Cauchy problem for the incompressible Navier-Stokes equation with an external force

TL;DR

This paper investigates the existence, long-term behavior, and uniqueness of solutions to the incompressible Navier-Stokes equations with a small rough external force, within critical Besov spaces.

Contribution

It establishes local-in-time existence, analyzes long-time behavior, and provides multiple uniqueness results for solutions under small external forces in critical Besov spaces.

Findings

Proved local-in-time existence for small external forces.

Analyzed the long-time behavior of solutions.

Established three types of uniqueness results.

Abstract

In this paper we focus on the Cauchy problem for the incompressible Navier-Stokes equation with a rough external force. If the given rough external force is small, we prove the local-in-time existence of this system for any initial data belonging to the critical Besov space $\dot{B}_{p,p}^{-1+\frac{3}{p}}$, where $3<p<\infty$. Moreover, We show the long-time behavior of the priori global solutins constructed by us. Also, we give three kinds of uniqueness results of the forced Navier-Stokes equations.