**Title:** Homogenization error of unsteady flow ruled by Darcy's law

**Speaker:** 徐强 (兰州大学)

**Meeting time: （Special time!）**10:00-11:00, April 15, 2022.

**Zoom Meeting ID**: 427 154 2002, Passcode: BIMSA

**Abstract:** This talk is devoted to studying homogenization error for non-stationary Stokes equations on perforated domains, which originally developed by J.-L. Lions. We now present a sharp error estimate in the sense of energy norms, where the main challenge is to control the boundary layers caused by the incompressibility condition. To obtain the optimal error, we first introduce some refined regularity estimates for corrector without compatibility conditions between initial and boundary data, as well as, the well posedness of the effective equations in Bochner space. Then, we further explain how we handle the boundary-layer correctors associated with Bogovskii's operator. This work is cooperated with Dr. Li Wang and Prof. Zhifei Zhang in Peking University.

**Upcoming Talks**

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** Past Talks**

**Title: Homogenization of elliptic systems with multiscales**

Speaker: 钮维生 (安徽大学)

Meeting time (时间): 14:00-15:00, October 15, 2021.

**Abstract** (摘要): We review some quantitative results in the homogenization of second-order elliptic systems with periodic coefficientsthat oscillate on multiple separated scales. The results include convergence rates and some uniform regularity estimates we obtained recently.

**Title: Babuska Problem in Composite Materials and Some Applications.**

Speaker: 李海刚 （北京师范大学）

Meeting time: 14:00-15:00, October 22, 2021.

**Abstract**: In high-contrast fiber-reinforced composite materials, the stress concentration between two adjacent inclusions is a common phenomenon, which always causes damage initiation. For the original problem proposed by Ivo Babuska concerning the system of linear elasticity, we develop an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principle in PDE theory and obtain the blow-up asymptotic expressions of the gradients of solutions to the Lame system with partially infinite coefficients in the narrow region between inclusions when they are close to touch. Our results hold for convex inclusions with arbitrary shape and in all dimensions. As an application, we recently proved an extended Flaherty-Keller formula on the effective elastic property of a periodic composite with densely packed fibers, which is related to the “Vigdergauz microstructure” in the shape optimizition of fibers. On the other hand, we recently applied our results to deal with the resonant behavior between two close-to-touching convex acoustic subwavelength resonators.

About the speaker: 李海刚，北京师范大学教授、博士生导师，主要从事材料科学中的偏微分方程理论研究。在复合材料中的Babuska问题及其相关领域做出一系列深刻原创成果，在Adv. Math.、ARMA、JMPA、JFA等杂志发表论文30余篇。2016年获得教育部霍英东青年教师基金，2018年获得教育部自然科学二等奖，2020年入选教育部长江学者奖励计划青年学者。

**Title**: Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable.

Speaker: 吕勇（南京大学）

Meeting time: 14:00-15:00, October 29, 2021.

**Abstract**: We prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which is slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of constant state nor of small energy. This talk is based on a joint paper with P. Zhang (AMSS).

**Title: Layer potential approach to homogenization of sound-absorbing and resonant acoustic metamaterials**

Speaker: Florian Feppon （ETH Zürich）

Meeting time: 15:00-16:00, November 12, 2021.

**Abstract**: In this presentation, I will present some new results about the derivation of an effective medium theory for sound-absorbing acoustic metamaterials. We determine an effective equation satisfied by the leading order asymptotic of the acoustic field scattered by many tiny obstacles filling a given volume and provide quantitative error estimates. One originality of our work is that we consider a fully non-periodic setting where the centers of the obstacles are randomly and independently distributed. Our analysis relies crucially on two new ingredients. First, using layer potentials, we show that the leading order asymptotic of the scattered field is determined by the solution to an algebraic linear system of size the number of obstacles, which can be physically interpreted as the usual ``Foldy-Lax approximation'' of the heterogeneous medium. The second ingredient is the convergence in some mean-square sense of this linear system towards a ``homogenized'' integral equation, from which the effective physics of the medium is inferred. We obtain as such that sound-absorbing media behave up to some critical size as metamaterials with positive effective refractive index. The case of high-contrast acoustic obstacles will then be briefly discussed, which lead to metamaterials exhibiting a positive or negative refractive index for an incident field whose frequency is respectively slightly smaller or slightly larger than resonant frequencies.

