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The existence and stability of stationary solutions of the inflow problem for full compressible Navier-Stokes-Poisson system

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Abstract

In this paper, we consider an inflow problem for the non-isentropic Navier-Stokes-Poisson system in a half line (0, ∞). For the general gas including ideal polytropic gas, we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4 × 4 system of autonomous ordinary differential equations. We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.

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References

  1. Caar J. Applications of Center Manifold Theory. Springer-Verlag, 1981

  2. Cui H, Gao Z, Yin H Y, Zhang F. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: existence stability and convergence rate. Discrete Contin Dyn Sys, 2016, 36(9): 4839–4870

    Article  MathSciNet  Google Scholar 

  3. Duan R J, Yang X F. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Comm Pure Appl Anal, 2013, 12: 985–1014

    Article  MathSciNet  Google Scholar 

  4. Duan R J, Liu S Q. Stability of rarefaction waves of the Navier-Stokes-Poisson system. J Differ Equ, 2015, 258: 2495–2530

    Article  MathSciNet  Google Scholar 

  5. Duan R J, Liu S Q, Yin H Y, Zhu C Z. Stability of the rarefaction wave for a two-fluid Plasma model with diffusion. Science China: Math, 2016, 59(1): 67–84

    Article  MathSciNet  Google Scholar 

  6. Fan L, Liu H, Wang T, Zhao H J. Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J Differ Equ, 2014, 257(10): 3521–3553

    Article  MathSciNet  Google Scholar 

  7. Hong H H, Shi X, Wang T. Stability of stationary solutions to the inflow problem for the two-fluid nonisentropic Navier-Stokes-Poisson system. J Differ Equ, 2018, 265: 1129–1155

    Article  Google Scholar 

  8. Hong H H, Wang T. Stability of stationary solutions to the inflow problem for full compressible Navier-Stokes equations with a large initial perturbation. SIAM J Math Anal, 2017, 49(3): 2138–2166

    Article  MathSciNet  Google Scholar 

  9. Hong H H, Wang T. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations with large perturbation. Nonlinearity, 2017, 30: 3010–3039

    Article  MathSciNet  Google Scholar 

  10. Huang F M, Li J, Shi X. Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun Math Sci, 2010, 8: 639–654

    Article  MathSciNet  Google Scholar 

  11. Huang F M, Matsumura A, Shi X. Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Comm Math Phys, 2009, 239: 261–285

    Article  MathSciNet  Google Scholar 

  12. Jiang M, Lai S, Yin H Y, Zhu C J. The stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. Acta Math Sci, 2016, 36B(4): 1098–1116

    Article  MathSciNet  Google Scholar 

  13. Liu S Q, Yin H Y, Zhu C J. Stability of contact discontinuity for Navier-Stokes-Poisson system with free boundary. Commun Math Sci, 2016, 14(2): 1859–1887

    Article  MathSciNet  Google Scholar 

  14. Matsumura A, Nishihara K. Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Comm Math Phys, 2001, 222(3): 449–474

    Article  MathSciNet  Google Scholar 

  15. Nakamura T, Nishibata S. Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas. J Hyper Differ Equ, 2011, 8(4): 657–670

    MathSciNet  MATH  Google Scholar 

  16. Qin X, Wang Y. Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2009, 41: 2057–2087

    Article  MathSciNet  Google Scholar 

  17. Qin X, Wang Y. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2011, 43: 341–346

    Article  MathSciNet  Google Scholar 

  18. Ruan L, Yin H Y, Zhu C J. Stability of the superposition of rarefaction wave and contact discontinuity for the non-isentropic Navier-Stokes-Poisson system. Math Meth Appl Sci, 2017, 40: 2784–2810

    Article  MathSciNet  Google Scholar 

  19. Wang L, Zhang G J, Zhang K J. Existence and stability of stationary solution to compressible Navier-Stokes-Poisson equations in half line. Nonlinear Analysis, 2016, 145: 97–117

    Article  MathSciNet  Google Scholar 

  20. Yin H Y, Zhang J S, Zhu C J. Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system. Nonlinear Anal Real World Appl, 2016, 31: 492–512

    Article  MathSciNet  Google Scholar 

  21. Zhou F, Li Y P. Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line. Bound Value Probl, 2013: 1–22

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Authors and Affiliations

  1. Institute of Mathematics, State Academy of Sciences, Pyongyang, Korea

    Hakho Hong

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  1. Hakho Hong
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Correspondence to Hakho Hong.

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Hong, H. The existence and stability of stationary solutions of the inflow problem for full compressible Navier-Stokes-Poisson system. Acta Math Sci 41, 319–336 (2021). https://doi.org/10.1007/s10473-021-0119-z

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  • DOI: https://doi.org/10.1007/s10473-021-0119-z

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