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Acta Mathematica Scientia

, Volume 39, Issue 2, pp 403–412 | Cite as

Time-Periodic Isentropic Supersonic Euler flows in One-Dimensional Ducts Driving by Periodic Boundary Conditions

  • Hairong YuanEmail author
Article
  • 1 Downloads

Abstract

We show existence of time-periodic supersonic solutions in a finite interval, after certain start-up time depending on the length of the interval, to the one space-dimensional isentropic compressible Euler equations, subjected to periodic boundary conditions. Both classical solutions and weak entropy solutions, as well as high-frequency limiting behavior are considered. The proofs depend on the theory of Cauchy problems of genuinely nonlinear hyperbolic systems of conservation laws.

Key words

supersonic flow isentropic compressible Euler equations duct time-periodic solution initial-boundary-value problem 

2010 MR Subject Classification

35B10 35L04 76G20 

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References

  1. [1]
    Li T. Global Classical Solutions for Quasilinear Hyperbolic Systems. Paris: Masson; Chichester: John Wiley & Sons, Ltd, 1994zbMATHGoogle Scholar
  2. [2]
    Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Fourth ed. Berlin: Springer-Verlag, 2016zbMATHGoogle Scholar
  3. [3]
    Li T, Yu L. One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws. J Math Pures Appl, 2017, 107(1): 1–40MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bruce P J K, Babinsky H. Unsteady shock wave dynamics. J Fluid Mech, 2008, 603: 463–473CrossRefzbMATHGoogle Scholar
  5. [5]
    Cai H, Tan Z. Time periodic solutions to the compressible Navier-Stokes-Poisson system with damping. Commun Math Sci, 2017, 15(3): 789–812MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chen S H, Hsia C H, Jung C Y, et al. Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers’ equation. J Math Anal Appl, 2017, 445(1): 655–676MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Jin C. Time-periodic solutions of the compressible Navier-Stokes equations in ℝ4. Z Angew Math Phys, 2016, 67(1): Art 5, 21 ppMathSciNetCrossRefGoogle Scholar
  8. [8]
    Jin C, Yang T. Time periodic solution to the compressible Navier-Stokes equations in a periodic domain. Acta Math Sci, 2016, 36B(4): 1015–1029MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Luo T. Bounded solutions and periodic solutions of viscous polytropic gas equations. Chinese Ann Math, Ser B, 1997, 18(1): 99–112MathSciNetzbMATHGoogle Scholar
  10. [10]
    Matsumura A, Nishida T. Periodic solutions of a viscous gas equation//Recent Topics in Nonlinear PDE, IV (Kyoto, 1988). North-Holland Math Stud 160. Amsterdam: North-Holland, 1989: 49–82Google Scholar
  11. [11]
    Dafermos C M. Periodic BV solutions of hyperbolic balance laws with dissipative source. J Math Anal Appl, 2015, 428(1): 405–413MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Frid H. Periodic solutions of conservation laws constructed through Glimm scheme. Trans Amer Math Soc, 2001, 353(11): 4529–4544MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Frid H. Decay of almost periodic solutions of conservation laws. Arch Ration Mech Anal, 2002, 161(1): 43-64MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Frid H, Perepelitsa M. Spatially periodic solutions in relativistic isentropic gas dynamics. Comm Math Phys, 2004, 250(2): 335–370MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Glimm J, Lax P D. Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, 1970, (101)Google Scholar
  16. [16]
    Nishida T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc Japan Acad, 1968, 44: 642–646MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Qu P, Xin Z. Long time existence of entropy solutions to the one-dimensional non-isentropic Euler equations with periodic initial data. Arch Ration Mech Anal, 2015, 216(1): 221–259MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Wang Z, Zhang Q. Periodic solutions to p-system constructed through Glimm scheme. J Math Anal Appl, 2016, 435(2): 1088–1098MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Takeno S. Time-periodic solutions for a scalar conservation law. Nonlinear Anal: TMA, 2001, 45(8): 1039–1060MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Temple B, Young R. A Nash-Moser framework for finding periodic solutions of the compressible Euler equations. J Sci Comput, 2015, 64(3): 761–772MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Bianchini S, Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann Math, 2005, 161(1): 223–342MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematical Sciences; Shanghai Key Laboratory of Pure Mathematics and Mathematical PracticeEast China Normal UniversityShanghaiChina

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