EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM

doi:10.1007/s10473-022-0108-x

Abstract

Abstract: In this paper, we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space. First, we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane. Based on these properties, we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme, and prove that there exists an entropy solution $V(x,t)$ which belongs to $L^{\infty}\cap {\rm BV}_{\rm loc}(\mathbb{R}\times\mathbb{R}_+)$.

Key words: isothermal relativistic Euler system, Glimm scheme, Riemann problem, BV space, periodicity

CLC Number:

76Y05

Fei WU, Zejun WANG. EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM[J].Acta mathematica scientia,Series B, 2022, 42(1): 155-171.

Trendmd

References

[1] Nishida T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc Jap Acad, 1968, 44(7):642-646
[2] Bakhvalov N S. The existence in the large of a regular solution of a quasilinear hyperbolic system. Ussr Comp Math Math Phys, 1970, 10:969-980
[3] Diperna R J. Global solutions to a class of nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1973, 26:1-28
[4] Diperna R J. Existence in the large for quasilinear hyperbolic conservation laws. Arch Ration Mech Anal, 1973, 52(3):244-257
[5] Ding S S, Chang T, Wang C H, et al. A study of the global solutions for quasi-linear hyperbolic systems of conservation laws. Sci China Ser A, 1973, 16(3):317-335
[6] Nishida T, Smoller J. Solutions in the large for some nonlinear hyperbolic conservation laws. Comm Pure Appl Math, 1973, 26(2):183-200
[7] Frid H. Periodic solutions of conservation laws constructed through Glimm scheme. Trans Amer Math Soc, 2001, 353:4529-4544
[8] Wang Z, Zhang Q. Periodic solutions to p-system constructed through Glimm scheme. J Math Anal Appl, 2016, 435(2):1088-1098
[9] Smoller J, Temple B. Global solutions of the relativistic Euler equations. Commun Math Phys, 1993, 156(1):67-99
[10] Chen J. Conservation laws for relativistic fluid dynamics. Arch Rational Mech Anal, 1997, 139(4):377-398
[11] Li Y, Feng D, Wang Z. Global entropy solutions to the relativistic Euler equations for a class of large initial data. Z Angew Math Phy, 2005, 56(2):239-253
[12] Makino T, Ukai S. Local smooth solutions of the relativistic Euler equation. J Math Kyoto Univ, 1995, 3:365-375
[13] Ruan L, Zhu C. Existence of global smooth solution to the relativistic Euler equations. Nonlinear Anal, 2005, 60(6):993-1001
[14] Hsu C H, Lin S S, Makino T. On the relativistic Euler equation. Meth Appl Anal, 2001, 8(1):159-208
[15] Hsu C H, Makino T. Spherically symmetric solutions to the compressible Euler equation with an asymptotic γ-law. Japan J Indust Appl Math, 2003, 20(1):1-15
[16] Geng Y. Steady state solutions of relativistic Euler equations with spherical symmetry. Acta Mathematica Scientia, 2014, 34A(4):841-850
[17] Frid H, Perepelitsa M. Spatially periodic solutions in relativistic isentropic gas dynamics. Commun Math Phys, 2004, 250(2):335-370
[18] Chen G Q, Li Y. Relativistic Euler equations for isentropic fluids:stability of Riemann solutions with large oscillation. Z Angew Math Phys, 2004, 55(6):903-926
[19] Min L, Ukai S. Nonrelativistic global limits of weak solutions of the relativistic Euler equation. J Math Kyoto Univ, 1998, 38:525-537
[20] Li Y, Geng Y. Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations. Z Angew Math Phy, 2006, 57(6):960-983
[21] Li Y, Wang L. Global stability of solutions with discontinuous initial data containing vacuum states for the relativistic Euler solutions. Chin Ann Math, 2005, 26(4):491-510
[22] Li Y, Wang A. Global entropy solutions of the Cauchy problem for nonhomogeneous relativistic Euler system. Chin Ann Math, 2006, 27(5):473-494
[23] Kunik M, Qamar S, Warnecke G. Second-order accurate kinetic schemes for the ultra-relativistic Euler equations. J Comput Phys, 2003, 192:695-726
[24] Safronov A V. Kinetic schemes for the gas dynamics equations. Vychisl Metody Programm, 2009:62-74
[25] Poupaud F, Rascle M, Vila J P. Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. J Diff Eqs, 1995, 123(1):93-121
[26] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. Berlin:Springer-Verlag, 2005
[27] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York:Springer-Verlag, 1994
[28] Wang D, Wang Z. Large BV solutions to the compressible isothermal Euler-Poisson equations with spherical symmetry. Nonlinearity, 2006, 19(8):1985-2004