We study the axisymmetric surface diffusion (ASD) flow, a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2+α)-little-Holder regular surfaces of revolution embedded in R3 and satisfying periodic boundary conditions. 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.
Copyright © 2013 Society for Industrial and Applied Mathematics. This article first appeared in SIAM Journal on Mathematical Analysis 45:5 (2013), 2834-2869.
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LeCrone, Jeremy and Gieri Simonett. "On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow." SIAM Journal on Mathematical Analysis 45, no. 5 (2013): 2834-2869. doi:http://dx.doi.org/10.1137/120883505.