## Seminar 26.8.2015: Andrej Zlatos

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

Moderator: sevcovic

### Seminar 26.8.2015: Andrej Zlatos

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.
quittner

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