| Abstract: |
| In this talk, we will discuss a (PDE-ODE) Cauchy problem in the presence of a moving bottleneck and discontinuity in flux. In particular, we consider the following Cauchy problem:
\[(P):
\begin{cases}\partial_t \rho + \partial_x[f(\gamma,\rho)] = 0 ; \
\rho(0, x) = \rho_\circ(x); \
f(\gamma, \rho) - \dot y \rho \le F(y);
\ \dot y = w(y, \rho), \quad y(0) = y_\circ;
\end{cases}
\]
In a macroscopic presentation of traffic flow, the existence of moving bottlenecks, e.g. slow-moving vehicles, can significantly affect traffic behavior by influencing the road capacity and consequently dynamics of the flow and is mathematically modeled by imposing a capacity constraint.
The discontinuities in the flux function arise as a result of the spatial dependence of flux functions, for instance, the variable speed limits in different regions of the road.
The presence of such discontinuities introduces new types of waves which imply the need for defining a new Riemann solution. We prove the existence and uniqueness of the solution to the Cauchy problem (P) through a rigorous analysis of the interaction of these waves. |
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