Weak and stationary solutions to a Cahn-Hilliard-Brinkman model with singular potentials and source terms

TL;DR

This paper establishes the existence of weak and stationary solutions for a complex tumor growth model coupling Cahn-Hilliard-Brinkman equations with singular potentials and source terms, advancing mathematical understanding of such systems.

Contribution

It provides the first proof of weak and stationary solutions for the CHB model with singular potentials, addressing challenges from solution-dependent source terms and boundary conditions.

Findings

Existence of weak solutions for the CHB model with singular potentials.

Existence of stationary solutions under certain conditions.

Extension of results to a Darcy variant model.

Abstract

We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn-Hilliard-Brinkman (CHB) system with a elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenodial, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures the phase field variable stay in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which may be of independent interests. As a consequence, included in our analysis is a weak existence result for a Darcy variant, and our work serve to generalise recent results on weak and stationary solutions to the Cahn-Hilliard inpainting model with singular potentials.