Stability of the linearized MHD-Maxwell free interface problem

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method. An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.


Introduction
Plasma-vacuum interface problems appear in the mathematical modeling of plasma confinement by magnetic fields in thermonuclear energy production (as in Tokamaks; see, e.g., [13]). In this model, the plasma is confined inside a perfectly conducting rigid wall and isolated from it by a region containing very low density plasma, which may qualify as vacuum, due to the effect of strong magnetic fields. This subject is very popular since the 1950-70's, but most of theoretical studies are devoted to finding stability criteria of equilibrium states. The typical work in this direction is the classical paper of Bernstein et al. [3]. In astrophysics, the plasma-vacuum interface problem can be used for modeling the motion of a star or the solar corona when magnetic fields are taken into account. Once again, the interface can be described as a tangential discontinuity, i.e. the magnetic fields do not intersect the interface.
In [31,32], the authors studied the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD), by considering the pre-Maxwell dynamics for the magnetic field in the vacuum region, as usually assumed in the classical formulation. The relativistic case has been addressed by Trakhinin in [35], in the case of plasma expansion in vacuum. In this paper we consider again the non-relativistic case, but, instead of the pre-Maxwell dynamics, in the vacuum region we don't neglect the displacement current and consider the complete system of Maxwell equations for the electric and the magnetic fields.
The introduction of this model aims at investigating the influence of the electric field in vacuum on the well-posedness of the problem, as in the classical pre-Maxwell dynamics such an influence is hidden. For the relativistic plasma-vacuum problem, Trakhinin [35] has shown the possible ill-posedness in the presence of a sufficiently strong vacuum electric field. Since relativistic effects play a rather passive role in the analysis of [35], it is natural to expect the same for the non-relativistic problem under consideration here. On the contrary, we will show that a sufficiently weak vacuum electric field, under the stability condition (89), precludes ill-posedness and gives the well-posedness of the linearized problem (see Theorem 4.1), thus somehow justifying the practice of neglecting the displacement current in the classical pre-Maxwell formulation when the vacuum electric field is weak enough.
We discuss different equivalent formulations of the problem and study the stability of the linearized problem with variable coefficients for nonplanar plasma-vacuum interfaces. The main result is given in Theorem 4.1, Section 4, where we derive a basic a priori estimate for solutions to the linearized problem in the Sobolev space H 1 tan with conormal regularity, under suitable stability conditions satisfied at each point of the plasma-vacuum interface. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method. The approach is similar to that in [31,35].
Observe that one important difficulty of the plasma-vacuum problem, as for other free-boundary problems in MHD, is that we cannot test the Kreiss-Lopatinski condition [14,2,7] analytically. On the other hand, since the number of dimensionless parameters for the constant coefficients linearized problem is big, a complete numerical test of the Kreiss-Lopatinski condition seems unrealizable in practice.
The free interface, which is an unknown of the problem, turns out to be nonuniformly characteristic, that is characteristic of variable multiplicity, so that a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa), are needed in different parts of the boundary. To overcome this huge obstacle, we introduce the new vacuum variable W, defined in (23), which makes the interface characteristic of constant multiplicity, with no need of further assumptions such as plasma expansion (as in [35,18]). This can be considered the main novelty of the paper.
First we consider the equations of ideal compressible MHD in the plasma region: ∂ t ρe + 1 2 |H| 2 + div (ρe + p)v + H×(v×H) = 0 , where ρ denotes density, v ∈ R 3 plasma velocity, H ∈ R 3 magnetic field, p = p(ρ, S) pressure, q = p + 1 2 |H| 2 total pressure, S entropy, e = E + 1 2 |v| 2 total energy, and E = E(ρ, S) internal energy. With a state equation of gas, ρ = ρ(p, S), and the first principle of thermodynamics, (1) is a closed system. System (1) is supplemented by the divergence constraint (2) div H = 0 on the initial data. As it is known, taking into account (2), we can easily symmetrize system (1) by rewriting it in the nonconservative form where ρ p ≡ ∂ρ/∂p and d/dt = ∂ t + (v · ∇). A different symmetrization is obtained if we consider q instead of p. In terms of q, the equation for the pressure in (3) takes the form where it is understood that now ρ = ρ(q − |H| 2 /2, S) and similarly for ρ p . Then we derive div v from (4) and rewrite the equation for the magnetic field in (3) as Substituting (4), (5) in (3) then gives the following symmetric system (6)  where 0 = (0, 0, 0). Given this symmetrization, as the unknown we can choose the vector U = U (t, x) = (q, v, H, S). For the sake of brevity we write system (6) in the form which is symmetric hyperbolic provided the hyperbolicity condition A 0 > 0 holds, i.e.
