In the first part of the talk, we study highly oscillatory problems for the incompressible 3D Navier-Stokes and prove existence on infinite time intervals of regular strong solutions; 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. In the second part of the talk, we analyze regularity and dynamics of highly oscillatory problems for the 3D Euler equations.
Detailed proofs can be found in the following references:
Asymptotic Analysis, vol. 15, No. 2, p. 103-150, 1997 (with A. Babin and B. Nicolaenko).
Indiana University Mathematics Journal, vol. 48, No. 3, p. 1133-1176, 1999 (with A. Babin and B. Nicolaenko).
Indiana University Mathematics Journal, vol. 50, p. 1-35, 2001 (with A. Babin and B. Nicolaenko).
Russ. Mathematical Surveys, vol. 58, No. 2 (350), p. 287-318, 2003 (with B. Nicolaenko).
Methods and Applications of Analysis, vol. 11, No. 4, p. 605-633, 2004 (with B. Nicolaenko, C. Bardos and F. Golse). Hokkaido Mathematical Journal, vol. 35, No. 2, p. 321-364, 2006 (with Y. Giga, K. Inui and S. Matsui).
Annals of Math. Studies, Princeton University Press (Editors: J. Bourgain and C. Kenig), In Press (with Y. Giga and B. Nicolaenko).