Commutator estimates in Besov-Morrey spaces with applications to the well-posedness of the Euler equations and ideal MHD system

TL;DR

This paper develops commutator estimates in Besov-Morrey spaces and applies them to establish local well-posedness and blow-up criteria for the Euler equations and ideal MHD system.

Contribution

It introduces new commutator estimates in Besov-Morrey spaces and applies them to analyze well-posedness of fluid dynamics equations.

Findings

Established local well-posedness in Besov-Morrey spaces

Derived blow-up criteria for Euler and MHD equations

Utilized Littlewood-Paley and Bony's para-product techniques

Abstract

We develop commutator estimates in the framework of Besov-Morrey spaces, which are modeled on Besov spaces and the underlying norm is of Morrey space rather than the usual $L^{p}$ space. As direct applications of commutator estimates, we establish the local well-posedness and blow-up criterion of solutions in Besov-Morrey spaces for the incompressible Euler equations and ideal MHD system. Main analysis tools are the Littlewood-Paley decomposition and Bony's para-product formula.