**Title: Extrapolation, weighted Good-$\lambda$ type bounds on Lipschitz domain, with application to Homogenization of Systems of Elasticity.**

Speaker: 耿俊 （兰州大学）

Meeting time: 14:00-15:00. Friday, November 19, 2021.

**Abstract**: For a family of systems of linear elasticity with rapidly oscillating periodic, bounded measurable coefficients, we give a sufficient condition for the weighted $W^{1,2}$ estimates for weak solutions of Neumann problems in a fixed bounded Lipschitz domains by using a weighted real variable method.

**Title: A Semilinear Interface Problem Arising from Population Genetics**

Meeting Time: 14:00-15:00. Friday, November 26, 2021.

Speaker: 苏琳琳 （南方科技大学）

**Abstract**: We study a semilinear parabolic problem that models the evolution of gene frequencies under the joint action of migration and selection with a geographical barrier. The barrier locates at an embedded interface and is of Kedem-Katchalsky type. Our main goal is to investigate the effects of the barrier on the global dynamics and on the existence, stability, and profile of spatially nonconstant equilibria.

**Title: 多组分渗流数值模拟开源软件开发**

Meeting Time: 15:00-16:00. Friday, November 26, 2021.

Speaker: 张晨松（中科院数学与系统科学研究院）

**Abstract**: 由于油气资源的成分复杂，油藏工程师在描述原油的性质时，需要使用不同的数学模型。除了常用的黑油模型之外，还有多种组分模型，如常平衡系数K模型、EOS状态方程组分模型等。一些复杂的驱油技术常基于组分模型的框架来进行数值模拟，例如聚合物驱油剂、表面活化剂驱油剂、碱驱油剂、微生物驱油剂、低盐度水驱等。一些在多孔介质中的流体运动，如二氧化碳的埋藏、地下水污染的处理、地热资源的开采等，也可使用组分模型来模拟。这些重要的应用使得组分模型逐渐成为油藏数值模拟的一个重要研究方向，但组分模型求解变量多、方程复杂，计算工作量也更大。我们基于一个通用组分模型框架，面向并行计算机体系结构，来构建快速求解方法及其开源软件平台。我们将讨论多组分渗流力学的工业应用、物理背景、数学模型、离散方法、快速求解、并行计算及其软件开发。

About the speaker: 张晨松，2002年获南京大学计算数学硕士学位，2007年获得美国马里兰大学应用数学博士学位，此后在宾州州立大学数学系从事博士后研究工作，于2011年进入中国科学院数学与系统科学研究院计算数学所，任副研究员。主要研究兴趣：自适应有限元方法、多层迭代法及其在流固耦合、油藏模拟等问题中的应用。2012年在第二十一届国际区域分解法会议（法国）做大会特邀报告，2013年在第16届全国流体力学数值方法会议做大会特邀报告。曾在Numer Math、SINUM、SIAM MMS、M3AS、JCP、Handbook of Numerical Analysis等国际一流期刊发表了多篇学术论文。

**Title: Convergence rates in homogenization of parabolic systems with locally periodic coefficients.**

Meeting Time: 14:00-15:00. Friday, December 3, 2021.

Speaker: 徐侥 （中国科学院大学）

**摘要**: 见图中信息。

**Title: Compactness and stable regularity in multi-scale homogenization**.

Meeting time：(Special) 中午12:00-13:00, Friday, December 17, 2021

Speaker: Jinping Zhuge （The University of Chicago）

**摘要**：见图中信息。

**Title: Some recent developments on numerical homogenization**

Meeting time: 14:00-15:00. Friday, December 31, 2021.

线上：Zoom Meeting ID: 361 038 6975, Passcode: BIMSA

Zoom link: https://zoom.us/j/98193841924?pwd=L2p5cTgxU1ZLUGRucllFSWNlZkl2UT09

Speaker**:** 张镭 （上海交通大学）

**Abstract:**Problems with a wide range of coupled temporal and spatial scales are ubiquitous in many phenomena and processes of materials science and biology. Numerical homogenization concerns the approximation of the high dimensional solution space of multiscale PDEs by a low dimensional approximation space with optimal error control, and furthermore, the efficient construction of such an approximation space, e.g., the localization of the basis on a coarse patch. There has been a vast amount of work concerning the design and analysis of numerical homogenization type methods for multiscale problems, such as asymptotic homogenization, numerical upscaling, heterogeneous multi-scale methods, multi-scale finite element methods, variational multi-scale methods, flux norm homogenization, rough polyharmonic splines (RPS), generalized multi-scale finite element methods, localized orthogonal decomposition, etc.Surprisingly, numerical homogenization has deep connections with Bayesian inference, kernel learning and probabilistic numerics. The Bayesian homogenization approach provides a unified framework for the construction of a proper coarse space with desired approximation and localization properties, and practically it corresponds to a variational problem with functional constraints, which is close to in this sense.In this talk, I will give a brief introduction of the recent development in numerical homogenization, its connection with Bayesian inference, fast solvers, and machine learning. I will also discuss some possible further directions.

