Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 79, pp. 1-25.

Title: Stability of Leray weak solutions to 3D Navier-Stokes equations

Authors: Zujin Zhang (Gannan Normal Univ., Jiangxi, China)
         Weijun Yuan (Gannan Normal Univ., Jiangxi, China)
         Zhengan Yao (Sun Yat-sen Univ., Guangzhou, China)

Abstract:
In this article, we show that if the Leray weak solution $u$ of the three-dimensional
Navier-Stokes system satisfies
$$
\nabla u\in L^p(0,\infty;\dot B^0_{q,\infty}(\mathbb{R}^3)),\quad
\frac{2}{p}+\frac{3}{q} =2,\quad \frac{3}{2}<q<\infty,
$$
or
$$
\nabla u\in L^\frac{2}{2-r}(0,\infty;\dot B^{-r}_{\infty,\infty}(\mathbb{R}^3)),\quad 0<r<1,
$$
then $u$ is uniformly stable, under small perturbation of initial data and external force, is asymptotically stable in the $L^2$ sense, is unique amongst all the Leray weak solutions, and satisfies some energy type equalities.
Also under spectral condition on the initial perturbation,
we obtain optimal upper and lower bounds of convergence rates.
Our results extend the results in [6,11]

Submitted March 25, 2025 Published July 24, 2025.
Math Subject Classifications: 35Q30, 76D03.
Key Words: Navier-Stokes equations; stability; energy equality; weak-strong uniquenes.