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Talks Series

Title: An Introduction to Elliptic Homogenization

Speaker: Fanghua Lin

Abstract: I shall started with some examples and problems that concern with partial differential equations in highly oscillating medium. Then we proceed with various classical results of G- and H-convergences, div-curl, oscillating test function and viscosity methods as well as correctors in the periodic case. Applications and further questions will be addressed at end.
Further developments including, in particular, various a priori estimates, rates of convergences and behavior of the Green and Poisson kernels would be discussed in Professor Zhongwei Shen's lectures.

Title: Analytic aspects on the heat flow of harmonic maps

Speaker: Changyou Wang

Abstract: In this series of lectures, I plan to discuss some classical theorems and further issues on the heat flow of harmonic maps, which include Eells-Sampson theorem on global smooth solutions into non-positively curvature manifold, Struwe's almost "regular" global solutions in dimension two, and Chen-Struwe, Chen-Lin's existence of partially regular solutions in higher dimensions, and some results on the blow-up analysis by Lin and Wang.

Title: Introduction to Almost-Periodic Homogenization

Speaker：Zhongwei Shen

Abstract: In this series of lectures we will introduce the quantitative homogenization theory of second-order linear elliptic equations and systems in divergence form with almost-periodic coefficients. We will start with the definition and properties of almost-periodic functions, and present the classical qualitative homogenization theory. The main body of the lectures will deal with the key issues in the quantitative theory: estimates of approximate correctors, convergence rates, uniform regularity estimates (H"older, Lipschitz, W^{1, p}). The lectures are designed for advanced graduate students as well as junior researchers in the general areas of analysis and PDEs. We assume that the audience is familiar with the basic material usually covered in the first-year graduate courses on real analysis, functional analysis, and linear elliptic PDEs.

Title: Gluing methods and finite time blow-ups for harmonic map flows into $S^2$

Speaker：Juncheng Wei

Abstract: I will explain the main ideas in the gluing constructions of Type II blow-ups for harmonic map flows into $S^2$: $$u_t=\Delta u+ |\nabla u|^2 u, u:\Omega \to S^2$$ where $\Omega$ is a smooth two-dimensional domains.

Title: Convergence rates for elliptic homogenization problems in Lipschitz domain

Speaker: Qiang Xu, Peking University

Abstract: In this talk, I plan to study convergence rates in $L^2$ norm for elliptic homogenization problems in Lipschitz domains. It involves some new weighted-type inequalities for the smoothing operator at scale $\varepsilon$, as well as, layer and co-layer type estimates, and the related details will be touched. In order to obtain a sharp result, a duality argument will be imposed. Here we do not require any smoothness assumption on the coefficients, and the main ideas may be extended to other models, such as Stokes systems and parabolic systems, arising in the periodic homogenization theory.

Title: Infinite time blow-up for half-harmonic map flow from into S1

Speaker: Youquan Zheng, Tianjin University

Abstract: We construct infinite time blow-up solutions for the half-harmonic map heat flow by the gluing method. This is a join work with Yannick Sire and Juncheng Wei.

Title: Transition threshold for the 3D Couette flow in Sobolev space

Speaker: Dongyi Wei, Peking University

Abstract: In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number {Re}.  It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.

Workshop on Nonlinear Partial Differential Equations

N204, 2018.05.26-05.27

2018年5月26日

8:50-9:00  主持人：张立群 研究员

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09:00-09:45  魏军城（University of British Columbia

目：Gluing methods and finite time blow-ups for harmonic map flows into S2

要：I will explain the main ideas in the gluing constructions of Type II blow-ups for harmonic map flows into $S^2$: $$u_t=\Delta u+ |\nabla u|^2 u, u:\Omega \to S^2$$ where $\Omega$ is a smooth two-dimensional domains.

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09:45-10:30  王克磊（Wuhan University

目：Scaling limits of interacting diffusions

要：In this talk I will discuss the scaling limit of N interacting Brownian motions through a finite range potential, a problem first considered by Varadhan. By establishing a Gamma convergence result on the free energy, and using the general mechanism on gradient flows of probability measures and the Gamma convergence of gradient flows, we establish a general convergence from this interacting particle system to a nonlinear diffusion equation. This approach bypasses the possible non-uniqueness of Gibbs measures. Hence the convergence holds in the phase transition region.

