On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

TL;DR

This paper analyzes the behavior of non-axisymmetric surface diffusion flows on cylinders, establishing existence, uniqueness, and stability conditions for perturbations based on cylinder radius and boundary conditions.

Contribution

It introduces a rigorous mathematical framework for analyzing non-axisymmetric perturbations of cylinders under surface diffusion flow, including stability criteria based on radius.

Findings

Cylinders with radius r > 1 are normally stable under periodic perturbations.

Cylinders with radius 0 < r < 1 are unstable under the same conditions.

Existence and uniqueness of solutions are established for bounded perturbations.

Abstract

We study the surface diffusion flow acting on a class of general (non--axisymmetric) perturbations of cylinders $\mathcal{C}_r$ in ${\rm I \! R}^3$. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially--unbounded) surfaces defined over $\mathcal{C}_r$ via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that $\mathcal{C}_r$ is normally stable with respect to $2 \pi$--axially--periodic perturbations if the radius $r > 1$,and unstable if $0 < r < 1$. Stability is also shown to hold in settings with axial Neumann boundary conditions.