Consider weak solutions u of the 3D Navier–Stokes equations in the critical space

Firstly, we show that although the initial perturbations \(w_0\) from u are large, every perturbed weak solution v satisfying the strong energy inequality converges asymptotically to u as \(t\rightarrow \infty \). Secondly, by virtue of the characterization of \(w_0\), we examine the optimal upper and lower bounds of the algebraic convergence rates for \(\Vert v(t)-u(t)\Vert _{L^2}\). It should be noted that the above results also hold if \(u\in C([0,\infty ); {\dot{B}}^{\frac{3}{q}-1}_{q,\infty }({\mathbb {R}}^3))\) with sufficiently small norm and \(2\le q\le 3\). The proofs are mainly based on some new estimates for the trilinear form in Besov spaces, the generalized energy inequalities and developed Fourier splitting method.

考虑临界空间中 3D Navier-Stokes 方程的弱解u

首先，我们证明尽管来自u的初始扰动\(w_0\)很大，但满足强能量不等式的每个扰动弱解v渐近收敛到u作为\(t\rightarrow \infty \)。其次，凭借\(w_0\)的表征，我们检查\(\Vert v(t)-u(t)\Vert _{L^2}\的代数收敛率的最佳上下界)。需要注意的是，如果\(u\in C([0,\infty ); {\dot{B}}^{\frac{3}{q}-1}_{q,\ infty }({\mathbb {R}}^3))\)具有足够小的范数和\(2\le q\le 3\). 证明主要基于对 Besov 空间中的三线性形式的一些新估计、广义能量不等式和发达的傅立叶分裂方法。