PDE Real Analysis Seminar Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity

2004-09-29 10:30 - 2004-09-29 11:30
Graduate School of Mathematical Sciences the University of Tokyo, Room #117
Alex Mahalov (Arizona State University)
We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity with periodic boundary conditions and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold.The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. Using Lemmas on restricted convolutions, we establish the global regularity of the latter without any restriction on the size of 3D initial data.With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity.