Singular integrals and a problem on mixing flows

TL;DR

This paper addresses Bressan's mixing problem by establishing bounds on Bianchini semi-norms under divergence-free flows, introducing a bilinear singular integral operator, and providing related theoretical insights.

Contribution

It introduces a new inequality for Bianchini semi-norms, constructs a bilinear singular integral operator, and offers novel observations and a toy model related to Bressan's mixing problem.

Findings

Bounded a bilinear singular integral operator on Hardy spaces.

Established an inequality for Bianchini semi-norms under divergence-free flows.

Provided a discrete toy model of Bressan's mixing problem.

Abstract

We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.