Author Archives: jzhao

Wellposedness of the electron MHD without resistivity for large perturbations of the uniform magnetic field

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The HADES seminar on Tuesday, March 18th, will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: The electron magnetohydrodynamics equation (E-MHD) is a fluid description of plasma in small scales where the motion of electrons relative to ions is significant. Mathematically, (E-MHD) is a quasilinear dispersive equation with nondegenerate but nonelliptic second-order principal term. In this talk, I’ll discuss a recent proof, joint with In-Jee Jeong, of the local wellposedness of the Cauchy problems for (E-MHD) without resistivity for possibly large perturbations of nonzero uniform magnetic fields. While the local wellposedness problem for (E-MHD) has been extensively studied in the presence of resistivity (which provides dissipative effects), this seems to be the first such result without resistivity.

More specifically, my goal is to explain the main new ideas introduced in this work and the related work of Pineau-Taylor on quasilinear ultrahyperbolic Schrodinger equations, which also have nondegenerate but nonelliptic principal terms. Both works significantly improve upon the classical work of Kenig-Ponce-Rolvung-Vega on such PDEs, in the sense that the regularity and decay assumptions on the initial data are greatly weakened to the level analogous to the recent work of Marzuola-Metcalfe-Tataru in the case of an elliptic principal term.

Integral formulas for under/overdetermined linear differential operators

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The HADES seminar on Tuesday, March 4th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: Solving an underdetermined PDE such as a divergence equation plays a central role in problems like general relativistic gluing. Starting from divergence equations on Euclidean spaces, I will introduce a method of constructing integral solution operators for a wide class of underdetermined differential operators with prescribed support properties. By duality, this will also produce integral representation formulas for overdetermined differential operators. The method extends various ideas from Bogovskii, Oh-Tataru, and Reshetnyak. The construction is based on an assumption called the recovery on curves condition (RC) imposed on the operators. I will also give an algebraic sufficient condition of RC that is easier to verify, which is called the finite-codimensional cokernel condition (FC). At the end, I will show some examples that satisfy FC on space forms and derive their integral formulas in the flat case. This is joint work with Philip Isett, Sung-Jin Oh, and Zhongkai Tao.

Dispersive quantisation in KdV

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The HADES seminar on Tuesday, February 25th, will be at 3:30pm in Room 740.

Speaker: Jason Zhao

Abstract: It has been observed both experimentally and mathematically that solutions to linear dispersive equations, such as the Schrodinger and Airy equations, posed on the torus exhibit dramatically different behaviors between rational and irrational times. For example, the evolution of piecewise constant data remains so at rational times, while it becomes continuous and fractalised at irrational times. A natural question to ask is whether this Talbot effect, as it broadly known, persists under non-linear dispersive flows. Focusing on the KdV equation, we will present two perspectives which follow in the spirit of the seminal works of Bourgain (1993) and Babin-Ilyin-Titi (2011): the first is the non-linear smoothing effect observed by Erdogan-Tzirakis (2013), and the second is the numerical work of Hofmanova-Schratz (2017) and Rousset-Schratz (2022}.

Lossless Strichartz estimates on manifolds with trapping

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The HADES seminar on Tuesday, February 18th, will be at 3:30pm in Room 740.

Speaker: Zhongkai Tao

Abstract: The Strichartz estimate is an important estimate in proving the well-posedness of dispersive PDEs. People believed that the lossless Strichartz estimate could not hold on manifolds with trapping (for example, the local smoothing estimate always comes with a logarithmic loss in the presence of trapping). Surprisingly, in 2009, Burq, Guillarmou, and Hassell proved a lossless Strichartz estimate for manifolds with trapping under the “pressure condition”. I will talk about their result as well as our recent work with Xiaoqi Huang, Christopher Sogge, and Zhexing Zhang, which goes beyond the pressure condition using the fractal uncertainty principle and a logarithmic short-time Strichartz estimate.

Characterizing the support of semiclassical measures for quantum cat maps

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The HADES seminar on Tuesday, December 3rd, will be at 2:00pm in Room 740.

Speaker: Elena Kim

Abstract: We consider a quantum cat map $M$ associated to a symplectic matrix $A$ acting on the torus $\mathbb{T}^{2n}$, a popular model in quantum chaos. The semiclassical limit of the mass of eigenfunctions of $M$ is characterized by the semiclassical measure.

For the analogous model on hyperbolic manifolds, the quantum unique ergodicity conjecture posits that the Liouville measure is the only semiclassical measure; however, the corresponding statement for quantum cat maps is known to be false. It is thus an open question to otherwise describe semiclassical measures for quantum cat maps.

In this talk, I will explain how the higher-dimensional fractal uncertainty principle of Cohen can be used to characterize the supports of semiclassical measures $\mu$, including cases where $\mu$ has full support.

Improved bounds for intermediate curved Kakeya sets in $\mathbb R^3$

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The HADES seminar on Tuesday, November 19th, will be at 2:00pm in Room 740.

Speaker: Arian Nadjimzadah (UCLA)

Abstract: In this talk, we describe the deep connection between oscillatory integrals and curved Kakeya problems that was observed by Bourgain. Then we sketch some of the key discoveries in the study of the classical Kakeya problem in $\mathbb R^3$, and see how they can inform an approach to solving curved Kakeya problems. The results we will discuss are Wolff’s hairbrush bound, the $SL_2$ Kakeya set bound of Katz-Wu-Zahl, and the multilinear Kakeya inequality of Bennett-Carbery-Tao.

The Yang-Mills equations from a microlocal perspective

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The HADES seminar on Tuesday, November 12th, will be at 2:00pm in Room 740.

Speaker: Jose Lopez

Abstract: The Yang-Mills equations are important in physics because the equations of motion of the fundamental forces of particle physics are described by quantized versions of these equations. As an example, we will study the work of Mazzeo-Swoboda-Weiss-Witt on the asymptotics of Hitchin’s equations, a dimensional reduction of Yang-Mills.

Stability of catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensions

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The HADES seminar on Tuesday, November 5th, will be at 2:00pm in Room 740.

Speaker: Ning Tang

Abstract: The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart of the minimal surface equation in Euclidean space. This talk will concern the modulated nonlinear asymptotic stability of the $1+4$ dimensional hyperbolic catenoid, viewed as a stationary solution to the HVMC equation. This stability result is under a “codimension-one” assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by Lührmann-Oh-Shahshahani, proving catenoid stability in $n = 4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties for the $n = 3$ case, the strong Huygens principle, as well as a miracle cancellation in the source term, plays an important role in the work of Oh-Shahshahani to obtain strong late time tails. Without these special structural advantages in $n = 4$ dimensions, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy with higher $r^p$-weights so that an improved pointwise decay can be established.

In this talk, I will first give an outline of the proof and then focus on a model case to illustrate our approach to achieving an improved decay in $4$ dimensions.

 

Some applications of interval arithmetic

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The HADES seminar on Tuesday, October 29th, will be at 2:00pm in Room 740.

Speaker: James Rowan

Abstract: Computers cannot do exact arithmetic with arbitrary real numbers. But as analysts, that shouldn’t bother us too much since we usually deal in inequalities; often, we only need upper and lower bounds. I will give an introduction to interval arithmetic and survey some ways to make rigorous use of numerics in PDE proofs. First, I will discuss an interval arithmetic version of Newton’s method for root-finding and illustrate its use to compute a threshold value of a physical parameter in a solitary water waves problem (from joint work with Lizhe Wan). Then I will discuss the use of interval arithmetic to “upgrade” approximate solutions to nearby exact solutions and give a survey of some recent developments in this area that have inspired some recent projects I have been working on.