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

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.

References in corpus (1)

• Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth

Cited by in corpus (4)

• Optimal distributed control of a diffuse interface model of tumor growth
• Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
• An accurate front capturing scheme for tumor growth models with a free boundary limit
• The multicomponent diffuse-interface model and its application to water/air interfaces