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

TL;DR

This paper establishes the existence of a broad class of smooth solutions to the 3D incompressible Navier-Stokes equations, highlighting new regularity conditions related to Sobolev embeddings and large initial data.

Contribution

It introduces a novel class of regular solutions for the Navier-Stokes equations with large Sobolev norms, advancing understanding of flow regularity.

Findings

Existence of smooth solutions with large Sobolev norms

New regularity criteria based on Sobolev embedding

Insights into the behavior of fast incompressible flows

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