Minimum principle of the temperature in compressible Navier-Stokes equations with application to the existence theory

Abstract

This thesis is on the Navier-Stokes equations which model the motion of compressible viscous fluid. A minimum principle on the temperate variable is established. Under the thermo-insulated boundary conditions and some reasonable assumptions on the solution, the minimum of the temperature does not increase. To our best knowledge, that's the first result on the minimum principle of the temperature variable in the compressible Navier-Stokes equation.

As an application of the minimum principle, global in time existence of the weak solution for the Navier-Stokes equations is established when the viscosities and heat conductivity are power functions of the temperature. In this model the temperature is coupled with density which may have vacuum or concentration and the heat conductivity has possible degeneracy. However the temperature is proved to obey the minimum principle, which secured the dissipative mechanism of the system, and paved the road to the existence theory.