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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

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 [3]); 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.

[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

Paper:
Available as PDF (623 Kbytes).
Author(s):
Boris Andreianov,
Kenneth H. Karlsen,
Nils H. Risebro,
Publishing information:
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Submitted by:
; 2009-12-02.