We use ∂ j , j = 1, 2, 3, to denote the partial derivative with respect to x j ; in the sequel we will use ∂ 0 = ∂ t to denote the partial derivative with respect to t (see Appendix A).
Let Ω + (t) and Ω − (t) be space-time domains occupied by the plasma and the vacuum respectively. That is, in the domain Ω + (t) we consider system (7) governing the motion of an ideal plasma and in the domain Ω − (t) we have the Maxwell system (9) ε∂ t H + ∇× E = 0 , ε∂ t E − ∇× H = 0 , describing the vacuum magnetic and electric fields H, E ∈ R 3 . Here, the equations are written in nondimensional form through a suitable scaling (see ), and ε =v c , wherev is the velocity of a uniform flow and c is the speed of light in vacuum. If we choosev to be the speed of sound in vacuum, we have that ε is a small, even though fixed parameter. System (9) is supplemented by the divergence constraints div H = div E = 0 on the initial data. We write system (9) in the form: Let us assume that the interface between plasma and vacuum is given by a hypersurface Γ(t) = {F (t, x) = 0}. It has to be determined and moves with the velocity of plasma particles at the boundary: (13) dF dt = 0 on Γ(t) (for all t ∈ [0, T ]). Since F is an unknown of the problem, this is a free-boundary problem. The plasma variable U is connected with the vacuum magnetic and electric fields H, E through the relations [3,13] (14) where N = ∇F and [q] = q| Γ − 1 2 |H| 2 |Γ + 1 2 |E| 2 |Γ denotes the jump of the total pressure across the interface. These relations together with (13) are the boundary conditions at the interface Γ(t). Let us note that, in particular, the magnetic fields on both sides are tangent to the free interface.
As in [16,34,31,32], we will assume that for problem (7), (10)-(14) the hyperbolicity conditions (8) hold true in Ω + (t) up to the boundary Γ(t), i.e., the plasma density does not go to zero continuously, but has a jump (clearly in the vacuum region Ω − (t) the density is identically zero). This assumption is compatible with the continuity of the total pressure in (14). For instance, in the case of ideal polytropic gases one has p = Aρ γ e S with A > 0, γ > 1. Then the continuity of the total pressure at Γ requires (Aρ γ e S + 1 2 |H| 2 ) |Γ + = ( 1 2 |H| 2 − 1 2 |E| 2 ) |Γ − , which may be obtained also for densities ρ discontinuous across Γ. Differently, in the absence of the magnetic field, the continuity of the pressure yields the continuity of the density so that ρ |Γ+ = 0.
Since the interface moves with the velocity of plasma particles at the boundary, by introducing the Lagrangian coordinates one can reduce the original problem to that in a fixed domain. This approach has been recently employed with success in a series of papers on the Euler equations in vacuum, see [12,9,10,11,16]. However, as for tangential discontinuities in various models of fluid dynamics (e.g. [4,6,20,21,22,33]), this approach seems hardly applicable to problem (7), (10)- (14). Therefore, we will work in the Eulerian coordinates and for technical simplicity we will assume that the space-time domains Ω ± (t) have the following form.
Let us assume that the moving interface Γ(t) takes the form With our parametrization of Γ(t), an equivalent formulation of the boundary conditions (13), (14) at the interface is where In particular, (17) consists of the third and second components of N × E = εH∂ t ϕ, whereas the first component of N × E = εH∂ t ϕ follows as a direct consequence of (17) and The analysis of the problem shows that the number of boundary conditions that should be imposed to (7), (10) varies with the sign of the front velocity ∂ t ϕ, see more details in Appendix C and [35]. This number should correspond to the number of incoming characteristics for both (7) and (10), plus one for the determination of the front. The correct number of boundary conditions is four if ∂ t ϕ < 0 (the plasma expands into vacuum). Thus the four boundary conditions (16), (17) are in agreement with this necessary condition for well-posedness. (It will be shown that the conditions in (15) may be considered just as a restriction on the initial data.) On the contrary, if ∂ t ϕ > 0 (the vacuum expands into plasma) the correct number is six and problem (7), (10), (16), (17) is formally underdetermined because it is missing two boundary conditions. In view of that, let us denote and supplement the system with the boundary conditions (18) div for all t ∈ [0, T ]. These additional boundary conditions, which are chosen according to [35], are unnecessary on Γ − (t). System (7), (9), (15)-(18) is supplemented with initial conditions Due to the particular boundary conditions, problem (7), (10), (15)- (19) is an initial boundary value problem with characteristic boundary. Moreover, since we prescribe a different number of conditions on different portions of the boundary (see (18)), the boundary is nonuniformly characteristic, that is characteristic with variable multiplicity. A satisfactory theory for such nonuniformly characteristic boundary value problems is still lacking, see [25,29,30] and references therein included. From the mathematical point of view, a natural wish is to find conditions on the initial data providing the existence and uniqueness on some time interval [0, T ] of a solution (U, H, E, ϕ) to problem (7), (10), (15)- (19) in Sobolev spaces. Since (7) is a system of nonlinear hyperbolic conservation laws that can produce shock waves and other types of strong discontinuities (e.g., current-vortex sheets [5,33]), it is natural to expect to obtain only local-in-time existence theorems. Morever, it is natural to expect in general a local-in time existence for problem (7), (10), (15)- (19), even under suitable stability conditions for well-posedness, because of the occurrence of instabilities coming from the interface in finite time.
