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

• Published in: Electronic journal of differential equations Vol. 2025; no. 1-??; pp. 79 - 25
• Main Authors: Zhang, Zujin, Yuan, Weijun, Yao, Zhengan
• Format: Journal Article
• Language: English
• Published: Texas State University 24.07.2025
• Subjects: energy equality; navier-stokes equations; stability; weak-strong uniqueness
• ISSN: 1072-6691; 1072-6691
• DOI: 10.58997/ejde.2025.79

In this article, we show that if the Leray weak solution \(\mathbf{u}\) of the three-dimensional Navier-Stokes system satisfies $$\nabla \mathbf{u}\in L^p(0,\infty;\dot B^0_{q,\infty}(\mathbb{R}^3)),\quad\frac{2}{p}+\frac{3}{q} =2,\quad \frac{3}{2}< q< \infty, $$ or $$\nabla \mathbf{u}\in L^\frac{2}{2-r}(0,\infty;\dot B^{-r}_{\infty,\infty}(\mathbb{R}^3)),\quad 0<r < 1,$$ then \(\mathbf{u}\) 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] For more information see https://ejde.math.txstate.edu/Volumes/2025/79/abstr.html