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

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 + α)-little-Hölder 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.

37 pages, 6 figures, To Appear in SIAM J. Math. Anal

References in corpus (5)

• Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension
• Surface diffusion flow near spheres
• Lifespan theorem for constrained surface diffusion flows
• Lifespan theorem for simple constrained surface diffusion flows
• Elliptic operators and maximal regularity on periodic little-Hölder spaces

Cited by in corpus (3)

• A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows
• Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus
• On the flow of non-axisymmetric perturbations of cylinders via surface diffusion