PDE Seminar Fall 2020

Date: Friday, Sept 18, 2020.

Speaker: Lan-Hsuan Huang, University of Connecticut.
Title: Bartnik’s quasi-local mass conjectures.
Abstract: In 1989 Bartnik defined the notion of quasi-local mass for a compact manifold with boundary by minimizing the ADM masses among admissible extensions. Among several proposed conjectures, Bartnik’s stationary conjecture asserts that a minimizing initial data set must be vacuum and admit a timelike Killing vector. We make partial progress toward this conjecture by showing that a minimizing initial data set must sit in a “null dust” spacetime carrying a global Killing vector. On the other hand, we find pp-wave counterexamples to Bartnik’s stationary and strict positivity conjectures in dimensions greater than 8. In our proof, we introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability using a new infinite-dimensional family of deformation to the Einstein constraint operator. This talk is based on a joint work with Dan Lee.
Zoom link: https://vanderbilt.zoom.us/j/92365165657.

Date: Friday, Sept 25, 2020.

Speaker: Sung-Jin Oh, University of California Berkeley.
Title: On the Cauchy problem for the Hall magnetohydrodynamics.
Abstract:In this talk, I will describe a recent series of work with I.-J. Jeong on the incompressible 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 the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. Our illposedness mechanism is sharp, in that it remains true under fractional dissipation of any subcritical order. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bicharacteristic flow associated with the appropriate principal symbol.
Zoom link: https://vanderbilt.zoom.us/j/97256715872.

Date: Friday, Oct 16, 2020.

Speaker: John Anderson, Princeton University.
Title: Stability and instability of traveling wave solutions to nonlinear wave equations.
Abstract: In this talk, I will discuss work with Samuel Zbarsky studying the stability of certain special solutions to a class of nonlinear wave equations. The null condition, introduced by Klainerman in the 1980s, is an algebraic condition on quadratic nonlinearities in 3 + 1 dimensions which, among other things, guarantees global stability of the trivial solution. These null forms also annihilate plane waves. We thus consider plane wave solutions to semilinear systems of wave equations satisfying the null condition and ask when they are stable. The linearization around the background plane wave solution introduces first order terms. When a certain condition is met, we are able to renormalize the equations and control the interaction between the background plane wave and the perturbation, resulting in global nonlinear stability. The proof of global stability is purely physical space. When the condition is not met and a certain genericity condition is true, we are able to exploit the first order terms in a geometric optics argument to show linear instability.
Zoom link: https://vanderbilt.zoom.us/j/99475454669

Date: Friday, Oct 23, 2020.

Speaker: Jameson Graber, Baylor University.
Title: The Master Equation in Mean Field Games: Recent Approaches.
Abstract: Mean field game theory is a relatively new development in mathematics that helps us model situations with large numbers of rational agents. Its many applications are found in economics and finance, networks and cybersecurity, and even biology. The Master Equation is fundamental in mean field game theory. Unlike classical PDE, it involves derivatives on an infinite dimensional space, namely a space of measures. As a result, any theory of solutions, including well-posedness and regularity, requires new techniques. In this talk I will give an introduction to the Master Equation and discuss recent approaches to proving existence of solutions.
Zoom link: https://vanderbilt.zoom.us/j/93722404091

Date: Friday, Dec 4, 2020. (Cancelled)

PDE Research Group