Type of Document Dissertation Author LeCrone, Jeremy URN etd-05222012-212749 Title On the axisymmetric surface diffusion flow Degree PhD Department Mathematics Advisory Committee
Advisor Name Title Gieri Simonett Committee Chair Akram Aldroubi Committee Member Alexander Powell Committee Member Emmanuele DiBenedetto Committee Member Zhaohua Ding Committee Member Keywords
- fourier multipliers
- nonlinear stability
- maximal regularity
- periodic boundary conditions
- surface diffusion
Date of Defense 2012-05-04 Availability unrestricted AbstractIn this thesis, we establish analytic results for the axisymmetric surface
diffusion flow (ASD), a fourth-order geometric evolution law. In the first part
of the work, we develop a general theory establishing maximal regularity results
for a broad class of abstract, higher-order elliptic operators, in the setting
of periodic little-Hölder spaces. These results are then applied, in the
second part of the thesis, to prove well-posedness results for ASD.
In particular, we prove that ASD generates a real analytic semiflow in the space of
(2 + alpha)-little-Hölder regular surfaces of revolution embedded in R^3.
Further, we give conditions for global existence of solutions and we prove that
solutions are real analytic in time and space.
We also investigate the dynamic properties of solutions to ASD in the second part of the thesis.
Utilizing a connection to axisymmetric surfaces with constant mean curvature,
we characterize the equilibria of ASD. We focus on the family of cylinders
as equilibria of ASD and we establish results regarding the stability, instability
and bifurcation behavior of cylinders with the radius acting as
a bifurcation parameter for the problem.
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