On a relaxed Cahn-Hilliard tumour growth model with single-well potential and degenerate mobility
Cecilia Cavaterra, Matteo Fornoni, Maurizio Grasselli, Benoît Perthame
TL;DR
The paper addresses a degenerate Cahn–Hilliard tumour-growth model with a singular single-well potential, degenerate mobility, nutrient diffusion, and chemotaxis. It introduces a relaxed, δ-regularized formulation by adding an elliptic term to the chemical potential, and proves existence of weak solutions for δ>0 using a Faedo–Galerkin approximation with regularized data. It then passes to the limit δ→0, establishing convergence to a weak solution of the original nonlocal-to-local system, preserving an energy–entropy structure that yields uniform bounds. The work combines nonlocal–to–local analysis for the chemical potential and compactness arguments to handle degeneracies, providing a rigorous foundation for the relaxation approach and potential numerical schemes. Overall, the results validate the δ-relaxation as a consistent approximation for modelling tumour growth with nutrient interaction and chemotaxis within a phase-field framework.
Abstract
We consider a phase-field system modelling solid tumour growth. This system consists of a Cahn-Hilliard equation coupled with a nutrient equation. The former is characterised by a degenerate mobility and a singular potential. Both equations are subject to suitable reaction terms which model proliferation and nutrient consumption. Chemotactic effects are also taken into account. Adding an elliptic regularisation, depending on a relaxation parameter $δ>0$, in the equation for the chemical potential, we prove the existence of a weak solution to an initial and boundary value problem for the relaxed system. Then, we let $δ$ go to zero, and we recover the existence of a weak solution to the original system.