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September  2010, 5(3): 617-633. doi: 10.3934/nhm.2010.5.617

On vanishing viscosity approximation of conservation laws with discontinuous flux

1. 

Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

2. 

Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway, Norway

Received  January 2010 Revised  April 2010 Published  July 2010

We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form

$ u_t + $div$ \mathfrak{f}(x,u)=0, \quad u|_{t=0}=u_0 $

in the domain $\mathbb R^+\times\mathbb R^N$. The flux $\mathfrak{f}=\mathfrak{f}(x,u)$ is assumed locally Lipschitz continuous in the unknown $u$ and piecewise constant in the space variable $x$; the discontinuities of $\mathfrak{f}(\cdot,u)$ are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of $\mathbb R^N$. We define "$\mathcal G_{VV}$-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 $\mathcal G_{VV}$-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation

$ u^\varepsilon_t + $div$ (\mathfrak{f}(x,u^\varepsilon)) =\varepsilon \Delta u^\varepsilon, \quad u^\varepsilon|_{t=0}=u_0, \quad \varepsilon\downarrow 0, $

of the conservation law. We show that, provided $u^\varepsilon$ enjoys an $\varepsilon$-uniform $L^\infty$ bound and the flux $\mathfrak{f}(x,\cdot)$ is non-degenerately nonlinear, vanishing viscosity approximations $u^\varepsilon$ converge as $\varepsilon \downarrow 0$ to the unique $\mathcal G_{VV}$-entropy solution of the conservation law with discontinuous flux.

Citation: Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
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