On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

TL;DR

This paper analyzes the mathematical properties of the axisymmetric surface diffusion flow, establishing well-posedness, regularity, and bifurcation behavior, especially focusing on stability and instability of cylindrical solutions.

Contribution

It proves well-posedness and regularity of ASD, characterizes equilibria, and analyzes bifurcation and stability of cylindrical solutions based on radius.

Findings

ASD generates a real analytic semiflow in suitable function spaces.

Conditions for global existence of solutions are provided.

Bifurcation analysis reveals stability and instability of cylinders depending on radius.

Abstract

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.