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It is shown that the L_3,∞-solutions of the Cauchy problem for the three-dimensional Navier-Stokes equations are smooth.

This paper presents the main results concerning solubility of the basic initial-boundary value problem and the Cauchy problem for the three-dimensional non-stationary Navier-Stokes equations, together with a list of what to prove in order to solve the sixth problem of the "seven problems of the millennium" proposed on the Internet at the site http://claymath.org/.

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  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 . 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 is proved by using the regularity of weak solutions and the strong convergence.

This paper is a discussion of some open problems concerning generalizations of Navier-Stokes fluids and other non-linear fluids, which are in need of thorough investigation. The first topic is an interesting generalization of the Navier-Stokes model which is appropriate, in particular, in the description of fluids like water subjected to processes in which the normal stresses are very large. Discussed next is the modelling of non-Newton fluids at a distance from the boundary of the volume occupied by the fluid. Attention is then given to the necessity of a careful revision of conditions obtained at the boundary of the fluid volume, and finally, the modelling of "turbulent" flows is discussed.

In this paper, which is mainly of a survey nature, a coercive estimate is proved in Sobolev spaces with a mixed norm to solve the non-stationary Stokes problem (with non-zero divergence) in bounded and exterior domains, and from the first estimate an estimate is proved for the resolvent of the Stokes operator. The latter proof uses the explicit representation of the solution of the problem in a half-space in terms of the Green's matrix; pointwise estimates are derived for the elements of this matrix.