On large‐time energy concentration in solutions to the Navier‐Stokes equations in the whole 3D space

In the first part of the paper we study the large‐time behavior of the higher‐order space derivatives of solutions to the Navier‐Stokes equations in ℝ 3 . Specifically, we show that if w is a nonzero global weak solution to the Navier‐Stokes equations satisfying the strong energy inequality and 0 < α < β < ∞, then there exist C = C(α,β) > 1, δ 0 = δ 0 (α,β) ∈ (0,1) and t 0 = t 0 (α,β) > 0 such that\documentclass{article}\usepackage{amssymb}\footskip=0pc\pagestyle{empty}\begin{document}$$\frac{||A^\beta w(t)||} {||A^\alpha w(t+\delta)||} \le C $$\end{document}for every t > t 0 and every δ ∈ [0,δ 0 ]. A denotes the Stokes operator and A β its powers. In the second part of the paper we derive several consequences of the above inequality concerning the large‐time energy concentration in w ; we show, for example, that for any positive ε and α\documentclass{article}\usepackage{amssymb}\usepackage{amsmath}\footskip=0pc\pagestyle{empty}\begin{document}$$\lim_{t \rightarrow \infty} \frac {\int_{\mathbf{R}^3 \setminus B_{\sqrt{a+\varepsilon}}(0)}|\xi|^{4\alpha}|F(w(t))(\xi)|^2 d\xi} {\int_{\mathbf{R}^3} |F(w(t))(\xi)|^2 d\xi} = 0, $$\end{document}where a = lim{ t → ∞ ||A 1/2 w(t)|| 2 /||w(t)|| 2 is a nonnegative real number and F denotes the Fourier transform in L 2 (ℝ 3 ) 3 . If 0 < \varepsilonε < a , then\documentclass{article}\usepackage{amssymb}\usepackage{amsmath}\footskip=0pc\pagestyle{empty}\begin{document}$$ \lim_{t \rightarrow \infty} \frac {\int_{B_{\sqrt{a-\varepsilon}}(0)}|\xi|^{4\alpha}|F(w(t))(\xi)|^2 d\xi} {\int_{\mathbf{R}^3} |F(w(t))(\xi)|^2 d\xi} = 0. $$\end{document}