Journal of Nonlinear Science

, Volume 27, Issue 3, pp 847–872 | Cite as

A Rigorous Sharp Interface Limit of a Diffuse Interface Model Related to Tumor Growth

Article

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 \(\varepsilon \), 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 Hawkins-Daruud et al. (Int J Numer Math Biomed Eng 28:3–24, 2011) (cf. also Hilhorst et al. Math Models Methods Appl Sci 25:1011–1043, 2015). 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.

Keywords

Sharp interface limit Gamma convergence Gradient-flow Diffuse interface models Cahn–Hilliard equation Reaction–diffusion equation Nonlocal operators Tumor growth 

Mathematics Subject Classification

49J40 82B24 35K46 35K57 35R11 92B05 

Notes

Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and their precious suggestions. The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) is gratefully acknowledged by the authors. The present paper also benefits from the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM) and the IMATI-C.N.R. Pavia.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.Erwin Schrödinger Institute, University of ViennaViennaAustria

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