Dissertations, Theses, and Capstone Projects

Date of Degree

5-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Stephen Preston

Committee Members

Enrique Pujals

Christina Sormani

Azita Mayeli

Subject Categories

Analysis | Geometry and Topology

Keywords

Euler-Arnold equation, Camassa-Holm equation, quasi-geostrophic equation, local well-posedness, global weak solution, sectional curvature

Abstract

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In this thesis, we investigate two Euler-Arnold equations; the Camassa-Holm equation from the shallow water equation theory and quasi-geostrophic equation from geophysical fluid dynamics.

First, we will prove the local-wellposedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms, satisfying certain asymptotic conditions at infinity. Motivated by the work of Misio{\l}ek, we will re-express the equation in Lagrangian variables, by which the PDE becomes an ODE on a Banach manifold with a locally Lipschitz right-side. Consequently, we obtain the existence and uniquenss of the solution, and the topological group property of the diffeomorphism group ensures the continuous dependence on the initial data.

Second, we will construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We will use a simple Lagrangian change of variables, which removes the wave breaking singularity of the original equation and allows the weak continuation. Furthermore, we obtain the global spatial smoothness of the Lagrangian trajectories via this construction. This work was motivated by Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.

Lastly, we will study some geometric aspects for the quasi-geostrophic equation, which is the geodesic on the quantomorphism group, a subgroup of the contactomorphism group. We will derive an explicit formula for the sectional curvature and discuss the nonpositive curvature criterion, which extends the work of Preston on two dimensional incompressible fluid flows.

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