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.


35L65 Hyperbolic conservation laws
35K65 Degenerate parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Miguel Escobedo, Juan Luis Vázquez, and Enrike Zuazua, A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J. 42 (1993), no. 4, 1413 – 1440. · Zbl 0791.35059
[2] Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in \?^{\?}, J. Funct. Anal. 100 (1991), no. 1, 119 – 161. · Zbl 0762.35011
[3] Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. · Zbl 0768.35059
[4] Tosio Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 135 – 148 (1973). · Zbl 0246.35025
[5] S. Kruzhkov, First-order quasilinear equations in several independent variables, Mat. USSR-Sb. 10 (1970), 217-243. · Zbl 0215.16203
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