Stephen Preston - Euler Equations as Geodesics on Diffeomorphism Groups (Nov 30 2020)

I will discuss Arnold’s perspective on the Euler equations as geodesics on the group of volume-preserving diffeomorphisms, with an eye to generalizations on other diffeomorphism groups. I’ll talk about the general notion of vorticity conservation and what it looks like in 2D, 3D, and in the 3D axisymmetric case. I will also describe how this infinite-dimensional geometry approach allows for simpler proofs of the local existence result. In addition we’ll look at curvature computations and what they say about stability of fluids, along with conjugate points along geodesics and how they differ in 2D and 3D.
In the second half of the talk I will describe other PDEs that can be described as geodesics on diffeomorphism groups, including 1D models that can be understood more easily, including the breakdown mechanisms and the global existence of weak solutions.