Long time estimate of solutions to 3d Navier-Stokes equations coupled with the heat convection

Abstract: We examine the Navier-Stokes equations with homogeneous slip boundary conditions coupled with the heat equation with homogeneous Neumann conditions in a bounded domain in $R^3$. The considered domain is a cylinder with $x_3$-axis. The aim of this paper is to show long time estimates without smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of $L_2$ norms of derivatives with respect to $x_3$ of the initial velocity, the initial temperature and the external force.