A priori tame estimates for free boundary plasma–vacuum problem

We study the free boundary problem for the plasma–vacuum interface in ideal compressible magnetohydrodynamics. Unlike the classical statement, when the vacuum magnetic field obeys the ${\rm div}$-${\rm rot}$ system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. This work is a continuation of the previous analysis by Mandrik and Trakhinin in 2014, where a sufficient condition on the vacuum electric field that precludes violent instabilities was found and analyzed, the well-posedness of the linearized problem in anisotropic weighted Sobolev spaces was proved under the assumption that this condition is satisfied at each point of the unperturbed nonplanar plasma-vacuum interface. Since the free boundary is characteristic, the functional setting is provide by weighted anisotropic Sobolev spaces $H^s_*$. The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivaties in a priori estimates. Assuming that the mentioned above condition is satisfied at each point of the unperturbed nonplanar plasma-vacuum interface, we prove that tame estimates in $H^s_*$ holds for the linearized problem. In future we are going to use those estimates to prove the existence of solutions of the nonlinear problem.