A Center Manifold Analysis for the Mullins-Sekerka Model

Journal of Differential Equations

Volumn 143, Issue 2, 1998, Pages 267-292

  Escher, Joachim ^a   Simonett, Gieri ^b  

^a UNIVERSITY OF BASEL   (Switzerland)

^b VANDERBILT UNIVERSITY   (United States)

Author keywords

Mullins Sekerka model; mean curvature; free boundary problem; generalized motion by mean curvature; center manifold

Indexed keywords

EID: 0001025274     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1997.3373     Document Type: Article

References (35)

1. Alikakos N., Fusco G. The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J. 42:1993;637-674.
2. Alikakos N., Bates P. W., Chen X. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal. 128:1994;164-205.
3. Alikakos N., Fusco G., Stefanopoulos V. Critical spectrum and stability for a class of reaction-diffusion equations. J. Differential Equations. 126:1996;106-167.
4. N. Alikakos, G. Fusco, G. Simonett, The two phase Hele-Shaw problem for many almost spherical particles in a bounded domain.
5. Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function Spaces, Differential Operators and Nonlinear Analysis. 1993;Teubner, Stuttgart/Leipzig. p. 9-126.
6. Amann H. Linear and Quasilinear Parabolic Problems. 1995;Birkhäuser, Basel.
7. Angenent S. B. Nonlinear analytic semiflows. Proc. Roy. Soc. Edinburgh A. 115:1990;91-107.
8. Bates P. W., Chen X., Deng X. A numerical scheme for the two phase Mullins-Sekerka problem. Electron. J. Differential Equations. 11:1995;1-28.
9. Chen X. The Hele-Shaw problem and area-preserving curve shortening motion. Arch. Rational Mech. Anal. 123:1993;117-151.
10. Chen X., Hong J., Yi F. Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations. 21:1996;1705-1727.
11. Constantin P., Pugh M. Global solutions for small data to the Hele-Shaw problem. Nonlinearity. 6:1993;393-415.
12. Crank J. Free and Moving Boundary Problems. 1984;Clarendon, Oxford.
13. Da Prato G., Grisvard P. Equations d'évolution abstraites nonlinéaires de type parabolique. Ann. Mat. Pura Appl. (4). 120:1979;329 - 396.
14. Da Prato G., Lunardi A. Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space. Arch. Rational Mech. Anal. 101:1988;115-144.
15. Duchon J., Robert R. Evolution d'une interface par capillarité et diffusion de volume I. Existence locale en temps. Ann. Inst. H. Poincaré, Anal. Non Linéaire. 1:1984;361-378.
16. Escher J. The Dirichlet-Neumann operator on continuous functions. Ann. Scuola Norm. Sup. Pisa (4). 21:1994;235-266.
17. Escher J., Simonett G. Analyticity of the interface in a free boundary problem. Math. Ann. 305:1996;439-459.
18. Escher J., Simonett G. Classical solutions of multi-dimensional Hele-Shaw models. SIAM J. Math. Anal. 28:1997;1028-1047.
19. Escher J., Simonett G. On Hele-Shaw models with surface tension. Math. Res. Lett. 3:1996;467-474.
20. Escher J., Simonett G. Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations. 2:1997;619-642.
21. Gage M., Hamilton R. The shrinking of convex curves by the heat equation. J. Differential Geom. 23:1986;69-96.
22. Gurtin M. E. Thermomechanics of Evolving Phase Boundaries in the Plane. 1993;Clarendon, Oxford.
23. Hanzawa E. I. Classical solutions of the Stefan problem. Tôhoku Math. J. 33:1981;297-335.
24. Harrell E. M. II. On the second eigenvalue of the Laplace operator penalized by curvature. Differential Geom. Appl. 6:1996;397-400.
25. Huisken G. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20:1984;237-266.
26. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. 1995;Birkhäuser, Basel.
27. Mayer U. F. One-sided Mullins-Sekerka flow does not preserve convexity. Electron. J. Differential Equations. 08:1993;1-7.
28. U. F. Mayer, Two-sided Mullins-Sekerka flow does not preserve convexity.
29. Mullins W. W., Sekerka R. F. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Math. 34:1963;323-329.
30. Papanicolaou V. G. The second periodic eigenvalue and the Alikakos-Fusco conjecture. J. Differential Equations. 130:1996;321-332.
31. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. 1983;Springer-Verlag, New York.
32. Pego R. L. Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A. 422:1989;261-278.
33. Simonett G. Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems. Nonlinear Anal. 23:1994;515-544.
34. Simonett G. Center manifolds for quasilinear reaction-diffusion systems. Differential Integral Equations. 8:1995;753-796.
35. Triebel H. Höhere Analysis. 1980;Verlag Harri Deutsch, Frankfurt.