The 1st PDE Real Analysis Seminar
(COE Partner Seminar, Department of Mathematics, Hokkaido University )
Outline
- Organizers :
- H.Arai (University of Tokyo), Y. Giga (Hokkaido University)
- Period :
- September 29, 2004 (Wednesday)
- Place :
- Graduate School of Mathematical Sciences the University of Tokyo, Room #117
- Programme :
-
September 29, 2004 (Wednesday)
- 10:30-11:30 Alex Mahalov (Arizona State University)
- Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity
- ABSTRACT:
- 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.