题目：Transition threshold for 2D Taylor-Couette flow in the exterior domain

报告人：李特 副研究员 （中国科学院）

摘要：In this paper, we consider the stability of the incompressible Navier-Stokes equations in the exterior domain of the two-dimensional Taylor-Couette (TC) flow. The linearized operator corresponding to 2D TC flow is $-\nu(\partial_r^2+\f{1}{r}\partial_r+\f{1}{r^2}\partial_{\theta

}^2)w+(A+\f{B}{r^2})\partial_{\theta}w$ with $(r, \theta)\in [1, +\infty)\times \mathbb{T}$ in the exterior domain and $A, B$ being constants. The previous work addressed the stability near shear flows, requiring the background flow $u$ to satisfy condition $u'\geq c_0>0$, that is, $u'$ has a uniform lower bound. However, the shear effects of TC flow $(A+\f{B}{r^2})'=-\f{2B}{r^3}$ in the exterior region gradually weaken and tend towards zero at infinity. Earlier studies on the stability near the Lamb–Oseen vortex with vanishing conditions at infinity introduced a self-similar transformation by Gallay and Wayne, turning the Laplacian operator into a harmonic oscillator $-\partial_r^2+r^2$. Here, the function $r^2$ in the harmonic oscillator is used in a sense to balance the vanishing shear effects at infinity. This paper employs weighted resolvent estimates directly within the original Sobolev space to address the stability problem, resulting in the linearized enhanced dissipation and integrable inviscid damping in the exterior region. The weighted space here, in a sense, is designed to match the weakening shear effects at infinity. Finally, based on these linear results, we also provide the estimate for the transition threshold of the fully nonlinear equations. This is a joint work with P. Zhang and Y. Zhang.

报告时间：2024年6月25日（周二）上午10:00-11:00

报告地点：教二楼612

联系人：牛冬娟