PDE Seminar Spring 2020

Date: Friday, Jan 24, 2020.

Speaker: Yakov Shlapentokh-Rothman, Princeton University.
Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations.
Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations. Then we will discuss our previous work on the asymptotically self-similar regime for the Einstein equations and the corresponding connection to the “ambient metric” of Fefferman and Graham. Finally, we will explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to the exterior region of a naked singularity. This is all joint work with Igor Rodnianski.

Date: Friday, Feb 7, 2020.

Speaker: Geng Chen, University of Kansas.
Title: Shock formation and large solutions of compressible Euler equations.
Abstract: Compressible Euler equations (introduced by Euler in 1757) model the motion of compressible inviscid fluids such as gases. It is well-known that solutions of compressible Euler equations often develop discontinuities, i.e. shock waves. Successful theories have been established in the past 150 years for small solutions in one space dimension. The theory on 1-d large solutions and multi-d solutions is widely open for a long time.There are many recent exciting progresses on the shock formation. In the first part of this talk, I will discuss our complete resolution of 1-d shock formation problem, which extends the celebrated work of Peter Lax in 1964, then the recent work on radially symmetric solutions.In the second part, I will briefly introduce the joint work with Bressan and Zhang on the negative example concerning the failure of current available frameworks on approximate solutions in order to establish large BV (bounded total variation) theory.

Date: Friday, Feb 21, 2020.

Speaker: Lan-Hsuan Huang, University of Connecticut.
Title: Improvability of the dominant energy scalar and Bartnik’s stationary conjecture.
Abstract: We will introduce the concept of improvabilty of the dominant energy scalar and discuss strong consequences of non-improvability. This concept is important to find perturbations of initial data sets that preserve or reinstate the dominant energy condition. We introduce infinite-dimensional families of deformations of the modified Einstein constraint operator and show that, generically, their adjoint linearizations are either injective, or else one can prove that kernel elements satisfy a “null-vector equation”. Combined with a conformal argument, we make substantial progress toward Bartnik’s stationary conjecture. More specifically, we prove that a Bartnik minimizing initial data set can be developed into a spacetime that both satisfies the dominant energy condition and carries a global Killing field. We also show that this spacetime is vacuum near spatial infinity. This talk is based on the joint work with Dan Lee.

Date: Friday, Feb 28, 2020.

Speaker: Leonardo Abbrescia, Michigan State University.
Title: Geometric analysis of 1+1 dimensional quasilinear waves.
Abstract: We will present a series of geometric ideas that are helpful to study the initial value problem of quasilinear wave equations satisfying the null condition on the (1+1)-dimensional Minkowski space. Using a double-null geometric formulation, we show how the conformal invariance of the equation semilinearizes it into a system that is decoupled from the equations governing the null geometry. This allows us to solve the wave equations independently, which we exploit to show that the null geometry is sufficiently regular to guarantee global existence. If time permits, I will explain how this ties into a global wellposedness result with “large” initial data. This is joint work with Willie Wong

All seminars below are canceled due to the coronavirus outbreak. See:

Coronavirus (COVID-19)

for more information.

Date: Friday, April 10, 2020.

Date: Friday, April 24, 2020.

Date: Friday, May 1, 2020.

Speaker: Ravshan Ashurov, Institute of Mathematics, National University of Uzbekistan.
Title: Generalized localization for spherical partial sums of Fourier series.
Abstract: Historically progress with solving the Luzin conjecture has
been made by considering easier problems. For multiple Fourier series one of such easier problems is to investigate convergence almost everywhere of the spherical sums on T^N\ supp(f) (so called the generalized localization principle).
For the spherical partial integrals of multiple Fourier integrals the generalized localization principle in classes L_p(R^N) was investigated by many authors. In particular, in the remarkable paper of A. Carbery and F. Soria the validity of the generalized localization was proved in L_p(R^N) when 2 <= p < 2N/(N – 1). In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in L₂ – class is proved. It was previously known that the generalized localization was not valid in classes L_p(T^N) when 1 <= p < 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in classes L_p(T^N), p >= 1: if p >= 2 then we have the generalized localization and if p < 2, then the generalized localization fails.

PDE Research Group