Conservation Laws with Discontinuous Gradient-Dependent Flux: the Stable Case

The paper is concerned with a scalar conservation law with discontinuous gradient- dependent flux. Namely, the flux is described by two different functions f(u) or g(u), when the gradient ux of the solution is positive or negative, respectively. We study here the stable case where f(u) < g(u) for all u ∈ R, with f,g smooth but possibly not convex. A front tracking algorithm is introduced, proving that piecewise constant approximations converge to the trajectories of a contractive semigroup on L1(R). In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations.