Classical flows of vector fields with exponential or sub-exponential summability

TL;DR

This paper investigates the regularity and well-posedness of flows generated by vector fields with derivatives satisfying certain Orlicz summability conditions, extending classical results to sub-exponential and exponential cases.

Contribution

It establishes new regularity and well-posedness results for vector fields with derivatives in Orlicz spaces, including Sobolev regularity of flows without bounded divergence assumptions.

Findings

Flows are spatially continuous under Orlicz summability.

Flows satisfy a Lusin (N) condition in the sub-exponential case.

Flows have Sobolev regularity under exponential summability.

Abstract

We show that vector fields $b$ whose spatial derivative $D_xb$ satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if $D_xb$ satisfies a suitable exponential summability condition then the flow associated to $b$ has Sobolev regularity, without assuming boundedness of ${\rm div}_xb$. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations.