Existence of weak solutions to across-diffusion Cahn-Hilliard type system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:286

Article

Ehrlacher, V¹  Marino, G.²  Pietschmann, J-F² 

[1] MATHERIALS Team Project, CERMICS, INRIA, Ecole Ponts, 6&8 Av Blaise Pascal, F-77455 Marne La Vallee, France

[2] Tech Univ Chemnitz, Fak Math, Reichenhainer Str 41, D-09126 Chemnitz, Germany

关键词: Cahn-Hilliard;    Cross-diffusion;    Weak solutions;    Global existence;    Degenerate Ginzburg-Landau;   

DOI  :  10.1016/j.jde.2021.02.025

来源: Elsevier

【 摘 要 】

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal gradient flow structure of the system and an extension of the boundedness by entropy method which involves a careful analysis of an auxiliary variational problem. This allows to obtain solutions to an approximate, time-discrete system. Letting the time step size go to zero, we recover the desired weak solution where, due to their low regularity, the Cahn-Hilliard terms require a special treatment. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free