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

Abstract

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 $\int |\nabla v|^2 dx$.