21世紀COEプログラム : 特異性から見た非線構造の数学

Alex Mahalov:COE シンポジウム「非線型分散方程式」一般講演

Fast Singular Oscillating Limits, Restricted Convolutions and Global Regularity of the 3D Navier-Stokes Equations of Gephysics

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 and for the full 3D Navier-Stokes Equations of Gephysics in the regime of strong stratification and rotation; 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, lemmas on restricted convolutions 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. 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 for above classes of fully 3D initial data.

21世紀COEプログラム : 特異性から見た非線構造の数学