## Cosmological post-Newtonian expansions to arbitrary order.(English)Zbl 1195.35286

Summary: We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter $${\varepsilon=v_T/c}$$ $${(0< \varepsilon < \varepsilon_0)}$$, where $$c$$ is the speed of light, and $$v _{T }$$ is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab $${M\cong [0,T)\times \mathbb {T}^3}$$, and converge as $${\varepsilon \searrow 0}$$ to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter $${\varepsilon}$$ to any specified order with expansion coefficients that satisfy $${\varepsilon}$$-independent (nonlocal) symmetric hyperbolic equations.

### MSC:

 35Q76 Einstein equations 35C20 Asymptotic expansions of solutions to PDEs 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 76Y05 Quantum hydrodynamics and relativistic hydrodynamics
Full Text:

### References:

 [1] Blanchet, L.: Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries. Living Rev. Relativity 9, (2006), 4, available at http://www.livingreviews.org/lrr-2006-4 · Zbl 1316.83004 [2] Brauer U., Karp L.: Local existence of classical solutions of the system using weighted sobolev spaces of fractional order. Les Comptes l’Académie des Sciences / Série Math. 345, 49–54 (2007) · Zbl 1122.35144 [3] Brauer U., Rendall A., Reula O.: The cosmic no-hair theorem and the nonlinear stability of homogeneous Newtonian cosmological models. Class. Quant. Grav. 11, 2283–2296 (1994) · Zbl 0815.53092 [4] Browning G., Kreiss H.O.: Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math. 42, 704–718 (1982) · Zbl 0506.35006 [5] Chugreev Y.V.: Post-Newtonian approximation of the relativistic theory of gravitation on a cosmological background. Theor. Math. Phys. 82, 328–333 (1990) · Zbl 0703.53066 [6] Deimling K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin (1998) · Zbl 1257.47059 [7] Futamase T.: Averaging of a locally inhomogeneous realistic universe. Phys. Rev. D 53, 681–689 (1996) [8] Futamase, T., Itoh, Y.: The Post-Newtonian Approximation for Relativistic Compact Binaries. Living Rev. Relativity 10 (2007), 2, available at http://www.livingreviews.org/lrr-2007-2 · Zbl 1255.83005 [9] Hwang J., Noh H.: Newtonian versus relativistic nonlinear cosmology. Gen. Rel. Grav. 38, 703–710 (2006) · Zbl 1093.83052 [10] Hwang J., Noh H., Puetzfeld D.: Cosmological nonlinear hydrodynamics with post-Newtonian corrections. JCAP 3, 10 (2008) [11] Heilig U.: On the Existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995) · Zbl 0813.53058 [12] Iriondo M.S., Leguizamón E.O., Reula O.A.: Fast and slow solutions in general relativity: the initialization procedure. J. Math. Phys. 39, 1555–1565 (1998) · Zbl 1056.83504 [13] Ishibashi A., Wald R.M.: Can the acceleration of our universe be explained by the effects of inhomogeneities?. Class. Quant. Grav. 23, 235–250 (2006) · Zbl 1148.83027 [14] Klainerman S., Majda A.: Compressible and incompressible fluids. Comm. Pure Appl. Math. 35, 629–651 (1982) · Zbl 0489.76081 [15] Kreiss H.O.: Problems with different time scales for partial differential equations. Comm. Pure Appl. Math. 33, 399–439 (1980) · Zbl 0439.35043 [16] Kreiss H.O.: Problems with different time scales. Acta Numerical 1, 101–139 (1991) · Zbl 0770.65045 [17] Künzle H.P.: Covariant Newtonian limit of Lorentz space-times. Gen. Rel. Grav. 7, 445–457 (1976) · Zbl 0349.53025 [18] Lottermoser M.: A convergent post-Newtonian approximation for the constraints in general relativity. Ann. Inst. Henri Poincaré 57, 279–317 (1992) · Zbl 0762.53053 [19] Matarrese S., Terranova D.: Post-Newtonian cosmological dynamics in Lagrangian coordinates. Mon. Not. Roy. Astron. Soc. 283, 400–418 (1996) [20] Oliynyk T.A.: The Newtonian limit for perfect fluids. Commun. Math. Phys. 276, 131–188 (2007) · Zbl 1194.83018 [21] Oliynyk T.A.: Post-Newtonian expansions for perfect fluids. Commun. Math. Phys. 288, 847–886 (2009) · Zbl 1175.83027 [22] Oliynyk, T.A.: The fast Newtonian limit for perfect fluids. http://arxiv.org/abs/0908.4455u1[gr-qc] , 2009 · Zbl 1175.83027 [23] Rendall A.D.: On the definition of post-Newtonian approximations. Proc. R. Soc. Lond. A 438, 341–360 (1992) · Zbl 0754.53062 [24] Rendall A.D.: The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system. Commun. Math. Phys. 163, 89–112 (1994) · Zbl 0816.53058 [25] Rüede C., Straumann N.: On Newton-Cartan cosmology. Helv. Phys. Acta 70, 318–335 (1997) · Zbl 0868.53071 [26] Schochet S.: Symmetric hyperbolic systems with a large parameter. Comm. Part. Diff. Eqs. 11, 1627–1651 (1986) · Zbl 0651.35047 [27] Schochet S.: Asymptotics for symmetric hyperbolic systems with a large parameter. J. Diff. Eqs. 75, 1–27 (1988) · Zbl 0685.35014 [28] Shibata M., Asada H.: Post-Newtonian equations of motion in the flat universe. Prog. Theor. Phys. 94, 11–31 (1995) [29] Takada M., Futamase T.: Post-Newtonian Lagrangian perturbation approach to the large-scale structure formation. Mon. Not. R. Astron. Soc. 306, 64–88 (1999) · Zbl 1081.85501 [30] Taylor M.E.: Partial differential equations III, nonlinear equations. Springer, New York (1996) · Zbl 0869.35004
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.