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

Abstract

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 𝜌 is a positive constant), the equations in a dimensionless form read 𝜕_𝑡()𝜕𝑢+∇⋅𝑢⊗𝑢−𝐻⊗𝐻+∇𝑞=0,_𝑡𝐻−∇×(𝑢×𝐻)=0,div𝑢=0,div𝐻=0, (1) where, using 𝑇 for transposition, 𝑢=(𝑢₁,𝑢₂,𝑢₃)^𝑇 denotes the plasma velocity, 𝐻=(𝐻₁,𝐻₂,𝐻₃)^𝑇 is the magnetic field (in Alfvén velocity units), 𝑞=𝑝+|𝐻|²/2 is the total pressure, and 𝑝 is the pressure. For smooth solutions, system (1) can be written in an equivalent form as 𝜕_𝑡()()𝜕𝑢+𝑢⋅∇𝑢−𝐻⋅∇𝐻+∇𝑞=0,_𝑡𝐻+(𝑢⋅∇)𝐻−(𝐻⋅∇)𝑢=0,div𝑢=0,div𝐻=0. (2)

We are interested in weak solutions, in a suitable sense, to (1) that are smooth on either side of a smooth hypersurface Γ(𝑡)={𝑥₃=𝑓(𝑡,𝑥^′)} in [0,𝑇]×Ω, where Ω⊂ℝ³, 𝑥^′=(𝑥₁,𝑥₂), and that satisfy suitable jump conditions at each point of the front Γ(𝑡). For notational simplicity, we assume that the density is the same constant on either side of Γ(𝑡), so that we can take, with no loss of generality, 𝜌≡1. 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 Ω^±(𝑡)={𝑥₃≷𝑓(𝑡,𝑥^′)}, where Ω=Ω⁺(𝑡)∪Ω⁻(𝑡)∪Γ(𝑡); given any function 𝑔, we set 𝑔^±=𝑔 in Ω^±(𝑡) and denote by [𝑔]=𝑔⁺_|Γ(𝑡)−𝑔⁻_|Γ(𝑡) the jump across Γ(𝑡).

We look for current-vortex sheets solutions, that is, smooth solutions (𝑞^±,𝑢^±,𝐻^±) of (2) in Ω^±(𝑡) such that Γ(𝑡) is a tangential discontinuity, namely, the plasma does not flow through the discontinuity front, and the magnetic field is tangent to Γ(𝑡), see, for example, Landau and Lifshitz [1]; thus, the boundary conditions take the form 𝜎_front=𝑢^±⋅𝑛,𝐻^±⋅𝑛=0,𝑞=0onΓ(𝑡), (3) where 𝑛=𝑛(𝑡) denotes the outward unit normal on 𝜕Ω⁻(𝑡) and 𝜎_front denotes the velocity of propagation of the interface front Γ(𝑡). With the given parametrization of Γ(𝑡), an equivalent formulation of these jump conditions is 𝜕_𝑡𝑓=𝑢^±⋅𝑁,𝐻^±⋅𝑁=0,𝑞=0onΓ(𝑡), (4) with 𝑁∶=(−𝜕₁𝑓,−𝜕₂𝑓,1)^𝑇. Notice that the function 𝑓 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 𝑢^±(0,𝑥)=𝑢^±₀(𝑥),𝐻^±(0,𝑥)=𝐻^±₀(𝑥),𝑥∈Ω^±(0),𝑓0,𝑥^′=𝑓₀𝑥^′,𝑥^′∈Γ(0), (5) where div𝑢^±₀=div𝐻^±₀=0 in Ω^±(0).

Current-vortex sheets have various interesting applications in geophysics and astrophysics. For instance, an accepted model in the literature for the interface region between the unperturbed flows of the interstellar plasma and the supersonic solar wind plasma is given by a current-vortex sheet separating the interstellar plasma compressed at the bow shock from the solar wind plasma compressed at the termination shock, see Ruderman and Fahr [2] and references therein. This current-vortex sheet is called the heliopause and in some sense can be considered as the outer boundary of the solar system. Similarly, the boundary separating the shocked solar wind plasma from the plasma of the magnetosphere of planets equipped with an intrinsic magnetic field (such as the Earth, Jupiter …) is a current-vortex sheet called magnetopause.

In order to prove the local existence of solutions to the current-vortex sheets (nonlinear) problem (2), (4), and (5), it is useful to prove existence for the linearized problem and the strong stability, that is, an a priori estimate for the solutions with no loss of derivatives (see later on). We will perform a further simplification by assuming a linearization around a constant-coefficient basic state. This must be seen as a preliminary step before considering a linearization around a general variable-coefficient basic state, in order to prove existence, uniqueness and stability for the solution to problem (2), (4), and (5) without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.

