Well-posedness analysis of the Cahn-Hilliard-Biot model

TL;DR

This paper establishes the existence and uniqueness of weak solutions for the complex, nonlinear, coupled Cahn-Hilliard-Biot PDE model, leveraging its gradient flow structure and discretization techniques.

Contribution

It provides the first rigorous well-posedness analysis of the coupled Cahn-Hilliard-Biot model, including existence, uniqueness, and continuous dependence results.

Findings

Existence of weak solutions proved using discretization and compactness arguments.

Uniqueness established under certain parameter conditions.

Solutions depend continuously on initial data and parameters.

Abstract

We investigate the well-posedness of the recently proposed Cahn-Hilliard-Biot model. The model is a three-way coupled PDE of elliptic-parabolic nature, with several nonlinearities and the fourth order term known to the Cahn-Hilliard system. We show existence of weak solutions to the variational form of the equations and uniqueness under certain conditions of the material parameters and secondary consolidation, adding regularizing effects. Existence is shown by discretizing in space and applying ODE-theory (the Peano-Cauchy theorem) to prove existence of the discrete system, followed by compactness arguments to retain solutions of the continuous system. In addition, the continuous dependence of solutions on the data is established, in particular implying uniqueness. Both results build strongly on the inherent gradient flow structure of the model.