On the Regularity of the Solutions for Cauchy Problem of Incompressible 3D Navier-Stokes Equation

TL;DR

This paper proves that vorticity solutions to the 3D incompressible Navier-Stokes equations are integrable over time and space, leading to the existence of global smooth solutions for the problem.

Contribution

It introduces a novel approach using auxiliary problems to establish the regularity and global existence of solutions for the 3D Navier-Stokes equations.

Findings

Vorticity belongs to L1(0; T; L2(R3))

Existence of global smooth solutions established

New approximation method for vorticity equations

Abstract

In this paper we will prove that the vorticity belongs to L1(0; T ; L2(R3)) for the Cauchy problem of 3D incompressible Navier-Stokes equation, then the existence of a global smooth solution is obtained. Our approach is to construct a set of auxiliary problems to approximate the original one of vorticity equation.