On the Cauchy Problem for Vortex Rings
Speaker: Vladimir Sverak, University of Minnesota
Location: Warren Weaver Hall 1302
Date: Thursday, September 29, 2016, 11 a.m.
We consider the initial-value problem for the 3d Navier-Stokes equation when the initial vorticity is supported on a circle. Such initial datum is in certain function spaces where perturbation theory works for small data, but not for large data, even for short times, and there are good reasons to believe that this is not just a technicality. We prove global existence and uniqueness for large data in the class of axi-symmetrix solutions. The main tools are Nash-type estimates and certain monotone quantities.Uniqueness in the class of solutions which are not necessarily axi-symmetric remains a difficult open problem, which we plan to discuss briefly. Joint work with Thierry Gallay.