On vanishing viscosity approximation of conservation laws with discontinuous flux. (English) Zbl 1270.35305

Summary: We characterize the vanishing viscosity limit for multi-dimensional conservation laws of the form \[ u_t + \text{div} f(x, u) = 0, u|_{t=0} = u_0 \] in the domain \(\mathbb R^+ \times \mathbb R^N\). 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 \(f(\cdot, u)\) 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”, 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 \[ u^\varepsilon_t + \text{div} \left( f \left( x, u^\varepsilon \right) \right) = \varepsilon \Delta u^\varepsilon, u^\varepsilon|_{t=0} = u_0, \varepsilon \downarrow 0, \] of the conservation law. We show that, provided \(u^\varepsilon\) enjoys an \(\varepsilon\)-uniform \(L^\infty\) bound and the flux \(f(x, \cdot)\) 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.


35L65 Hyperbolic conservation laws
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