CONSERVATION LAWS WITH REGULATED FLUXES

Scalar conservation laws ∂tu + ∂x f (t, x, u) = 0 where the flux f is discontinuous w.r.t. the time and space variables t, x arise in many applications, related to physical models in rough media. Typical examples include traffic flow with variable road conditions and polymer flooding in porous media. An extensive body of recent literature has dealt with fluxes that are discontinuous along a finite number of curves in the t-x plane. Here we are interested in the existence and uniqueness of solutions obtained via vanishing viscosity approximations i.e. solutions to ∂t u + ∂x f (t, x, u) = ε∂xx u when ε → 0+ , for more general discontinuous fluxes. We first give a definition of regulated functions in two variables. After recalling some results about parabolic equations with discontinuous coefficients, we show how the knowledge of the existence and uniqueness of the vanishing viscosity limit for fluxes with a single discontinuity at x = 0 can be used as a building block to prove the existence and uniqueness of the vanishing viscosity limit for regulated fluxes.