Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

Carrillo, J. A.¹  Huang, Y.¹  Santos, M. C.²  Vazquez, J. L.³ 

[1] Univ London Imperial Coll Sci Technol & Med, Dept Math, London SW7 2AZ, England

[2] Univ Estadual Campinas, Dept Matemat IMECC, BR-13083859 Campinas, SP, Brazil

[3] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain

关键词: Porous medium equation;    Fractional operators;    Asymptotic behaviour;    Entropy dissipation;    Functional inequalities;   

DOI  :  10.1016/j.jde.2014.10.003

来源: Elsevier

【 摘 要 】

We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vazquez [1,2], where the pressure is obtained as a Riesz potential associated with the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states. Crown Copyright (C) 2014 Published by Elsevier Inc. All rights reserved.

【 授权许可】

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