Two-Point Boundary Value Problems on Diffeomorphism Groups

Doctoral Dissertation

Abstract

In this thesis we study the Riemannian geometry of diffeomorphism groups equipped with a variety of Sobolev-type metrics. Most notably, we consider the group of volume-preserving diffeomorphisms equipped with a weak L^2 metric, whose geodesics correspond to Lagrangian solutions to the Euler equations.

A well known result of Ebin and Marsden states that Lagrangian solutions to the Euler equations, when framed as an initial value problem, are exactly as smooth as their initial conditions. We investigate a similar regularity property for Lagrangian solutions to the Euler equations when framed as a two-point boundary value problem. In particular, we prove that these L^2 geodesics are exactly as smooth as their boundary conditions.

We achieve like results in an array of other settings including 3D axisymmetric ideal fluids, symplectic Euler equations, Euler-alpha equations, and one dimensional integrable systems including the mu-CH and Hunter-Saxton equations.

Attributes

Attribute NameValues
Author Patrick Heslin
Contributor Alex Himonas, Committee Member
Contributor Matt Gursky, Committee Member
Contributor Nick Edelen, Committee Member
Contributor Gerard K. Misiolek, Research Director
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Banner Code
  • PHD-MATH

Defense Date
  • 2021-04-29

Submission Date 2021-04-29
Record Visibility Public
Content License
  • All rights reserved

Departments and Units
Catalog Record

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