Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper, the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied. The solutions to the system (1.6)-(1.8) are in the class of radius-dependent solutions, i.e., independent of the axial variable and the angular variable. In particular, the expanding rate of the moving boundary is obtained. The main difficulty of this problem lies in the strong coupling of the magnetic field, velocity, temperature and the degenerate density near the free boundary. We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates, and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variabler(x)by weighted estimates, and also the uniform-in-time weighted estimates of the higher order derivatives of solutions by delicate analysis.