Date of Award
Doctor of Philosophy (PhD)
Chi Hin Chan
In fluid mechanics, the vorticity provides a valuable alternative perspective of the behavior of flows. Constantin-Lax-Majda approached studying the 3D vorticity equation by proposing a 1D model equation with significant analytic similarities, the Constantin-Lax-Majda equation. This has been followed by a collection of model equations in both 1D and 2D whose behaviors capture many aspects of the full 3D equations.
This thesis contains many new results for several of these equations. We begin by outlining the original analytic theory as well as the Euler-Arnold theory which studies these equations as geodesic equations on infinite dimensional manifolds. We build on the work of Castro-Cόrdoba and Bauer-Kolev-Preston to show that every solution to the Wunsch equation, a special case of the generalized Constantin-Lax-Majda equation, blows up in finite time. This result also applies to the Constantin-Lax-Majda equation itself. We also investigate the Euler-Weil-Petersson equation which has significant links to the Wunsch equation in the context of Teichmüller theory.
Additionally, we lay the foundations for a geometric theory of the surface quasi-geostrophic equation (SQG). Originally discovered in the context of geophysical fluid mechanics (see Pedlosky), SQG was proposed by Constantin-Majda-Tabak as a 2D version of the 1D Constantin-Lax-Majda equation. In a blog post, Tao showed that SQG arises as the critical point of a functional. This discovery naturally leads to the formulation of SQG as an Euler-Arnold equation. In this thesis we show that the associated geometric space has a smooth, non-Fredholm Riemannian exponential map, and has arbitrarily large curvature of both signs.
Finally we discuss the geometric setting for the Axi-symmetric Euler equations. Here we consider a 3D analogue of the 2D flows considered in Preston. Surprisingly, while the 2D flows exhibit negative curvature, we show that the corresponding 3D flows exhibit positive curvature and a rich structure of conjugate points. Such a result may have significant ramifications for our understanding of the nature of stability in 2D and 3D fluids.
Washabaugh, Pearce C., "The Diffeomorphism Group Approach to Vorticity Model Equations" (2017). Mathematics Graduate Theses & Dissertations. 48.