On the Sobolev space of functions with derivative of logarithmic order

TL;DR

This paper explores two concepts of derivatives of logarithmic order in functions, relevant for understanding regularity in flows and solutions of transport equations with weakly differentiable drifts.

Contribution

It introduces and analyzes two notions of logarithmic order derivatives, advancing the mathematical understanding of regularity in weakly differentiable contexts.

Findings

Characterization of the two notions of logarithmic derivatives

Applications to regularity of flows

Implications for renormalized solutions in transport equations

Abstract

Two notions of "having a derivative of logarithmic order" have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.