Long time estimate of solutions to 3d Navier–Stokes equations coupled with heat convection
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 $\mathbb R^3$. The domain is a cylinder along the $x_3$ axis. The aim of this paper is to show long time estimates without assuming smallness of the initial velocity, the initial temperature and the external force. To prove the estimate we need however smallness of the $L_2$ norms of the $x_3$-derivatives of these three quantities.