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

2022/03/21 by David J. Meyer, Meyer, David, Christian Seis +1 · 3 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2203.10860

openalex publication_date 2022/03/21 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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.

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