We must regard the boundary conditions on H, H in (15) as the restriction on the initial data (19). Similarly for div H = div H = div E = 0. More precisely, we can prove that a solution of (7), (16) (if it exists for all t ∈ [0, T ]) satisfies div H = 0 in Ω + (t) and H N = 0 on Γ(t), for all t ∈ [0, T ], if the latter were satisfied at t = 0, i.e., for the initial data (19). Similarly, we can prove that a solution of (9), (17), (18) for all t ∈ [0, T ], if the latter were satisfied at t = 0, i.e., for the initial data (19).
The content of the paper is as follows: the rest of this introduction is devoted to an equivalent formulation of the free boundary problem (7), (10), (16)- (19) on the fixed domain with flat boundary. In Section 2 we introduce the linearized problem with some useful reductions and equivalent formulations. After introducing some function spaces in Section 3, we present our main result in Section 4, Theorem 4.1. The proof of the main result is given in Section 5. Appendix A collects the main notations, hoping that it may help the reader, Appendix B contains the proofs of some technical lemmas, and in Appendix C we discuss the number of boundary conditions. 1.1. An equivalent formulation in the fixed domain. Let us denote We want to reduce the free boundary problem (7), (10), (16)- (19) to the fixed domains Ω ± . For this purpose we introduce a suitable change of coordinates that is inspired by Lannes [15] (see also [6,31]). In all what follows, H s (ω) denotes the Sobolev space of order s on a domain ω.
Notation. We use the notation Ψ j to denote ∂ j Ψ, being Ψ a scalar function, whereas by H j , E j we denote the j-th component of H, E, being them vector functions (see Appendix A).
We introduce the change of independent variables defined by (20) by setting and define the new dependent variables Moreover, we set Notice that Let us define the matrix which is invertible by virtue of the smallness of Ψ 1 (see Lemma 1.1). Then: We notice that W = JW, where with B ′ 1 as in (12). The matrix J is not invertible, in general, since However, this is prevented if ε is sufficiently small, which is true for physical problems. Otherwise, this difficulty can be overcome by assuming the smallness of the normal velocity of the plasma at the boundary.
Notice that the relation W = JW corresponds to the non-relativistic version of the Joule-Bernoulli equation connecting the magnetic and the electric fields in two inertial frames moving at relative velocity equal to the interface speed when the interface curvature is neglected. Hypothesis 1.2. We assume that εv N is sufficiently small on Γ, namely By virtue of (16), (21), from Hypothesis 1.2 it follows that sup |εΨ t (t, x)| < 1 in [0, T ] × R 3 , and J becomes invertible. Thus we can recoverW from W, bỹ W = J −1 W. We remark that Ψ t (t, x 1 , x ′ ) = 0 for |x 1 | sufficiently large, namely x 1 ∈ supp χ (see Appendix B for the definition of χ), hence it is not possible to derive the invertibility of J by assuming |εΨ t (t, For the reader's convenience, we define In [31,32], assuming pre-Maxwell dynamics in vacuum, i.e. ∇× H = 0, div H = 0 in Ω − (t), the authors obtained the equation ∇× H = 0 in the fixed domain, with the variable H given by H : Having this in mind, one may also write as well as The variable W is also used in [35].
By resorting to the variableW, the equations (10) in Ω − (t) can be recast on the fixed domain Ω − as The new set of dependent variables w, W defined in (23) is convenient for the reformulation of equations on the fixed domain Ω − .
