勷勤数学•专家报告

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

报  告  人：许孝精 教授  (邀请人：黄锐)

                                              北京师范大学       

时      间：3月30日  14：30-15：30

地     点：数科院西楼111报告厅

报告人简介:

        许孝精， 理学博士，北京师范大学学数学科学学院教授，博士生导师。主要研究来自流体动力学中的偏微分方程组的数学理论，主持多项国家自然科学基金项目和省部级科研项目，在不可压缩流体力学的数学理论研究中取得系列重要进展，发表在 JMPA、 SJMA、 JNLS、 JDE、 Nonlinearity 等国际期刊上。曾多次访问法国、美国、加拿大、波兰和香港等地知名高校，进行科研合作。

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.

          欢迎老师、同学们参加、交流！