# PDE Seminar Spring 2021

### Fridays, 4:10pm, through Zoom.

Date: **Friday, Feb 5, 2021.**

- Speaker:
**Annalaura Stingo, University of California Davis.** - Title: Almost-global well-posedness for 2d strongly-coupled wave-Klein-Gordon systems.
- Abstract: In this talk we discuss the almost-global well-posedness of a wide class of coupled Wave-Klein-Gordon equations in 2+1 space-time dimensions, when initial data are assumed to be small and localized. The Wave-Klein-Gordon systems arise from several physical models especially related to General Relativity, but few results are know at present in lower space-time dimensions. Compared with prior related results, our novel contributions include a strong quadratic quasilinear coupling between the wave and the Klein-Gordon equation, and no restriction is made on the support of the initial data which are supposed to only have a mild decay at infinity and very limited regularity. Our proof relies on a combination of energy estimates localized to dyadic space-time regions, and pointwise interpolation type estimates within the same regions. This is akin to ideas previously used by Metcalfe-Tataru-Tohaneanu in a liner setting, and is also related to Alinhac’s ghost weight method. A refinement of these estimates through different techniques will allow us to pass, in a future work, from almost global existence to global existence of solutions under the same hypothesis on the initial data. This is a joint work with M. Ifrim.
- Zoom link: https://vanderbilt.zoom.us/j/95612594074

Date: **Friday, Feb 12, 2021.**

- Speaker:
**Jason Metcalfe, University of North Carolina at Chapel Hill.** - Title: Local energy in the presence of degenerate trapping.
- Abstract: Trapping is a known obstruction to local energy estimates for the wave equation and local smoothing estimates for the Schrödinger equation. When this trapping is sufficiently unstable, it is known that estimates with a logarithmic loss can be obtained. On the other hand, for very stable trapping, it is known that all but a logarithmic amount of local energy decay is lost. Until somewhat recently, explicit examples of scenarios where an algebraic loss (of regularity) was both necessary and sufficient for local energy decay had not be constructed. We will review what is known in these specific examples. We will also examine the relationship between the trapping and the existence of a boundary. In this highly symmetric case, a relatively simple proof showing a bifurcation in the behavior of local energy as the boundary passes through the trapping is available. This is related, e.g., to the instability of ultracompact neutrino stars.
- Zoom link: https://vanderbilt.zoom.us/j/93820116803

Date: **Friday, Feb 26, 2021.**

- Speaker:
**Stephen Taylor, New Jersey Institute of Technology.** - Title: Peer-to-Peer Risk Sharing with an Application to Flood Risk Pooling.
- Abstract: In contrast with classic centralized risk sharing, a novel peer-to-peer risk sharing framework is proposed. The presented framework aims to devise a risk allocation mechanism that is structurally decentralized, Pareto optimal, and mathematically fair. An explicit form for the pool allocation ratio matrix is derived, and convex programming techniques are applied to determine the optimal pooling mechanism in a constrained variance reduction setting. A tiered hierarchical generalization is also constructed to improve computational efficiency. As an illustration, these techniques are applied to a flood risk pooling example. It is shown that peer-to-peer risk sharing techniques provide an economically viable alternative to traditional flood policies.
- Zoom link: https://vanderbilt.zoom.us/j/93404060425

Date: **Friday, Mar 5, 2021.**

- Speaker:
**Stefan Czimek, Brown University.** - Title: The spacelike-characteristic Cauchy problem of general relativity in low regularity.
- Abstract: In this talk I will present my recent results (in collaboration with O. Graf) on the spacelike-characteristic initial value problem for the Einstein equations, where the initial data is posed on a spacelike ball and the outgoing null cone emanating from its boundary. We prove that if the initial data is sufficiently close to Minkowski (measured in the L2-norm of the curvature) then the associated solution to the Einstein equations exists until time T=1. Our result extends the bounded L2-curvature theorem of Klainerman-Rodnianski-Szeftel to initial data on spacelike-characteristic hypersurfaces.In the beginning of the talk, I will shortly recapitulate the Einstein equations and their initial value problem. Then I will discuss the role of continuation results in general relativity and state our result. Subsequently, I will outline its proof, explaining in particular how our setting introduces new geometric boundary terms. To estimate the latter, we extend the methods of Klainerman-Rodnianski to provide a novel analysis of the so-called “canonical foliation” on null hypersurfaces which admits stronger estimates in the direction transversal to the null hypersurface.
- Zoom link: https://vanderbilt.zoom.us/j/98740771837

Date: **Friday, Mar 12, 2021.**

- Speaker:
**Alexandru Ionescu, Princeton University.** - Title: On the nonlinear asymptotic stability of vortices and shear flows.
- Abstract: I will talk about some recent work on the global nonlinear asymptotic stability of two families of solutions of the 2D Euler equations: monotonic shear flows on bounded channels and point vortices in the plane. This is joint work with Hao Jia.
- Zoom link: https://vanderbilt.zoom.us/j/99273048235

Date: **Friday, Mar 26, 2021.**

- Speaker:
**Igor Kukavica, University of Southern California.** - Title: On the inviscid problem for the Navier-Stokes equations.
- Abstract: The question of whether the solution of the Navier-Stokes equation converges to the solution of the Euler equation as the viscosity vanishes is one of the fundamental problems in fluid dynamics. In the talk, we will review current results on this problem. We will also present a recent result, joint with Vlad Vicol and Fei Wang, which shows that the inviscid limit holds for the initial data that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the interior.
- Zoom link: https://vanderbilt.zoom.us/j/93727990819

Date: **Friday, Apr 16, 2021.**

- Speaker:
**Georgios Moschidis, University of California Berkeley.** - Title: The instability of Anti-de Sitter spacetime for the Einstein–scalar field system.
- Abstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.

In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. I will also discuss possible paths for extending these ideas to the vacuum case. - Zoom link: https://vanderbilt.zoom.us/j/93342418975

Date: **Friday, Apr 30, 1:10pm (note the different time), 2021.**

- Speaker:
**Zoe Wyatt, University of Cambridge.** - Title: Global Stability of Spacetimes with Supersymmetric Compactifications.
- Abstract: Spacetimes with compact directions which have special

holonomy, such as Calabi-Yau spaces, play an important role in

supergravity and string theory. I will discuss a recent work with Lars

Andersson, Pieter Blue and Shing-Tung Yau, where we show the global,

nonlinear stability of a spacetime which is a cartesian product of a high

dimensional Minkowski space with a compact Ricci flat internal space

with special holonomy. This stability result is related to a conjecture

of Penrose concerning the validity of string theory. Our proof uses the

intersection of methods for quasilinear wave and Klein-Gordon equations,

and so towards the end of the talk I will comment more generally on

coupled wave–Klein-Gordon equations. - Zoom link: https://vanderbilt.zoom.us/j/91985317432

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