Long‐time existence of solutions to the Navier–Stokes equations with inflow–outflow and heat convection

Abstract

Long time existence of regular solutions to the Navier–Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in cylindrical pipe with inflow and outflow is shown. We assume the slip boundary conditions for velocity and the Neumann conditions for temperature. First, an appropriate estimate is shown, and next the existence of solutions is proved by the Leray–Schauder fixed point theorem. The estimate is obtained for a long time, which is possible because L2 norms of derivatives in the direction along the cylinder of the initial velocity, initial temperature and the external force are sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.