On the inviscid Boussinesq system with rough initial data

TL;DR

This paper establishes local well-posedness for the 2D inviscid Boussinesq system with rough initial data, including vortex patches, by addressing critical terms involving Riesz transforms.

Contribution

It provides the first well-posedness results for the inviscid Boussinesq system with Yudovich-type rough initial data, including singular vortex patches.

Findings

Well-posedness for smooth vortex patches.

Extension to singular vortex patches with constant density near singularities.

Handling of critical Riesz transform terms in the analysis.

Abstract

We deal with the local well-posedness theory for the two-dimensional inviscid Boussinesq system with rough initial data of Yudovich type. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. For the latter case we assume in addition that the initial density is constant around the singular part of the patch boundary.