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Nonlocal-to-Local Convergence for a Cahn-Hilliard Tumor Growth Model

Christoph Hurm, Maximilian Moser

TL;DR

For sufficiently smooth bounded domains in three dimensions, 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 is proved.

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 towards 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 towards the Laplacian due to Abels, Hurm arXiv:2307.02264.

Nonlocal-to-Local Convergence for a Cahn-Hilliard Tumor Growth Model

TL;DR

For sufficiently smooth bounded domains in three dimensions, 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 is proved.

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 towards 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 towards the Laplacian due to Abels, Hurm arXiv:2307.02264.
Paper Structure (7 sections, 5 theorems, 30 equations)

This paper contains 7 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Assumptions
  5. Inequalities
  6. Existence and Uniqueness Results
  7. Convergence Result

Key Result

Lemma 2.2

For every $\delta>0$, there exist constants $C_\delta>0$ and $\varepsilon_\delta>0$ such that for every sequence $(f_\varepsilon)_{\varepsilon>0}\subset L^2(\Omega)$ there holds for all $\varepsilon_1,\varepsilon_2\in(0,\varepsilon_\delta)$.

Theorems & Definitions (12)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4: Well-posedness of the local model
  • proof
  • Theorem 2.5: Well-posedness of the non-local model
  • proof
  • Theorem 3.1
  • ...and 2 more