The Surface Diffusion and the Willmore Flow for Uniformly Regular Hypersurfaces

Authors

Jeremy LeCrone, University of RichmondFollow
Yuanzhen Shao
Gieri Simonett

DOI

10.3934/dcdss.2020242

Abstract

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are C^1+α–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are C^1+α–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

Document Type

Post-print Article

Publication Date

12-2020

Publisher Statement

Copyright © 2021, American Institute of Mathematical Sciences.

DOI: https://doi.org/10.3934/dcdss.2020242

The definitive version is available at: Discrete and Continuous Dynamical Systems-Series S 13, no. 12.

Recommended Citation

LeCrone, Jeremy, Yuanzhen Shao, and Gieri Simonett. “The Surface Diffusion and the Willmore Flow for Uniformly Regular Hypersurfaces.” Discrete and Continuous Dynamical Systems-Series S 13, no. 12 (December 2020): 3503–24. https://doi.org/10.3934/dcdss.2020242.