### 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.

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.

#### Keywords

Navier-Stokes equations; heat convection; slip boundary condition; Neumann condition; global existence; Leray-Schauder fixed point theorem

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