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

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+εσa0(xh,εx3),(εu0h(xh,εx3),u03(xh,εx3))) for some σ>0σ>0 and ε being sufficiently small. We remark that initial data of this type does not satisfy the smallness conditions in [11] and [18] no matter how small ε is.