On vanishing viscosity approximation of conservation laws with discontinuous flux

Abstract

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 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.