On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow

On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow

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-Hölder 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 axisymmetr...

On the Axisymmetric Surface Diffusion Flow

On the Global Well-posedness for the Axisymmetric Euler Equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces B 1+3/p p,1 . In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity.

Stability and Bifurcation of Equilibria for the Axisymmetric Averaged Mean Curvature Flow

We study the averaged mean curvature flow, also called the volume preserving mean curvature flow, in the particular setting of axisymmetric surfaces embedded in R3 satisfying periodic boundary conditions. We establish analytic well–posedness of the flow within the space of little-Hölder continuous surfaces, given rough initial data. We also establish dynamic properties of equilibria, including ...

Well-posedness, Regularity, and Finite Element Analysis for the Axisymmetric Laplace Operator on Polygonal Domains

Let L := −r−2(r∂r)2 − ∂2 z . We consider the equation Lu = f on a bounded polygonal domain with suitable boundary conditions, derived from the three-dimensional axisymmetric Poisson’s equation. We establish the well-posedness, regularity and Fredholm results in weighted Sobolev spaces, for possible singular solutions caused by the singular coefficient of the operator L, as r → 0, and by non-smo...