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1 2017 Workshop on Mathematical Study of Boundary Layers and Related Problems September 22 nd 24 th, 2017 Objectives: The goal of this workshop is to bring together experts in the field of mathematical theory of boundary layers and related problems to exchange recent progress and discuss challenging problems in this field. This is the annual meeting in 2017 for a key project supported by the NNSFC. Organizing Committee: Beixiang Fang (Shanghai Jiao Tong University, China) Daiwen Huang (Institute of Applied Physics and Computational Mathematics, China) Song Jiang (Institute of Applied Physics and Computational Mathematics, China) Ya-Guang Wang (Shanghai Jiao Tong University, China) Chunjing Xie (Shanghai Jiao Tong University, China) Liqun Zhang (Academy of Mathematics and System Sciences, CAS, China) Sponsors: National Natural Science Foundation of China School of Mathematical Sciences, Shanghai Jiao Tong University Institute of Natural Sciences, Shanghai Jiao Tong Unviersity MOE Key Lab in Scientific and Engineering Computing Shanghai Committee of Science and Technology Shanghai Key Lab in Mathematical Modeling, Analysis, and Computation for Multi-Physical Process Workshop Homepage:

2 Workshop Venue: Sep. 22 nd -24 th, 2017 Large Lecture Room, Math Building, SJTU Minhang Campus 上海交通大学闵行校区数学楼大会议室 Accommodation: Manhattan Hotel( 曼哈顿酒店 ( 闵行店 )) Bldgs 5-6, 900 Heqing Road, Minhang District, Shanghai 上海市闵行区鹤庆路 900 号 5-6 号楼 ( 近碧江路 ) Transportation: To SJTU Minhang Campus By Taxi ( 乘出租车 ) Directions from Pudong Airport/ Hongqiao Airport/ Hongqiao Railway Station/ Shanghai Railway Station/ Shanghai South Railway Station: It is recommended to take a taxi directly to Minhang Campus, Shanghai Jiao Tong University. If necessary, you can print the following information and show it to taxi driver: (To Minhang Campus of SJTU) 请载我到上海市闵行区东川路 800 号上海交通大学闵行校区 ( 东川路永平路 1 号门进校区 ) 谢谢! 从浦东机场虹桥机场虹桥火车站出发, 推荐乘出租车直达上海交通大学闵行校区, 告知出租车司机 : 请载我到上海市闵行区东川路 800 号上海交通大学闵行校区 ( 东川路永平路 1 号门进校区 ) 谢谢! By Subway( 乘地铁 ) Directions from Pudong Airport/ Hongqiao Airport/ Hongqiao Railway Station: Please take Metro Line 2, transfer to Line 1 at People s Square Station, and then transfer to Line 5 at Xinzhuang Station and get off at Dongchuan Road Station. Then take a taxi to Minhang Campus.

3 Or take airport shuttle bus No. 9 and get off at the terminal Xinzhuang; walk to Xinzhuang Metro Station to take Line 5. 从浦东机场虹桥机场虹桥火车站出发, 可搭乘地铁 2 号线至人民广场站换乘地铁 1 号线, 乘至莘庄站换乘地铁 5 号线,5 号线东川路站下车下车后乘出租车至闵行校区或者乘坐机场 9 线至终点站莘庄下车, 步行至莘庄地铁站乘坐 5 号线 To Hotels By Taxi ( 乘出租车 ) Directions from Pudong Airport/ Hongqiao Airport/ Hongqiao Railway Station/ Shanghai Railway Station/ Shanghai South Railway Station: It is recommended to take a taxi directly to the hotels. If necessary, you can print the following information and show it to taxi driver: (To Manhattan Hotel) 请载我到上海市闵行区鹤庆路 900 号碧江广场 5-6 号楼曼哈顿酒店, 谢谢! By Subway( 乘地铁 ) Directions from Pudong Airport/ Hongqiao Airport/ Hongqiao Railway Station: (To Manhattan Hotel): Please take Metro Line 2, transfer to Line 1 at People s Square Station, and then transfer to Line 5 at Xinzhuang Station and get off at Jinping Road Station. Then walk to the hotel. Or take airport shuttle bus No. 9 and get off at the terminal Xinzhuang; walk to Xinzhuang Metro Station to take Line 5. 从浦东机场虹桥机场虹桥火车站出发, 可搭乘地铁 2 号线至人民广场站换乘地铁 1 号线, 乘至莘庄站换乘地铁 5 号线,5 号线金平路站下车下车后步行至酒店或者乘坐机场 9 线至终点站莘庄下车, 步行至莘庄地铁站乘坐 5 号线

