Global existence for full compressible Navier-Stokes equations around the Couette flow with a temperature gradient in an infinite channel

TL;DR

This paper proves the global existence of strong solutions for the full compressible Navier-Stokes equations around a Couette flow with a temperature gradient in an infinite channel, under low Mach and Reynolds numbers and small temperature differences.

Contribution

It establishes the global existence of solutions near Couette flow with temperature gradient and analyzes the low Mach number limit in this setting.

Findings

Global strong solutions exist under specified conditions.

Solutions persist for all time near the Couette flow.

Low Mach number limit is validated for equal wall temperatures.

Abstract

This paper is concerned with the two-dimensional full compressible Navier-Stokes equations between two infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.