Global well-posedness of 3-D inhomogeneous Navier–Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field

Motivated by [21], we consider the global wellposedness to the 3-D incompressible inhomogeneous Navier–Stokes equations with large horizontal velocity. In particular, we proved that when the initial density is close enough to a positive constant, then given divergence free initial velocity field of...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2018-09, Vol.465 (1), p.459-499
Main Authors: Chen, Yuhui, Huang, Jingchi
Format: Article
Language:English
Subjects:
Anisotropic Littlewood–Paley Theory
Inhomogeneous Navier–Stokes systems
Large solutions
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Summary:Motivated by [21], we consider the global wellposedness to the 3-D incompressible inhomogeneous Navier–Stokes equations with large horizontal velocity. In particular, we proved that when the initial density is close enough to a positive constant, then given divergence free initial velocity field of the type (v0h,0)(xh)+(w0h,w03)(xh,x3), we shall prove the global wellposedness of (1.1). The main difficulty here lies in the fact that we will have to obtain the L1(R+;Lip(R3)) estimate for convection velocity in the transport equation of (1.1). Toward this and due to the strong anisotropic properties of the approximate solutions, we will have to work in the framework of anisotropic Littlewood–Paley theory here.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2018.05.011