4 Maps:

5 Shuttle Buses: There will be a shuttle bus commuting between Manhattan Hotel and Math Building in the Minhang Campus, and it will leave the Hotel at 8:00am in the Moring of Sep. 23 rd and Sep. 24 th.

6 Speakers: 1. 丁玉涛大连理工大学 2. 李维喜武汉大学 3. 刘成杰上海交通大学 4. 罗天文清华大学 5. 陶涛山东大学 6. 王术北京工业大学 7. 王亚光上海交通大学 8. 谢春景上海交通大学 9. 辛周平香港中文大学 10. 徐超江武汉大学 11. 杨彤香港城市大学 12. 张剑文厦门大学 13. 张立群中国科学院数学与系统科学研究院 14. 章志飞北京大学

7 Sep. 22 nd : Registration 2017 Workshop on Mathematical Study of Boundary Layers and Related Problems Sep. 22 nd Sep. 24 th, 2017 Sep. 23 rd Morning Sep. 23 rd Afternoon Sep. 24 th Morning 08:30-08:45 Opening Ceremony Chair: 王亚光 Chair: 杨彤 Chair: 江松 08:45-09:20 辛周平 Global well-podedness of 2D Prandtl's boundary layer equations 09:20-09:55 章志飞 Well-posedness of the Prandtl equation and inviscid limit 14:00-14:35 徐超江 Boundary layer analysis for the fast horizontal rotating fluids 14:35-15:10 张剑文 On the non-resistive limit of 1D compressible MHD equations 08:30-09:05 张立群 On the regularity of weak solution of parabolic equations 09:05-09:40 谢春景 Some studies on steady flows in channels 09:55-10:15 拍照茶歇 15:10-15:30 茶歇 09:40-10:00 茶歇 Chair: 辛周平 Chair: 徐超江 Chair: 张立群 10:15-10:50 杨彤 Well-posedness of Prandtl equations in Gevrey function spaces 10:50-11:25 李维喜 Well-posedness in Gevrey function space for the three-dimensional Prandtl equations 11:25-12:00 罗天文 Some Results on The Three-Dimensional Prandtl Equations 15:30-16:05 王术 Boundary Layer Problem and Zero Viscosity-Diffusion Limit of the 2D/3D Incompressible MHD System with Dirichlet Boundary Conditions 16:05-16:40 陶涛 On the zero-viscosity limit of the Navier-Stokes equation 16:40-17:15 丁玉涛 Blow up criterion for the Prandtl equation 3 R 10:00-10:35 刘成杰 Prandtl boundary layer theory for MHD equations in Sobolev spaces without monotonicity 10:35-11:15 王亚光关于边界层理论问题的一些分析 11:15-12:00 自由讨论 12:00-14:00 午餐 17:30- 晚餐 12:00- 午餐

8 Titles and Abstracts Blow up criterion for the Prandtl equation Yutao Ding ( 丁玉涛 ) Dalian University of Technology Abstract: We discuss the blow up criterion for the Prandtl equation. There are two conditions in special Sobelev Space, one is the regularity condition, the other is the monotone and decay condition. The blow up criterion pays attention to the both conditions. Well-posedness in Gevrey function space for the three-dimensional Prandtl equations Weixi Li ( 李维喜 ) Wuhan University Abstract: We will talk about the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ]1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting. Joint work with Tong Yang.

9 Prandtl boundary layer theory for MHD equations in Sobolev spaces without monotonicity Chengjie Liu ( 刘成杰 ) Shanghai JiaoTong University Abstract: In the joint work with Feng Xie and Tong Yang, we study the validity of the Prandtl boundary layer theory for MHD equations in the half plane with no-slip boundary conditions on the velocity vector and the perfect conducting boundary conditions on the magnetic field. In the case that viscosity and magnetic diffusion tend to zero simultaneously, we derive the boundary layer problem and establish the well-posedness result in Sobolev spaces under the assumption of nonzero tangential magnetic field, without any monotonicity assumption on the velocity. Then, when the initial tangential magnetic field of MHD doesn't vanish at the boundary, we justify the validity of corresponding Prandtl boundary layer expansions in the Sobolev framework. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics.

10 Some Results on The Three-Dimensional Prandtl Equations Tianwen Luo ( 罗天文 ) Tsinghua University Abstract: We obtain a class of weak solutions of the three-dimensional Prandtl equations, which are related to the secondary flows in the three-dimensional boundary layers. This is a joint work with Prof. Zhouping Xin. 3 On the zero-viscosity limit of the Navier-Stokes equation R Tao Tao ( 陶涛 ) Shandong University Abstract: In this talk, we consider the zero viscosity limit of the incompressible Navier-Stokes equations with non-slip boundary 2 condition in R for the initial vorticity located away from the boundary. By using the vorticity formulation and Cauchy-Kowaleskaya theorem, Maekawa proved the local in time convergence of the Navier-Stokes 2 equations in R to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. We develop a direct energy 3 method to generalize Maekawa s result to R. This is a joint work with Mingwen Fei and Zhifei Zhang.

11 Boundary Layer Problem and Zero Viscosity-Diffusion Limit of the 2D/3D Incompressible MHD System with Dirichlet Boundary Conditions Shu Wang ( 王术 ) Beijing University of Technology Abstract: In this talk, we study the boundary layer problem, zero viscosity-diffusion limit and zero magnetic diffusion limit of the initial boundary value problem for the 2D/3D incompressible viscous and diffusive MHD system with Dirichlet boundary conditions. The main difficulties overcome here are to deal with the effects of the difference between the viscosity and diffusion coefficient on the error estimates and to control the boundary layer resulted by the Dirichlet boundary condition for the velocity and magnetic field. Firstly, we establish the result on the stability of the Prandtl boundary layer of MHD system with a class of non-trivial initial data and prove rigorously the solution of incompressible viscous and diffusive MHD system converges to the sum of the solution to the ideal MHD system and the approximating solution to Prandtl boundary layer equation by using the elaborate energy methods and the special structure of the solution to ideal MHD system, which yields that there exists the cancellation between the boundary layer of the velocity and the one of the magnetic field. Next, we obtain the stability result on the boundary layer for the magnetic field and zero magnetic diffusion limit of viscous and diffusive MHD system with the general initial data when the magnetic diffusion coefficient goes to zero. Finally, for general initial data, we consider the boundary layer problem of the incompressible viscous and diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with the different speeds. We prove rigorously the convergence to the ideal MHD system and the anisotropic MHD system from the incompressible viscous and diffusion MHD system by constructing the exact boundary layers and using the elaborate energy methods.

12 关于边界层理论问题的一些分析 Ya-Guang Wang ( 王亚光 ) Shanghai Jiao Tong University Abstract: 在此报告中, 我们将首先就 Prandtl 边界层方程的适定性理论进行一些回顾, 然后分析 Prandtl 边界层方程解在有限时间内的 Blowup 问题, 以及边界层分离现象的数学理论 Some studies on steady flows in channels Chunjing Xie ( 谢春景 ) Shanghai Jiao Tong University Abstract: We will first address the well-posedness for steady inviscid compressible flows in nozzles. Then we study the stability of some special viscous flows in channels.

