Regularity, Uniqueness and the Relative Size of Small and Large Scales in SQG Flows

Authors

• Z. Akridge, University of Arkansas, Fayetteville
• Z. Bradshaw, University of Arkansas, FayettevilleFollow

Document Type

Article

Publication Date

6-30-2025

Keywords

3D, Navier-Stokes, Quasi-Geostrophic Equations, Weak Solutions, Gevery Regularity, Criterion, Nonuniqueness, Existence, Time

Abstract

The problems of regularity and uniqueness are open for the supercritically dissipative surface quasi-geostrophic equations in certain classes. In this note we examine the extent to which small or large scales are necessarily active both for the temperature in a hypothetical blow-up scenario and for the error in hypothetical non-uniqueness scenarios, the latter understood within the class of Marchand's solutions. This extends prior work for the 3D Navier-Stokes equations. The extension is complicated by the fact that mild solution techniques are unavailable for supercritical SQG. This forces us to develop a new approach using energy methods and Littlewood-Paley theory.

Comments

Web of Science

Springer

Citation

Akridge, Z., Bradshaw, Z. Regularity, Uniqueness and the Relative Size of Small and Large Scales in SQG Flows. J. Math. Fluid Mech. 27, 51 (2025). https://doi.org/10.1007/s00021-025-00947-x

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.