The Stability of the dust-Einstein System with a Positive Cosmological Constant

PDE Seminar
Tuesday, April 2, 2013 - 15:05
1 hour (actually 50 minutes)
Skiles 006
 We study small perturbations of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant on a spatially periodic background. These solutions model a quiet fluid in a spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our result extends the stability results of Rodnianski and Speck for the Euler-Einstein system with positive cosmological constant to the case of dust (i.e. a pressureless fluid). The main difficulty that we overcome is the degenerate nature of the dust model that loses one degree of differentiability with respect to the Euler case. To resolve it, we commute the equations with a well-chosen differential operator and develop a new family of elliptic estimates that complement the energy estimates. This is joint work with J. Speck.