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

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.