## 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.

### MSC:

 35L65 Hyperbolic conservation laws
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