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

Abstract We 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 ∈ L T q ( B ˙ p , ∞ 0 ) with 2 q + 3 p ⩽ 1 and b ∈ L T 1 ( B ˙ ∞ , ∞ 0 ) or ( ω , J ) ∈ L T q ( B ˙ p , ∞ 0 ) with 2 q + 3 p ⩽ 2 , where ω ( t ) = ∇ × u denotes the vorticity of the velocity and J = ∇ × b stands for the current density.

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