分析和PDE线上讨论班

组织者:安歆亮,郭紫华,苏庆堂,张瑞祥,张芷媛


   报告的PPT和视频在dropbox和百度网盘

2020-07-11/12, 8:30am-12:15pm (Beijing), Workshop, chair: Ruixiang Zhang

7月11日
08:30-09:30 王虹 (Institute for Advanced Study)
Incidence estimates for tubes with application to Fourier analysis
Abstract: If \mathbb{T} is a collection of distinct tubes of length N, radius 1 and P is a collection of disjoint unit balls, what is the largest size of their incidences I(P, \mathbb{T})=\{(p, T): |p\cap T|\geq 1/10, p\in P, T\in \mathbb{T}\}? We will use some basic Fourier analysis to give upper bounds on the number of incidences. Then we discuss how to use these upper bounds to solve some problems in Fourier analysis. This includes joint work with Larry Guth, Noam Solomon, and Ciprian Demeter.

09:45-10:45 赵子慧 (University of Chicago)
Boundary regularity of area-minimizing currents: a linear model with analytic interface
Abstract: Given a curve \Gamma, what is the surface T that has least area among all surfaces spanning \Gamma? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction and my joint work with C. De Lellis.

11:00-12:00 郭少明 (University of Wisconsin-Madison)
某些振荡积分的变分估计以及在离散算子中的应用
摘要: 报告的内容是关于Stein-Wainger型振荡积分的最优变分范数估计。结合圆法,这些估计可以推出某些离散算子的有界性.



7月12日
08:30-09:30 金龙 (清华大学)
Control of eigenfunctions on surfaces of negative curvature
Abstract: In this talk, we present a uniform lower bound for the mass in any fixed nonempty open set of normalized Laplacian eigenfunctions on negatively curved surfaces, independent of eigenvalues. The result extends previous joint work with Semyon Dyatlov on surfaces with constant negative curvature. The proof relies on microlocal analysis, chaotic behavior of the geodesic flow and a new ingredient from harmonic analysis called Fractal Uncertainty Principle by Jean Bourgain and Semyon Dyatlov. Further applications include control for Schr\"{o}dinger equation and exponential decay of energy for damped waves. This is based on joint work with Semyon Dyatlov and St\'{e}phane Nonnenmacher.

09:45-10:45 郏浩 (University of Minnesota)
Long time dynamics and inviscid damping of 2d Euler equations
Abstract: In this talk, we will first survey some recent development in the study of long time behavior of the two dimensional incompressible Euler equations, and then focus on the inviscid damping for vortex and shear flows. We will explain in some detail the linear inviscid damping in Gevrey spaces and how to combine the precise linear analysis with the nonlinear perturbation techniques. Joint with A. Ionescu.

11:00-12:00 安歆亮 (National University of Singapore)
Polynomial blow-up upper bounds for the Einstein-scalar field system under spherical symmetry
Abstract: In gravitational collapse of the Einstein-scalar field system, with the focusing initial data, a black hole region could form. Within the black hole region, singularities at $r=0$ could arise. It is quite mysterious on how strong these singularities could be. In this talk, I will present two new results in this direction within spherical symmetry. i) With Ruixiang Zhang we show that even in the most singular scenario, along the singular boundary $r=0$, the curvature (Kretschmann scalar) would obey polynomial blow-up upper bounds $O(1/r^N)$. This improves previously best-known double-exponential upper bounds $O(\exp\exp(1/r))$. Our result is sharp in the sense that there are known examples showing that no sub-polynomial upper bound could hold. ii) With Dejan Gajic, we extend the aforementioned result to a global one and calculate the precise polynomial rate-$N$. We find that, when it is close to the timelike infinity, the blow-up rates of Kretschmann scalar could be different from the Schwarzschild value. In particular, the blow-up rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism.

2020-07-04, 10-11am (Beijing)

Speaker: Jiaqi Liu (U of Toronto)

Title: Long time asymptotics for the sine-Gordon equation

Abstract: We calculate the long time asymptotic formula for the sine-Gordon equation on the real line. Radiation term and kink/breather stability will be discussed. Our approach is based on the nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. This is joint work with Gong Chen and Bingying Lu.

