Full compressible Navier-Stokes equations with the Robin boundary condition on temperature

Abstract

We consider full compressible Navier-Stokes equations with the Robin boundary condition on temperature. Note that the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. It is shown that a unique global strong solution existed if the initial data belongs to $H^1$. Subsequently, we find that the strong solution is nonlinearly exponentially stable as time tends to infinity. This result could be viewed as the first one on the global well-posedness of the strong solution to full Navier-Stokes equations in a bounded domain with the degenerate heat conductivity and the Robin boundary condition on temperature. The proofs are mainly based on the energy method and a special inequality.

Keywords:

• Degenerate heat conductivity
• Navier-Stokes equations
• Robin boundary value problem
• strong solution

AMS Classifications:

• 35B40
• 35Q30
• 76N06
• 76N10

Acknowledgments

The author would like to thank the referee for his/her careful reading and helpful suggestions on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).