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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Li T. Global Classical Solutions for Quasilinear Hyperbolic Systems. Paris: Masson; Chichester: John Wiley & Sons, Ltd, 1994
Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Fourth ed. Berlin: Springer-Verlag, 2016
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): 1–40
Bruce P J K, Babinsky H. Unsteady shock wave dynamics. J Fluid Mech, 2008, 603: 463–473
Cai H, Tan Z. Time periodic solutions to the compressible Navier-Stokes-Poisson system with damping. Commun Math Sci, 2017, 15(3): 789–812
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): 655–676
Jin C. Time-periodic solutions of the compressible Navier-Stokes equations in ℝ4. Z Angew Math Phys, 2016, 67(1): Art 5, 21 pp
Jin C, Yang T. Time periodic solution to the compressible Navier-Stokes equations in a periodic domain. Acta Math Sci, 2016, 36B(4): 1015–1029
Luo T. Bounded solutions and periodic solutions of viscous polytropic gas equations. Chinese Ann Math, Ser B, 1997, 18(1): 99–112
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: 49–82
Dafermos C M. Periodic BV solutions of hyperbolic balance laws with dissipative source. J Math Anal Appl, 2015, 428(1): 405–413
Frid H. Periodic solutions of conservation laws constructed through Glimm scheme. Trans Amer Math Soc, 2001, 353(11): 4529–4544
Frid H. Decay of almost periodic solutions of conservation laws. Arch Ration Mech Anal, 2002, 161(1): 43-64
Frid H, Perepelitsa M. Spatially periodic solutions in relativistic isentropic gas dynamics. Comm Math Phys, 2004, 250(2): 335–370
Glimm J, Lax P D. Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, 1970, (101)
Nishida T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc Japan Acad, 1968, 44: 642–646
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): 221–259
Wang Z, Zhang Q. Periodic solutions to p-system constructed through Glimm scheme. J Math Anal Appl, 2016, 435(2): 1088–1098
Takeno S. Time-periodic solutions for a scalar conservation law. Nonlinear Anal: TMA, 2001, 45(8): 1039–1060
Temple B, Young R. A Nash-Moser framework for finding periodic solutions of the compressible Euler equations. J Sci Comput, 2015, 64(3): 761–772
Bianchini S, Bressan A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann Math, 2005, 161(1): 223–342
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by the National Natural Science Foundation of China (11371141 and 11871218); Science and Technology Commission of Shanghai Municipality (STCSM) under Grant No. 18dz2271000.
Rights and permissions
About this article
Cite this article
Yuan, H. Time-Periodic Isentropic Supersonic Euler flows in One-Dimensional Ducts Driving by Periodic Boundary Conditions. Acta Math Sci 39, 403–412 (2019). https://doi.org/10.1007/s10473-019-0206-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-019-0206-6
Key words
- supersonic flow
- isentropic
- compressible Euler equations
- duct
- time-periodic solution
- initial-boundary-value problem