Global existence of bounded smooth solutions for the compressible ideal MHD system with planar symmetry

This paper studies the global existence of bounded smooth solutions for the compressible ideal magnetohydrodynamic (MHD) system. We assume that the solutions have planar symmetry. Then, the MHD system can be reduced to a \(7\times 7\) $7\times 7$ first-order quasilinear hyperbolic system. We find a sufficient condition on the initial data to ensure the global existence of bounded smooth solutions. The main difficulty for the global existence is to establish a priori estimates for the derivatives of the solution. To this end, we derive a group of delicate characteristic decompositions for the MHD system. These characteristic decompositions can be seen as a system of “nonlinear ordinary differential equations” for \(C_{+}\) $C_{+}$ and \(C_{-}\) $C_{-}$ characteristic directional derivatives of the unknown functions. Owing to the good structures of the characteristic decompositions, we use “maximum principle” to obtain some uniform a priori estimates for the derivatives of the solution.