走向现代数学学术报告 - 高金城副教授（No. 854)

报告题目：Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near a background magnetic field

报告时间：2025年11月7日 10:30

报告地点：东海岸校区-D实209

报 告 人：高金城 副教授（中山大学）

邀 请 人：吴忠二 博士

报告摘要：This paper resolves the global regularity problem for the three-dimensional incompressible magnetohydrodynamics (MHD) equations in the upper half-space with slip boundary conditions, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of conormal derivatives with the decay of tangential derivatives. These global conormal regularity estimates, together with sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of a magnetic field, the global-in-time vanishing viscosity limit for the 3D incompressible Navier-Stokes equations with anisotropic dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the vanishing viscosity limit.

报告人简介：高金城，博士研究生导师，中山大学逸仙学者，科技部重点研发青年科学家项目资助等，主要从而流体力学相关方程的理论与应用研究，在时间衰减估计、适定性和粘性消失极限方程取得了一些好的成果。