A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation

Marie Henry; Danielle Hilhorst; Masayasu Mimura

doi:10.3934/dcdss.2011.4.125

Article Contents

2011, Volume 4, Issue 1: 125-154. Doi: 10.3934/dcdss.2011.4.125

This issue Previous Article Heteroclinic connections for multidimensional bistable reaction-diffusion equations Next Article Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence

A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation

1.
CMI, Université de Provence, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13
2.
CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11, F-91405 Orsay Cedex
3.
Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1 Higashi Mita, Tama-ku, Kawasaki, 214-8571

Received: February 28, 2009

Revised: November 30, 2009

Published: January 2011

Abstract Related Papers Cited by

Abstract

Motivated by the motion of an alcohol droplet, we derive a simplified phenomenological free boundary model which consists of an area preserving mean curvature flow coupled with a bulk equation. Our aim is to introduce a nonlocal reaction-diffusion system with a small parameter $\e$ which converges to the original model as $\e$ tends to zero. This approximation enables us to overcome the technical difficulty of the free boundary problem arising in the original model.

Keywords:

reaction-diffusion system of conservative type,
singular limits,
non local motion by mean curvature,
asymptotic expansions.

Mathematics Subject Classification: 35K57, 35B25, 53C44, 35C20.

Citation:

Related Papers

Cited by

References

[1]	N. D. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal, 128 (1994), 165-205.doi: doi:10.1007/BF00375025.
[2]	L. Bronsard and B. Stoth, Volume preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.doi: doi:10.1137/S0036141094279279.
[3]	X. Chen, Spectrums for the Allen-Cahn, Cahn-Hilliard and phase field equations for generic interface, Comm. P.D.E., 19 (1994), 1371-1395.doi: doi:10.1080/03605309408821057.
[4]	X. Chen and G. Caginalp, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445.doi: doi:10.1017/S0956792598003520.
[5]	X.-F. Chen, S.-I. Ei and M. Mimura, Self-motion of camphor discs model and analysis, to appear in Networks and Heterogeneous Media 4, 1 (2009), 1-18.
[6]	X. Chen, D. Hilhorst and E. Logak, Mass conserved Allen-Cahn equation and volume preserving mean curvature flow, to appear, (2010).
[7]	C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7 (1997), 467-490.
[8]	J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.doi: doi:10.1090/S0002-9939-98-04727-3.
[9]	Y. Hayashima, M. Nagayama and S. Nakata, A camphor grain oscillates while breaking symmetry, in J. Phys. Chem. B, 105 (2001), 5353-5357.doi: doi:10.1021/jp004505n.
[10]	G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48.doi: doi:10.1515/crll.1987.382.35.
[11]	O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, Providence, R. I., 1967.
[12]	K. Nagai, Spontaneous irregular motion of an alcohol droplet, RIMS Kokyuroku B, 3 (2007), 139-147.
[13]	K. Nagai, Y. Sumino, H. Kitahata and K. Yoshikawa, Model selection in the spontaneous motion of an alcohol droplet, Phys. Rev. E., 71 (2005), 065301.doi: doi:10.1103/PhysRevE.71.065301.
[14]	K. Nagai, H. Sumino, H. Kitahata and K. Yoshikawa, Change in the mode of spontaneous motion of an alcohol droplet caused by a temperature change, Prog. Theor. Phys., 161 (2006), 286-289.doi: doi:10.1143/PTPS.161.286.
[15]	J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. of Appl. Math., 48 (1992), 249-264.doi: doi:10.1093/imamat/48.3.249.