Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

Well-Posedness for the Free-Boundary Ideal Compressible Magnetohydrodynamic Equations with Surface Tension

“…For the free-boundary ideal incompressible MHD without surface tension, the a priori estimates and local wellposedness have been respectively proved by Hao-Luo [14] and Gu-Wang [13] under the generalized Rayleigh-Taylor sign condition on the total pressure, while a counterexample to well-posedness has been provided by Hao-Luo [15] when the generalized sign condition fails. Regarding the ideal compressible MHD equations (1.1), the authors [35] have recently established the local well-posedness for the case of zero surface tension s = 0 under the generalized sign condition. It would expect that the surface tension can have a stabilization effect on the evolution as for the cases without magnetic fields.…”

“…Without loss of generality we set κ ♯ = 1. As in [22,[33][34][35], we employ the cut-off function χ to avoid assumptions about compact support of the initial data (shifted to an equilibrium state).…”

“…For the linearized problem around a basic state, if the generalized Rayleigh-Taylor sign condition on the total pressure were imposed as in our previous work [35] for the case of zero surface tension, then we could deduce a uniform-in-s L 2 energy estimate with no loss of derivatives from the source term to the solution (cf. (2.22)), allowing us to construct the unique solution by a direct application of the classical duality argument in Lax-Phillips [19] and Chazarain-Piriou [3, Chapter 7].…”

Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

“…See also Lee [21,22] for an alternative proof by using the vanishing viscosity-resistivity method, Sun-Wang-Zhang [36] for the incompressible MHD current-vortex sheets, Sun-Wang-Zhang [37] and the first author [9,10] for the plasma-vacuum interface model, and the second and the third authors [26] for the minial regularity estimates in a small fluid domain. For the compressible ideal MHD, we refer to [2,38,41,30,39,25] and references therein.…”

“…To the best of our knowledge, our previous works [27,12] are the only available results on the free-boundary incompressible ideal MHD with surface tension. In addition, we also refer to Trakhinin-Wang [40] for the compressible case.…”