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∈LTr1˜FB˙p,q2−3p+2r1 and b∈LTr2˜FB˙p,q2−3p+2r2 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)∈LTr˜FB˙p,q2−3p+2r only.