Long Time Decay of Leray Solution of 3D-NSE With Damping

TL;DR

This paper investigates the long-term decay behavior of Leray solutions to the 3D Navier-Stokes equations with damping, establishing decay rates and uniqueness under certain conditions using Fourier analysis.

Contribution

It proves the uniqueness, continuity in L^2, and large-time decay of solutions for the damped 3D Navier-Stokes equations when eta > 3.

Findings

Proves solution decay for eta  10/3.

Establishes uniqueness and continuity in L^2 for eta > 3.

Uses Fourier analysis and standard techniques.

Abstract

In \cite{CJ}, the authors show that the Cauchy problem of the Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u(\alpha>0,\;\beta\geq1)$ has global weak solutions in $L^2(\R^3)$. In this paper, we prove the uniqueness, the continuity in $L^2$ for $\beta>3$, also the large time decay is proved for $\beta\geq\frac{10}3$. Fourier analysis and standard techniques are used.