Caltech/UCLA Joint Analysis Seminar
Given a map S, the H1 projection problem seeks a measure preserving bijection Z that is as close as possible to S in the H1 norm. This problem is closely related to Arnold's geometric interpretation of fluid mechanics. Indeed, the Euler-Lagrange equation associated to the H^1 projection problem can be viewed as a discrete-in-time approximation to the Lagrangian formulation of the Navier-Stokes equations. If the H1 norm is replaced by the L2 norm, the existence and uniqueness of minimizers follows from Brenier's celebrated polar factorization theorem, however the corresponding results for the H1 projection problem in 3-dimensions have long remained open. In this talk, I will present an argument establishing existence and uniqueness of minimizers for the H1 projection problem under some regularity assumptions on S. This talk is joint work with Wilfrid Gangbo and Inwon Kim.