In the last years, there has been a renewed interest for the analysis of free interface problems in fluid dynamics, especially for the Euler equations in the vacuum and the water waves problem, see Coutand and Shkoller [3, 4] and the references thereinto. This fact has produced different methodologies for obtaining a priori estimates and the proof of the existence of solutions. If the interface moves with the velocity of fluid particles, a natural approach consists in the introduction of the Lagrangian coordinates that reduces the original problem to a new one on a fixed domain. This approach has been recently employed with success in a series of papers by Coutand and Shkoller on the incompressible and compressible Euler equations in the vacuum, see [3, 4]. However, this method seems hardly applicable to problem (2), (4), and (5).

For incompressible current-vortex sheets, the precise description of the region of weak stability is known, meaning that for states outside this region the problem is ill posed. In this region, Morando et al. [5], for the constant-coefficient linearized equations, have shown an a priori estimate for solutions with loss of regularity with respect to the data. In a subset of the region of weak stability, that is, under a more restrictive stability condition (i.e., (6), introduced in the following), Trakhinin [6] has shown a similar result with no loss of regularity for the solutions to the linearized problem with respect to the source terms, but with loss of derivatives with respect to the coefficients of the problem. Moreover, in a recent paper, Coulombel et al. [7] have shown that the same more restrictive stability condition is sufficient for an a priori estimate of solutions to the nonlinear problem with no loss of regularity (here, space periodicity is assumed). However, in all the previous cases, existence of the solution is missing. Let us note that we are looking for the existence and an a priori estimate with no loss of derivatives for the linearized incompressible current-vortex sheet problem. Reference [6] already provides an estimate with no loss of regularity in the constant-coefficient case, but such a loss manifests for variable coefficients, and existence is not proved. The main novelty of our paper is the existence result; however, we hope that it is possible to extend the techniques here used in order to obtain an a priori estimate with no loss of regularity even for the variable-coefficient linearized problem. We point out that the estimate in [7] is optimal, with no loss of derivatives, but refers to the nonlinear problem and does not help in proving existence for the linearized system.

In [8], Secchi and Trakhinin have established existence and stability for the linearized compressible plasma-vacuum (free boundary) problem. In that case, the interface separates a plasma region governed by the usual MHD equations from a vacuum region with pre-Maxwell dynamics for the magnetic field, as in the case of magnetic confinement in nuclear fusion processes. This work has been employed by Morando et al. [9] to obtain the analogous result for the incompressible problem. However, the techniques used in these papers cannot be easily adapted to current-vortex sheets, since they rely on a secondary symmetrization in the vacuum part that cancels problematic boundary terms and cannot be reproduced in plasma part.

In the present paper, we follow an approach different from the one of Coutand and Shkoller. To reduce our free boundary problem to the fixed domain, we consider a change of variables inspired from Lannes [10] and performed in Secchi and Trakhinin [8]. The control of the function describing the free interface follows from a stability condition introduced by Trakhinin in [11], that is, the strong stability condition max𝐻⁺×[𝑢],|𝐻⁻×[𝑢]|<𝐻⁺×𝐻⁻, (6) which implies in particular that 𝐻⁺×𝐻⁻ is not the zero vector.

We use a compressible approximation and perform some changes of variables in order to obtain a symmetric hyperbolic problem with conditions in a more suitable form, so that we can apply the results by Lax and Phillips [12] and Secchi [13, 14] for maximally nonnegative linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. Another fundamental tool is the so-called secondary symmetrization, which is a linear combination of the equations of the hyperbolic system that allows a simplification of the border term necessary to obtain the a priori estimate (see Section 4). By exploiting such techniques, we obtain the existence of a unique solution to the incompressible current-vortex sheets free boundary problem linearized about a suitable constant-coefficient state and show that the solution satisfies a priori estimates with no loss of derivatives with respect to data. As mentioned above, this is a preliminary step in order to prove existence and stability for the current-vortex sheet nonlinear problem.

1.2. The Reference Domain Ω

To avoid using local coordinate charts necessary for arbitrary geometries, and for simplicity, we will assume that the space domain Ω occupied by the fluid is given by Ω=ℝ³. We also set Ω^±∶=Ω∩𝑥₃≷0,Γ∶=Ω∩𝑥₃=0. (7) The moving discontinuity front is given by Γ(𝑡)∶=Ω∩𝑥₃=𝑓𝑡,𝑥^′. (8)

1.3. An Equivalent Formulation in the Fixed Domain Ω

To reduce the free boundary problem (2), (4), and (5) to the fixed domains Ω^±, we introduce a suitable change of variables. In all what follows, 𝐻^𝑠(𝜔) denotes the Sobolev space of order 𝑠 on a domain 𝜔.

The diffeomorphism that reduces the free boundary problem (2), (4), and (5) to the fixed domains Ω^± is given in the following lemma (see [8]).

