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We investigate a perturbation problem for the three-dimensional compressibleisentropic viscous magnetohydrodynamic system with zero resistivity in thepresence of a modified gravitational force in a vertical strip domain in whichthe velocity of the fluid is non-slip on the boundary, and focus on thestabilizing effect of the (equilibrium) magnetic field through the non-slipboundary condition. We show that there is a discriminant $\Xi$, depending onthe known physical parameters, for the stability/instability of theperturbation problem. More precisely, if $\Xi<0$, then the perturbation problemis unstable, i.e., the Parker instability occurs, while if $\Xi>0$ and theinitial perturbation satisfies some relations, then there exists a global(perturbation) solution which decays algebraically to zero in time, i.e., theParker instability does not happen. The stability results in this paper revealthe stabilizing effect of the magnetic field through the non-slip boundarycondition and the importance of boundary conditions upon the Parkerinstability, and demonstrate that a sufficiently strong magnetic field canprevent the Parker instability from occurring. In addition, based on theinstability results, we further rigorously verify the Parker instability underSchwarzschild's or Tserkovnikov's instability conditions in the sense ofHadamard for a horizontally periodic domain.

On the Dynamical Stability and Instability of Parker Problem