Differentiability properties of the flow of 2d autonomous vector fields

TL;DR

This paper explores the regularity inheritance of flows generated by 2D autonomous vector fields, focusing on Sobolev and BV properties, and provides a counterexample showing limitations of BV regularity.

Contribution

It establishes conditions under which the flow inherits Sobolev or BV regularity and presents a novel example where BV regularity fails despite Sobolev divergence-free vector fields.

Findings

Flow inherits Sobolev regularity under certain conditions

Flow may lack BV regularity even for Sobolev divergence-free fields

Hamiltonian structure aids in analyzing regularity properties

Abstract

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.