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

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 $t\to \infty$.

References in corpus (2)

• A Lemma and a Conjecture on the Cost of Rearrangements
• On geometric and analytic mixing scales: comparability and convergence rates for transport problems

Cited by in corpus (5)

• Mixing and diffusion for rough shear flows
• Properties of Mixing BV vector fields
• Initial-boundary value problems for merely bounded nearly incompressible vector fields in one space dimension
• Differentiability properties of the flow of 2d autonomous vector fields
• Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields