A priori tame estimates for free boundary plasma–vacuum problem

N. V. Mandrik

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 16 (2016) no. 4, pp. 72-96

Résumé

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.

Détail

Keywords: tame estimates, ideal compressible magnetohydrodynamics, vacuum Maxwell equations, plasma-vacuum interface.