For the proof of Proposition 1, see Appendix B. Proposition 1 implies that the additional boundary conditions (18) (17) can be written as

Remark 1. The two boundary conditions in
Dropping for convenience tildes inŨ andW, problem (7), (10), (16)- (19) can be reformulated on the fixed reference domains Ω ± as where P(U, Ψ) = P (U, Ψ)U , To avoid an overload of notation, we have denoted by the same symbol v N here above and v N as in (16) (16). Similarly, we will also denote H N := h 1 , E N := e 1 . We also define For the proof of Proposition 2, see Appendix B. As a consequence of (30) and Proposition 2, the equation (33) in vacuum can be rewritten as
The new formulation (41)-(44) can be used for the resolution of the problem with variable sign of ∂ t ϕ (which yields the change of multiplicity for (32)-(36)), that is with both expansion and contraction of the plasma region in vacuum. This extends the analysis of [18,35], where only the case of expansion of plasma into vacuum is considered, i.e. it is assumed ∂ t ϕ < 0 throughout the whole boundary.
For a detailed discussion on the number of boundary conditions in (32) be a given sufficiently smooth vector-function withÛ = (q,v,Ĥ,Ŝ) andŴ = (Ĥ,Ê), where κ > 0 is a constant. Corresponding to the givenφ, we constructΨ and the diffeomorphismΦ as in Lemma 1.1 such that We assume that the basic state (46) satisfies the following conditions (which are less restrictive in the vacuum side than in [35]). We have, for some positive ρ 0 , ρ 1 ∈ R, where all the "hat" functions are determined like for the corresponding values of (U, W, ϕ), i.e.p It follows from (49) that the constraints (53) divĥ = 0 in Q + T ,Ĥ N = 0 on ω T are satisfied for the basic state (46) if they hold at t = 0 (see [33] for the proof). Thus, for the basic state we also require the fulfillment of conditions (53) at t = 0. Other assumptions on the basic state needed for the stability analysis will be given in the statement of Theorem 4.1.
Note that (47) yields where ∇ t,x = (∂ t , ∇) and C = C(κ) > 0 is a constant depending on κ. We also remark that thanks to (52) and taking into account the definition of h, E (and recalling thatΨ 1 = 0 on ω T ), equation (51) may be written in the form: It follows from the first equation in (54), that the constraint (55)ĥ 1 = 0 on ω T is satisfied for the basic state (46) if it holds at t = 0. We remark that if we strengthen assumption (51) to the following T , then also (50) follows as a consequence of (56), provided that it holds for t = 0.
Notation. From now on, we denote by H, E, h, e variables defined using the basic stateΨ (instead of Ψ) and W (without "hat"), while "hat"-variablesĤ,Ê,ĥ,ê are defined using the basic state for all terms (Ψ andŴ). For instance,

2.2.
Linearized problem. We want to linearize (41)-(43) (or (32)-(34)), and in particular (42) in vacuum, since in this formulation we have the main advantage of a characteristic boundary of constant multiplicity. However, the linearization can be performed more easily, by resorting to standard techniques, if we recast (42) in terms of W (clearly, the multiplicity remains the same). We recall that K has positive eigenvalues, hence it is invertible and the multiplication by K does not alter the number of incoming characteristics of a system, so that we may consider (42) multiplied on the left by K −1 . Noticing that (30) The linearized equations for (41), (57), (43) read: Here we introduce the source terms f = (f 1 , . . . , f 8 ), χ = (χ 1 , . . . χ 6 ) and g = (g 1 , g 2 , g 3 , g 4 ) to make the interior equations and the boundary conditions inhomogeneous.
2.2.1. Vacuum part. First, we compute the exact form of the linearized equations in Q − T (below we drop δ). By exploiting (50), it is standard to obtain (see [31]) Note that V(Ψ)Ψ denotes the matrix obtained from V(W, Ψ) withΨ instead of Ψ and where all derivatives are applied to Ψ. Since V ′ (Ŵ,Ψ)(W, Ψ) is a first-order differential operator in Ψ, as in [1], the linearized problem is rewritten in terms of the "good unknown" so that, again by standard computations, we deduce We can consider Ψ as an error term for the nonlinear analysis that we will address in a future work, so that we get the system V(Ẇ,Ψ) = χ , which has the same form of the starting system, but with coefficients depending on Ψ instead of Ψ.

Plasma part.
Proceeding similarly as in the vacuum part, we linearize, introduce the good unknown and remove error terms to obtain (see [8,31]) where the matrix C(Û ,Ψ) is determined as follows: . . , y 8 ).
We assume that the source term χ of (62b) satisfies the constraint that the source terms f, χ and the boundary datum g vanish in the past, and we consider the case of zero initial data. We postpone the case of nonzero initial data to the nonlinear analysis of a future work (see e.g. [8,32,33]).

2.3.
