讲座题目：Liouville-type Theorems for steady solutions to the Navier-Stokes system in a slab

主讲人：桂长峰 教授 澳门大学

讲座时间：2023年6月12日下午15:00 -16:00

讲座地点：良乡校区数统楼311

主讲人简介：

  桂长峰，澳门大学数学系讲座教授，数学系主任，博士生导师。1991年在美国明尼苏达大学获博士学位。桂长峰教授曾入选国家级人才计划和海外高层次人才，于2013年当选美国数学会首届会士，获得过IEEE最佳论文奖、加拿大太平洋数学研究所研究成果奖、加拿大数学中心Andrew Aisensdadt 奖等荣誉。桂长峰教授现致力于非线性偏微分方程的研究，特别是在Allen-Cahn方程的研究、Moser-Trudinger不等式最佳常数的猜想、De Giorgi 猜想和Gibbons 猜想等方面取得了一系列在国际上有影响的工作，在Ann. of Math., Invent. Math., Comm. Pure Appl. Math., Arch. Ration. Mech. Anal., Adv. Math.等国际顶级期刊上发表论文80余篇。

主讲内容：

       In this talk, I will present recent results on Liouville-type theorems for the steady incompressible Navier-Stokes system in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl velocity is independent of the angular variable, or $ru^r$ decays to zero as $r$ tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions. The talk is based on a recent joint work with Jeaheang Bang, Yun Wang and Chunjing Xie.