Local Well-Posedness of the Cahn-Hilliard-Biot System

TL;DR

This paper proves short-time well-posedness for a coupled diffuse interface model describing fluid flow in deformable porous media, integrating Biot's poroelasticity and Cahn-Hilliard equations with different visco-elastic assumptions.

Contribution

It introduces a novel mathematical analysis demonstrating well-posedness of a complex coupled system using advanced functional analytic techniques.

Findings

Established short-time existence and uniqueness of solutions.

Developed a fixed-point framework using maximal regularity theory.

Handled different visco-elastic assumptions with tailored mathematical approaches.

Abstract

We show short-time well-posedness of a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including phase-field dependent material properties, with the Cahn-Hilliard equation to model the evolution of the solid, where we further distinguish between the absence and presence of a visco-elastic term of Kelvin-Voigt type. While both problems will be reduced to a fixed-point equation that can be solved using maximal regularity theory along with a contraction argument, the first case relies on a semigroup approach over suitable Hilbert spaces, whereas treating the second case under minimal assumptions with respect to spatial regularity necessitates the application of Banach scales.