Long time existence of regular solutions to 3d Navier–Stokes equations coupled with heat convection
Volume 39 / 2012
Abstract
We prove long time existence of regular solutions to the Navier–Stokes equations coupled with the heat equation. We consider the system in a non-axially symmetric cylinder, with the slip boundary conditions for the Navier–Stokes equations, and the Neumann condition for the heat equation. The long time existence is possible because the derivatives, with respect to the variable along the axis of the cylinder, of the initial velocity, initial temperature and external force are assumed to be sufficiently small in the $L_2$ norms. We prove the existence of solutions such that the velocity and temperature belong to $W_\sigma ^{2,1}(\varOmega \times (0,T))$, where $\sigma >{5/3}$. The existence is proved by using the Leray–Schauder fixed point theorem.
