Well-Posedness and Stability for the Generalized Incompressible Magneto-Hydrodynamic Equations in Critical Fourier-Besov-Morrey Spaces

Abstract

This paper concerns the Cauchy problem of the 3D generalized incompressible magneto-hydrodynamic (GMHD) equations. By using the Fourier localization argument and the Littlewood-Paley theory, we get local well-posedness results of the GMHD equations with large initial data (u0, b0) belonging to the critical Fourier-Besov-Morrey spaces $${\cal F}\dot {\cal N}_{p,{\rm{\lambda }},q}^{1 - 2\alpha + {3 \over {p\prime }} + {{\rm{\lambda }} \over p}}\left( {{\mathbb{R}^3}} \right)$$ . Moreover, stability of global solutions is also discussed.