Author Archives: gmoschidis

Sung-Jin Oh (UCB)

The APDE seminar on Monday, 09/16 will be given by Sung-Jin Oh in Evans 939 from 4:10 to 5pm.

Title: On the Cauchy problem for the Hall-magnetohydrodynamics equations

Absract:
In this talk, I will describe a recent series of work with I.-J. Jeong on the Cauchy problem for the Hall-MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field). Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character.

Jeffrey Galkowski (Northeastern)

The APDE seminar on Monday, 05/20 will be given by Jeffrey Galkowski in Evans 740 from 4:10 to 5pm.

Title:
Concentration and Growth of Laplace Eigenfunctions.

Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration including Weyl laws; in each case obtaining quantitative improvements over the known bounds.

Dejan Gajic (Cambridge)

The APDE seminar on Monday, 04/29 will be given by Dejan Gajic in Evans 740 from 4:10 to 5pm.

Title:
Resonances on asymptotically flat black holes 

Abstract:
A fundamental problem in the context of Einstein’s equations of general relativity is to understand the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late times and carry information about the nature of the black hole, much like how the normal frequencies of a vibrating guitar string play an important role in the resulting sound wave. These frequencies are called quasinormal frequencies or resonant frequencies and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will consider the linear wave equation on black hole backgrounds as a toy model for Einstein’s equations and give an introduction to resonances in this setting. Then I will discuss a new method of defining and studying resonances on asymptotically flat spacetimes, developed from joint work with Claude Warnick, which puts resonances on the same footing as normal modes by showing that they are eigenfunctions of a natural operator acting on a Hilbert space.

Martin Taylor (Princeton)

The APDE seminar on Monday, 04/22 will be given by Martin Taylor in Evans 740 from 4:10 to 5pm.

Title:
The nonlinear stability of the Schwarzschild family of black holes .

Abstract:
I will present a theorem on the full finite codimension asymptotic stability of the Schwarzschild family of black holes.  The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family.  This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

Annalaura Stingo (UC Davis)

The APDE seminar on Monday, 04/15 will be given by Annalaura Stingo in Evans 740 from 4:10 to 5pm.

Title:
Global existence of small amplitude solutions for a model quadratic quasilinear wave-Klein-Gordon system in 2D.

Abstract:

In this talk we discuss the problem of global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small, smooth and mildly decaying at infinity. Several physical models related to general relativity have shown the importance of studying such systems but very few results are known at present in low space dimension, where linear solutions slowly decay in time.

We study here a model quadratic quasilinear two-dimensional system, in which the nonlinearity writes in terms of “null forms”, and prove global existence by propagating a-priori energy estimates and optimal uniform estimates on the solution. In proving such estimates one has to deal with several issues such as the quasilinear nature of the problem, the very low decay in time of quadratic nonlinearities, the fact that initial data are not compactly supported…

We will show how to obtain energy estimates by using systematically quasilinear normal forms, in their para-differential version. Uniform estimates will instead be recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system by means of a semiclassical microlocal analysis of the problem.