司靈得 (Daniel Spector)

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SPRING 2022 Nonlinear Analysis Seminar Series

*To view the video, click the title of each lecture.
Date: Tuesday, 8th March 2022, 09:00–11:00 am CST (GMT+8), online on Webex

Speaker: Ángel D. Martinez, University of Toronto

Title: On the symmetry conjecture for eigenfunctions

Abstract:

In this talk we will provide a brief introduction to the eigenfunctions of the Laplace-Beltrami operator. We will then focus on the so-called symmetry conjecture and a natural variant raised by Nadirashvili and Jakobson. In particular we will present certain results (positive and negative) obtained in collaboration with F. Torres de Lizaur, and extended by a group of students during last summer at the Fields Undergraduate Summer Research Program under our supervision. Time permitting we will comment on work in progress or open questions.




Date: Tuesday, 15th March 2022, 15:00–16:00 pm CST (GMT+8), online on 
Webex

Speaker: Giorgio Stefani, Scuola Internazionale di Studi Superiori Avanzati

Title: An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

Abstract: 

We revisit Yudovich's well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid. Existence of global-in-time weak solutions holds in suitable uniformly-localized versions of the Lebesgue space and of the Yudovich space, respectively, for the vorticity, with explicit modulus of continuity for the associated velocity. Uniqueness of weak solutions, in contrast to Yudovich's energy method, follows from a Lagrangian comparison. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions. This is a joint work in collaboration with Gianluca Crippa.



Date: Tuesday, 22nd March 2022, 15:00–16:00 pm CST (GMT+8), online on Webex

Speaker: Giovanni Eugenio Comi, Universität Hamburg

Title: Distributional fractional Sobolev and BV spaces: fine properties and Leibniz rules

Abstract: 

Differently from their integer versions, the fractional Sobolev spaces \( W^{\alpha,p}(\mathbb{R}^n) \) do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and divergence already existing in the literature, we introduce via a distributional approach the new spaces \( BV^{\alpha,p}(\mathbb{R}^n) \) of \( L^p \) functions with bounded \( \alpha \)-fractional variation in \( \mathbb{R}^n \), for \( \alpha \in (0,1) \) and \( p \in [1, \infty] \). In addition, we define in a similar way the distributional fractional Sobolev spaces \( S^{\alpha,p}(\mathbb{R}^n) \), which extend naturally \( W^{\alpha,p}(\mathbb{R}^n) \). The properties of these spaces have been analyzed in a series of papers in collaboration with Giorgio Stefani, Daniel Spector, Elia Bruè, and Mattia Calzi. In this talk, we shall focus ourselves on the absolute continuity property of the fractional variation with respect to a suitable Hausdorff measure and on the existence of precise representatives for \( S^{\alpha,p} \) and \( BV^{\alpha,p} \) functions. Subsequently, we shall derive fractional Leibniz rules and Gauss-Green formulas involving \( BV^{\alpha,p} \) and \( S^{\alpha,p} \) functions. As an application of these results, we will show the well-posedness of the boundary-value problem for a general \( 2\alpha \)-order fractional elliptic operator in divergence form.


Date: Tuesday, 29th March 2022, 15:00–16:00 pm CST (GMT+8), online on Webex

Speaker:
Rahul Garg,  IISER Bhopal

Title: 
Grushin pseudo-multipliers

Abstract: 

I In this seminar, I shall talk about analogues of pseudo-differential operators (pseudo-multipliers) associated with the joint functional calculus of the Grushin operator. In particular, I shall discuss some sufficient conditions on a symbol function, those which imply $L^2$-boundedness of the associated Grushin pseudo-multiplier. This talk is based on a joint work (arXiv:2111.10098) with Sayan Bagchi. 



Date: Tuesday, 12th April 2022, 09:00–10:00 am CST (GMT+8), online on Zoom


Speaker: Andreas Seeger, University of Wisconsin-Madison

Title: Lp improving bounds for spherical maximal operators

Abstract:

Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the Lp improving estimates for the associated maximal operators and related objects.  Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some surprising  statements on the shape of the possible type sets.  Joint work with J. Roos and with T. Anderson, K. Hughes and J. Roos.



Date: Tuesday, 26th April 2022, 15:00–16:00 pm CST (GMT+8), online on 
Zoom

Speaker: Ritva Hurri-Syrjanen,  University of Helsinki

Title: On Choquet integrals and John domains

Abstract: 

We consider integrals in the sense of Choquet with respect to the Hausdorff content. In particular, we discuss the validity of Poincare, Poincare-Sobolev, and Trudinger type inequalities for functions defined on John domains in this context. This is joint work with Petteri Harjulehto.



Date: Tuesday, 3rd May 2022, 15:00–16:00 pm CST (GMT+8), online on 
Zoom

Speaker: Sean McCurdy, Carnegie Mellon University

Title: Elimination of Cusps in a Free Boundary Problem for Water Waves

Abstract: 

The study of a 2-dimensional wave profile of an inviscid, incompressible fluid acted upon by gravity traveling at constant speed goes back to 1847 and the work of Stokes, who conjectured that there should exists a one-parameter family of smooth wave profiles and an extreme wave of maximal height which had a peak with a 120 degree angle. It turns out that a wave profile in this setting is the free-boundary of a solution to a related free-boundary problem. The first half of this talk will serve as an incomplete and biased introduction to the study of this free boundary problem. Particularly, it will focus upon the ``density methods" developed by Varvaruca and Weiss. The second half of this talk will focus upon the issue of cusps. The aforementioned ``density methods" are unable to eliminate cusps, but recent work (joint with Lisa Naples) has recently found a rather simple approach to eliminating cusps.



Date: Tuesday, 10rd May 2022, 15:00–16:00 pm CST (GMT+8), online on 
Zoom

Speaker: Jann-Long Chern, National Taiwan Normal University

Title: The Heat Equation With A Dynamic Hardy-Type Singular Potential

Abstract: 

Motivated by the celebrated paper of Baras and Goldstein (1984), in this talk, we study the heat equation with a dynamic Hardy- type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. This is a joint work with Profs. G. Hwang, J. Takahashi and E. Yanagida.


Date: Tuesday, 17rd May 2022, 15:00–16:00 pm CST (GMT+8), online on 
Zoom

Speaker: Hideyuki Miura, Tokyo Institute of Technology

Title: Local regularity conditions on initial data for weak solutions to the incompressible Navier-Stokes equations

Abstract: 

We consider the regularity of weak solutions to the three dimensional incompressible Navier-Stokes equations. We prove that the suitable weak solution is locally regular if a local scaled energy of the initial data is sufficiently small. As an application, it is shown that if a global weighted L^2 norm of the initial data is finite, then the weak solution is regular in a region confined by space-time hypersurfaces determined by the weight. Our result is also applied to studying energy concentration near a possible blow-up time. This talk is based on a joint work with Kyungkeun Kang (Yonsei Univ.) and Tai-Peng Tsai (Univ. British Columbia). 

 


Date: Tuesday, 24th May 2022, 15:00–16:00 am CST (GMT+8), online on 
Zoom

Speaker: Neal Bez, Saitama University

Title: Stability of the Brascamp-Lieb inequality

Abstract: 

In this talk I will discuss multilinear Kakeya estimates and the nonlinear Brascamp-Lieb inequality. Both may be viewed as perturbed versions of the classical version of the Brascamp-Lieb inequality and emerged around 2005 in connection with the multilinear restriction problem. Progress on these problems has relied heavily on induction-on-scales and I will try to give an overview of the main ideas.


 
 
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