The well-posedness of the incompressible Magnetohydro Dynamic equations in the framework of Fourier–Herz space

In this paper, we study the well-posedness and the blow-up criterion of the mild solution for the 3D incompressible MHD equations in the framework of Fourier–Herz space involving highly oscillating function. First, we study the well-posedness of the incompressible MHD equations by establishing the smoothing effect in the mixed time–space Fourier–Herz space, which include the local in time for large initial data as well as the global well-posedness for small initial data. Next, we prove the blow-up criterion, that is, if u∈L_T^r₁˜FB˙_p,q^{2−[Formula presented]+[Formula presented]} and b∈L_T^r₂˜FB˙_p,q^{2−[Formula presented]+[Formula presented]} for 1≤r₁,r₂<∞, the mild solution to the MHD equations can be extended beyond t=T. More importantly, we give a better blow-up criterion in which we require velocity field u(t)∈L_T^r˜FB˙_p,q^{2−[Formula presented]+[Formula presented]} only.