Propagation of regularity for transport equations. A Littlewood-Paley approach

TL;DR

This paper introduces a novel Littlewood-Paley based method to analyze the regularity of solutions to transport equations, improving understanding of their behavior and convergence properties in the zero-diffusivity limit.

Contribution

It develops a new approach using Besov norms for regularity analysis, extending existing results to diffusive cases with optimal bounds.

Findings

Recovered known regularity propagation results

Extended analysis to diffusive transport equations

Derived sharp convergence rates in zero-diffusivity limit

Abstract

It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We propose a new approach to the study of regularity that is based on Littlewood-Paley theory, thus measuring regularity in terms of Besov norms. We recover the results that are available in the literature and extend these optimally to the diffusive setting. As a consequence, we derive sharp bounds on rates of convergence in the zero-diffusivity limit.