Lemma 1. Let 𝑚≥3 be an integer. Then, for all 𝑇>0 and for all 𝑓∈⋂^𝑚−1_𝑗=0𝒞^𝑗([0,𝑇];𝐻^{𝑚−𝑗−0.5}(ℝ²)) satisfying without loss of generality ‖𝑓‖_{𝒞([0,𝑇];𝐻}²_(ℝ²₎₎≤1, there exists a function 𝜓∈^𝑚−1_𝑗=0𝒞^𝑗[0,𝑇];𝐻^𝑚−𝑗ℝ³ (9) such that the function Ψ(𝑡,𝑥)∶=𝑥^′,𝑥₃+𝜓(𝑡,𝑥),(𝑡,𝑥)∈[0,𝑇]×Ω (10) defines an 𝐻^𝑚-diffeomorphism of Ω for all 𝑡∈[0,𝑇]. Moreover, there holds 𝜕^𝑗_𝑡(Ψ−𝐼𝑑)∈𝒞[0,𝑇];𝐻^𝑚−𝑗(Ω)Ψ𝑓𝑜𝑟𝑗=0,…,𝑚−1,𝑡,𝑥^′,0=𝑥^′,𝑓𝑡,𝑥^′,𝜕₃Ψ𝑡,𝑥^′,0=(0,0,1). (11)

We set 𝐴∶=[𝐷Ψ]⁻¹(),[),,inverseoftheJacobianmatrix𝐽∶=det𝐷Ψ](determinantoftheJacobianmatrix𝑎∶=𝐽𝐴transposeofthecofactormatrix (12) and we compute −𝜕𝐴=100010₁𝜓𝐽−𝜕₂𝜓𝐽1𝐽,𝐽=1+𝜕₃𝜓,𝑎=𝐽000𝐽0−𝜕₁𝜓−𝜕₂.𝜓1 (13) Now we may reduce the free boundary problem (2), (4), and (5) to a problem in the fixed domains Ω^± by the change of variables (10). Let us set 𝑣^±(𝑡,𝑥)∶=𝑢^±(𝑡,Ψ(𝑡,𝑥)),𝐵^±(𝑡,𝑥)∶=𝐻^±((,𝑄𝑡,Ψ𝑡,𝑥))_±(𝑡,𝑥)∶=𝑞^±(𝑡,Ψ(𝑡,𝑥)). (14) Then, the system (2), (4), and (5) can be reformulated on the fixed reference domains Ω^± as 𝜕_𝑡𝑣^±+𝑣^∼±⋅∇𝑣^±−𝐵^∼±⋅∇𝐵^±+𝐴^𝑇∇𝑄^±𝜕=0,_𝑡𝐵^±+𝑣^∼±⋅∇𝐵^±−𝐵^∼±⋅∇𝑣^±𝐴=0,_𝑇∇⋅𝑣^±=0,𝐴^𝑇∇⋅𝐵^±=0,in[0,𝑇]×Ω^±,[𝑄]=0,𝜕_𝑡𝑓=𝑣^±⋅𝑁,𝐵^±[]𝑣⋅𝑁=0,on0,𝑇×Γ,_±|𝑡=0=𝑣^±₀,𝐵^±_|𝑡=0=𝐵^±₀,onΩ^±,𝑓_|𝑡=0=𝑓₀,onΓ. (15) In (15), we have set 𝑁∶=−𝜕₁𝜓,−𝜕₂𝜓,1^𝑇,_∼𝑣𝜕∶=𝐴𝑣−0,0,^𝑡𝜓𝐽^𝑇=𝑣₁,𝑣₂,𝑣⋅𝑁−𝜕_𝑡𝜓𝐽^𝑇_∼,𝐵∶=𝐴𝐵=𝐵₁,𝐵₂𝑁,𝐵⋅𝐽^𝑇. (16) Notice that 𝐽=1,𝑁=−𝜕₁𝑓,−𝜕₂𝑓,1^𝑇,𝑣^∼₃=𝐵^∼₃=0onΓ. (17) We warn the reader that the notation 𝐴^𝑇 is used to denote the transpose of 𝐴 and has nothing to do with the time interval [0,𝑇] on which the smooth solution is sought; we hope that this does not create any confusion. Vectors are written in columns.

If we set 𝑄^±_𝑇=[0,𝑇]×Ω^±, 𝜔_𝑇=[0,𝑇]×Γ, and 𝑉^±=(𝑄^±,𝑣^±,𝐵^±)^𝑇, problem (15) can be written in short form as ℙ𝑉^±,𝜓=0in𝑄^±_𝑇,𝔹𝑉^±,𝑓=0on𝜔_𝑇,𝑣^±,𝐵^±,𝑓^±_𝑡=0=𝑣^±₀,𝐵^±₀,𝑓^±₀inΩ^±2×Γ, (18) where 0 is the zero vector.