**Title: ****Homogenization of Linear Elliptic Equations in Nondivergence-Form: Characterizations of Good Diffusion Matrices**

Meeting Time: 14:00-15:00. Friday, January 7, 2022.

线上：Zoom Meeting ID: 361 038 6975, Passcode: BIMSA

Speaker**:** Timo Sprekeler （National University of Singapore）

**Abstract: **In this talk, we discuss the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a Dirichlet boundary condition. We characterize good diffusion matrices $A$, i.e., those for which the sequence of solutions converges at a rate of $\mathcal{O}(\varepsilon^2)$ in the $L^{\infty}$-norm to the solution of the homogenized problem. Such diffusion ma/trices are considered “good” as the optimal rate of convergence in the generic case is only $\mathcal{O}(\varepsilon)$. First, we provide a class of good diffusion matrices, confirming a conjecture posed by Guo and Tran in 2020. Then, we give a complete characterization of diagonal diffusion matrices in two dimensions and a systematic study in higher dimensions. This talk is based on joint work with Xiaoqin Guo (University of Cincinnati) and Hung V. Tran (University of Wisconsin Madison).

**Title: ****Stochastic homogenization for reaction-diffusion equations and G-equations**

**Meeting time:** 14:00-15:00, March 4, 2022.

**Speaker****:** Yuming Paul Zhang (University of California - San Diego)

**Abstract: ** Homogenization is a general phenomenon when physical processes in periodic or random environments exhibit homogeneous long time dynamics due to large space averaging of the variations in the environment. While this area of Mathematics saw a slew of remarkable developments in the last 20 years, the progress in the case of reaction-diffusion equations, which model many important physical phenomena, has been somewhat limited due to the homogenized dynamic involving discontinuous solutions to different (first-order) equations. In this talk I will discuss the first proofs of homogenization for reaction-diffusion equations with random not necessarily isotropic reactions in several spatial dimensions. These include the cases of both time-independent and time-dependent reactions, with the later proof employing a new subadditive ergodic theorem for time-dependent environments. We obtain analogous results for G-equations with random flame speeds and incompressible background advections.

This talk is based on joint works with Andrej Zlatoš.

**Title: ****近壁湍流有界吗？**

**Meeting time:** (**Special time**)10:30-11:30, March 18, 2022.

**Speaker****:** 陈曦 （北航大学)

**摘要: ** 探索湍流的终极统计不变性(the ultimate statistical invariance of turbulence)是一个经典的科学问题。特别是对于自然界和工业界广泛存在的湍流边界层而言，因为流体的能量主要在近壁面被耗散掉，因此预测近壁湍流在无穷大雷诺数时的渐近态具有重要的科学和工程意义。一个世纪前，Prandtl提出了经典的壁面律（Law-of-wall）理论并在平均流速的预测上获得极大成功，但是对于湍流的脉动平均量所言有限。最近六十年的实验和计算数据显示，多达二十余项的近壁湍流脉动量（如脉动能极值、压力极值、涡量极值、耗散极值、扩散极值、速度高阶矩极值，以及壁面剪应力和壁面压力等等，通记为Φ）随着雷诺数（Re）增加而显著增长。对此，学术界一个普遍性的观点认为Φ∝ln(Re)，意味着随着Re增加Φ将发散性增长从而壁面律将失效。与此不同的是，本报告将介绍本人最近发展的有限耗散率理论（Law-of-bounded-dissipation）以及普适雷诺数相似律：Φ_{∞}-Φ∝Re^{-1/4}，新的结论是在Re→∞时诸Φ各为其有限Φ_{∞}所界（见Chen & Sreenivasan, 2021 JFM; 2022 JFM）。目前，这一观点引发了国际学者的密切关注和讨论 (如：Monkewitz 2022 JFM；Smits *et al* 2021 JFM；Pirozzoli *et al* 2021 JFM)，相关进展在此也将有所介绍。