The APDE seminar on Monday, 11/09, will be given by Jared Speck online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
Title: Stable Big Bang formation in general relativity: The complete sub-critical regime.
Abstract: The celebrated theorems of Hawking and Penrose show that under appropriate assumptions on the matter model, a large, open set of initial data for Einstein’s equations lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness due to lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). Despite the “general ambiguity,” in the mathematical physics literature, there are heuristic results, going back 50 years, suggesting that whenever a certain “sub-criticality” condition holds, the Hawking–Penrose incompleteness is caused by the formation of a Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with I. Rodnianski and G. Fournodavlos, we have given a rigorous proof of the heuristics. More precisely, our results apply to Sobolev-class perturbations – without symmetry – of generalized Kasner solutions whose exponents satisfy the sub-criticality condition. Our main theorem shows that – like the generalized Kasner solutions – the perturbed solutions develop Big Bang singularities. In this talk, I will provide an overview of our result and explain how it is tied to some of the main themes of investigation by the mathematical general relativity community, including the remarkable work of Dafermos–Luk on the stability of Kerr Cauchy horizons. I will also discuss the new gauge that we used in our work, as well as intriguing connections to other problems concerning stable singularity formation.
The APDE seminar on Monday, 10/19, will be given by Maxime van de Moortel online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
Title: Nonlinear interaction of three impulsive gravitational waves for the Einstein equations.
Abstract: An impulsive gravitational wave is a weak solution of the Einstein vacuum equations whose metric admits a curvature delta singularity supported on a null hypersurface; the spacetime is then an idealization of a gravitational wave emanating from a strongly gravitating source. In the presence of multiple sources, their impulsive waves eventually interact and it is interesting to study the spacetime up to and after the interaction.
For such singular solutions, the classical well-posedness results (such as the bounded L^2 curvature theorem) are not applicable and it is not even clear a priori whether the initial regularity propagates or a worse singularity occurs from the nonlinear interaction.
I will present a local existence result for U(1)-polarized Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each other. The proof is achieved with the help of localization techniques inspired from Christodoulou’s short pulse method and new tools in Harmonic Analysis, notably anisotropic estimates that are tailored to the problem.
This is joint work with Jonathan Luk.
The APDE seminar on Monday, 10/12, will be given by Jérémie Szeftel online via Zoom from 9:10 to 10am (note the time change). To participate, email Georgios Moschidis (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
Title: Finite time blow up for focusing supercritical NLS and compressible fluids
Abstract: I will present recent results in collaboration with Frank Merle, Pierre Raphaël and Igor Rodnianski concerning finite time blow up for the focusing supercritical NLS equation, for compressible Euler equations, and for compressible Navier-Stokes equations.
The APDE seminar on Monday, 11/02, will be given by Bjoern Bringmann online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
Title: Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity.
Abstract. In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity.
In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutel continuous with respect to the Gaussian free field.
In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.
The APDE seminar on Monday, 10/05, will be given by Federico Pasqualotto online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis (firstname.lastname@example.org) or Federico Pasqualotto (email@example.com).
Title: Global stability for nonlinear wave equations with multi-localized initial data.
Abstract: The classical global existence theory for nonlinear wave equations requires initial data to be small and localized around a point. In this work, we initiate the study of the global stability of nonlinear wave equations with non localized data.
In particular, we extend the classical theory to data localized around several points. This is achieved by generalizing the vector field method to the multi-localized case.
The core of our argument lies in a close inspection of the geometry of two interacting waves emanating from different localized sources. We show trilinear estimates to control such interaction, by means of a physical space method. This is joint work with John Anderson (Princeton University).