On the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin domain

Abstract

In this work, we will show the global existence of the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin three dimensional domain $\Omega=\mathbb R^2\times [0,\epsilon]$, with Dirichlet boundary condition on the top and bottom boundary: the global well-posedness may hold for large initial data when the vertical size $\epsilon$ is sufficiently small. Furthermore, when $\epsilon\rightarrow 0$ the velocity tends to vanish away from the initial time. The analysis relies on the a priori $H^2$-estimate for the solutions (similar as in [4, 5, 21]) and one pays attention to the dependence on the vertical size $\epsilon$.