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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alinhac S. Blowup for nonlinear hyperbolic equations. Progr Nonlinear Differential Equations Appl, 2006, 17: 327–333
Alinhac S. Hyperbolic Partial Differential Equations. New York: Springer, 2009
Alinhac S. Geometric Analysis of Hyperbolic Differential Equations: An Introduction. Cambridge: Cambridge University Press, 2010
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: 2283–2296
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: 500–519
Courant R, Fridriches K O. Supersonic Flow and Shock Waves. New York: Interscience Publishers, 1948
Gordon W B. On the diffeomorphisms of Euclidean space. Amer Math Monthly, 1972, 79: 755–759
Hörmander L. Lectures on Nonlinear Hyperbolic Differential Equations. New York: Springer, 1997
John F. Nonlinear Wave Equations: Formation of Singularities. Providence: Amer Math Soc, 1990
Kong D X, Wei C H. Lifespan of smooth solutions for timelike extremal surface equation in de Sitter spacetime. J Math Phys, 2017, 58: 425–451
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: 2136–2158
LeFloch P G, Wei C H. The global nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FRW geometry. ArXiv:1512.03754, 2015
Majda A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer-Verlag, 1984
Oliynyk T. Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. Comm Math Phys, 2016, 346: 293–312
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: 633–715
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: 535–579
Wei C H, Lai N A. Global existence of smooth solutions to exponential wave maps in FLRW spacetimes. Pacific J Math, 2017, 289: 489–509
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huo, S., Wei, C. Classical solutions to relativistic Burgers equations in FLRW space-times. Sci. China Math. 63, 357–370 (2020). https://doi.org/10.1007/s11425-017-9309-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-017-9309-7