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arXiv · 1905.00156

Global well-posedness of $3$-D anisotropic Navier-Stokes system with small unidirectional derivative

Yanlin Liu·Marius Paicu·Ping Zhang
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Abstract

In \cite{LZ4}, the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier-Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier-Stokes system $(ANS)$ with only horizontal dissipation. More precisely, given initial data $u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12},$ $(ANS)$ has a unique global solution provided that $|D_\h|^{-1}\pa_3u_0$ is sufficiently small in the scaling invariant space $\cB^{0,\f12}.$

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Yanlin Liu, Marius Paicu, Ping Zhang. 2019-05-01. Global well-posedness of $3$-D anisotropic Navier-Stokes system with small unidirectional derivative. https://doi.org/10.1007/s00205-020-01555-x

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