Solution operators for divergence-type equations with prescribed support properties

The HADES seminar on Tuesday, November 2nd, will be given by Sung-Jin Oh at 5 pm in 740 Evans.

Speaker: Sung-Jin Oh

Abstract: An alternative title for this talk could be “What I wish I knew about the divergence equation in graduate school.” Equations that resemble the prescribed divergence equation arise from many places in physics, such as the incompressibility condition in fluid mechanics, the Gauss law in electromagnetism and the (linearized) constraint equations in general relativity. I will describe a construction of solution operators for these equations with certain support properties based on a few simple ideas, such as manipulation of delta distributions, smooth averaging and standard harmonic analysis. Then I will discuss how such a construction leads to simplification (and improvement) of some theorems for the Yang-Mills and Einstein equations.

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

The HADES seminar on Tuesday, October 26th, will be given by Benjamin Pineau at 5 pm in 740 Evans.

Speaker: Benjamin Pineau

Abstract: In this talk, we study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS)
for small powers $\sigma$. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in $H^s$ when $s\in [1,4\sigma)$ and $\sigma \in (\frac{\sqrt{3}}{2},1)$. Our result when $s=1$ is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu, Simpson and Sulem (~2013) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable $H^1$ well-posedness theory.

To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and is of lower than cubic order. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools we developed are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity well-posedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold $s<4\sigma$ is twice as high as one might naively expect, given that the function $z\mapsto |z|^{2\sigma}$ is only $C^{1,2\sigma-1}$ Hölder continuous. Moreover, although we cannot prove $H^1$ well-posedness when $\sigma\leq \frac{\sqrt{3}}{2}$, we are able to establish $H^s$ well-posedness in the high regularity regime $s\in (2-\sigma,4\sigma)$ for the full range of $\sigma\in (\frac{1}{2},1)$. This considerably improves the known local results, which had only been established in either $H^2$ or in weighted Sobolev spaces. This is joint work with Mitchell Taylor.

Observability for Schrodinger equation on the torus

The HADES seminar on Tuesday, October 19th, will be given by Zhongkai Tao at 5 pm in 740 Evans.

Speaker: Zhongkai Tao

Abstract: The Schrodinger equation describes the motion of a particle on a manifold. It is quite nice that the distribution of the particle is closely related to classical dynamics. I will introduce the observability estimate, the control result and describe how they are related to classical dynamics. At the end, I will talk about my attempt to make the estimates quantitative. No prerequisite in microlocal analysis is needed. This work comes from my undergraduate research mentored by Semyon Dyatlov.

Well-Posedness For The Dispersive Hunter-Saxton Equation

The HADES seminar on Tuesday, October 12th, will be given by Ovidiu-Neculai Avadanei at 5 pm in 740 Evans.

Speaker: Ovidiu-Neculai Avadanei

Abstract: This talk represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and its non-dispersive version is known to be completely integrable. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character, with only continuous dependence on initial data. Here, we prove the local and global well-posedness of the Cauchy problem using a normal form approach to construct modified energies, and frequency envelopes in order to prove the continuous dependence with respect to the initial data. This is joint work with Albert Ai.

A stationary set method for estimating oscillatory integrals

The HADES seminar on Tuesday, October 5th, will be given by Ruixiang Zhang at 5 pm in 740 Evans.

Speaker: Ruixiang Zhang (UC Berkeley)

Abstract: Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a “stationary set” method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry’s problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

Streak artifacts from non-convex metal objects in X-ray tomography

The HADES seminar on Tuesday, September 28th, will be given by Joey Zou at 5 pm in 740 Evans.

Speaker: Joey Zou (University of California, Santa Cruz)

Abstract: In X-ray CT scans with metallic objects, streak artifacts in the computed image may arise due to beam hardening effects, where the attenuation coefficient of metallic objects vary strongly with energy. A mathematical description of these artifacts using the notion of wavefront sets was given by Choi, Park, and Seo in 2014, followed by the work of Palacios, Uhlmann, and Wang, who gave quantitative descriptions of the artifacts that recovered qualitative observations from CT scans when the metallic objects are strictly convex. In this talk, I will discuss joint work with Yiran Wang which builds on the previous work by using microlocal analysis to study artifacts generated by non-convex metallic objects, as well as artifacts associated to a broader class of attenuation variations than was considered before. The problem relies on the analytic behavior of a nonlinear function composed with the image of the X-ray transform applied to certain functions, for which we use the work of Melrose, Ritter, Sa Barreto et al. on semilinear wave equations via the usage of iterated regularity spaces in which both the X-ray transform image and its nonlinear composition live.

The Benjamin-Ono approximation for low frequency gravity water waves with constant vorticity

The HADES seminar on Tuesday, September 21st, will be given by James Rowan from at 5 pm in 740 Evans.

Speaker: James Rowan (University of California, Berkeley)

Abstract: It is well-known that the cubic nonlinear schrodinger equation gives a good approximation for frequency-localized solutions to the irrotational 2D gravity water waves equations, at least on a cubic timescale.  Replacing the assumption of irrotationality with one of constant vorticity allows the model to apply to waves in settings with countercurrents, but the new terms introduced by the vorticity break the scaling symmetry, and in the low-frequency regime, they should have a large effect.  We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good approximation to the 2D gravity water waves equations with constant vorticity.  This work is joint with Mihaela Ifrim, Daniel Tataru, and Lizhe Wan.  Along the way to this result, I will give a brief introduction to some topics in nonlinear dispersive PDE and fluid dynamics.

On linear inviscid damping around monotone shear flows and singularity structures near boundaries

The HADES seminar on Tuesday, May 11 will be given by Wenjie Lu via Zoom from 3:40 to 5 pm.

Speaker: Wenjie Lu (University of Minnesota)

Abstract: Hydrodynamic stability is one of the oldest problems studied in PDEs. In this talk, I will introduce results related to the linear stability of monotone shear flows with boundaries. If the vorticity vanishes near boundaries, one can obtain optimal decay estimates in Gevery spaces. However, the boundary effect is significant and can be an obstruction for the scattering of the vorticity in high regularity spaces. In order to understand the asymptotic behavior more clearly, we need to have a full picture of the singularity structure of the generalized eigenfunctions. It turns out that we can actually track singularities of arbitrary derivatives of the generalized eigenfunctions. With this, we can get arbitrary many terms in the asymptotic, not only the main term. This is a recent joint work with Hao Jia.

The shock formation problem: an overview

The HADES seminar on Tuesday, April 27 will be given by Federico Pasqualottovia Zoom from 3:40 to 5 pm.

Speaker: Federico Pasqualotto

Abstract: Shock waves are a fundamental phenomenon which appears in the context of compressible fluid flow.In this talk, we will review the problem of shock formation, focusing on various techniques which are suitable to study the problem in one and several space dimensions.

The instability of Anti-de Sitter spacetime for the Einstein–scalar field system

The HADES seminar on Tuesday, March 30 will be given by Georgios Moschidis via Zoom from 3:40 to 5 pm.

Speaker: Georgios Moschidis

Abstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. According to this conjecture, there exist arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting  boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.  In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric  Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.