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Acta Mathematica Scientia

, Volume 39, Issue 4, pp 1195–1212 | Cite as

Asymptotic Stability of the Rarefaction Wave for the Non-Viscous and Heat-Conductive Ideal Gas in Half Space

  • Meichen Hou (侯美晨)Email author
Article
  • 4 Downloads

Abstract

This article is concerned with the impermeable wall problem for an ideal poly-tropic model of non-viscous and heat-conductive gas in one-dimensional half space. It is shown that the 3-rarefaction wave is stable under some smallness conditions. The proof is given by an elementary energy method and the key point is to do the higher order derivative estimates with respect to t because of the less dissipativity of the system and the higher order derivative boundary terms.

Key words

Non-viscous impermeable problem rarefaction wave 

2010 MR Subject Classification

00A69 35B40 35M33 

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References

  1. [1]
    Hong H, Huang F. Asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave for compressible Navier-Stokes equations with free boundary. Acta Mathematica Scientia, 2012, 32B(1): 389–412MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Fan L, Matsumura A. Asymptotic stability of a composite wave of two viscous shock waves for a one-dimensional system of non-viscous and heat-conductive ideal gas. J Differential Equations, 2015, 258(4): 1129–1157MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Hou C, Fan L. Asymptotic stability for the inflow problem of the heat-conductive ideal gas without viscosity. PreprintGoogle Scholar
  4. [4]
    Huang F, Wang Y, Zhai X, Stability of viscous contact wave for compressible Navier-Stokes system of general gas with free boundary. Acta Mathematica Scientia, 2010, 30B(6): 1906–1916MathSciNetzbMATHGoogle Scholar
  5. [5]
    Huang F, Li J, Matsumura A. Stability of the combination of the viscous contact wave and the rarefaction wave to the compressible Navier-Stokes equations. Arch Rat Mech Anal, 2010, 197(1): 89–116CrossRefzbMATHGoogle Scholar
  6. [6]
    Huang F, Li J, Shi X. Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun Math Sci, 2010, 8: 639–654MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Huang F, Matsumura A. Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compresible Navier-Stokes Equation. Comm Math Phys, 2009, 289(3): 841–861MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Huang F, Matsumura A, Shi X. Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Comm Math Phys, 2003, 239(1/2): 261–285MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Huang F, Matsumura A, Xin Z. Stability of Contact discontinuties for the 1-D Compressible Navier-Stokes equations. Arch Ration Mech Anal, 2005, 179(1): 55–77CrossRefzbMATHGoogle Scholar
  10. [10]
    Huang F, Shi X, Wang Y. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinet Relat Models, 2010, 3(3): 409–425MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Huang F, Xin Z, Yang T. Contact discontinuity with general perturbations for gas motions. Adv in Math, 2008, 219(4): 1246–1297MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Huang F, Zhao H. On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend Sem Mat Univ Padova, 2003, 109: 283–305MathSciNetzbMATHGoogle Scholar
  13. [13]
    Kawashima S, Nakamura T, Nishibata S, et al. Stationary waves to viscous heat-conductive gases in half space:existence, stability and convergence rate. Math Models Methods Appl Sci, 2010, 20(12): 2201–2235MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Kawashima S, Nishibata S, Zhu P. Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. Commun Math Phys, 2003, 240(3): 483–500MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Kawashima S, Zhu P. Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space. Arch Ration Mech Anal, 2009, 194(1): 105–132MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Kwon B, Suzuki M, Takayama M. Large-time behavior of solutions to an outflow problem for a shallow water model. J Differential Equations, 2013, 255: 1883–1904MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Liu H, Yang T, Zhao H, et al. One-dimensional compressible Navier-Stokes equtions with temperature dependent transport coefficients and large data. SIAM J Math Anal, 2014, 46(3): 2185–2228MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Liu T, Shock waves for Compresible Navier-Stokes Equations are stable. Comm Pure Appl Math, 1986, 39(5): 565–594MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Matsumura A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl Anal, 2001, 8(4): 645–666MathSciNetzbMATHGoogle Scholar
  20. [20]
    Matsumura A. Large-time behavior of solutions for a one-dimensional system of non-viscous and heat-conductive ideal gas. Private Communication, 2016Google Scholar
  21. [21]
    Matsumura A, Mei M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch Ration Mech Anal, 1999, 146(1): 1–22MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Matsumura A, Nishihara K. Global asymptotics toward the rarefaction waves for solutions of viscous p-system with boundary effect. Quart Appl Math, 2000, 58(1): 69–83MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    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): 449–474MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Min L, Qin X. Stability of rarefaction wave for compressible Navier-Stokes equations on the half line. Acta Math Appl Sin Engl Ser, 2016, 32(1): 175–186MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Nakamura T, Nishibata S. Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half-line. Math Models Methods Appl Sci, 2017, 27(11): 2071–2110MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Nakamura T, Nishibata S, Usami N. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinet Relat Models, 2018, 11(4): 757–793MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Nishihara K, Yang T, Zhao H. Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. SIAM J Math Anal, 2004, 35(6): 1561–1597MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Qin X. Large-time behaviour of solution to the outflow problem of full compressible Navier-Stokes equations. Nonlinearity, 2011, 24(5): 1369–1394MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Qin X, Wang Y. Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2009, 41(5): 2057–2087MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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(1): 341–366MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Rauch J, Massey F. Differentiability of solutions to hyperbolic initial-boundary value problems. Trans Amer Math Soc, 1974, 189: 303–318MathSciNetzbMATHGoogle Scholar
  32. [32]
    Zheng S. Nonlinear parabolic equations and hyperbolic-parabolic coupled systems. Longman Group, 1995zbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.Institute of Applied MathematicsAMSSBeijingChina
  3. 3.China Academy of Mathematics and Systems ScienceAcademia SinicaBeijingChina

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