma_+$ are given by the intersections between the previous curves. Similarly in the $(\rho, \rho v)$ plane, the curves $\rho v= \check w \psi(\rho)\rho $, $\rho v= \hat w \psi(\rho)\rho $ and the densities $\sigma_-$ and $\sigma_+$ are represented
Figure 2. The case $(\bar \rho,\bar \eta)\in C$. The set $\mathcal T_{inc}\left(\bar \rho, \bar \eta\right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{inc}^f \left(\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 3. The case $(\bar \rho,\bar \eta)\in F$. The set $\mathcal T_{inc}\left(\bar \rho, \bar \eta\right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{inc}^f \left(\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 4. The case $(\bar \rho,\bar \eta)\in F$. The set $\mathcal T_{out}\left(w, \bar \rho, \bar \eta \right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{out}^f \left(w,\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 5. The case $(\bar \rho,\bar \eta)\in C$. The set $\mathcal T_{out}\left(w, \bar \rho, \bar \eta \right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{out}^f \left(w,\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 6. The case $(\bar \rho,\bar \eta)\in F$ in an outgoing road for the approach in Subsection 4.1. The set of all the possible traces it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The corresponding set of flows is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 7. The case $(\bar \rho,\bar \eta)\in C$ in an outgoing road for the approach in Subsection 4.1. The set of all the possible traces it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The corresponding set of flows is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
Figure 8. The situation in the outgoing road related to the approach of Subsection 4.1. Left, in the $\left(\rho, \eta\right)$-plane, the states $\left(\rho_3^\ast, \eta_3^\ast\right)$ and $\left(\bar \rho_3, \bar \eta_3\right)$, connected through the middle state $\left(\rho^m, \eta^m\right)$. Right, in the $(t,x)$-plane, the waves generated by the Riemann problem. Note that the first wave has negative speed, so that it is not contained in the feasible region of the outgoing road
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The situation in the outgoing road related to the approach of Subsection 4.1. Left, in the
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