On the well-posedness of the Cahn-Hilliard-Biot model and its applications to tumor growth

Abstract

We study the Cahn-Hilliard-Biot model with respect to its mathematical well-posedness. The system models flow through deformable porous media in which the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg-Landau energy functional, with additional influence from both viscoelastic and fluid effects. The flow-deformation coupling in the system is governed by Biot's theory. This results in a three-way coupled system that can be viewed as an extension of the Cahn-Larche equations by adding a fluid flowing through the medium. We distinguish the cases between a spatially dependent and a state-dependent Biot-Willis function. In the latter case, we consider a regularized system. In both cases, we use a Galerkin approximation to discretize the system and derive suitable energy estimates. Moreover, we apply compactness methods to pass to the limit in the discretized system. In the case of Vegard's law and homogeneous elasticity, we show that the weak solution depends continuously on the data and is unique. Lastly, we present some numerical simulations to highlight the features of the system as a tumor growth model.