偏微分方程研讨班：Geometric regularity criteria and inviscid limit of incompressible Navier--Stokes Equations under Navier boundary condition

中科院数学与系统科学研究院

数学研究所

学术报告

偏微分方程研讨班

报告人：李思然 (上海交通大学)

题  目：Geometric regularity criteria and inviscid limit of incompressible Navier--Stokes Equations under Navier boundary condition

时  间：2023. 11.23（星期四）14:30-16:30

地  点：数学院南楼N226

摘  要：We report our recent works on the analysis of 3D incompressible Navier--Stokes Equations subject to the Navier slip boundary condition, i.e., tangential components of the stress exerted by the normal direction are proportional to the tangential components of the velocity. Our first topic is the (boundary) regularity criteria: if the vorticity directions remain 1/2-Holder in space up to time T, then the velocity remain smooth before T. This was first proved by Constantin--Fefferman on the whole space $\mathbb{R}^3$ and later extended by Beirao da Veiga--Berselli on $\mathbb{R}^3_+$. We discuss our ongoing work to extend it to any C^{3,\alpha}-domain under the Navier boundary condition, exploiting the Green's matrix and geodesic normal coordinates. Then, we briefly discuss our (nearly completed, joint with Gui-Qiang Chen and Zhongmin Qian) work on the inviscid limit of Navier--Stokes Equations under the Navier slip boundary condition. We justify a boundary layer expansion in conormal Sobolev spaces, thus providing a refined characterisation of the uniform estimates obtained by Masmoudi--Rousset and the boundary layer expansion obtained by Iftimie--Sueur.