Variational approach to nonlinear gravity-driven instabilities in a MHD setting

摘要

We establish a variational framework for nonlinear instabilities in a setting of the ideal magnetohydrodynamic (MHD) equations. We apply a variational method to unstable smooth steady states for both incompressible and compressible ideal MHD equations. The destabilizing effect of compressibility is justified along with the stabilizing effect of magnetic field lines arising in the MHD dynamics. This generalizes the result of the Rayleigh-Taylor instability for incompressible fluids in the absence of magnetic field lines; see Hwang and Guo, On the dynamical Rayleigh-Taylor instability, Arch. Rational Mech. Anal. 167 (2003), 235-253.