# The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2369-2462
- ISSN: 1435-9855

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topRodnianski, Igor, and Speck, Jared. "The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant." Journal of the European Mathematical Society 015.6 (2013): 2369-2462. <http://eudml.org/doc/277534>.

@article{Rodnianski2013,

abstract = {In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1+3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p=c^2\rho ,0<c^2<1/3$, the background metric + fluid solutions are globally future-stable under small irrotational perturbations of their initial data. In particular, we prove that the perturbed spacetime solutions, which have the topological structure $[0,\infty ]XT^3$, are future causally geodesically complete. Our analysis is based on a combination of energy estimates and pointwise decay estimates for quasilinear wave equations featuring dissipative inhomogeneous terms. Our main new contribution is showing that when $0<c^2<1/3$, exponential spacetime expansion is strong enough to suppress the formation of fluid shocks. This contrasts against a well-known result of Christodoulou, who showed that in Minkowski spacetime, the corresponding constant-state irrotational fluid solutions are unstable.},

author = {Rodnianski, Igor, Speck, Jared},

journal = {Journal of the European Mathematical Society},

keywords = {cosmological constant; energy dissipation; expanding spacetime; geodesically complete; global existence; irrotational fluid; relativistic fluid; wave coordinates; Einstein equation; cosmological constant; background solutions; energy dissipation; expanding spacetime; geodesically complete; global existence; irrotational fluid; relativistic fluid; wave coordinates},

language = {eng},

number = {6},

pages = {2369-2462},

publisher = {European Mathematical Society Publishing House},

title = {The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant},

url = {http://eudml.org/doc/277534},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Rodnianski, Igor

AU - Speck, Jared

TI - The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 6

SP - 2369

EP - 2462

AB - In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1+3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p=c^2\rho ,0<c^2<1/3$, the background metric + fluid solutions are globally future-stable under small irrotational perturbations of their initial data. In particular, we prove that the perturbed spacetime solutions, which have the topological structure $[0,\infty ]XT^3$, are future causally geodesically complete. Our analysis is based on a combination of energy estimates and pointwise decay estimates for quasilinear wave equations featuring dissipative inhomogeneous terms. Our main new contribution is showing that when $0<c^2<1/3$, exponential spacetime expansion is strong enough to suppress the formation of fluid shocks. This contrasts against a well-known result of Christodoulou, who showed that in Minkowski spacetime, the corresponding constant-state irrotational fluid solutions are unstable.

LA - eng

KW - cosmological constant; energy dissipation; expanding spacetime; geodesically complete; global existence; irrotational fluid; relativistic fluid; wave coordinates; Einstein equation; cosmological constant; background solutions; energy dissipation; expanding spacetime; geodesically complete; global existence; irrotational fluid; relativistic fluid; wave coordinates

UR - http://eudml.org/doc/277534

ER -

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