Regularity estimates for the flow of BV autonomous divergence free vector fields in $\mathbb{R}^2$

TL;DR

This paper establishes regularity estimates for the flow of divergence-free BV vector fields in 2D, showing linear growth of Lipschitz constants over time and bounds on mixing rates.

Contribution

It provides the first regularity results for the Lagrangian flow associated with BV divergence-free vector fields in two dimensions.

Findings

Lusin-Lipschitz regularity for the flow

Lipschitz constant grows at most linearly in time

Lower bounds on mixing rates of order t^{-1}

Abstract

We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order $t^{-1}$ as $t \to \infty$.