Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. (English) Zbl 1346.83023

Summary: 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 \(p=K\rho\). 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 \(0<K\leq 1/3\) in a uniform manner. It also determines the asymptotic behavior of the perturbed solutions in the far future.


83C15 Exact solutions to problems in general relativity and gravitational theory
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
83F05 Relativistic cosmology
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
Full Text: DOI arXiv


[1] Adams R.A., Fournier J.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003) · Zbl 1098.46001
[2] Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford University Press, Oxford (2009) · Zbl 1157.83002
[3] Friedman A.: Partial Differential Equations. Krieger Publishing Company, Malabar (1976)
[4] Friedrich, H., On the hyperbolicity of einstein’s and other gauge field equations, Commun. Math. Phys., 100, 525-543, (1985) · Zbl 0588.35058
[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) · Zbl 0659.53056
[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) · Zbl 0737.53070
[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) · Zbl 1333.35281
[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) · Zbl 1263.83188
[9] Oliynyk, T.A., Cosmological post-Newtonian expansions to arbitrary order, Commun. Math. Phys., 295, 431-463, (2010) · Zbl 1195.35286
[10] Oliynyk, T.A., The cosmological Newtonian limit on cosmological scales, Commun. Math. Phys., 339, 455-512, (2015) · Zbl 1352.35120
[11] Ringstöm, H., Future stability of the Einstein-non-linear scalar field system, Invent. Math., 173, 123-208, (2008) · Zbl 1140.83314
[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) · Zbl 1294.35164
[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) · Zbl 1251.83071
[14] Taylor M.E.: Partial Differential Equations III: Nonlinear Equations. Springer, Berlin (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.