2. The Linearized Problem

2.1. Basic State

Let 𝑉^±,𝑓 (19) be such that 𝑉⁺ and 𝑉⁻ are constant vectors, 𝑉=(𝑄,𝑣,𝐵)^𝑇, and 𝑓≡0. (20) We can then take 𝜓≡0, so that 𝐽̂=𝜕₃Ψ₃=1+𝜕₃𝜓=1. Similarly, we determine all the “hat” functions as for the corresponding functions obtained for (𝑉^±,𝑓). Let us note, in particular, that 𝐴=𝑎=𝐼₃, 𝑁^𝑇=(0,0,1), 𝑣=𝑣^∼, and 𝐵=𝐵^∼. We assume that 𝑄=0,𝑣^±₃=𝐵^±₃=0. (21) In particular, 𝑄⁺=𝑄⁻. Under these hypotheses, we have that the basic state (𝑉^±,𝑓) is a solution of problem (15).

Moreover, we assume that the basic state satisfies max𝐵⁺×[𝑣],𝐵⁻×[𝑣]<𝐵⁺×𝐵⁻. (22) This is the analogue of condition (6) and implies 𝐵⁺×𝐵⁻>0. (23)

2.2. Linearized Problem

The linearized equations for (15) are 𝑑ℙ𝑉𝑑𝑠^±𝑠,𝜓_𝑠𝑠=0=𝖥^±,𝑑𝔹𝑉𝑑𝑠^±𝑠,𝑓_𝑠𝑠=0=𝑔^±, (24) where 𝑉_𝑠=𝑉+𝑠𝛿𝑉,  𝑓_𝑠=𝑓+𝑠𝛿𝑓,  𝜓_𝑠=𝜓+𝑠𝛿𝜓, and 𝛿𝑉,𝛿𝑓,𝛿𝜓 are the unknowns. Here, we consider the source terms in order to make the system inhomogeneous.

If we omit 𝛿 and write explicitly the linearized equations, we get, in 𝑄^±_𝑇, 𝜕_𝑡𝑣^±+𝑣^±⋅∇𝑣^±−𝐵^±⋅∇𝐵^±+∇𝑄^±=𝖥^±₁,𝜕_𝑡𝐵^±+𝑣^±⋅∇𝐵^±−𝐵^±⋅∇𝑣^±=𝖥^±₂,div𝑣^±=𝐹^±_𝑣,div𝐵^±=𝐹^±_𝐵. (25) The linearized boundary conditions are obtained in a similar way (let us recall that we are omitting 𝛿): [𝑄]=𝑔₀,𝜕_𝑡𝑓−𝑣^±₃+𝑣^′±⋅∇^′𝑓=𝑔^±₁,−𝐵^±₃+𝐵^′±⋅∇^′𝑓=𝑔^±₂ (26) on 𝜔_𝑇, where, as usual, if 𝑤=(𝑤₁,𝑤₂,𝑤₃)^𝑇 is a vector, we set 𝑤^′=(𝑤₁,𝑤₂)^𝑇, and moreover,∇^′=(𝜕₁,𝜕₂)^𝑇.

2.3. Reduction to Homogeneous Boundary Conditions

Let 𝑢^±,  ℎ^± denote vector-valued functions which solve, respectively, the elliptic problems div𝑢^±=𝐹^±_𝑣in𝑄^±_𝑇,𝑢^±₃=−𝑔^±₁on𝜔_𝑇,divℎ^±=𝐹^±_𝐵in𝑄^±_𝑇,ℎ^±₃=−𝑔^±₂on𝜔_𝑇 (27) and set 𝑢=𝑣−𝑢, ℎ = 𝐵−ℎ, and 𝑞=𝑄−𝑞, where 𝑞⁺=𝑔₀ on 𝜔⁺_𝑇 and 𝑞⁻=0 on 𝜔⁻_𝑇 (here, letters 𝑢 and 𝑞 represent different variables from the ones introduced in the original formulation (1)).

The equations in (25) can be recast in terms of (𝑞,𝑢,ℎ): 𝜕_𝑡𝑢^±+𝑣^±⋅∇𝑢^±−𝐵^±⋅∇ℎ^±+∇𝑞^±=𝐹^±₁,𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±−𝐵^±⋅∇𝑢^±=𝐹^±₂,div𝑢^±=0,divℎ^±=0, (28) with new source terms 𝐹_𝑖=𝖥_𝑖+𝖿_𝑖(𝑣,𝐵,∇𝑞,∇_𝑡,𝑥𝑢,∇_𝑡,𝑥ℎ),  𝑖=0,1,2, with boundary conditions 𝑞𝜕=0,_𝑡𝑓−𝑢^±₃+𝑣^′±⋅∇^′𝑓=0,−ℎ^±₃+𝐵^′±⋅∇^′𝑓=0 (29) and initial data 𝑢^±,ℎ^±,𝑓^±_𝑡=0=𝑣^±₀−𝑢^±(0),𝐵^±₀−ℎ^±(0),𝑓^±₀. (30)

