Global well-posedness of 3-D inhomogeneous Navier-Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field
type: Item
instance of: scholarly article
title: Global well-posedness of 3-D inhomogeneous Navier-Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field
zbMATH Open document ID: 1397.35167
author: Yuhui Chen
author: Jingchi Huang
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-05-31T00:00:00Z
review text: This very interesting paper is concerning the global wellposedness for 3-D anisotropic Navier-Stokes equations with a particular divergence free initial velocity, in the framework of Besov spaces. The results of \textit{M. Paicu} and \textit{P. Zhang} [Commun. Math. Phys. 307, No. 3, 713--759 (2011; Zbl 1237.35129)] and \textit{M. Paicu} and \textit{P. Zhang} [J. Funct. Anal. 262, No. 8, 3556--3584 (2012; Zbl 1236.35112)] are improved, for the case when we have two large components of velocity- see Theorem 1.2. Some new and interesting elements are used in the proof. The main point is following: ``\dots The horizontal components of velocity can be arbitrary large. But we fail to estimate it in any three dimensional function space'' (Remark 1.1). Some results concerning the case when the horizontal components of space variables are large compared with the vertical one are mentioned. The uniqueness result is given in Section 4, based on a logarithmic estimate combined with Osgood's lemma. In order to obtain a better description of the regularizing effect of the transport-diffusion equation, the Chemin-Lerner type spaces and a generalization of the weighted Chemin-Lerner type norm are used. The main tools are the product laws in Besov space based on different Lebesgue spaces, the logarithmic interpolation, Bonny's decomposition (related with Littlewood-Paley decomposition), Gronwall's lemma. Some specific results obtained in [\textit{H. Bahouri} et al., Fourier analysis and nonlinear partial differential equations. Berlin: Heidelberg (2011; Zbl 1227.35004)] are also used.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 30H25
Mathematics Subject Classification ID: 76D08
zbMATH DE Number: 6879565
zbMATH Keywords: inhomogeneous Navier-Stokes systems
zbMATH Keywords: anisotropic Littlewood-Paley theory
zbMATH Keywords: large solutions
reviewed by: Gelu Paşa
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.05.011
OpenAlex ID: W2803946379
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Wikidata QID: Q129735491
DOI: 10.1016/J.JMAA.2018.05.011
description: scientific article; zbMATH DE number 6879565