Convergence rate of stationary solutions to outflow problem for full Navier–Stokes equations

ABSTRACT

This paper is concerned with the large-time behavior of solutions to the outflow problem of full compressible Navier–Stokes equations for the general gas including ideal polytropic gas in the half line $R_{+}=(0,\infty )$, where the gas flows out through the boundary. We first give some necessary and sufficient conditions for the existence of the stationary solution with the aid of center manifold theory. We also show the time asymptotic stability of the nondegenerate stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the exponential and algebraic decay of the solution toward the supersonic stationary solution is obtained, provided that the initial perturbation belongs to the Sobolev space with exponential and algebraic weight respectively.

KEYWORDS:

• Compressible Navier–Stokes equations
• outflow problem
• stationary solution
• stability
• convergence rate

AMS SUBJECT CLASSIFICATIONS:

• 35Q30
• 35B35
• 35L65
• 76D33
• 74J40

Acknowledgements

The authors would like to thank the referees for their valuable comments.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially supported by National Natural Sciences Foundation of China [grant number 11671027], [grant number 11471321], [grant number 11371348].