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10:30-10:45  Tea Break

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10:45-11:30  麻希南（ University of Science and Technology of China

目：曲率方程Neumann边值问题的梯度估计

要： 我们研究平均曲率方程以及对应的平均曲率流的Neumann边值问题的梯度估计，从而得到其到平移解的收敛性。我们也研究高阶曲率方程Neumann问题以及预定夹角的梯度估计。它是与徐金菊、王培合、韦韡以及邓斌的合作工作。

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12:00 - 14:00 Lunch    Wuke hotel (the fourth floor)

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14:00-14:45  申仲伟（University of Kentucky

目：Regularity Results in Elliptic Homogenization

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14:45-15:30  熊金钢（Beijing Normal University

目：On nonlocal Nirenberg problem

要：The Nirenberg problem asks: Which function $K$ is the Gaussian or Scalar curvature of a conformal metric to the standard one on the unit sphere? This problem is equivalent to solving a critical nonlinear elliptic equation with variational structure. I will give a brief review of its developments over the past 50 years, and report our recent progresses on the nonlocal cases. A major new input is a blow up analysis approach for critical nonlinear integral equations. This is joint work with Tianling Jin and Yanyan Li.

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15:30-15:45  Tea Break

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15:45-16:30 章志飞（Peking University

目：Linear inviscid damping for shear flows

要：Landau damping can be defined as damping of a collective mode of oscillations in a collisionless plasma. Analogues of Landau damping has been observed in the sheared hydrodynamic flows. In this talk, I will talk about our recent results on the damping of the linearized 2-D Euler equations around shear flows including monotone flows and non-monotone flows.

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16:30-17:15    震（Fudan University

目：Global Vanishing Viscosity Limit of PDEs Arising from Fluid Mechanics

要：Vanishing viscosity limit is a key topics in both the theory of fluid mechanics and the analysis of partial differential equations. In general it is expected to be true locally in time for Cauchy problem. However, as long as the time is global, the verification of such a theory is highly nontrivial and is thus open for most fluid systems. In this talk, we report our results on incompressible viscoelasticity and MHD. Those are joint works woth Yuan Cai, Fanghua Lin and Nader Masmoudi.

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18:00 - 20:00  Dinner   Wuke hotel (the fourth floor)

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2018年5月27日（星期日）

8:50-9:00  主持人：张 平 研究员

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09:00-09:45  王立河（IOWA State University

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09:45-10:30 张会春（SUN YAT-SEN UNIVERSITY

目：Geometric analysis on Alexandrov spaces

要：In this talk, we shall survey some improvements about geometric analysis on a class of singular spaces metric spaces with curvature bounded from below in the sense of Alexandrov. The main results include: (1) the Lott-Sturm-Villani's problem about generalized Ricci lower bounds, (2) the Weyl's law of the asymptotic behavior of eigenvalues on Alexandrov spaces, (3) a Lipschitz regularity of harmonic maps between Alexandrov spaces, which resolves a problem of Fanghua Lin, and some quantitative gradient estimates for harmonic maps into singular spaces, which resolves a problem of Jost. This talk is based on some joint works with Prof. Xi-Ping Zhu.

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10:30-10:45  Tea Break

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10:45-11:30      龙（Nanjing University of Science and Technology

目：Nodal sets of bi-­‐harmonic functions on a !C Riemannian manifold

要：In this talk,we Consider about the upper bound for the nodal set for a bi-­‐harmonic function !u on an !n dimensional !C∞ Riemannian manifold !M . We define the frequency function and the doubling index for such bi-­‐harmonic function, and then establish the monotonicity formulas and the corresponding doubling conditions. Then by using the smallness propagation and partitions, we will show the upper bound for the measure of nodal set of !u in some ball !Br(x0) with the radius !r small enough, it holds that ! Hn−1({x ∈Br/2(x0)|u(x)= 0})≤CNα rn−1 ,for some !α >1/2. Here !N is the frequency function of !u on !Br(x0) .

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12:00 - 14:00 Lunch    Wuke hotel (the fourth floor)

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