Large-time behavior of the weak solution to 3D Navier-Stokes equations

TL;DR

This paper proves the uniqueness and exponential decay over time of weak solutions to the 3D Navier-Stokes equations in bounded domains, under certain conditions on the force term.

Contribution

It establishes the uniqueness of weak solutions with an additional condition and demonstrates their exponential decay in bounded domains.

Findings

Weak solutions are unique under an extra requirement.

Solutions exist for all time $t \\geq 0$.

Solutions decay exponentially if the force term decays suitably.

Abstract

The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a bounded domain $D$ the solution decays exponentially fast as $t\to \infty$ if the force term decays at a suitable rate.