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

Abstract

This paper concerns the Cauchy problem of the n-dimensional generalized incompressible magneto-hydrodynamic (GMHD) equations with β∈(12,1]. 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 (u0,b0) belongs to the critical Fourier–Herz spaces B˙q−(2β−1) with q∈[1,2]. In addition, for 2<q≤∞, ill-posedness for the case β=1 in B˙q−1 is also established.