The Navier-Stokes equations. Regularity of fast flows and Sobolev imbedding

The paper proves existence of a large class of smooth solutions to the incompressible Navier-Stokes equations in the three dimensional space. The viscosity coefficient is put to be $1$. Our result points a new class of regular solutions with arbitrary large Cauchy integral $\int \mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^{2}dx$.