Long time existence of solutions to 2D Navier-Stokes equations with inflow-outflow and heat convection

Abstract

Global 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 in the two-dimensional case is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence of solutions is proved by the Leray-Schauder fixed point theorem.