2020-06-27, 9-10am (Beijing)

Speaker: 赖力(清华大学)

Title: 关于zeta(2k+1)的无理性的一些进展

Abstract: Riemann zeta函数在奇数处的值的算术性质是有趣的问题。1978年Apéry证明了zeta(3)是无理数,至今我们仍不知道zeta(5)是否是无理数。此次报告中,我们将讨论 Ball-Rivoal (2000),Fischler-Sprang-Zudilin (2018),Lai-Yu (2020) 的工作的共同点: 利用一些具体构造的有理函数作为辅助函数,我们能对zeta(3),zeta(5),...,zeta(s)中的无理数的个数给出下界估计。我们将介绍这些具体的有理函数的特点。

2020-06-20, 10-11am (Beijing)

Speaker: Yuan Gao (Duke University)

Title: Curved dislocation and nonlocal Ginzburg-Landau systems: existence, 1D symmetry and asymptotic stability

Abstract: Dislocations are important line defects in crystalline materials and play essential roles in understanding materials properties like plastic deformation. In this talk, I will first talk about the static Peierls-Nabarro (PN) models for a single straight/curved dislocation line, which can be reduced to a Ginzburg-Landau equation/systems involving "anisotropic half-Laplacian". The existence of local minimizers, uniqueness of stable solution and the exponential relaxation of dynamic solutions to the global minimizer (uniquely determined) will be discussed. This talk is based on joint works with Hongjie Dong, Jian-Guo Liu and Zibu Liu.

2020-06-13, 10-11am (Beijing)

Speaker: Hui Zhu (University of Michigan, Ann Arbor)

Title: Microlocal smoothing effect for gravity-capillary water waves

Abstract: The surface tension makes free surfaces of fluids instantaneously smooth. For 2D gravity-capillary water waves, this phenomenon has been justified by Christianson–Hur–Staffilani and Alazard–Burq–Zuily as local smoothing effects. In this talk, I will present a microlocal justification of this phenomenon in arbitrary dimensions.

水波的微局部光滑效应

表面张力使液体的自由表面瞬间光滑。对于二维重力毛细波,这个现象被Christianson–Hur–Staffilani 和 Alazard–Burq–Zuily 用“局部”光滑效应解释。在此报告中,我将对任意维度的重力毛细波给出光滑效应的“微局部”刻画。

2020-06-06, 10-11am (Beijing)

Speaker: Jiqiang Zheng (IAPCM)

Title: Laplacian operator with Hardy potential and applications to dispersive equations

Abstract: In this talk, we first talk about the Sobolev space theory and harmonic analysis tools for the Laplacian opeartor associated with Hardy potential. And then we consider the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold. This work is jointed with Rowan Killip, Changxing Miao, Jason Murphy, Monica Visan and Junyong Zhang.

2020-05-30, 10-11am (Beijing)

Spleaker: Chenyun Luo (Vanderbilt University & The Chinese University of Hong Kong)

Title: Survey on the Recent Developments of the Compressible Water Waves

Abstract: I will discuss some recent advancements in the theory of compressible water waves. First, we establish the local well-posedness for the motion of a compressible gravity water wave in 3D with vorticity taken into account. It is well-known that the existence for the free-boundary problem is not a direct consequence of the apriori estimate, since the approximate problems destroy the symmetry enjoyed by the original problem. We adapt the tangential smoothing introduced by Coutand-Shkoller to construct the approximation system with energy estimates uniform in the smoothing parameter. It should be emphasized that, when doing the a priori estimates, we need neither the higher-order wave equation of the pressure and delicate elliptic estimates nor the higher regularity assumption on the initial vorticity. Instead, motivated by Gu-Wang we generalize the Alinhac good unknown method to the estimates of full spatial derivatives. This technique is widely used in the study of free-boundary problems of incompressible fluids but seldomly for compressible fluids before. Second, if time permits, I will discuss a new approach that leads to the long time existence for the motion of a compressible gravity water wave.

2020-05-23, 10-11am (Beijing)

Speaker: Fei Wang (University of Maryland, College Park)

Title: The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary

Abstract: I will talk about the inviscid limit for the Navier-Stokes equations in a half space (in both 2D and 3D case), with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the solution of the Navier-Stokes equations converges in the vanishing viscosity limit to the solution of the Euler equation, on a constant time interval.

2020-05-16, 10-11am (Beijing)

Speaker: Liding Yao (University of Wisconsin-Madison)

Title: Frobenius theorem on log-Lipschitz subbundles

Abstract: In differential geometry, Frobenius theorem says that if a (smooth) real tangential subbundle is involutive, i.e. that X,Y are sections implies that [X,Y] is also a section, then this subbundle is spanned by some coordinate vector fields. Recently we prove that the theorem in the log-Lipschitz setting with distributional involutive condition. In the talk I will go over the formulation of the theorem and the sketch of the proof. If time permit I can also talk about the complex analogy of Frobenius theorems.

