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Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow |
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| Authors: | Sahbi Boussandel Ralph Chill Eva Fa?angová |
| Affiliation: | 1. Faculté des sciences de Bizerte, Département de Mathématiques, 7021, Jarzouna Bizerte, Tunisie 2. Laboratoire de Mathématiques et Applications de Metz et CNRS, UMR 7122, Bat. A, ?le de Saulcy, Université Paul Verlaine-Metz, 57045, Metz Cedex 1, France 4. Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic 5. Institut für Angewandte Analysis, Universit?t Ulm, 89069, Ulm, Germany |
| Abstract: | Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and L 2-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented in the special situation only. |
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