On The Existence of Multi-dimensional Compressible MHD Contact Discontinuities

• 主讲人：Zhouping Xin (The Chinese University of Hong Kong)
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2023-03-30 20:00 - 2023-03-30 21:00
• 地点： online

Recording: https://disk.pku.edu.cn:443/link/FE999300C34E3C58E76EAA978AC3313F
Valid Until: 2027-04-30 23:59

 

Abstract: Contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) are the most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for systems of hyperbolic conservations laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversally, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have discontinuities. Some of the major difficulties for the existence of the Multi-dimensional ideal MHD contact discontinuities are the possible nonlinear Rayleigh-Taylor instability and loss of derivatives due to the non-ellipticity of the associated linearized problem. In this talk, I will present the recent work where we have proved the local existence and uniqueness of MHD contact discontinuities in both 2D and 3D in Sobolev spaces without any additional constraints such as Rayleigh-Taylor sign condition or with surface tensions. The key ingredients of our analysis are the Cauchy formula for MHD, the transversality of the magnetic field, and an elaborate viscous approximation. This talk is based on a joint work with Professor Yanjin Wang of Xiamen University.

 

Bio: Professor Zhouping Xin is an expert in the areas of partial differential equations, mathematical physics, fluid mechanics, nonlinear waves, numerical analysis and numerical methods for PDEs and applied mathematics. He has made some substantial contributions to the stability theory of linear and nonlinear waves, boundary layer theory, multi-dimensional shock wave theory, transonic flows, interfacial wave motions and free boundary problems in fluid dynamics, vacuum dynamics, vortex methods and relaxation methods, transonic flows, and multi-dimensional compressible and incompressible Navier-Stokes systems with more than one hundred eighty publications in leading international research journals. After getting his Ph. D in mathematics from the University of Michigan (Ann Arbor) in 1988, Professor Xin became a Courant instructor at The Courant Institute of New York University, where he was promoted to be a full professor of mathematics in 1995. In 2000, Professor Xin moved from the New York University to the Chinese University of Hong Kong where he has been the William M. W. Mong Professor of Mathematics and the executive director of the Institute of Mathematical Sciences. Professor Xin got many honors including: Sloan Research Fellow (1991-1993, USA), Presidential Fellow (1993, NYU, USA); ICM invited speaker (2002); and Morningside Gold Medalist in Mathematics (2004). Professor Xin is the co-editor-in-chief of MAA, associated editor of JMP and M2AS, and member of the editorial boards for many journals such as M3AS, Kyoto J. Math, Sciences China Mathematics, JHDE, etc.. http://www.ims.cuhk.edu.hk/people/staff/zpxin/