On vanishing viscosity approximation of conservation laws with discontinuous flux

Abstract

We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form ut + div f(x,u) = 0, u|t=0=u0 in the domain R+ × RN. The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities of f(·, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of RN. We define " GV V -entropy solutions" (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the L1 contraction principle for the GV V -entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation, of the conservation law. We show that, provided uε enjoys an ε-uniform L∞ bound and the flux f(x, ·) is non-degenerately nonlinear, vanishing viscosity approximations uε converge as ε ↓ 0 to the unique GVV-entropy solution of the conservation law with discontinuous flux.