Stability and existence of relativistic plasma--vacuum interfaces

We consider the free boundary problem for relativistic plasma--vacuum interfaces in two and three spatial dimensions. The plasma flow is governed by the equations of ideal relativistic magnetohydrodynamics, while the vacuum magnetic and electric fields satisfy Maxwell's equations. The plasma and vacuum magnetic fields are tangential to the interface, which moves with the plasma flow. This yields a nonlinear, multidimensional hyperbolic problem with a free boundary that is characteristic of variable multiplicity. We identify a quantitative stability condition and establish the linear stability of three-dimensional relativistic plasma--vacuum interfaces in the sense that the variable-coefficient linearized problem satisfies energy estimates in anisotropic Sobolev spaces. In estimating tangential derivatives, we exploit an intrinsic cancellation effect to convert the boundary term into an instant integral. We then separate the estimate involving spatial derivatives from that involving time derivatives, so that the instant integral can be mainly absorbed by the instant energy under the stability condition. Moreover, we prove the local-in-time existence and uniqueness of solutions to the nonlinear problem in two-dimensional space, provided that the plasma and vacuum magnetic fields do not vanish simultaneously at any point of the initial interface. The proof combines the solvability and tame estimate of the linearized problem with a suitable modified Nash--Moser iteration. In particular, to establish its solvability, the two-dimensional linearized problem is reduced to a transport equation for the interface function and a hyperbolic boundary problem with maximally nonnegative boundary conditions.