Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system

Virginie Ehrlacher (1) , Greta Marino (2) , Jan-Frederik Pietschmann (2)

1 MATHERIALS - MATHematics for MatERIALS
2 Chemnitz University of Technology / Technische Universität Chemnitz

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.

Mots clés: Cahn-Hilliard; Cross-diffusion; Weak solutions; Global existence; Degenerate Ginzburg-Landau

Domaines: Equations aux dérivées partielles [math.AP]

Virginie Ehrlacher, Greta Marino, Jan-Frederik Pietschmann. Existence of weak solutions to a cross-diffusion Cahn-Hilliard type system. Journal of Differential Equations, 2021, 286, pp.578-623. ⟨hal-02888479⟩