Local Well-Posedness of the Cahn-Hilliard-Biot System
Helmut Abels, Jonas Haselböck
TL;DR
The paper addresses local well-posedness of a diffuse-interface Cahn–Hilliard–Biot model describing two-phase flow through a deformable porous medium, incorporating both elastic and Kelvin–Voigt visco-elastic solid responses. The authors reformulate the coupled system into a fixed-point problem by eliminating auxiliary variables and proving maximal $L^p$-regularity for the linearized operators; the analysis hinges on analytic semigroup theory and Banach scales to handle the sum of high-order diffusion-like operators. They prove two main theorems: a local-in-time existence and uniqueness result for the purely elastic case, and a local-in-time result for the visco-elastic case under minimal spatial regularity assumptions, with a detailed contraction argument supported by Lipschitz bounds on nonlinear terms. The results provide a rigorous mathematical foundation for CH–Biot models with potential applications to tumor growth and porous-media flows, and open paths toward studying global behavior via advanced inequalities and energy methods.
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.