On vanishing viscosity approximation of conservation laws with discontinuous flux
Boris Andreianov, Kenneth H. Karlsen and Nils H. Risebro
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 “GVV-entropy solutions” (this formulation is a particular case of the one of ); the definition readily implies the uniqueness and the L1 contraction principle for the GVV-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation
uεt+div(f(x,uε)) = εΔuε, uε|t=0 = u0, ε↓0,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.
 B. Andreianov, K.H. Karlsen, and N.H. Risebro. A theory of L1-dissipative solvers for scalar conservation laws with discontinuous ﬂux. In preparation