On some large global solutions to 3-D density-dependent Navier–Stokes system with slow variable: Well-prepared data

Paicu, Marius; Zhang, Ping

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 813-832

In this paper, we consider the global wellposedness of 3-D incompressible inhomogeneous Navier–Stokes equations with initial data slowly varying in the vertical variable, that is, initial data of the form $(1+{ϵ}^{\sigma }{a}_0(x_h,ϵx_3),(ϵu_0^h(x_h,ϵx_3),u_0^3(x_h,ϵx_3)))$ for some $\sigma >0$ and ε being sufficiently small. We remark that initial data of this type does not satisfy the smallness conditions in [11,18] no matter how small ε is.

Classification : 35Q30, 76D03
Keywords: Inhomogeneous Navier–Stokes equations, Littlewood–Paley theory, Wellposedness

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