Zujin Zhang, Weijun Yuan, Zhengan Yao
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.
DOI: 10.58997/ejde.2025.79
Show me the PDF file (470 KB), TEX file for this article.
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Zujin Zhang School of Mathematics and Computer Science Gannan Normal University Ganzhou 341000, Jiangxi, China email: zhangzujin361@163.com |
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| Weijun Yuan School of Mathematics and Computer Science Gannan Normal University Ganzhou 341000, Jiangxi, China email: 1453540745@qq.com | |
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Zhengan Yao School of Mathematics Sun Yat-sen University Guangzhou 510275, China email: mcsyao@mail.sysu.edu.cn |
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