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.
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.
Talk, UCSD Zoom for Thought, San Diego, California
Talk, UC San Diego, San Diego, California
Talk, KAIST Guerilla Seminar, Daejeon, Korea
Talk, NIMS Seminar, Daejeon, Korea