Download this article
 Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
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

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.

generalized SQG equation, patch solutions, splash singularity
Mathematical Subject Classification
Primary: 35Q35
Received: 5 December 2021
Revised: 15 May 2022
Accepted: 11 August 2022
Published: 24 April 2024
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

Open Access made possible by participating institutions via Subscribe to Open.