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arXiv · 1506.01006

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

Jeremy LeCrone·Gieri Simonett
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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.

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Jeremy LeCrone, Gieri Simonett. 2015-06-02. On the flow of non-axisymmetric perturbations of cylinders via surface diffusion. https://doi.org/10.1016/j.jde.2015.12.008

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