Received September 2015 Revised March 2016 Published July 2016
The present paper is devoted to the well-posedness issue of solutions of a full system of the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations. By means of Littlewood-Paley analysis we prove the global well-posedness of solutions in the Besov spaces $\dot{B}_{2,1}^\frac1{2}\times B_{2,1}^\frac3{2}\times B_{2,1}^\frac3{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} \big(\|u_0^h\|_{\dot{B}_{2,1}^\frac1{2}} +\|E_0\|_{B_{2,1}^\frac{3}{2}}+\|B_0\|_{B_{2,1}^\frac{3}{2}}\big)\exp \Big\{\frac{C_0}{\nu^2}\|u_0^3\|_{\dot{B}_{2,1}^\frac1{2}}^2\Big\}\leq c_0, \end{align*} for some sufficiently small constant $c_0.$
Keywords: Navier-Stokes-Maxwell system, Besov spaces., Littlewood-Paley theory, Well-posedness.
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76A0.
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