3. Notations and Results

3.1. Function Spaces

We denote by 𝐻^𝑚=𝐻^𝑚(Ω) the usual Sobolev space and by ||⋅||_𝑚 its norm, while ||⋅|| is simply the norm of 𝐿²=𝐿²(Ω). We introduce a smooth odd function 𝜎=𝜎(𝑥₃) defined in ℝ such that 𝜎(𝑥₃)=𝑥₃ for 0≤𝑥₃≤1/2 and 𝜎(𝑥₃)=1 for 𝑥₃≥1 and set 𝑍₁=𝜕_𝑥1, 𝑍₂=𝜕_𝑥2 and 𝑍₃=𝜎𝜕_𝑥3. For a multi-index 𝛼=(𝛼₁,𝛼₂,𝛼₃), |𝛼|=𝛼₁+𝛼₂+𝛼₃, we set 𝜕^𝛼_∗=𝑍^𝛼1₁𝑍^𝛼2₂𝑍^𝛼3₃, ‖𝑢‖²_𝑚,∗=_{|𝛼|+2𝛽≤𝑚}𝜕^𝛼_∗𝜕^𝛽_𝑥3𝑢²,𝐻^𝑚_∗=𝐻^𝑚_∗(Ω)=𝑢∈𝐿²∶‖𝑢‖_𝑚,∗<+∞, (31) where 𝛽≥0 is an integer number (see [14–17]).

If 𝑢(𝑡), defined on [0,𝑇] with values in 𝐻^𝑚_∗, is such that 𝜕^𝑘_𝑡𝑢(𝑡)∈𝐻_∗^𝑚−𝑘 for 𝑘=1,2,…,𝑚, we set |||𝑢(𝑡)|||²_𝑚,∗=^𝑚_𝑘=0𝜕^𝑘_𝑡𝑢(𝑡)²_{𝑚−𝑘,∗},ℒ^𝑝_𝑇𝐻^𝑚_∗=^𝑚_𝑘=0𝑊^𝑘,𝑝0,𝑇;𝐻_∗^𝑚−𝑘, (32) where 𝑊^𝑘,𝑝(0,𝑇;𝐻_∗^𝑚−𝑘) is the space of functions which are in 𝐿^𝑝 together with the time derivatives up to order 𝑘 on [0,𝑇], with values in 𝐻_∗^𝑚−𝑘. The spaces ℒ²_𝑇(𝐻^𝑚_∗) and ℒ^∞_𝑇(𝐻^𝑚_∗) are equipped with norms, respectively, [𝑢]²_{𝑚,∗,𝑇}=^𝑇₀|||𝑢(𝑡)|||²_𝑚,∗𝑑𝑡,|||𝑢|||_{𝑚,∗,𝑇}=sup_[]0,𝑇|||𝑢(𝑡)|||_𝑚,∗. (33)

We define the same function spaces and norms for a general space domain 𝐸 in the same way; we write, for instance, ||𝑔||_{𝑚,∗;𝐸} to specify the domain that we are considering.

3.2. Main Result

Now, we are able to state the main theorem. We set 𝑈=(𝑞,𝑢,ℎ)^𝑇 and consider problem 𝜕_𝑡𝑢^±+𝑣^±⋅∇𝑢^±−𝐵^±⋅∇ℎ^±+∇𝑞^±=𝐹^±₁,𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±−𝐵^±⋅∇𝑢^±=𝐹^±₂,div𝑢^±=0 (34) in 𝑄^±_𝑇, with boundary conditions 𝑞𝜕=0,_𝑡𝑓−𝑢^±₃+𝑣^′±⋅∇^′𝑓=0 (35) on 𝜔_𝑇, and initial data 𝑢^±,ℎ^±,𝑓_𝑡=0=𝑢^±₀,ℎ^±₀,𝑓₀inΩ^±2×Γ, (36) where div𝑢^±₀=divℎ^±₀=0onΩ^±. (37)

Theorem 2. Assume that the constant basic state (𝑉^±,𝑓) satisfies (20), (21), and (22), and moreover, assume that 𝑢^±₀,ℎ^±₀∈𝐻¹(Ω^±), div𝑢^±₀=divℎ^±₀=0 on Ω^±, and 𝑓₀∈𝐻^1/2(Γ). Assume that 𝐹^±=(𝐹^±₁,𝐹^±₂)∈ℒ²_𝑇(𝐻¹_∗(Ω^±)), div𝐹^±₂=0 in 𝑄^±_𝑇, and (𝐹^±₂)₃=0 on [0,𝑇]×Γ, where (𝐹^±₂)₃ denotes the third component of vector 𝐹^±₂. Finally, assume as well the compatibility conditions on {𝑡=0}×Γ: 𝜕_𝑡𝑓₀−𝑢^±₃+𝑣^′±⋅∇^′𝑓₀=0,−ℎ^±₃+𝐵^′±⋅∇^′𝑓₀=0. (38)
Then there exists a unique solution (𝑈^±,𝑓) to (34), (35), and (36) (𝑞^± unique up to additive constants) such that 𝑢^±,ℎ^±∈𝐿^∞([0,𝑇];𝐻¹_∗(Ω^±))∩𝑊^1,∞([0,𝑇];𝐿²(Ω^±)), 𝑞^±∈𝐿^∞([0,𝑇];𝐻̇₁(Ω^±)) (𝐻̇₁ is the homogeneous Sobolev space), and 𝑓∈𝒞⁰([0,𝑇];𝐻^1/2(Γ)); in addition, for each 𝑡∈[0,𝑇], the following a priori estimate |||(_±𝑢,ℎ)^±(𝑡)|||²_1,∗;Ω^±+𝑢₃,ℎ₃^±(𝑡)²_𝐻1Ω±+∇𝑞^±(𝑡)²_𝐿2Ω±+𝑓(𝑡)²_𝐻1/2(Γ)≤𝐶𝑒^𝐶𝑡_±|||𝑢₀,ℎ₀^±|||²_1,∗;Ω±+𝐹^±2_{1,∗,𝑡;Ω±}+𝐹^±₁(0)²_𝐿2Ω±+𝑓₀²_𝐻1/2(Γ) (39) is satisfied, where 𝐶=𝐶(𝑣,𝐵) is a positive constant.
Moreover, one has (𝑢^±₃,ℎ^±₃)|_Γ∈𝐿^∞([0,𝑇];𝐻^1/2(Γ)).

Remark 3. In order to consider the nonlinear problem, it might be useful to have higher regularity estimates, with no loss of derivatives, provided that data are sufficiently regular. The 𝐻¹ regularity can be obtained from the equations for rot𝑢 and rotℎ, which allow to improve the regularity for the normal derivatives. Taking further derivatives and proceeding as in the proof of Proposition 13, one can expect to get higher regularity. The details are postponed to a future paper.
As for the regularity assumption on 𝑢^±₀,  ℎ^±₀, see Section 4.5.
Note that divℎ^±=0 in 𝑄^±_𝑇 and −ℎ^±₃+𝐵^′±⋅∇^′𝑓=0 on [0,𝑇]×Γ follow from (34), (35), and (36) and the assumptions of Theorem 2 and, in particular, from the validity of the restrictions of such equations at time 𝑡=0. More precisely, we have the following result.

Proposition 4. Under the same assumptions of Theorem 2, the solution 𝑈^± satisfies divℎ^±=0𝑖𝑛𝑄^±_𝑇,−ℎ^±₃+𝐵^′±⋅∇^′𝑓=0𝑖𝑛𝜔_𝑇. (40)(41)

Proof. If we take the divergence in the second equation in (34) and use div𝑢^±=div𝐹^±₂=0 in 𝑄^±_𝑇, we get 𝜕_𝑡+𝑣^±⋅∇divℎ^±=0, (42) and hence divℎ^±=0 in 𝑄^±_𝑇, since at time 𝑡=0, we have divℎ^±=divℎ^±₀=0 on Ω^± by assumption.
On the other hand, let us take just the third component of the second equation in 𝜕_𝑡ℎ^±₃+𝑣^′±⋅∇^′ℎ^±₃−𝐵^′±⋅∇^′𝑢^±₃=0, (43) where we have used (𝐹^±₂)₃=0 and 𝑣^±₃=𝐵^±₃=0. If we substitute 𝑢^±₃ from the second equation into (35) and observe that (𝐵^′±⋅∇^′)(𝑣^′±⋅∇^′𝑓)=(𝑣^′±⋅∇^′)(𝐵^′±⋅∇^′𝑓), we deduce that 𝜕_𝑡+𝑣^′±⋅∇^′ℎ^±₃−𝐵^′±⋅∇^′𝑓=0 (44) in [0,𝑇]×Γ. Using the third equation in (38), which holds for 𝑡=0, we conclude that ℎ^±₃−𝐵^′±⋅∇^′𝑓=0 in [0,𝑇]×Γ.

