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

Speaker: Ryan Martinez

Abstract: We present work, still in progress, with Mihaela Ifrim and Daniel Tataru, which proves global well-posedness, global $L^6$ based Strichartz estimates, and global bilinear spacetime $L^2$ estimates for non-integrable 1D defocusing cubic NLS at the sharp regularity $H^{-1/2 + \epsilon}$ with mild regularity assumptions on the nonlinearity; taking for granted a suitable local well-posedness theory.

In $L^2$, this problem was well understood by Ifrim and Tataru, by using a modified energy method in a frequency localized setting. However, below $L^2$ there are several challenges. First, Christ, Colliander, and Tao show that the initial data-to-solution map fails to even be uniformly continuous locally in time below $L^2$. For the completely integrable problem Harrop-Griffiths, Killip, and Visan proved global (and local) well-posedness in the sense of continuous dependence and local smoothing estimates for the problem in the sharp space. Our work supplements their work by in addition providing global $L^6$ and bilinear $L^2$ estimates, but does not itself depend on complete integrability. To emphasize this, we prove the result for general nonlinearities, of course assuming the existence of a local theory, which at this time, seems out of reach.

The main challenge of this work is that the modified energy method used by Ifrim and Tataru at $L^2$ fails at high frequency below $s = -1/3$. To overcome this we use an infinite series of corrections.

The HADES seminar on Wednesday, October 22st, will be at 4:00pm in Room 732.

Speaker: Sanchit Chaturvedi

Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small-scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. We provide an asymptotic expansion in viscosity of the viscous solution via the help of matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. Furthermore, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. This is joint work with John Anderson and Cole Graham.

The HADES seminar on Tuesday, October 21st, will be at 3:30pm in Room 740.

Speaker: Sehyun Ji

Abstract: The global existence of smooth solution for the Landau-Coulomb equation remained elusive for a long time. Two years ago, Nestor Guillen and Luis Silvestre made a breakthrough by showing the Fisher information is monotone decreasing. As a consequence, they deduced the solutions do not blow up for C^1 initial data with Maxwellian tails. For a monotone quantity, It is very natural to ask for its dissipation estimate. In this talk, I will derive an a priori estimate for the dissipation of the Fisher information, which appears to be a higher-order analogue of the entropy dissipation estimate. As an application, I’ll show the global existence of smooth solutions for rough initial data in L^1_5 \cap L \log L. I will start from discussing the proof of Guillen and Silvestre.

The HADES seminar on Tuesday, October 14th, will be at 3:30pm in Room 740.

Speaker: Zachary Lee

Abstract: The Nonlinear Schrödinger Equation (NLS) arises in various physical contexts, notably in models of Bose–Einstein condensation and nonlinear optics. I will begin by outlining these motivations and by presenting several heuristics—scaling, dispersion, and symmetry—that shed light on the qualitative behavior of its solutions. I will then turn to a rigorous analysis based on the Duhamel formulation of the equation, together with Strichartz estimates, conservation laws, and Morawetz inequalities, which provide global control for data in the defocusing case. In the final part of the talk, I will describe how these techniques can be adapted below the energy space using almost conservation laws (the I-method), and present a new global existence result for the one-dimensional defocusing septic NLS for a class of discontinuous and unbounded initial data.

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

Speaker: Rui Liang

Abstract:In this talk, we will consider the Schrödinger equation with cubic nonlinearity on the circle, with initial data distributed according to the Gibbs measure.  We will discuss the challenges and strategies involved in establishing the Poincaré recurrence property with respect to the Gibbs measure in the full dispersive range. This work, using the theory of the random averaging operator developed by Deng-Nahmod-Yue ’19, addresses an open question proposed by Sun-Tzvetkov ’21. We will also explain why the Gibbs dynamics for the full dispersive range is sharp in some sense. Finally, we will see how the theory of random tensors works for extending this work to multi-dimensional settings.

The HADES seminar on Tuesday, September 23rd, will be at 3:30pm in Room 740.

Speaker: Sung-Jin Oh

Abstract: In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IHÉS) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for a broad class of underdetermined operators, including the divergence operator, linearized scalar curvature operator, and the linearized Einstein constraint operator. They are optimally regularizing and, more interestingly, have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). My goal is to (1) describe our approach, (2) demonstrate how it generalizes the well-known construction of Bogovskii, which has proved very useful in fluid dynamics, and (3) explain how it connects underdetermined PDEs with the rich literature on the dual problem on overdetermined differential operators.