13 Global well-podedness of 2D Prandtl's boundary layer equations Zhouping Xin ( 辛周平 ) Chinese University of Hong Kong Abstract: In this talk, I will discuss on the Global in time existence, uniqueness, and continuous dependence of regular solution to the two-dimensional unsteady Prandtl's boundary layer equation with monotonic initial data and favorable pressure. We will emphasize the regularity of the solution. This is a joint work with Liqun Zhang. Boundary layer analysis for the fast horizontal rotating fluids Chao-jiang Xu ( 徐超江 ) Wuhan University Abstract: It is well known that, for fast rotating fluids with the axis of rotation being perpendicular to the boundary, the boundary layer is of Ekman-type, described by a linear ODE system. In this work, we consider fast rotating fluids, with the axis of rotation being parallel to the boundary. We show that, for certain initial data, the corresponding boundary layer is described by a nonlinear, degenerated PDE system which is similar to the 2D Prandtl system. Finally, we prove the well-posedness of the governing system of the boundary layer in the space of analytic functions with respect to tangential variable. This is a joint work with Wei-Xi Li and Van-Sang Ngo.

14 Well-posedness of Prandtl equations in Gevrey function spaces Tong Yang ( 杨彤 ) City University of Hong Kong Abstract: After a brief review on the ill-posedness result about the Prandtl equations as perturbation of shear flow, we will present some well-posedness theories in Gevrey function spaces with index in (1,2] for both 2D and 3D problems. This is about the recent joint works with Wei-xi Li. On the non-resistive limit of 1D compressible MHD equations Jianwen Zhang ( 张剑文 ) Xiamen University Abstract: The non-resistive limit of one-dimensional compressible MHD equations with large data is justified. The convergence rates and the thickness of boundary-layer are obtained.

15 On the regularity of weak solution of parabolic equations Liqun Zhang ( 张立群 ) Academy of mathematic and system science Abstract: We review the proof of regularty of weak solutions for heat equations and then consider a class of ultraparabolic differential equations with measurable coefficients and lower order terms. It is proved that weak solutions to the equation are Holder continuous,which generalizes the classic results of parabolic equations with general coefficients. The main observation is a weak Sobolev inequality and Poincare inequality related to weak sub-solutions. This is a jointed work with Wang Wendong. Well-posedness of the Prandtl equation and inviscid limit Zhifei Zhang ( 章志飞 ) Beijing University Abstract: In this talk,i will present some results on the long time well-posedness and finite time blow-up of the Prandtl equation in the analytic class and the well-posedness in the Gevrey class for a class of data without monotonicity. If time is permitted, I also present a new proof based on energy method of inviscid limit of the Navier-Stokes equations in the analytic setting.

16 Participants: 1. 丁玉涛大连理工大学 2. 何成自然科学基金委 3. 黄代文北京应用物理与计算数学研究所 4. 江松北京应用物理与计算数学研究所 5. 雷天刚自然科学基金委 6. 李维喜武汉大学 7. 李自来北京应用物理与计算数学研究所 8. 罗天文清华大学 9. 陶涛山东大学 10. 王术北京工业大学 11. 王云苏州大学 12. 辛周平香港中文大学 13. 徐超江武汉大学 14. 杨彤香港城市大学 15. 臧爱彬宜春学院 16. 张剑文厦门大学 17. 张立群中国科学院数学与系统科学研究院 18. 章志飞北京大学 19. 邓师瑾上海交通大学 20. 方北香上海交通大学 21. 顾琪龙上海交通大学 22. 龚胜波上海交通大学 23. 郭正光上海交通大学

17 24. 何晓上海交通大学 25. 李亚纯上海交通大学 26. 刘成杰上海交通大学 27. 王海涛上海交通大学 28. 王维克上海交通大学 29. 王亚光上海交通大学 30. 谢春景上海交通大学 31. 杨雄锋上海交通大学 32. 赵勤上海交通大学

From Airports to SJTU There are two international airports in Shanghai, one is Hong Qiao Airport (SHA), and the other is Pu Dong Airport (PVG). Each o

From Airports to SJTU There are two international airports in Shanghai, one is Hong Qiao Airport (SHA), and the other is Pu Dong Airport (PVG). Each o Transportation Information Arrival by Flights: From PuDong International Airport (PVG) to SJTU Minhang Campus From Hongqiao Airport (SHA) to SJTU Minhang Campus Arriving by Train: From Shanghai Railway