Reduction to homogeneous constraints in the "vacuum part". We decomposeẆ in (62) asẆ = W ′ + W ′′ (and accordinglyḢ = H ′ + H ′′ , and similarly forĖ,ḣ,ė), where W ′′ is required to solve for each t The source term χ of the first equation should satisfy the constraint (63). By classical results on Maxwell's equations, we have the following.
Lemma 2.1. Assume that the data χ, g 3 , g 4 in (64), vanishing in appropriate way as x goes to infinity, satisfy the constraints (63). Then there exists a solution W ′′ of (64) vanishing at infinity.
In the statement of the lemma above we intentionally leave unspecified the description of the regularity and the behavior at infinity of the data and consequently of the solution. This point will be faced in the forthcoming paper on the resolution of the nonlinear problem.
From (62), (65), the new form of the reduced linearized problem with unknowns (U , W ′ ), dropping for convenience the prime sign in W ′ , g ′ 2 and the dot sign inU , readsÂ 2.4. Reduction to homogeneous constraints in the "plasma part". From problem (66) we can deduce nonhomogeneous equations associated with the divergence constraint div h = 0 and the "redundant" boundary conditions H N | x1=0 = 0 for the nonlinear problem. Proceeding as in [31], Proposition 7, we can reduce (66) to a problem with homogeneous constraints (68) and (69) in terms of a new variable U ♮ .
Dropping for convenience the indices ♮ , the new form of our reduced linearized problem now readsÂ and solutions should satisfy All the notations here for U and W (e.g., h, H, h, etc.) are analogous to the corresponding ones forU andẆ introduced above.

2.5.
An equivalent formulation of (67). In the following analysis it is convenient to make use of different "plasma" variables and an equivalent form of equations (67a). With the usual notation, we define the matrix It follows that Multiplying (67a) on the left side by the matrix after some calculations we get the symmetric hyperbolic system for the new vector of unknowns U = (q, u, h, S) (compare with (6), (67a)): whereâ 0 is the symmetric and positive definite matrix C ′ is a new zero-order term (a matrix whose precise form has no importance) and where we have set F = (1 +Ψ 1 )RF. We write system (71) in compact form as where E 1j+1 denotes the 8×8 matrix with 0 entries, exception given for the (1, j +1) and (j + 1, 1) entries, which assume value 1.
System (74b) can be written in terms of W = (H, E) ⊺ if |Ψ t | < ε −1 , i.e. the matrix J in (25) is invertible. Similarly to Hypothesis 1.2, we have the following assumption involving the plasma normal speed at the boundary of the basic state.
Since the basic state satisfies (79), we could write (74b) as a symmetric hyperbolic system: where B 0 , B 4 are as in (45), replacing η byη, as usual. Let us prove that B 0 > 0. If we define then the characteristic polynomial of B 0 is given by the square of hence all the eigenvalues are positive, by virtue of Descartes' sign rule, and using ε|Ψ t | < 1 ≤ |N |. We observe that (recalling thatΨ 1 = 0 on ω T , in particular H 1 = H 1 and Thus we may replace in (74d): Remark 2. The invertible part of the boundary matrix of a system allows to control the trace at the boundary of the so-called noncharacteristic component of the vector solution. Thus, with system (74a) (whose boundary matrix is −E 12 , because of (73)), we have the control of q, u 1 at the boundary; therefore the components of U appearing in the boundary conditions (74c), (74d) are well defined. The same holds true for (74b) where we can get the control of H 2 , H 3 , E 2 , E 3 , in particular. The control of e 1 (which appears in (83)) is not given by the system (74b), but by the constraint (76).
Before studying problem (74), we should be sure that the number of boundary conditions is in agreement with the number of incoming characteristics for the hyperbolic systems in (74). Since one of the four boundary conditions (74c)-(74g) is needed for determining the function ϕ(t, x ′ ), the total number of "incoming" characteristics should be three. Let us check that this is true. Proof. Consider first system (74a). In view of (73), the boundary matrix on ω T is −E 12 , which has one negative (incoming in the domain Q + T ) and one positive eigenvalue, while all other eigenvalues are zero. Now consider system (74b). The boundary matrix B 1 (see (12)) has eigenvalues λ ± = ±1, λ 0 = 0, each one with multiplicity 2. Thus, system (74b) has indeed two incoming characteristics in the domain Q − T .

Function Spaces
Now we introduce the main function spaces to be used in the following. Let us denote (84) Q ± := R t × Ω ± , ω := R t × Γ.
For functions defined over Q − T we will consider the weighted Sobolev spaces Similar weighted Sobolev spaces will be considered for functions defined on Q − .

3.2.
