# The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

## TL;DR

This paper studies the mathematical properties of surface diffusion and Willmore flows on hypersurfaces, proving well-posedness and long-term behavior, including convergence to spheres for certain initial conditions.

## Contribution

It establishes well-posedness of surface diffusion and Willmore flows on a broad class of hypersurfaces and proves convergence to spheres for the Willmore flow near spherical shapes.

## Key findings

- Well-posedness for initial surfaces with $C^{1+eta}$ regularity.
- Long-term existence and convergence to spheres for Willmore flow near spherical surfaces.
- Solutions become spherical as time approaches infinity.

## 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 $C^{1+\alpha}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are $C^{1+\alpha}$-close to a sphere, and we prove that these solutions become spherical as time goes to infinity.