On the well-posedness of the Hall-MHD system in a critical setting of Besov-Morrey type

In this paper, we address the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our objective is to provide local and global well-posedness results for initial velocity $u_0$, magnetic field $B_0$ and the current $J_0 := \nabla \times B_0$ in a new critical framework, namely critical Besov-Morrey spaces. These spaces combine typical characteristics from both Besov and Morrey spaces allowing a broader framework that encompasses the regularity properties inherent in Besov spaces with the Morrey space structure. Compared to previous works in Sobolev and Besov spaces, our approach accommodates a broader class of initial data, ensuring the construction of a unique solution over time.