The nonlinear future stability of the FLRW family of solutions to the Euler-Einstein system with a positive cosmological constant.(English)Zbl 1251.83071

Summary: In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the $$1+3$$ dimensional Euler-Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state $${p = c^2_s \rho}$$, $${0 < c^2_s < 1/3}$$, the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure $${[0,\infty) \times \mathbb{T}^3}$$, are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents.

MSC:

 83F05 Relativistic cosmology 35A01 Existence problems for PDEs: global existence, local existence, non-existence 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 35Q31 Euler equations 35Q76 Einstein equations 83C10 Equations of motion in general relativity and gravitational theory 83C75 Space-time singularities, cosmic censorship, etc. 83C25 Approximation procedures, weak fields in general relativity and gravitational theory 85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics 76E20 Stability and instability of geophysical and astrophysical flows
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