Commutator estimates and their applications to the transport-type equations

TL;DR

This paper develops new commutator estimates in Triebel-Lizorkin spaces and applies them to establish a general theory for transport equations, extending previous results and enabling analysis of various evolution equations.

Contribution

The paper introduces a unified approach to commutator estimates in Triebel-Lizorkin and Besov spaces, advancing the analysis of transport and evolution equations.

Findings

Established local well-posedness for the Euler-Poincaré system.

Derived blow-up criteria in sub-critical and critical spaces.

Unified proofs extend previous results to broader function spaces.

Abstract

In this paper, we derive new commutator estimates in the Triebel-Lizorkin spaces by employing Bony's para-product decomposition, the Nikol'skij representation, and the Fefferman-Stein vector-valued maximal function. These estimates are then applied to develop a general theory for transport equations. Although analogous results are already available in the setting of Besov spaces, the methods developed there do not carry over directly to the Triebel-Lizorkin case. Our approach works for Triebel-Lizorkin spaces and, as a byproduct, also yields the corresponding results in Besov spaces. All proofs are presented in a unified manner that applies to both scales of function spaces, thereby extending and sharpening previous results on transport equations in these frameworks. Furthermore, the general theory we obtain is widely applicable to evolution equations, including incompressible and compressible ideal fluid flows, shallow water waves, and related models. As an illustration, we consider the two-component Euler-Poincar\'e system. Using the theoretical framework developed herein, we establish its local well-posedness and a blow-up criterion in both sub-critical and critical Triebel-Lizorkin spaces.