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

Abstract

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 $(1+{\epsilon }^{\sigma }{a}_0(x_h,\epsilon x_3),(\epsilon u_0^h(x_h,\epsilon x_3),u_0^3(x_h,\epsilon x_3)))$ 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.