Regularity, uniqueness and the relative size of small and large scales in SQG flows

TL;DR

This paper investigates the roles of small and large scales in supercritically dissipative SQG flows, focusing on regularity, uniqueness, and scale activity, using new energy and Littlewood-Paley methods due to the lack of mild solutions.

Contribution

It introduces a novel approach employing energy methods and Littlewood-Paley theory to analyze scale activity in supercritical SQG flows, extending prior work on 3D Navier-Stokes.

Findings

Small and large scales are necessarily active in blow-up and non-uniqueness scenarios.

Develops new analytical techniques due to absence of mild solutions.

Extends understanding of scale dynamics in supercritical SQG flows.

Abstract

The problem 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.