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An improved regularity criterion and absence of splash-like singularities for g-SQG patches

Junekey Jeon and Andrej Zlatoš

Vol. 17 (2024), No. 3, 1005–1018
Abstract

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 α 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 α (0,1)). As a corollary, we obtain an improved global regularity criterion for H3 patch solutions when α 1 4, namely that finite time singularities cannot occur while the H3 norms of patch boundaries remain bounded.

Keywords
generalized SQG equation, patch solutions, splash singularity
Mathematical Subject Classification
Primary: 35Q35
Milestones
Received: 5 December 2021
Revised: 15 May 2022
Accepted: 11 August 2022
Published: 24 April 2024
Authors
Junekey Jeon
Department of Mathematics
University of California San Diego
La Jolla, CA
United States
Andrej Zlatoš
Department of Mathematics
University of California San Diego
La Jolla, CA
United States

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