Linear instability of Z-pinch in plasma: Inviscid case

The [Formula: see text]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [Formula: see text] direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [Formula: see text]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system.