Well-posedness for the surface quasi-geostrophic front equation

TL;DR

This paper proves the well-posedness of the surface quasi-geostrophic front equation for large and rough initial data, improving previous results by removing small data restrictions and lowering regularity thresholds.

Contribution

It establishes unconditional large data local well-posedness and global well-posedness in rough data regimes for the SQG front equation, advancing the mathematical understanding of its solutions.

Findings

Unconditional large data local well-posedness established.

Global well-posedness proven in rough data regimes.

Improved low regularity threshold for initial data.

Abstract

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter-Shu-Zhang [9] established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation's nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim-Tataru.