Preprint 2009-056

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

u_t+div f(x,u)=0, u|_t=0=u₀

in the domain R⁺×R^N. 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 R^N.

We define “G_VV-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 G_VV-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation

u^ε_t+div(f(x,u^ε)) = εΔu^ε, u^ε|_t=0 = u₀, ε↓0,

of the conservation law. We show that, provided u^ε enjoys an ε-uniform L^∞ bound and the flux f(x,·) is non-degenerately nonlinear, vanishing viscosity approximations u^ε converge as ε ↓ 0 to the unique G_VV-entropy solution of the conservation law with discontinuous flux.

[3] B. Andreianov, K.H. Karlsen, and N.H. Risebro. A theory of L¹-dissipative solvers for scalar conservation laws with discontinuous ﬂux. In preparation