Global well-posedness for the generalized magneto-hydrodynamic equations in the critical Fourier–Herz spaces

This paper concerns the Cauchy problem of the n-dimensional generalized incompressible magneto-hydrodynamic (GMHD) equations with β ∈ ( 1 2 , 1 ] . By using the Fourier localization argument and the Littlewood–Paley theory, we get the global well-posedness of the GMHD equations with small initial data ( u 0 , b 0 ) belongs to the critical Fourier–Herz spaces B ˙ q − ( 2 β − 1 ) with q ∈ [ 1 , 2 ] . In addition, for 2 < q ≤ ∞ , ill-posedness for the case β = 1 in B ˙ q − 1 is also established.