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Classical solutions to relativistic Burgers equations in FLRW space-times

  • Saisai Huo
  • Changhua WeiEmail author
Articles
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Abstract

We are interested in the classical solutions to the Cauchy problem of relativistic Burgers equations evolving in Friedmann-Lemat tre-Robertson-Walker (FLRW) space-times, which are spatially homogeneous, isotropic expanding or contracting universes. In such kind of space-times, we first derive the relativistic Burgers equations from the relativistic Euler equations by letting the pressure be zero. Then we can show the global existence of the classical solution to the derived equation in the accelerated expanding space-times with small initial data by the method of characteristics when the spacial dimension n = 1 and energy estimate when n ⩾ 2, respectively. Furthermore, we can also show the lifespan of the classical solution by similar methods when the expansion rate of the space-times is not so fast.

Keywords

FLRW space-times relativistic Burgers equations characteristics energy estimate 

MSC(2010)

53C44 53C21 58J45 35L45 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 91630311 and 11701517), Fundamental Research Funds for the Central Universities (Grant No. 2017XZZX007-02) and the Scientific Research Foundation of Zhejiang Sci-Tech University (Grant No. 16062021-Y).

References

  1. 1.
    Alinhac S. Blowup for nonlinear hyperbolic equations. Progr Nonlinear Differential Equations Appl, 2006, 17: 327–333Google Scholar
  2. 2.
    Alinhac S. Hyperbolic Partial Differential Equations. New York: Springer, 2009CrossRefzbMATHGoogle Scholar
  3. 3.
    Alinhac S. Geometric Analysis of Hyperbolic Differential Equations: An Introduction. Cambridge: Cambridge University Press, 2010CrossRefzbMATHGoogle Scholar
  4. 4.
    Brauer U, Rendall A, Reula O. The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models. Classical Quantum Gravity, 1994, 11: 2283–2296MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ceylan T, LeFloch P G, Okutmustur B. A finite volume method for the relativistic Burgers equation on a FLRW background spacetime. Commun Comput Phys, 2018, 23: 500–519MathSciNetGoogle Scholar
  6. 6.
    Courant R, Fridriches K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers, 1948Google Scholar
  7. 7.
    Gordon W B. On the diffeomorphisms of Euclidean space. Amer Math Monthly, 1972, 79: 755–759MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hörmander L. Lectures on Nonlinear Hyperbolic Differential Equations. New York: Springer, 1997zbMATHGoogle Scholar
  9. 9.
    John F. Nonlinear Wave Equations: Formation of Singularities. Providence: Amer Math Soc, 1990Google Scholar
  10. 10.
    Kong D X, Wei C H. Lifespan of smooth solutions for timelike extremal surface equation in de Sitter spacetime. J Math Phys, 2017, 58: 425–451MathSciNetzbMATHGoogle Scholar
  11. 11.
    LeFloch P G, Makhlof H, Okutmusur B. Relativistic Burgers equations on curved space-times: Derivation and finite volume approximation. SIAM J Numer Anal, 2012, 50: 2136–2158MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    LeFloch P G, Wei C H. The global nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FRW geometry. ArXiv:1512.03754, 2015Google Scholar
  13. 13.
    Majda A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer-Verlag, 1984Google Scholar
  14. 14.
    Oliynyk T. Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. Comm Math Phys, 2016, 346: 293–312MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Speck J. The nonlinear future stability of the FLRW family of solutions to the Euler-Einstein system with a positive cosmological constant. Selecta Math (NS), 2012, 18: 633–715MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Speck J. The stabilizing effect of space-times expansion on relativistic fluids with sharp results for the radiation equation of state. Arch Ration Mech Anal, 2013, 210: 535–579MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wei C H, Lai N A. Global existence of smooth solutions to exponential wave maps in FLRW spacetimes. Pacific J Math, 2017, 289: 489–509MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouChina

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