Abstract
We prove the long-time existence of solutions for the Mullins- Sekerka flow. We use a time discrete approximation which was introduced by Luckhaus and Sturzenhecker [Calc. Var. PDE 3 (1995)] and pass in a new weak formulation to the limit.
The research of the author was supported by DFG Sonderforschungsbereich 611 and by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00274, FRONTS-SINGULARITIES..
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F.J. Almgren. The theory of varifolds. Princeton notes, 1965.
L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, 2000.
X. Chen. Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differential Geom., 44(2):262–311, 1996.
X. Chen, J. Hong, and F. Yi. Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations, 21(11–12):1705–1727, 1996.
J. Escher and G. Simonett. Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations, 2(4):619–642, 1997.
S. Luckhaus and T. Sturzenhecker. Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Equ., 3, No.2:253–271, 1995.
P.I. Plotnikov and V.N. Starovoitov. The Stefan problem with surface tension as a limit of a phase field model. Diff. Eq., 29, No. 3:395–404, 1993.
M. Röger. Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation. Interfaces Free Bound., 6(1):105–133, 2004.
M. Röger. Weak solutions for the Mullins-Sekerka flow. Siam J. on Math. Analysis, to appear, 2005.
R. Schätzle. A counterexample for an approximation of the Gibbs-Thomson law. Advances in Math. Sciences and Appl., 7, No. 1:25–36, 1997.
R. Schätzle. Hypersurfaces with mean curvature given by an ambient Sobolev function. J. Differential Geometry, 58:371–420, 2001.
L. Simon. Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis Australian National University Vol. 3, 1983.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Röger, M. (2006). Existence of Weak Solutions for the Mullins-Sekerka Flow. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_35
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7719-9_35
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-7718-2
Online ISBN: 978-3-7643-7719-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)