February  2011, 4(1): 125-154. doi: 10.3934/dcdss.2011.4.125

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, France

2. 

CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11, F-91405 Orsay Cedex, France

3. 

Institute for Advanced Study of Mathematical Sciences, Meiji University, 1-1 Higashi Mita, Tama-ku, Kawasaki, 214-8571, Japan

Received  March 2009 Revised  December 2009 Published  October 2010

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.
Citation: Marie Henry, Danielle Hilhorst, Masayasu Mimura. A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 125-154. doi: 10.3934/dcdss.2011.4.125
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.  doi: doi:10.1007/BF00375025.  Google Scholar

[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.  doi: doi:10.1137/S0036141094279279.  Google Scholar

[3]

X. Chen, Spectrums for the Allen-Cahn, Cahn-Hilliard and phase field equations for generic interface,, Comm. P.D.E., 19 (1994), 1371.  doi: doi:10.1080/03605309408821057.  Google Scholar

[4]

X. Chen and G. Caginalp, Convergence of the phase field model to its sharp interface limits,, European J. Appl. Math., 9 (1998), 417.  doi: doi:10.1017/S0956792598003520.  Google Scholar

[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.   Google Scholar

[6]

X. Chen, D. Hilhorst and E. Logak, Mass conserved Allen-Cahn equation and volume preserving mean curvature flow,, to appear, (2010).   Google Scholar

[7]

C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws,, Adv. Math. Sci. Appl., 7 (1997), 467.   Google Scholar

[8]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres,, Proc. Amer. Math. Soc., 126 (1998), 2789.  doi: doi:10.1090/S0002-9939-98-04727-3.  Google Scholar

[9]

Y. Hayashima, M. Nagayama and S. Nakata, A camphor grain oscillates while breaking symmetry,, in J. Phys. Chem. B, 105 (2001), 5353.  doi: doi:10.1021/jp004505n.  Google Scholar

[10]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35.  doi: doi:10.1515/crll.1987.382.35.  Google Scholar

[11]

O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", American Mathematical Society, (1967).   Google Scholar

[12]

K. Nagai, Spontaneous irregular motion of an alcohol droplet,, RIMS Kokyuroku B, 3 (2007), 139.   Google Scholar

[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).  doi: doi:10.1103/PhysRevE.71.065301.  Google Scholar

[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.  doi: doi:10.1143/PTPS.161.286.  Google Scholar

[15]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation,, IMA J. of Appl. Math., 48 (1992), 249.  doi: doi:10.1093/imamat/48.3.249.  Google Scholar

show all references

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.  doi: doi:10.1007/BF00375025.  Google Scholar

[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.  doi: doi:10.1137/S0036141094279279.  Google Scholar

[3]

X. Chen, Spectrums for the Allen-Cahn, Cahn-Hilliard and phase field equations for generic interface,, Comm. P.D.E., 19 (1994), 1371.  doi: doi:10.1080/03605309408821057.  Google Scholar

[4]

X. Chen and G. Caginalp, Convergence of the phase field model to its sharp interface limits,, European J. Appl. Math., 9 (1998), 417.  doi: doi:10.1017/S0956792598003520.  Google Scholar

[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.   Google Scholar

[6]

X. Chen, D. Hilhorst and E. Logak, Mass conserved Allen-Cahn equation and volume preserving mean curvature flow,, to appear, (2010).   Google Scholar

[7]

C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws,, Adv. Math. Sci. Appl., 7 (1997), 467.   Google Scholar

[8]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres,, Proc. Amer. Math. Soc., 126 (1998), 2789.  doi: doi:10.1090/S0002-9939-98-04727-3.  Google Scholar

[9]

Y. Hayashima, M. Nagayama and S. Nakata, A camphor grain oscillates while breaking symmetry,, in J. Phys. Chem. B, 105 (2001), 5353.  doi: doi:10.1021/jp004505n.  Google Scholar

[10]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35.  doi: doi:10.1515/crll.1987.382.35.  Google Scholar

[11]

O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", American Mathematical Society, (1967).   Google Scholar

[12]

K. Nagai, Spontaneous irregular motion of an alcohol droplet,, RIMS Kokyuroku B, 3 (2007), 139.   Google Scholar

[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).  doi: doi:10.1103/PhysRevE.71.065301.  Google Scholar

[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.  doi: doi:10.1143/PTPS.161.286.  Google Scholar

[15]

J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation,, IMA J. of Appl. Math., 48 (1992), 249.  doi: doi:10.1093/imamat/48.3.249.  Google Scholar

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