# Seminars & Events for Analysis Seminar

##### New continuation criteria for the relativistic Vlasov-Maxwell System

We consider the relativistic Vlasov-Maxwell (VM) system with initial data of unrestricted size. In the 3D case, since the work of Glassey-Strauss in 1986, it has been known that as long as the 3D momentum support remains bounded then solutions can be continued and they will remain regular. We prove that as long as there exists a plane upon which the momentum support remains bounded then solutions can be continued and they will remain regular. We will also extend the Glassey-Strauss continuation criterion for the VM system to initial data with unbounded initial momentum support. Moreover, by using moment bounds and Strichartz estimates, we will show that it suffices to assume the kinetic energy density to be in $L^p$ for $p>2$ in order to guarantee that the solution remains regular. These are joint works with Jonathan Luk.

##### Random matrix and partial differential equation

It was known that Dyson Brownian motion is closely related to the local statistics of random matrices. In this lecture, I’ll explain that Dyson Brownian motion can be studied by a partial differential equation with random coefficients. From the regularity theory of this PDE, important properties of local spectral statistics of random matrices can be derived.

##### On the two-dimensional pressure-less Euler equations

We prove the existence of weak solutions for the two-dimensional pressure-less Euler equations. To this end we develop an L1 framework of dual solutions for such equations. Their existence is realized as vanishing viscosity limits. The limit follows from new BV estimates, derived by tracing the spectral dynamics of the velocity gradient matrix.

##### Extension and interpolation by smooth functions

We introduce the classical extension results formulated by Whitney and later by Brudnyi-Shvartsman and Fefferman in the setting of the function space C^m. We present our theorems on interpolation of functions in Sobolev spaces. We then present our interpolation algorithms based on a recent constructive proof of our main theorem. We hope to explain the main obstacles that arose in converting our original non-constructive proof into a constructive one. If time permits, we will explain our work related to extension problems for bandlimited functions. This talk is based partly on joint work with C. Fefferman and G. Luli.

##### On 1d models of certain 3d Euler and 2d Boussinesq regimes

Recently T. Hou and G. Lou suggested a blow-up scenario for axi-symmetric incompressible 3d Euler equations, as well as a 1d model capturing certain features of the situation. The model is related to equations previously introduced by Constantin-Lax-Majda and De Gregorio. We will discuss some rigorous results for the model. Joint work with K. Choi, T. Hou, A. Kiselev, G. Luo and Y. Yao.

##### Recent progress on the Landis conjecture

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. In the late 60's, E.M. Landis conjectured that if $\Delta u+Vu=0$ in $\mathbb{R}^n$ with $\|V\|_{L^{\infty}(\mathbb{R}^n)}\le 1$ and $\|u\|_{L^{\infty}(\mathbb{R}^n)}\le C_0$ satisfying $|u(x)|\le C\exp(-C|x|^{1+})$, then $u\equiv 0$.

##### The pointwise convergence of Fourier Series near $L^1$

In this talk we discuss some recent developments on the old question regarding the pointwise behavior of Fourier Series near $L^1$. We start with several brief historical remarks on the subject, describing the context and the formulation of the main problem(s). We then present the evolution of the main negative and positive results from early 20th century to present day. In the main part of our talk we provide a near-complete classification of the Lorentz spaces for which the sequence $\{Sigma_n\}_{n\in\mathbb{N}} of partial Fourier sums is almost everywhere convergent along lacunary subsequences. In particular, we resolve a conjecture stated by Konyagin in his 2006 ICM address.

##### Global well-posedness of incompressible elastodynamics in 2D

I will report our recent result on the global wellposedness of classical solutions to system of incompressible elastodynamics in 2D. The system is revealed to be inherently strong linearly degenerate and

automatically satisfies a ***strong*** null condition, due to the isotropic nature and the incompressible constraint.