Asymptotic stability of stationary waves to the Navier–Stokes–Poisson equations in half line

ABSTRACT

The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial perturbation by the elementary energy method. Finally, we discuss the convergence rate of the time-dependent solution towards the stationary solution, and give a new condition to ensure an algebraic decay or an exponential decay. The proof is based on a time and space weighted energy method by fully utilizing the self-consistent Poisson equation.

Keywords:

• Compressible Navier–Stokes–Poisson equations
• Boltzmann relation
• stationary solution
• asymptotic stability
• weighted energy method

2010 Mathematics Subject Classifications:

• 35B33
• 35B35
• 35B40

Acknowledgements

We would like to express our sincere appreciation to Professor Hailiang Li for his helpful discussion. Zhang’s work is partially supported by the National Science Foundation of China (No. 11771071). Wang’s work is partially supported by the “13th Five-Year” Science and Technology Project of the Education Department of Jilin Province (No. JJKH20200138KJ).

Disclosure statement

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

Additional information

Funding

Zhang's work is partially supported by the National Science Foundation of China [grant number 11771071]. Wang’s work is partially supported by the “13th Five-Year” Science and Technology Project of the Education Department of Jilin Province [grant number JJKH20200138KJ].