Existence and Stability for the 3D Linearized Constant-Coefficient Incompressible Current-Vortex Sheets

We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study of linearized variable-coefficient current-vortex sheets, a first step to prove the existence of solutions to the nonlinear problem. 1. Introduction 1.1. The Eulerian Description Let us consider the equations of magnetohydrodynamics (MHD) governing the motion of a perfectly conducting inviscid incompressible plasma in three-space dimension. In the case of a homogeneous plasma (i.e., the density is a positive constant), the equations in a dimensionless form read where, using for transposition, denotes the plasma velocity, is the magnetic field (in Alfvén velocity units), is the total pressure, and is the pressure. For smooth solutions, system (1) can be written in an equivalent form as We are interested in weak solutions, in a suitable sense, to (1) that are smooth on either side of a smooth hypersurface in , where , , and that satisfy suitable jump conditions at each point of the front . For notational simplicity, we assume that the density is the same constant on either side of , so that we can take, with no loss of generality, . In physical applications, the two densities can be very different, but such a difference intervenes only at the boundary, and it is taken into account by the jump condition for the total pressure (see below), so it does not alter the mathematical techniques applied in this paper. Let us set , where ; given any function , we set in and denote by the jump across . We look for current-vortex sheets solutions, that is, smooth solutions of (2) in such that is a tangential discontinuity, namely, the plasma does not flow through the discontinuity front, and the magnetic field is tangent to , see, for example, Landau and Lifshitz [1]; thus, the boundary conditions take the form where denotes the outward unit normal on and denotes the velocity of propagation of the interface front . With the given parametrization of , an equivalent formulation of these jump conditions is with . Notice that the function describing the discontinuity front is part of the unknowns of the problem, that is, this is a free boundary problem. System (2), (4) is supplemented with initial conditions where in . Current-vortex sheets have various interesting applications in