Abstract
In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces \({\mathcal{B}}\) and \({\mathcal{B}^{-\frac12,\frac12}_4}\) (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that (ANS ν ) has a unique global solution with initial data \({u_0 = (u_0^h, u_0^3)}\) which satisfies \({\|u_0^h\|_{\mathcal{B}} \exp\bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}}^4\bigr) \leq c_0\nu}\) or \({\|u_0^h\|_{\mathcal{B}^{-\frac12,\frac12}_{4}} \exp \bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}^{-\frac12,\frac12}_{4}}^4\bigr)\leq c_0\nu}\) for some c 0 sufficiently small. To overcome the difficulty that Gronwall’s inequality can not be applied in the framework of Chemin-Lerner type spaces, \({\widetilde{L^p_t}(\mathcal{B})}\), we introduced here sort of weighted Chemin-Lerner type spaces, \({\widetilde{L^2_{t, f}}(\mathcal{B})}\) for some apropriate L 1 function f(t).
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Paicu, M., Zhang, P. Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces. Commun. Math. Phys. 307, 713–759 (2011). https://doi.org/10.1007/s00220-011-1350-6
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DOI: https://doi.org/10.1007/s00220-011-1350-6