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Global Solutions of 3-D Navier–Stokes System with Small Unidirectional Derivative

Abstract

Given initial data \(u_0=(u_0^{\mathrm{h}},u_0^3)\in H^{\frac{1}{2}}({{\mathbb {R}}}^3)\cap B^{0,\frac{1}{2}}_{2,1}({{\mathbb {R}}}^3)\) with \(u^{{\mathrm{h}}}_0\) belonging to \(L^2({{\mathbb {R}}}^3)\cap L^\infty ({{\mathbb {R}}}_{\mathrm{v}}; H^{-\delta }({{\mathbb {R}}}^2_{\mathrm{h}}))\cap L^\infty ({{\mathbb {R}}}_{\mathrm{v}}; H^3({{\mathbb {R}}}^2_{\mathrm{h}}))\) for some \(\delta \in ]0,1[,\) if in addition \(\partial _3u_0\) belongs to the homogeneous anisotropic Sobolev space, \(H^{-\frac{1}{2},0},\) we prove that the classical 3-D Navier–Stokes system has a unique global Fujita–Kato solution provided that the \(H^{-\frac{1}{2},0}\) norm of \(\partial _3u_0\) is sufficiently small compared to \(\exp \left( -\,C\bigl (A_\delta (u^{{\mathrm{h}}}_0)+B_\delta (u_0)\bigr )\right) \) with \(A_\delta (u^{{\mathrm{h}}}_0)\) and \(B_\delta (u_0)\) being scaling invariant quantities of the initial data, which is scaling invariant with respect to the variable \(x_3\). This result provides some classes of large initial data which are large in Besov space \(B^{-1}_{\infty ,\infty }\) and which generate unique global solutions to the 3-D Navier–Stokes system. In particular, we extend the previous results in [5,7,10] for initial data with a slow variable to multi-scales slow variable initial data.

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Notes

  1. 1.

    Through out this paper, we always designate \(H^s\) to be homogeneous Sobolev spaces, and \(B^s_{p,r}\) to be homogeneous Besov spaces.

  2. 2.

    This is just for convenience of notations, and one should keep in mind that \({\bar{u}}^3=0.\)

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Acknowledgements

The authors would like to thank the referee for profitable comments and suggestions that improved the original version. P. Zhang would like to thank Professor J.-Y. Chemin for profitable discussions on this topic. P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, the Morningside Center of Mathematics of the Chinese Academy of Sciences and an innovation Grant from the National Center for Mathematics and Interdisciplinary Sciences.

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Correspondence to Ping Zhang.

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Liu, Y., Zhang, P. Global Solutions of 3-D Navier–Stokes System with Small Unidirectional Derivative. Arch Rational Mech Anal 235, 1405–1444 (2020). https://doi.org/10.1007/s00205-019-01447-9

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