Conormal Sobolev spaces. Let us introduce some classes of function spaces of Sobolev type, defined over the half-space Q + T . For j = 0, . . . , 3, we set Z 0 = ∂ t , Z 1 := σ(x 1 )∂ 1 , Z j := ∂ j , for j = 2, 3 , where σ(x 1 ) ∈ C ∞ (R + ) is a monotone increasing function such that σ(x 1 ) = x 1 in some neighborhood of the origin and σ(x 1 ) = 1 for x 1 large enough. Then, for every multi-index α = (α 0 , . . . , α 3 ) ∈ N 4 , the conormal derivative Z α is defined by . Given an integer m ≥ 1, the conormal Sobolev space H m tan (Q + T ) is defined as the set of functions u ∈ L 2 (Q + T ) such that Z α u ∈ L 2 (Q + T ), for all multi-indices α with |α| ≤ m (see [24,23]). Agreeing with the notations set for the usual Sobolev spaces, for γ ≥ 1, H m tan,γ (Q + T ) will denote the conormal space of order m equipped with the γ−depending norm and we have H m tan (Q + T ) := H m tan,1 (Q + T ). Similar conormal Sobolev spaces with γdepending norms will be considered for functions defined on Q + . We will use the same notation for spaces of scalar and vector-valued functions.

The main result
We are now in a position to state the main result of the paper. Recall that U = (q, u, h, S), where u and h were defined in (70).
We assume that the plasma velocity of the basic state at the boundary is smaller than the light speed at the boundary, namely and assume also where δ is an arbitrary fixed constant. Then there exists a constant µ * and there exists γ 0 ≥ 1 such that, for all |μ| ≤ µ * , γ ≥ γ 0 and for all F γ ∈ H 1 tan,γ (Q + T ), vanishing in the past, namely for t < 0, any solution , where we have set U γ = e −γt U, H γ = e −γt H, ϕ γ = e −γt ϕ and so on. Here C = C(κ, T, δ) > 0 is a constant independent of the data F and γ.
Remark 3. In [31], where the pre-Maxwell dynamics in the vacuum side is considered, instead of the Maxwell equations, it is shown that the stability condition (89) is sufficient by itself for the well-posedness of the problem. Here we also need to impose (88) and the smallness condition |μ| ≤ µ * . The present situation with more restrictive stability conditions is more similar to the relativistic case studied in [35], with the difference that, in order to prevent violent instabilities, we don't need to assume that the plasma expands into the vacuum, as in [35].
Let us note as well that, in real problems, ε is very small if compared with the velocity of the basic state, so that (88) is always satisfied. Alternatively, given the basic state, we can choosev (see the definition of ε) so that (88) holds.
Moreover, the smallness of ε says essentially thatμ is small provided that E 1 be small, i.e. the electric field be sufficiently weak. Thus, when the electric field is weak, it is justified to neglect the displacement current and consider the classical plasma-vacuum non-relativistic model with pre-Maxwell dynamics. On the other hand, when the electric field is sufficiently strong, it is natural to expect that the plasma-vacuum interface problem could be ill-posed, see [35].

Proof of Theorem 4.1
To prove the a priori estimate for problem (74) given in Theorem 4.1, we will extend our problem to the spaces Q ± , ω given in (84). 5.1. The extended boundary value problem. Assuming that all coefficients and data appearing in (74) and (80) are extended for all times to the whole real line (for such standard procedure we refer the reader to [17,19]), let us consider the boundary value problem (recall the definition of Q ± , ω in (84)) Since problem (92) looks similar to a corresponding one in relativistic MHD [35], for the deduction of estimate (90) we use the same ideas as in [35]. As in [31,35] we will need a secondary symmetrization of the transformed Maxwell equations in vacuum.

5.2.
A secondary symmetrization. In order to show how to get the secondary symmetrization, for the sake of simplicity we first consider a planar unperturbed interface, i.e. the caseφ ≡ 0. For this case (74b), (76) become that is, the classical Maxwell system (9). We write for system (93) the following secondary symmetrization (for a similar secondary symmetrization of the Maxwell equations in vacuum see [31,35]): The arbitrary functions ν i (t, x) will be chosen in appropriate way later on. It may be useful to notice that system (95) can also be written as with the vector-function ν = (ν 1 , ν 2 , ν 3 ). The symmetric system (95) (or (97)) is hyperbolic if B 0 > 0, i.e. for (98) | ν| < 1. Since the boundary is noncharacteristic for system (95) (or (97)) provided (98) and ν 1 = 0 hold. Consider now a nonplanar unperturbed interface, i.e., the general case whenφ is not identically zero. Dealing with the variable W, we may write the system 0, 2, 3), Since J is invertible (see Hypothesis 2.2), we may multiply (99) by on the lefthand side, obtaining the symmetric system The equivalence of systems (74b) and (103) for every ν = 0 follows as in [31]. This is the same as the equivalence of (92b) and (101). Proof. See [31].