2020-05-09, 10-11am (Beijing)

Speaker: Haitian Yue (University of Southern California)

Title: Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation.

Abstract: In this talk, we consider the periodic derivative nonlinear Schrodinger's equation, which is L^2 critical. We show local well-posedness in Fourier-Lebesgue spaces which scale like H^s(T) for s>0. In particular we close the existing gap in the subcritical theory by improving the result of Grunrock-Herr (08), which established local well-posedness in Fourier-Lebesgue spaces which scale like H^s(T) for s>1/4. We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space.

2020-05-02, 10-11am (Beijing)

Speaker: Zhiyuan Zhang (Brown University)

Title: Linear Stability and Magnetic Confinement of the Relativistic Vlasov-Maxwell System

Abstract: The talk consists of two parts. In the first part, we consider the Relativistic Vlasov-Maxwell System on a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings and then provide several explicit families of stable/unstable equilibria. In the second part, we verify, for the 1.5D relativistic Vlasov-Maxwell system on an interval 0, 1), that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near-boundary region.

2020-04-25, 10-11am (Beijing)

Speaker: Qingtang Su (University of Southern California)

Title: Long time behavior of rotational water waves

Abstract: The study of the water waves has been one of the central problems in applied mathematics for centuries, yet the rigorous mathematical analysis for the full water waves was quite recently. Indeed, the local wellposedness for large initial data in Sobolev spaces was open until the breakthrough works of S. Wu in the late 1990s, and the global wellposedness for small localized initial data were proved in the last decade. The aforementioned global wellposedness result assumes irrotionality. However, most fluids feature the vorticity.  Although the local wellposedness of rotational water waves is well understood, the study of its long time behavior is still largely open.  In this talk I will focus on my work on the long time dynamics of 2d water waves with concentrated vorticity. First, I will survey the results on the long time behavior of irrotational water waves,  in particular, I'll discuss the idea of proving local and long time existence. Then I'll formulate the rotational water waves as the coupling of the evolution of the free surface and the transport of the vorticity. In particular, if the vorticity is given by point vortices, then the problem is reduced to the evolution of the free boundary coupled to the motion of the point vortices. In general, we don't expect  such a system to remain smooth and small for a long time. However, for some special cases, such as for the case with a pair of couter-rotating point vortices  traveling away from the free boundary, we are able to obtain a long time existence and describe its long time behavior.

2020-03-07, 8:30am-11:50am (Beijing), Workshop, chair: Zihua Guo

08:30-09:10 Ruixiang Zhang (University of Wisconsin-Madison)
三维时空中波动方程的局部光滑(local smoothing)性质
摘要:1990年左右,Sogge猜想n+1维时空中波动方程的解满足如下的局部光滑性质:如果我们考虑局部时间区间的L^p时空范数估计,则比起固定时间的n维L^p估计可以几乎增加1/p阶光滑性(p≥2n/(n-1) ). 最近与Larry Guth和王虹合作,我们证明了n=2时猜想成立. 本次报告将介绍本猜想的一些历史,以及我们证明的一个指标最佳的平方函数估计. 从这个估计可以得到n=2时的局部光滑猜想. 一个关联几何的定理在证明中扮演了很重要的作用, 我们也会提到它.

09:20-10:00 Yu Deng (USC)
Invariant Gibbs measures and global strong solutions for 2D nonlinear Schrodinger equations
Abstract: We solve the long-standing problem of proving almost sure global well-posedness (i.e. existence with uniqueness) for the nonlinear Schrodinger equation (NLS) on T^2 on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is done by the new method of random averaging operator, which precisely captures the implicit randomness structure of the high-low interactions. This is joint work with Andrea R. Nahmod (UMass Amherst) and Haitian Yue (USC).

10:10-10:50 Gong Chen (University of Toronto)
Long-time asymptotics for the cubic NLS in 1d
Abstract: I will discuss the long-time asymptotics of small solutions to the 1d cubic NLS with a potential. Using distorted Fourier transforms, localized dispersive estimates, we obtain the long-time asymptotics for the 1d cubic NLS under very mill assumptions on potentials. This is joint work with Fabio Pusateri.

11:00-11:40 Shiwu Yang (BICMR)
Decay properties for defocusing semilinear wave equations
Abstract: In this talk, I will present recent progress on the asymptotic decay properties for energy subcritical defocusing semilinear wave equations. For certain range of the power p, containing part of subconformal class, the solution scatters in the critical Sobolev space and energy space in space dimension greater than 2. In addition, in space dimension 3, the solution also verifies pointwise decay properties.