学术报告报告主题一: Nonlinear stability of two-dimensional compressible current-vortex sheets
报 告 人: Paola Trebeschi (University of Brescia）
报告时间：2024年 4月23日（星期二）下午 3:00-4:00
报告地点：37号楼3A02
报告主题二: The two-dimensional plasma-vacuum interface problem in ideal MHD
报 告 人: Alessandro Morando (University of Brescia）
报告时间：2024年 4月23日（星期二）下午 4:00-5:00
报告地点：37号楼3A02
邀请人: 朱长江教授、温焕尧教授、洪广益副教授
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数学学院
2024年 4月22日
报告一摘要：In this talk we are concerned with nonlinear stability and existence of two dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash-Moser iteration, provided the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin-Helmholtz instabilities.
报告二摘要：In this talk we consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolicelliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We present our result about the local-in-time existence and uniqueness of solutions to the nonlinear free boundary problem, provided that the plasma magnetic field or the vacuum magnetic field is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash-Moser iteration.
报告人介绍:
Paola Trebeschi is Associate Professor at the University of Brescia (Italy). She received a Laurea Degree in Mathematics in 1991 at University of Pavia (Italy) and a PhD in Mathematics in 1998 at University of Pisa (Italy). Professor Paola Trebeschi’s research interests concern nonlinear system of hyperbolic equations with application to Fluid dynamics and Magnetohydrodynamics. In particular, she focuses on stability and local existence of piece-wise smooth weak solutions to compressible Euler system and compressible and incompressible Magnetohydrodynamics, which exibit a free interface of strong discontinuities.
Alessandro Morando is Associate Professor at University of Brescia (Italy). He is received a Laurea Degree in Mathematics in 1997 and a PhD in Mathematics in 2003. His research interests mainly address to nonlinear hyperbolic equations and systems, with applications to Fluid dynamics and Magnetohydrodynamics. In particular, his current research focuses on the solvability of free boundary problems for compressible Euler equations and compressible or incompressible Magnetohydrodynamics.