On the Existence and Stability of 2D Compressible Current-Vortex Sheets

We study the existence and the stability of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. Under a suitable stability condition for the background state, we show that the linearized current-vortex sheets problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Then we establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash–Moser iteration, provided the stability condition is satisfied at each point of the initial discontinuity. This survey paper presents our recent results in Morando et al. (Arch. Rational Mech. Anal. 247, 50 (2023)).