The existence of Prandtl-Batchelor flows on disk and annulus

Series
PDE Seminar
Time
Tuesday, October 4, 2022 - 3:00pm for 1 hour (actually 50 minutes)
Location
Speaker
Zhiwu Lin – Georgia Tech – zlin@math.gatech.eduhttps://zlin.math.gatech.edu/
Organizer
Gong Chen

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. 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 a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao.