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Future Stability of the FLRW Fluid Solutions in the Presence of a Positive Cosmological Constant

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Abstract

We introduce a new method for establishing the future non-linear stability of perturbations of FLRW solutions to the Einstein–Euler equations with a positive cosmological constant and a linear equation of state of the form p = Kρ. The method is based on a conformal transformation of the Einstein–Euler equations that compactifies the time domain and can handle the equation of state parameter values 0 < K ≤ 1/3 in a uniform manner. It also determines the asymptotic behavior of the perturbed solutions in the far future.

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References

  1. Adams R.A., Fournier J.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)

    MATH  Google Scholar 

  2. Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2009)

    MATH  Google Scholar 

  3. Friedman A.: Partial Differential Equations. Krieger Publishing Company, Malabar (1976)

    Google Scholar 

  4. Friedrich H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys. 100, 525–543 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Friedrich H.: On the existence of n-geodesically complete or future complete solutions of Einsteins field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587–609 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Friedrich H.: On the global existence and the asymptotic behavior of solutions to the Einstein–Maxwell–Yang–Mills equations. J. Differ. Geom. 34, 275–345 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Hadžić M., Speck J.: The global future stability of the FLRW solutions to the Dust–Einstein system with a positive cosmological constant. J. Hyper. Differ. Equ. 12, 87–188 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lübbe C., Valiente~Kroon J.A.: A conformal approach for the analysis of the non-linear stability of radiation cosmologies. Ann. Phys. 328, 1–25 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Oliynyk T.A.: Cosmological post-Newtonian expansions to arbitrary order. Commun. Math. Phys. 295, 431–463 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Oliynyk T.A.: The cosmological Newtonian limit on cosmological scales. Commun. Math. Phys. 339, 455–512 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Ringstöm H.: Future stability of the Einstein-non-linear scalar field system. Invent. Math. 173, 123–208 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  12. Rodnianski I., Speck J.: The stability of the irrotational Euler–Einstein system with a positive cosmological constant. J. Eur. Math. Soc. 15, 2369–2462 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Speck J.: The nonlinear future-stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Selecta Math. 18, 633–715 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Taylor M.E.: Partial Differential Equations III: Nonlinear Equations. Springer, Berlin (1996)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    Todd A. Oliynyk

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  1. Todd A. Oliynyk
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Correspondence to Todd A. Oliynyk.

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Communicated by P. T. Chruściel

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Oliynyk, T.A. Future Stability of the FLRW Fluid Solutions in the Presence of a Positive Cosmological Constant. Commun. Math. Phys. 346, 293–312 (2016). https://doi.org/10.1007/s00220-015-2551-1

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  • DOI: https://doi.org/10.1007/s00220-015-2551-1

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