Conservation laws with discontinuous gradient-dependent flux: The stable case

This paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions [Formula: see text] or [Formula: see text], when the gradient [Formula: see text] of the solution is positive or negative, respectively. We study here the stable case where [Formula: see text] for all [Formula: see text], with [Formula: see text] 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 [Formula: see text]. In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations.