应用数学系学术报告

--- 分析、偏微分方程与动力系统讨论班(2024秋季第15讲)

On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

许孝精 (北京师范大学)

时间：2025年1月10日(周五)下午 15:00-16:00

地点：沙河主楼E602

摘要: In this talk, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our aim is to determine the stability threshold index on $\mathbf{Re}$: the maximum range of perturbations within which the solution  remains stable. Initially, we examine the linear stability effects of a linearized perturbed system. Comparing our results with those obtained by Bedrossian, Germain, and Masmoudi [Ann.  Math. 185(2): 541–608 (2017)], we observe that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow while Coriolis force acts as a restoring force inducing a dispersion mechanism for inertial waves that cancels out lift-up effects occurred at zero frequency velocity. This dispersion mechanism exhibits favorable algebraic decay properties distinct from those observed in classical 3D Navier-Stokes equations. Consequently, we demonstrate that if initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}<\delta \mathbf{Re}^{-1}$   for any $\sigma>\frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$, then the solution to the 3D Navier-Stokes equations with rotation is  global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from $\frac{3}{2}$ to 1. This is a joint work with Wenting Huang and Ying Sun.

报告人简介: 许孝精，北京师范大学数学科学学院教授、博士生导师，“京师特聘”拔尖学者，数学建模教育中心执行主任。主要从事流体力学数学理论的研究，给出了一系列Boussinesq方程的适定性理论。曾获“吉林省优秀博士学位论文”，北京市优秀教学成果奖，北师大励耘优秀青年教师奖一等奖，校优秀博士生学位论文指导教师，校优质课程特等奖等。主持了多项国家自然科学基金。 发表学术论文60余篇，被引用1000余次。大部分成果发表在J. Math. Pures Appl., J. Funct. Anal., SIAM J. Math. Anal.，IUMJ，Nonlinearity等国际知名学术期刊。曾在法国、美国、加拿大、波兰和香港等地区进行学术访问十余次。

欢迎大家参加！