Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 35K57, 35B25, 53C44, 35C20.


    \begin{equation} \\ \end{equation}
  • [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.


    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.


    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.


    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.


    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.


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


    C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7 (1997), 467-490.


    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.


    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.


    G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), 35-48.doi: doi:10.1515/crll.1987.382.35.


    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.


    K. Nagai, Spontaneous irregular motion of an alcohol droplet, RIMS Kokyuroku B, 3 (2007), 139-147.


    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.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(113) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint