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 () or Federico Pasqualotto ().

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).

The APDE seminar on Monday, 09/21, will be given by Jonathan Luk online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: High-frequency limits, null dust shells, and the formation of trapped surfaces in general relativity.

Abstract: I will discuss the three problems in the title and some (surprising?) connections between them. This is a joint work with Igor Rodnianski.

The APDE seminar on Monday, 09/28, will be given by Maciej Zworski online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().

Title: Outgoing property via Gevrey regularity.

Abstract: One of the main issues in theoretical and numerical scattering theory is distinguishing outgoing parts of solutions modeling scattered waves. That is then closely related to defining scattering resonances. Motivated by the study of quasi-normal modes in general relativity, Gajic and Warnick have recently proposed an approach to characterising outgoing solutions based on Gevrey-2 regularity at infinity and introduced a new class of potentials for which resonances can be defined. In joint work with Galkowski we show that standard methods based on complex scaling apply to a slightly larger class of potentials and provide a definition of resonances in larger angles.

The APDE seminar on Monday, 05/18, will be given by Yakov Shlapentokh-Rothman online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Thibault de Poyferré ().

Title: Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution.

Abstract: We will start by recalling the weak cosmic censorship conjecture. Then we will review Christodoulou’s construction of naked singularities for the spherically symmetric Einstein-scalar field system. Finally, we will discuss joint work with Igor Rodnianski which constructs spacetimes corresponding to the exterior region of a naked singularity for the Einstein vacuum equations.

The APDE seminar on Monday, 05/11, will be given by Albert Ai online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Thibault de Poyferré ().

Title: Two dimensional gravity waves at low regularity.

Abstract: In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. This is joint work with Mihaela Ifrim and Daniel Tataru.

The APDE seminar on Monday, 04/27, will be given by Sung-Jin Oh online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Thibault de Poyferré ().

Title: Strong cosmic censorship and generic mass inflation for charged black holes in spherical symmetry.

Abstract: In this talk, I will first review a previous work with J. Luk, in which the C2-formulation of the strong cosmic censorship is proved for the Einstein-Maxwell-(real)-Scalar Field system in spherical symmetry for two-ended asymptotically flat data. More precisely, it was shown that a “generic” class of data for this model gives rise to maximal future developments which are future C2-inextendible. In the second part of the talk, I will present a new, complementary theorem (also joint with J. Luk) that for a further “generic” subclass of such data, the Hawking mass blows up identically along the Cauchy horizon. This result confirms, rigorously and unconditionally, the mass inflation scenario of Poisson-Israel and Dafermos for the model at hand.

The APDE seminar on Monday, 04/20, will be given by Daniel Tataru online via Zoom from 4:10 to 5pm. To participate, email Georgios Moschidis () or Thibault de Poyferré ().

Title: Multisolitons and their stability in 1-d cubic NLS.

Abstract: The aim of this talk is first to describe the multisoliton manifold for the 1-d cubic NLS flow, and then to consider their stability. This continues recent work, joint with Herbert Koch, on energy estimates for this and other related integrable evolutions.

The APDE seminar on Monday, 03/02, will be given by Wolf-Patrick Düll in Evans 939 from 4:10 to 5pm.

Title: Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension.

Abstract: We consider the two-dimensional water wave problem in an infinitely long canal of
finite depth both with and without surface tension. In order to describe the evolution
of the envelopes of small oscillating wave packet-like solutions to this problem the
Nonlinear Schrödinger equation can be derived as a formal approximation equation.
The rigorous justification of the Nonlinear Schrödinger approximation for the water
wave problem was an open problem for a long time. In recent years, the validity
of this approximation has been proven by several authors only for the case without
surface tension.
In this talk, we present the first rigorous justification of the Nonlinear Schrödinger approximation for the two-dimensional water wave problem which is valid for the
cases with and without surface tension by proving error estimates over a physically
relevant timespan in the arc length formulation of the water wave problem. Our
error estimates are uniform with respect to the strength of the surface tension, as the
height of the wave packet and the surface tension go to zero.

The APDE seminar on Monday, 03/09 will be given by Mihai Tohaneanu in Evans 939 from 4:10 to 5pm.

Title: Local energy estimates on black hole backgrounds.

Abstract: Local energy estimates are a robust way to measure decay of solutions to linear wave equations. I will discuss several such results on black hole backgrounds, such as Schwarzschild, Kerr, and suitable perturbations converging at various rates, and briefly discuss applications to nonlinear problems. The most challenging geometric feature one needs to deal with is the presence of trapped null geodesics, whose presence yield unavoidable losses in the estimates. This is joint work with Lindblad, Marzuola, Metcalfe, and Tataru.

The APDE seminar on Monday, 02/24 will be given by Mihaela Ifrim in Evans 939 from 4:10 to 5pm.

Title: Almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems in two space dimensions.

Abstract: We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We systematically investigate all the possible quadratic null form type quasilinear strong coupling nonlinearities, and provide a new, robust approach for the proof.