A rigorous sharp interface limit of a diffuse interface model related to tumor growth

TL;DR

This paper rigorously derives the sharp interface limit of a diffuse interface tumor growth model as the interface thickness parameter approaches zero, connecting the model to classical interface conditions.

Contribution

It provides a rigorous mathematical analysis of the limit, combining gradient-flow and gamma convergence techniques, to relate the diffuse model to sharp interface conditions.

Findings

Established the interface condition involving curvature and velocity.

Connected diffuse interface model to classical sharp interface models.

Validated the limit process through mathematical rigor.

Abstract

In this paper we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter $\epsilon$, representing the interface thickness between the tumorous and non tumorous cells, tends to zero. More in particular, we analyze here a gradient-flow type model arising from a modification of the recently introduced model for tumor growth dynamics in [A. Hawkins-Daruud, K. G. van der Zee, J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3--24.] (cf. also [D. Hilhorst, J. Kampmann, T. N. Nguyen, K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci. 25 (2015), 1011--1043.]). Exploiting the techniques related to both gradient-flows and gamma convergence, we recover a condition on the interface $\Gamma$ relating the chemical and double-well potentials, the mean curvature, and the normal velocity.