Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Wednesday 26.8.2015 at 10:30 Lecture room M-223

Andrej Zlatoš (University of Wisconsin):

Finite time blow-up for the $\alpha$-patch model

Abstract:

The global regularity vs finite time blow-up question remains open for many

fundamental equations of fluid dynamics. In two dimensions, the solutions

to the incompressible Euler equation have been known to be globally regular

since the 1930s. On the other hand, this question has not yet been resolved

for the less regular (by one derivative) surface quasi-geostrophic (SQG)

equation. The latter state of affairs is also true for a natural family

of PDE which interpolate between these two equations. They involve

a parameter $\alpha$, which appears as a power in the kernel of their

Biot-Savart laws and describes the degree of regularity of the equation,

with the values $\alpha=0$ and $\alpha=\frac 12$ corresponding

to the Euler and SQG cases, respectively.

In this talk I will present two results about the patch dynamics version

of these equations in the half-plane. The first is global-in-time regularity

for the Euler patch model, even if the patches initially touch the boundary

of the half-plane. The second is local regularity and existence of solutions

which blow up in finite time for the $\alpha$-patch model with any small

enough $\alpha>0$. The latter appears to be the first rigorous proof

of finite time blow-up in this type of fluid dynamics models.