Mathematical Physics Seminar
The surface quasi-geostrophic (SQG) equation and its variants appear in geophysics and describe, under certain simplifying assumptions, the evolution of the temperature on the surface. I will discuss a family of modified SQG equations that varies between 2D Euler and SQG with patch-like initial data defined on half-plane.
The family is modulated by a parameter \alpha that sets the degree of the kernel in the Biot-Savart law. The value \alpha=0 corresponds to the 2D Euler equation, while \alpha=1/2 to the SQG case. The main result I would like to describe is the phase transition in the behavior of solutions that happens at \alpha=0. Namely, for 2D Euler equation the patch solution stays globally regular, while for a range of small \alpha>0 there exist regular initial data that lead to finite time blow up. The finite time blow up involves either loss of regularity of the boundary, or touching of different patches, or patch self-intersection. This talk is based on joint works with Lenya Ryzhik, Yao Yao and Andrej Zlatos.