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：陶涛（山东大学商学院）
题 目：Prandtl-Batchelor flows on an annulus
时 间：2022.08.25（星期四）14:30-15:30
地 点：N208
摘 要：For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the limit of vanishing viscosity, the vorticity is constant in an inner region separated from the boundary layer. In this talk, we consider the generalized Prandtl-Batchelor theory on an annulus that is non-simply-connected. First, we observe that in the vanishing viscosity if steady Navier-Stokes solutions with nested closed streamlines on an annulus converge to steady Euler flows which are rotating shear flows, then the vorticity of Euler flows must be the form "a\ln r+b" and the associated velocity must be the form "(ar+\frac{b}{r}+cr\ln r,0)" in polar coordinates. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocity slightly different from the rigid-rotation. In particular, for the above boundary conditions, we prove that there is a continuous curve (i.e. infinitely many) of solutions to the steady Navier-Stokes equations when the viscosity is sufficiently small. This is a joint work with M.Fei, C.Gao, Z.Lin.
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