# 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}=u_{0}

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 L^{1} 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} = u_{0}, ε↓0,

^{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 L^{1}-dissipative solvers
for scalar conservation laws with discontinuous ﬂux. In preparation