Global weak solutions and asymptotic limits of a Cahn--Hilliard--Darcy system modelling tumour growth

TL;DR

This paper proves the existence of global weak solutions for a complex tumour growth model involving Cahn--Hilliard--Darcy equations, with new regularity results and analysis of different model variants in 2D and 3D.

Contribution

It introduces novel regularity results for the velocity and pressure fields in a coupled tumour growth system and establishes global weak solutions in multiple dimensions and model variants.

Findings

Existence of global weak solutions in 2D and 3D.

New regularity results for velocity and pressure fields.

Global solutions for models with and without chemotaxis and active transport.

Abstract

We study the existence of weak solutions to a Cahn--Hilliard--Darcy system coupled with a convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy's law. The system of equations arises from a mixture model for tumour growth accounting for transport mechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, the existence of global weak solutions in two and three dimensions, along with new regularity results for the velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivatives have lower regularity compared to systems without Darcy flow, but in the two dimensional case we employ a new regularity result for the velocity to obtain better integrability and temporal regularity for the time derivatives. Then, we deduce the global existence of weak solutions for two variants of the model; one where the velocity is zero and another where the chemotaxis and active transport mechanisms are absent.