Traveling Waves for Conservation Laws with Nonlocal Flux for Traffic Flow on Rough Roads

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.