In this talk, we address a scalar conservation law in one space dimension with discontinuous gradient-dependent flux, where the discontinuity is determined by the sign of the derivative u_x of the solution; namely, the flux is given by two different smooth functions $f(u)$ or $g(u)$, when $u_x$ is positive or negative, respectively. This problem is motivated by traffic flow modeling, under the assumption that drivers exhibit different behaviors in accelerating or decelerating mode.

A vanishing viscosity approximation identifies two different situations according to the mutual positions of the graphs of $f$ and $g$: a well-posed parabolic problem when $f(u)<g(u)$ for all $u$, or an ill-posed problem if $f(u)>g(u)$ for all $u$. We refer to these problems as the stable/unstable case respectively.

In the stable case, we prove that semigroup trajectories obtained as a limit of a suitable wave-front tracking algorithm coincide with the unique limits of vanishing viscosity approximations.

In the unstable case, examples show that infinitely many solutions can occur. For piecewise monotone initial data, we prove that a global solution of the Cauchy problem exists. The solution has a finite number of interfaces, where the flux switches between $f$ and $g$; such structure allows us to provide a uniqueness criterion based on minimizing the number of interfaces at $t=0+$ and all future times.

Joint work with Alberto Bressan and Wen Shen (Penn State University).