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.