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

Matthias Ebenbeck,Kei Fong Lam

doi:10.1515/anona-2020-0100

Matthias Ebenbeck, Kei Fong Lam

Open Access

https://doi.org/10.1515/anona-2020-0100

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Abstract

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 an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, 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 for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays 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 maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

Highlights

We study a phase eld model proposed recently in the context of tumour growth
In this paper we give a rst result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, speci cally the double obstacle potential and the logarithmic potential, which ensures that the phase eld variable stays in the physically relevant interval
Phase eld models [17, 45] have recently emerged as a new mathematical tool for tumour growth, which o er new advantages over classical models [11, 24] based on a free boundary description, such as the ability to capture metastasis and other morphological instabilities like ngering in a natural way

Summary

Introduction

Phase eld models [17, 45] have recently emerged as a new mathematical tool for tumour growth, which o er new advantages over classical models [11, 24] based on a free boundary description, such as the ability to capture metastasis and other morphological instabilities like ngering in a natural way. In addition, (C ) and (C ) hold, there exists a weak solution (φ, μ, σ, v, p) to the stationary CHB model with logarithmic potential ψlog in the sense of De nition 2.2 In both cases, ≤ σ ≤ a.e. in ΩT and the following regularity properties hold μ ∈ Hn, v ∈ W ,q, p ∈ W ,q, for q ∈ [ , ∞) if d = and q = if d =. In addition, (C )-(C ) hold, there exists a weak solution (φ, μ, σ, v, p) to the CHD model with logarithmic potential ψlog in the sense of De nition 2.3, which satis es, for a.e. t ∈ ( , T), the inequality (2.15) with η(φ) ≡ Proceeding as in the proof of Theorem 1, we can recover (2.7b), (2.7d) and (2.11) (resp. (2.12)) for the double obstacle (resp. logarithmic) case in the limit δ → , whereas recovery of (2.20a), (2.20b), the improved regularity p ∈ L ( , T; H ) and the boundary condition (2.19) follow from similar arguments outlined in [20, Sec. 4.2]

A Properties of approximation potentials

Weak solvability

Strong solvability