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PDE and Applied Math Seminar

Speaker: Jared Speck, MIT
Location: 1147 MSB
Start time: Tue, Apr 23 2013, 1:10PM

I will discuss some results that I recently obtained in collaboration with Igor Rodnianski. Our main result is a proof of stable Big Bang formation in small perturbations of the well-known Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) solution to the Einstein-scalar field system. The FLRW solution is a special case of a family of spatially homogeneous, isotropic solutions that arise in cosmology. It models a scalar field evolving in a spacetime that expands as t → ∞ and that col- lapses as t ↓ 0. In particular, the FLRW solution contains a “Big Bang” singularity at Σ0 := {t = 0}. To study the perturbed solutions, we place data on a Cauchy hypersurface Σ1 that are close to the FLRW data (at time 1) as measured by a Sobolev norm. No symmetry assumptions are made on the data. We then study the global behavior of the perturbed solution in the collapsing direction. We first show that the spacetime region of interest can be foliated by a family of spacelike Cauchy hypersurfaces Σt, t ∈ (0,1], of constant mean curvature −1t−1. We then analyze the behavior of the 3 solution as t ↓ 0 and provide a detailed description of its asymptotics. Our main conclusion is that the perturbed solution remains globally close to the FLRW solution and has approximately mono- tonic behavior. In particular, the perturbed solution also has a Big Bang singularity at Σ0. More precisely, as t ↓ 0, various curvature invariants uniformly blow-up and the volume of Σt collapses to 0. These blow-up results demonstrate the validity of Penrose’s Strong Cosmic Censorship conjecture for the past half of the perturbed spacetimes. We have also shown that the same results hold for the stiff fluid matter model. From the point of view of analysis, our main results can be viewed as a proof of stable blow-up for an open set of solutions to a highly nonlinear elliptic-hyperbolic system. The most important aspect of our analysis is our identification of a new L2−type energy almost-monotonicity inequality that holds for the solutions under consideration.