Relation (106) follows from (110), by virtue of the first part of (52), that is, v N =φ t .
Proof. For the sake of brevity, we omit the pedices. By virtue of Lemma 5.2 and (110), it holds As a consequence of (92e)-(92f), and using (78), we may replace using (92c), i.e. ϕ t +v 2 ϕ 2 +v 3 ϕ 3 + γϕ = ϕ∂ 1vN + u 1 , we may write where in the second equality we used (92d), and whereμ is as in (87). The proof of (112) immediately follows. 5.3. The a priori estimate. For the proof of our basic a priori estimate (90) we will apply the energy method to the symmetric hyperbolic systems (92a) and (101). In the sequel γ 0 ≥ 1 denotes a generic constant sufficiently large which may increase from formula to formula, and C is a generic constant that may change from line to line.
First of all we provide some preparatory estimates. In particular, to estimate the weighted conormal derivative Z 1 = σ∂ 1 of U (recall the definition (86) of the γ-dependent norm of H 1 tan,γ ) we do not need any boundary condition because the weight σ vanishes on ω. Applying to system (92a) the operator Z 1 and using standard arguments of the energy method 1 yields the inequality for γ ≥ γ 0 . On the other hand, directly from the equation (92a) we have where C is independent of γ. Thus from (114), (115) we get where C is independent of γ. Furthermore, using the special structure of the boundary matrix in (92a) (see (73)) and the divergence constraint (75), we may estimate the normal derivative of the noncharacteristic part U nγ = e −γt (q, u 1 , h 1 ) of the "plasma" unknown U γ : In a similar way we wish to express the normal derivative of W through its tangential derivatives. Here it is convenient to use system (92b) rather than (101). We find from (92b) an explicit expression for the normal derivatives of H 2 , H 3 , E 2 , E 3 . An explicit expression for the normal derivatives of 1 We multiply Z 1 (92a) by e −γt Z 1 Uγ and integrate by parts over Q + , then we use the Cauchy-Schwarz inequality.
H 1 , E 1 is found through the divergence constraints (76). Thus we can estimate the normal derivatives of all the components of W through its tangential derivatives: where C does not depend on γ.
As for the front function ϕ we easily obtain from (92c) the L 2 estimate where C is independent of γ. Furthermore, thanks to our basic assumption (89) 2 we can resolve (77), (78) and (92c) for the space-time gradient where the vector-functionsâ α = a α (Û |ω ,Ĥ |ω ) of coefficients can be easily written in explicit form. From (120) we get Now we prove an L 2 energy estimate for (U, W). We multiply (92a) by e −γt U γ and (101) by e −γt W γ , integrate by parts over Q ± , then we use the Cauchy-Schwarz inequality. If we set where A is as in (111) and the constant C in (122) is uniform with respect to γ. Now observe that, if we temporarily omit γ subscripts, for the term withμ in A, we haveμ where we have set ∇ ′φ := (1 + |φ 2 | 2 + |φ 3 | 2 ) 1/2 .

A, we obtain
where C is independent of γ. Now we want to derive the a priori estimate for tangential derivatives. Differentiating systems (92a) and (101) with respect to x 0 = t, x 2 or x 3 , using standard arguments of the energy method, and applying (117), (118), gives the energy inequality where ℓ = 0, 2, 3, and where we have denoted Using again (112), we obtain (for simplicity we drop again the index γ) The only exception being the first term, using (120) we reduce the above terms to those likeĉ terms as above with h 1 , u 1 instead of h 1 , or even "better" terms like Here and belowĉ is the common notation for a generic coefficient depending on the basic state (46). By integration by parts such "better" terms can be reduced to the above ones and terms of lower order. The terms likeĉ h 1 Z ℓ u 1|x1=0 are estimated by passing to the volume integral and integrating by parts: Terms like γĉϕZ ℓ ϕ can be handled similarly. Note that, passing to the volume integral and integrating by part yields, among other things, the term In the same way, we estimate the other similar termsĉ h 1 Z ℓ H j ,ĉ h 1 Z ℓ E j , etc. Notice that we only need to estimate normal derivatives either of components of U nγ or W γ . For terms likeĉ h 1 Z ℓ u 1 ,ĉ h 1 Z ℓ E j , etc. we use (118) instead of (117). We treat the terms likeĉ h 1|x1=0 Z ℓ ϕ by substituting (120) again: Combining (124), (126), (127), (128) and similar inequalities for the other terms of (125), and proceeding similarly as above for the term withμ, yields (we restore the index γ) where C is independent of γ. To treat the boundary integral in the lefthand side of (129), which has no definite sign, we argue as in [35]. Let us recall that µ * is a constant such that |μ| ≤ µ * . By substituting (120) and proceeding as for (126), we deal with Therefore, we obtain We write the systems for U and W (see (7) and (10)) as: We notice that (see (12)): After the change of coordinates introduced in Lemma 1.1 by and dropping the tilde for convenience, we derive the new set of variables in the vacuum region (see Proposition 1) We also denote In the plasma region, we similarly define We notice that The plasma and vacuum regions are now given by Ω ± = {x ∈ R 3 , x 1 ≷ 0}, whereas the boundary is given by Γ = {(0, x ′ ), x ′ ∈ R 2 }. We also define Q ± := (−∞, T ]×Ω ± and ω T := (−∞, T ] × Γ (see Section 2.1).
