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Long-time behaviors for the Navier–Stokes equations under large initial perturbation

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Abstract

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

$$\begin{aligned} u\in L^{p}\left( 0,\infty ; {\dot{B}}^{\frac{2}{p}+\frac{3}{q}-1}_{q,\infty }({\mathbb {R}}^3)\right) , \quad 2<p<\infty ,\ 2\le q<\infty \ \mathrm{and} \ \frac{1}{p}+\frac{3}{q}\ge 1. \end{aligned}$$

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.

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Acknowledgements

The authors would like to thank Professor Bo-Qing Dong for his helpful suggestions. Ye was supported by the NSFC (11701384, 11671155, 11871346) and Natural Science Foundation of SZU (2017057). Jia was supported by the NNSFC Grant No. 11801002 and the NSF of Anhui Province 1808085MA01.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Hailong Ye & Yan Jia

  2. Shenzhen Key Laboratory of Advanced Machine Learning and Applications, Shenzhen University, Shenzhen, 518060, China

    Hailong Ye

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  1. Hailong Ye
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Correspondence to Yan Jia.

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Ye, H., Jia, Y. Long-time behaviors for the Navier–Stokes equations under large initial perturbation. Z. Angew. Math. Phys. 72, 136 (2021). https://doi.org/10.1007/s00033-021-01569-9

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  • DOI: https://doi.org/10.1007/s00033-021-01569-9

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