Hamiltonian Systems

Event Information Some no-gain results for two-point boundary value problems on diffeomorphism groups
14:00 on Tuesday March 09, 2021
15:30 on Tuesday March 09, 2021
Patrick Heslin

University of Notre Dame

One of the central results in Ebin and Marsden's 1970 paper says that, roughly speaking, if one considers the Euler Equations of ideal fluid motion in Lagrangian coordinates as a second order ODE in $D^s_\mu(M)$, any solution of the corresponding Cauchy problem (i.e. an $L^2$ geodesic) is as smooth as its initial conditions.

A natural question one might ask is if an analogous result holds when the geodesic equation is framed as a two-point boundary value problem. More precisely, given an $L^2$ geodesic $\gamma(t)$ emanating from the identity in $D^s_\mu(M)$ and, at some later time $t_0>0$, passing through $\eta \in D^{s+k}_\mu(M)$, can it be shown that $\gamma(t)$ evolves entirely in $D^{s+k}_\mu(M)$, i.e., is an $L^2$ geodesic as smooth as its boundary conditions? In this talk we will show an affirmative answer to this question in the flat setting for 2D ideal fluids, 3D axisymmetric fluids with zero swirl as well as several other equations of interest in mathematical physics.

The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828

Passcode: 448487