To describe the relations between W, w, W, W, we use the matrices which give us (see (24) and (25)) We also define (see (28)) , which is given by System (30) may be written by means of variables w and W (see Subsection 1.1): After we linearize the problem (see Section 2), we use a "hat" to distinguish the variables corresponding to the basic state, in particularÛ ,Ŵ,φ and the related quantities, likeŵ,Ŵ,Ŵ,ĥ,N ,v N ,η (see Section 2.1). On the other hand, in the construction of the composite quantities without the "hat", like w, W, W, h, N, v N , the basic state only appears inΨ (see Notation 2.1). We set µ := (Ê 1 − εv 3Ĥ2 + εv 2Ĥ3 ) |x1=0 .
In (45), we define B 0 = KJ −1 , the matrix which allow us to write the system for W in the form concludes the proof of (21).
The following remark shows how the diffeomorphism in Lemma 1.1 influences our differential system.
Proof of Proposition 1. Thanks to Remark 5, for the second component of ε∂ t H + ∇× E = 0 in Ω − (t) we have: and similarly for the third one, and for the second and third components of ε∂ t E − ∇× H = 0 in Ω − (t). To deal with the first component of ε∂ t H+ ∇× E = 0 in Ω − (t) we need more computation. First we notice that We may use (9) in Ω − (t) to derive ∇Ψ · (∇×Ẽ) = ∇Ψ · (∇× E) • (t, Φ) = −ε∇Ψ · (∂ t H) • (t, Φ) , where the first equality follows from straightforward calculations. Therefore, replacing we obtain We proceed similarly for the second equation of (9) and this completes the proof of (30). Thanks again to Remark 5, we obtain: which proves the equivalence of div H = 0 in Ω − (t) and div h = 0 in the fixed domain. The equivalence of div E = 0 in Ω − (t) and div e = 0 in the fixed domain is identical.
Proof of Proposition 2. The proof of the part concerning div h, H N can be found in [33]. Let us consider the vacuum variables. Taking the divergence of the first equation in (30) yields Multiplying by div h and integrating by parts gives Using the first of the equations in (35) we obtain d dt x1<0 | div h| 2 dx ≤ C x1<0 | div h| 2 dx.
Appendix C. On the number of boundary conditions for problem (33) The correct number of boundary conditions that should be imposed to (33) for well-posedness is given by the number of incoming characteristics, i.e. the number of negative eigenvalues of the boundary matrix of (33). Evaluated at Γ (where Ψ = ϕ, Ψ 1 = 0) it reads This matrix has eigenvalues λ 1,2 = −εϕ t + 1 + ϕ 2 2 + ϕ 2 3 , λ 3,4 = −εϕ t , λ 5,6 = −εϕ t − 1 + ϕ 2 2 + ϕ 2 3 . In agreement with our choice of measure units with the speed of light in vacuum set to be 1, we may assume ε |ϕ t | < 1. Thus we have λ 1,2 > 0 and λ 5,6 < 0. If ϕ t < 0 (the plasma expands into vacuum) λ 3,4 > 0 and so the total number of negative eigenvalues ofB 1 (Ψ) |x1=0 is two. Considering that the boundary matrix A 1 (U, Ψ) |x1=0 has one negative and one positive eigenvalue, see Proposition 3, and that one more boundary condition needs to be imposed for the determination of the front ϕ, the correct number of boundary conditions for the resolution of (32), (33) is four, as in (34).
Differently, after the introduction of the new variables W, the negative eigenvalues of the boundary matrix B 1 of (42) are always two, see Proposition 3, so that the correct number of boundary conditions for the resolution of (41), (42) is four, as in (43).