Nonlocal-to-local convergence for a Cahn–Hilliard tumor growth model

Abstract

We consider a local Cahn–Hilliard-type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model toward strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall-type argument and a convergence result with rates for the nonlocal integral operator toward the Laplacian due to Abels and Hurm.