On the Regularity for 3D Navier-Stokes Equation

TL;DR

This paper proves that vorticity in 3D Navier-Stokes equations with periodic conditions belongs to a specific function space, leading to the existence of a global smooth solution, by using auxiliary problems for approximation.

Contribution

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

Findings

Vorticity belongs to L1(0,T;L2(5)) for 3D Navier-Stokes.

Existence of a global smooth solution is proven.

Method involves constructing auxiliary problems for approximation.

Abstract

In this paper we will prove that the vorticity belongs to L1(0; T ; L2(\Omega)) for 3D incompressible Navier-Stokes equation with periodic initial-boundary value conditions, 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.