Talks and presentations

Abstract Level-Set Dynamics of gSQG Equation

July 18, 2024

Talk, Seoul National University, Seoul, Korea

We develop an abstract measure-theoretic notion of solutions to gSQG equation based on the dynamics of level-sets of solutions, and prove an $H^{2}$ local-wellposedness result in this setting using a sequence of regularized contour equations.

An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches

July 06, 2022

Talk, Seoul National University, Zoom

We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter $\alpha \le 1/4$. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all $\alpha \in (0,1)$). As a corollary, we obtain an improved global regularity criterion for H3 patch solutions when $\alpha \le 1/4$, namely that finite time singularities cannot occur while the H3 norms of patch boundaries remain bounded.