On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

TL;DR

This paper investigates the local well-posedness of a family of logarithmically regularized generalized SQG equations in borderline Sobolev spaces, extending previous results to more singular velocity regimes with $eta ext{ in } (1,2)$.

Contribution

It introduces a novel linearized system that preserves the commutator structure, enabling well-posedness analysis in more singular regimes of the generalized SQG equations.

Findings

Established well-posedness for $eta ext{ in } (1,2)$

Identified a linearized system preserving the commutator structure

Ensured continuity of the flow map in borderline Sobolev spaces

Abstract

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $\theta$ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $u$ is of lower singularity, i.e., $u=-\nabla^{\perp}\Lambda^{\beta-2}p(\Lambda)\theta$, where $p$ is a logarithmic smoothing operator and $\beta \in [0,1]$. We complete this study by considering the more singular regime $\beta\in(1,2)$. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.