Existence and Uniqueness of Global Smooth Solution of Incompressible Navier-Stokes Equation

TL;DR

This paper proves the existence and uniqueness of global smooth solutions for the incompressible Navier-Stokes equations under certain initial conditions, advancing understanding of their well-posedness.

Contribution

It establishes global existence and uniqueness results for smooth initial data and bounded Fourier frequency initial data in the Navier-Stokes equations.

Findings

Global smooth solutions exist and are unique for infinite smooth initial data.

Solutions are globally well-posed when initial data has Fourier frequency in a bounded set.

The results apply to both Cauchy and spatially periodic problems.

Abstract

The Cauchy problem and spatially periodic problem of incompressible Navier-Stokes equation are considered. The existence and uniqueness of global solution for these two problem with infinite smooth initial data $u_0$, i.e. $u_0,\;(u_0\cdot\nab)u_0\in H^{\infty}$, are established. Moreover these two problem with initial data $u_0\in H^m$ ($m\ge1$) are globally well-posed provided the Fourier frequency of $u_0$ is contained in a bounded compact set.