  In this paper, we establish new $L^p$ gradient estimates of the solutions in
order to discuss Cauchy problem for the full compressible
magnetohydrodynamic(MHD) systems in $\mathrm{R}^3$. We use the
"$\rm{div}-\rm{curl}$" decomposition technique (see \cite{{HJR},{MR}}) and new
modified effective viscous flux and vorticity to calculate "$\Vert\nabla
\mathbf{u}\Vert_{L^3}$" and "$\Vert\nabla \mathbb{H}\Vert_{L^3}$".As a result,
we obtain global well-posedness for the solution with the initial data being in
a class of space with lower regularity, while the energy of which should be
suitably small.
