Global Stability of 3D Compressible Non-Resistive MHD: Hidden Damping and Rational Background Fields

We prove the global well-posedness and nonlinear stability of classical solutions to the three-dimensional compressible viscous, non-resistive MHD system on ${\mathbb{T}}^3$ near the equilibrium $(1,\mathbf{0},{\mathbf{e}}_3)$, for small $x_3$-symmetric perturbations and without any Diophantine condition on the background magnetic field. The central obstruction is the $x_3$-independent sector, in which the density and the magnetic field possess no dissipation, no damping, and no decay. We overcome it by exhibiting a hidden wave structure for the exact total pressure $\mathcal{D}=P(1+a)+B_3+\frac{1}{2}\mathrm{\mid }\mathbf{B}{\mathrm{\mid }}^2$, the averaged pair $(\bar{\mathbf{u}},\bar{\mathcal{D}})$ obeys a closed system, and $\bar{\mathcal{D}}$ satisfies a strongly damped wave equation whose principal wave part propagates at the fast magnetosonic speed and whose non-parabolic branch damps at a rate that involves no gain of derivatives. This hidden damping substitutes for the missing magnetic dissipation, and simultaneously absorbs the magnetic pressure $\frac{1}{2}\mathrm{\nabla }\mathrm{\mid }\mathbf{B}{\mathrm{\mid }}^2$, the main obstruction created by compressibility. Together with a damped wave structure for the oscillatory sector and space-time weighted energy functionals with shifted time weights, this yields global existence, uniform stability, and explicit polynomial decay rates.