Global solubility of the three-dimensional Navier--Stokes equations with uniformly large initial vorticity

A. S. Makhalov ; V. P. Nikolaenko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 2, pp. 287-318

Résumé

This paper is a survey of results concerning the three-dimensional Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity. The existence of regular solutions of the three-dimensional Navier–Stokes equations on an unbounded time interval is proved for large initial data both in $\mathbb R^3$ and in bounded cylindrical domains. Moreover, the existence of smooth solutions on large finite time intervals is established for the three-dimensional Euler equations. These results are obtained without additional assumptions on the behaviour of solutions for $t>0$. Any smooth solution is not close to any two-dimensional manifold. Our approach is based on the computation of singular limits of rapidly oscillating operators, non-linear averaging, and a consideration of the mutual absorption of non-linear oscillations of the vorticity field. The use of resonance conditions, methods from the theory of small divisors, and non-linear averaging of almost periodic functions leads to the limit resonant Navier–Stokes equations. Global solubility of these equations is proved without any conditions on the three-dimensional initial data. The global regularity of weak solutions of three-dimensional Navier–Stokes equations with uniformly large vorticity at $t=0$ is proved by using the regularity of weak solutions and the strong convergence.