Global well-posedness of 3-D inhomogeneous Navier-Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field
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.
scientific article; zbMATH DE number 6879565
scientific article