Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equation in R3

AbstractWe first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, and then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's paraproduct decomposition, it is proved that the strong solution (u,b) can be extended after t=T if either u∈LTq(B˙p,∞0) with 2q+3p⩽1 and b∈LT1(B˙∞,∞0) or (ω,J)∈LTq(B˙p,∞0) with 2q+3p⩽2, where ω(t)=∇×u denotes the vorticity of the velocity and J=∇×b stands for the current density