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Entropy solutions for diffusion-convection equations with partial diffusivity. (English) Zbl 0803.35084

Summary: We consider the Cauchy problem for the following scalar conservation law with partial viscosity \[ u_ t = \Delta_ x u + \partial_ y \bigl( f(u) \bigr), \quad (x,y) \in \mathbb{R}^ N,\;t>0. \] The existence of solutions is proved by the vanishing viscosity method. By introducing a suitable entropy condition we prove uniqueness of solutions. This entropy condition is inspired by the entropy criterion introduced by Kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion.

MSC:

35L65 Hyperbolic conservation laws
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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