35:37
publiée le 8 avril 2024
On the Radon-Carleman problem in uniformly rectifiable domains
De Irina Mitrea
Regularity of vortex and SQG patches
Apparaît dans la collection : MathFlows
The patch solutions of the 2D Euler and (modified) SQG equations have form $\omega(x, t)=\chi_{\Omega(t)}(x)$ of a characteristic function of a domain $\Omega(t)$ evolving in time according to the Biot-Savart law $u=\nabla^{\perp}(-\Delta)^{-1+\alpha} \omega$, here $\alpha=0$ corresponds to the Euler case and $0<\alpha<1$ to the modified SQG family. For the Euler case, the first proof of global regularity for pathes was given by Chemin in Hölder spaces $C^{k, \beta}, 0<\beta<1$. For the modified SQG family, the problem remains largely open - with the only finite time singularity formation result available in the presence of boundary and for small $\alpha[5,2]$. I will discuss some recent conditional results on the possible scenarios for finite time blow up. Also, for the Euler patch case, I will describe a construction of patches that are $C^{2}$ at the initial and all integer times, but lack this regularity for all other times - without being time periodic. This result is based on the analysis of the curvature evolution equation, which may also be useful for other applications.
