Year:    2021

CAM-Net Digest, Vol. 18 (2021), Iss. 13 : p. 5

Abstract

国家天元数学东北中心将于2021年7月8日-7月15日在举办“Convex Integration and Its Applications in PDE”暑期短课。该课程由BaishengYan，Michigan State University开授，旨在为从事该领域研究的青年教师和研究生提供系统性讲解。

一、课程基本信息
授课人姓名：BaishengYan
授课人单位：Michigan State University
课程名称：Convex Integration and Its Applications in PDE  
课程形式：腾讯会议
开课时间段：2021年7月8日至7月15日
7月8日，  7月9日， 7月10日  10：00-11：00
7月13日，7月14日，7月15日  10：00-11：00
预备知识：Partial differential equation, 
Real analysis and Convex analysis

天元数学东北中心将开通B站转播，链接如下：https://space.bilibili.com/393390076。

非常感谢老师和同学们的关注，欢迎大家在线听课。

二、课程介绍

Many problems have different types of solutions in different spaces. Solutions in some spaces with high regularity may become more restricted (rigid or hard) so that they are stable under certain perturbation, while solutions in other spaces with low regularity can be less restricted (flexible or soft) so that they can be perturbed into a larger class. We focus on the flexibility question about approximating certain functions by the solutions of partial differential equations or inclusions. In geometry and topology, such a flexibility is known as the h-principle by Gromov. Convex integration is a set of approaches to prove the h-principle. The idea is initiated in the celebrated work of Nash and Kuiper on $C^1$ isometric embeddings of Riemann manifolds into a Euclidean space. The method of convex integration has recently found remarkable success in many important PDE problems, such as the counter-examples to regularity of elliptic and parabolic systems, the incompressible Euler equation, active scalar equations, porous medium equations, Boussinesq equations, Monge-Ampère equations, magneto-hydrodynamic equations, and Navier-Stokes equations.

In this series of lectures, I will discuss the basic ideas of convex integration with an easiest example and then explain Nash's original ideas in proving the h-principle of $C^1$ isometric embeddings. More general partial differential inclusions and the constrained inclusions in Tartar’s compensated compactness framework will be discussed together with some sufficient conditions for the convex integration of these problems. I will then discuss the celebrated work of De Lellis and Szèkelyhidi on the $h$-principle of the incompressible Euler equation. In the last two lectures, I will discuss some work (including my joint work with Seonghak Kim) on the Perona-Malik and general forward-backward diffusion equations and certain general diffusion systems including the $L^2$-gradient flows of some strictly polyconvex functionals.

Part I: Introduction of Convex Integration
Part II: Nash-Kuiper Theorem on $C^1$ Isometric Embeddings
Part III: Partial Differential Inclusions and Constrained Inclusions
Part IV: The Incompressible Euler Equation
Part V: Perona-Malik and Forward-Backward Diffusion Equations and Certain Diffusion System
Part VI: Examples of Some Polyconvex Gradient Flows

三、授课人介绍

Yan Baisheng is full-time professor at the Michigan state University. He got his PhD at University of Minnesota. He spent one year at Institute for Advanced Study.

His main research interests are partial differential equations, calculus of variations, nonlinear elasticity and applications in continuum mechanics and materials science. Mainly, well-posedness (existence, uniqueness and stability of solutions).

四、联系方式

若您对该门课程感兴趣，欢迎联系我们！

联系人：王老师
办公电话：0431-85167375
邮箱：tianyuanmath@jlu.edu.cn
地址：吉林大学数学学院315办公室