On vanishing viscosity approximation of conservation laws with discontinuous flux

We characterize the vanishing viscosity limit for multi-dimensionalconservation laws of the formuₜ +divf(x,u)=0, u|ₜ₌₀=u₀in the domain R⁺× RN. The flux f=f(x,u) is assumed locallyLipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities off(·,u) are contained in the union of a locally finite number of sufficiently smoothhypersurfaces of RN. We define 'GVV-entropy solutions'' (this formulation is aparticular case of the one of [3]); the definition readily implies the uniqueness and the L¹contraction principle for the GVV-entropy solutions. Our formulation is compatible with thestandard vanishing viscosity approximationu^εₜ +div(f(x,u^ε)) =ε Δ u^ε, ^ε|ₜ₌₀=u₀, ε↓ 0,of the conservation law. We show that, provided u^ε enjoys an ε-uniform L^∞ boundand the flux f(x,·) is non-degenerately nonlinear, vanishing viscosity approximationsu^ε converge as ε ↓ 0 to the unique GVV-entropy solution of theconservation law with discontinuous flux.