Title: On the stationary and uniformly rotating solutions of active scalar equations

Time: Tuesday, April 6th at 11:00am

Location: bluejeans.com/984226566

Committee: 

Yao Yao (Georgia Tach School of Mathematics, Advisor) 

Chongchun Zeng (Georgia Tech, School of Mathematics)

Zhiwu Lin (Georgia Tech, School of Mathematics)

Ronghua Pan (Georgia Tech, School of Mathematics)

Javier Gomez-Serrano (University of Barcelona, Departament de Matemátiques i Informática)

Abstract

We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.

   For qualitative properties, we aim to establish sufficient conditions for such solutions to be radially symmetric. The proof  is based on variational argument, using the fact that a uniformly-rotating solution can be formally thought of as  a critical point of an energy functional. It turns out that if positive vorticity is rotating with angular velocity, not in (0,1/2), then the corresponding energy functional has a unique critical point, while radial ones are always critical points. We apply similar ideas to more general active scalar equations (gSQG) and vortex sheet equation. We also prove that for rotating vortex sheets, there exist  non-radial rotating vortex sheets, bifurcating from radial ones. This work is based on the joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao. 

    It is well-known that there are non-radial rotating patches with angular velocity in (0,1/2). Using the variational argument, we derive some quantitative estimates for their angular velocities and the difference from the radial ones.