4. Proof

4.1. Hyperbolic Regularization

Now, let us consider the following hyperbolic compressible regularization (see Secchi [18]), where 𝜀>0:𝜕_𝑡𝑞^±+𝑣^±⋅∇𝑞^±−𝜀𝐵^±⋅𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±+1𝜀div𝑢^±𝜕=0,_𝑡𝑢^±+𝑣^±⋅∇𝑢^±−𝐵^±⋅∇ℎ^±+1𝜀∇𝑞^±=𝐹^±₁,𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±−𝐵^±⋅∇𝑢^±−𝜀𝐵^±𝜕_𝑡𝑞^±+(𝑣⋅∇)𝑞^±−𝜀𝐵^±⋅𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±=𝐹^±₂, (45)(46)(47) with boundary conditions (35).

Here, the solution 𝑈=(𝑞,𝑢,ℎ)^𝑇 depends on 𝜀 but, in this moment, we will not use the more precise notation 𝑈^𝜀=(𝑞^𝜀,𝑢^𝜀,ℎ^𝜀)^𝑇 to denote the solution to the previous system (with suitable initial data). Let us note that, at least formally, if we project the system (45), (46), and (47) on the subspace of divergence-free functions, it converges, as 𝜀→0, to system (34) projected on the same subspace; it is sufficient to multiply (45) by 𝜀 and take the limit. We will make this statement more precise in Section 4.7.

Equations (45)–(47) in 𝑄^±_𝑇 form a symmetric system: 10^𝑇−𝜀𝐵^±𝑇0𝐼₃𝑂₃−𝜀𝐵^±𝑂₃𝐼₃+𝜀²𝐵^±⊗𝐵^±𝜕_𝑡𝑞^±𝑢^±ℎ^±+1𝜀div𝑢^±∇𝑞^±0+𝑣^±⋅∇0^𝑇−𝜀𝐵^±𝑇𝑣^±0⋅∇𝑣^±⋅∇−𝐵^±𝐵⋅∇−𝜀^±𝑣^±⋅∇−𝐵^±⋅∇𝐼₃+𝜀²𝐵^±⊗𝐵^±𝑣^±𝑞⋅∇^±𝑢^±ℎ^±=0𝐹^±₁𝐹^±₂, (48) or in a compact form, 𝐿^±𝑈^±≐𝐴^±₀𝜕_𝑡𝑈^±+³_𝑗=1𝐴^±_𝑗+1𝜀ℰ_1,𝑗+1𝜕_𝑗𝑈^±=𝐹^±, (49) where each matrix 𝐴^±_𝑗 is bounded in 𝜀 and ℰ_ℎ,𝑖=(𝑒^(ℎ,𝑖)_𝑗,𝑘)_{1≤𝑗,𝑘≤7}, where 𝑒^(ℎ,𝑖)_𝑗,𝑘=1 if (𝑗,𝑘)=(ℎ,𝑖) or (𝑗,𝑘)=(𝑖,ℎ), 𝑒^(ℎ,𝑖)_𝑗,𝑘=0 otherwise. Notice that 𝐴^±₃=𝑂₇ because of (21).

One can easily check that, as 𝜀→0, this system converges to (34), at least formally. Moreover, the system is hyperbolic, since (omitting ±) 𝐴₀𝑈⋅𝑈=𝑞−𝜀𝐵⋅ℎ²+|𝑢|²+|ℎ|²>0 (50) for each (𝑞,𝑢,ℎ)≠(0,0,0).

4.2. Secondary Symmetrization

This new formulation is obtained by a linear combination of (45), (46), and (47) in the previous step and aims to simplify boundary terms in the following computations (see Remark 5).

More precisely, let us consider the system ()()−𝜆45+𝜀𝜆𝐵⋅46𝜀𝐵()+()()divℎ=0,𝜀𝜆4546−𝜆54=0,−𝜆𝐼₃+𝜀²𝐵⊗𝐵(46)+(47)=0, (51)(52)(53) where 𝜆=𝜆^± will be chosen conveniently later on and (54) denotes the equation obtained by substituting (45) into (47), that is, 𝜕_𝑡ℎ^±+𝑣^±⋅∇ℎ^±−𝐵^±⋅∇𝑢^±+𝐵^±div𝑢^±=𝐹^±₂. (54) We get 𝑑_𝑡𝑞−𝜀𝐵⋅𝑑_𝑡1ℎ+𝜀div𝑢+𝜀𝜆𝐵⋅𝑑_𝑡1𝑢−𝐵⋅∇ℎ+𝜀−𝜆∇𝑞𝜀divℎ=𝜀𝜆𝐵⋅𝐹₁,𝜀𝜆𝐵𝑑_𝑡𝑞−𝜀𝐵⋅𝑑_𝑡ℎ+𝑑_𝑡1𝑢−𝐵⋅∇ℎ+𝜀𝑑∇𝑞−𝜆_𝑡ℎ−𝐵⋅∇𝑢=𝐹₁−𝜆𝐹₂,−𝜆𝐼₃+𝜀²𝐵⊗𝐵𝑑_𝑡1𝑢−𝐵⋅∇ℎ+𝜀∇𝑞+𝑑_𝑡𝑢𝐵𝑑ℎ−𝐵⋅∇−𝜀_𝑡𝑞−𝜀𝐵⋅𝑑_𝑡ℎ=−𝜆𝐼₃+𝜀²𝐵⊗𝐵𝐹₁+𝐹₂, (55) where 𝑑_𝑡≐𝜕_𝑡+(𝑣⋅∇). This system is symmetric; indeed, the coefficient of 𝑑_𝑡 is 𝒜₀=1𝜀𝜆𝐵^𝑇−𝜀𝐵^𝑇𝜀𝜆𝐵𝐼₃−𝜆𝐼₃+𝜀²𝐵𝐼𝐵⊗−𝜀𝐵−𝜆₃+𝜀²𝐵⊗𝐵𝐼₃+𝜀²𝐵⊗𝐵, (56) the singular term is given by 1𝜀0∇^𝑇−𝜆∇^𝑇∇𝑂₃𝑂₃−𝜆∇𝑂₃𝑂₃, (57) and the remaining derivatives are included in 𝜆𝐵⋅∇0^𝑇−𝜀𝜆𝐵^𝑇0𝐵⋅∇𝜆𝐵𝐼𝐵⋅∇−𝐵⋅∇−𝜀𝜆𝐵⊗∇−𝐵⋅∇𝜆₃+𝜀²𝐵⊗𝐵𝐵⋅∇, (58) where 𝐼₃ and 𝑂₃ denote, respectively, the identity and null 3×3 matrix. All these matrices are symmetric, and the system can be recast as ℒ^±𝑈^±≐𝒜^±₀𝜕_𝑡𝑈^±+³_𝑗=1𝒜^±_𝑗+1𝜀ℰ_1,𝑗+1−𝜆𝜀ℰ_1,𝑗+4𝜕_𝑗𝑈^±=ℱ^±, (59) where each matrix 𝒜^±_𝑗 is bounded in 𝜀 provided 𝜆^± is bounded and ℰ_ℎ,𝑖=(𝑒^(ℎ,𝑖)_𝑗,𝑘)_{1≤𝑗,𝑘≤7}, where 𝑒^(ℎ,𝑖)_𝑗,𝑘=1 if (𝑗,𝑘)=(ℎ,𝑖) or (𝑗,𝑘)=(𝑖,ℎ),  𝑒^(ℎ,𝑖)_𝑗,𝑘=0 otherwise. Notice that 𝒜^±₃=𝑂₇.

Remark 5. The secondary symmetrization is inspired by [11, 19], which we refer to for further details. The main idea is to consider the following variant of (49), which takes into account the divergence constraint for the magnetic field 𝐷^±𝐴^±₀𝜕_𝑡𝑈^±+³_𝑗=1𝐷^±𝐴^±_𝑗+1𝜀ℰ_1,𝑗+1𝜕_𝑗𝑈^±+𝜆^±𝐸^±divℎ^±=𝐹^±, (60) where the matrix 𝐷^± and the vector 𝐸^± depend on the basic state. 𝐷 is then chosen in order to obtain 𝒜₀, whose form is useful to get the energy a priori estimate, while 𝐸 is chosen to maintain the symmetry of the system. As pointed out in [19] (Remark 5), such a choice is related to the conservation of cross-helicity: 𝑑/𝑑𝑡∫(𝑣⋅𝐵)𝑑𝑥=0. On the other hand, one main feature of the secondary symmetrization is that 𝜆^± can be chosen widely arbitrarily, so that in the following we will be able to find a suitable 𝜆^± which simplifies singular boundary terms in the a priori estimate.

The following lemma guarantees that, under our assumptions and a suitable choice of 𝜆^±, the matrices 𝒜^±₀ are positive definite and (59) is symmetric hyperbolic.

Lemma 6. If 𝜆^±2<11+𝜀𝐵^±2, (61) then the symmetric matrices 𝒜^±₀ are positive definite.

Proof. Omitting “±”, we have 𝒜₀=𝐼𝑈⋅𝑈𝑞+𝜀𝜆𝐵⋅𝑢−𝜀𝐵⋅ℎ𝑞+𝜀𝜆𝐵𝑞+𝑢−𝜆₃+𝜀²𝐵ℎ+𝐼𝐵⊗⋅𝑢−𝜀𝐵𝑞−𝜆₃+𝜀²𝐵⊗𝐵𝑢+𝐼₃+𝜀²𝐵ℎ=𝐵⊗⋅ℎ𝑞−𝜀𝐵⋅ℎ²≥+𝑢⋅𝑢+ℎ⋅ℎ+2𝜀𝜆𝐵⋅𝑢𝑞−𝜀𝐵⋅ℎ−2𝜆ℎ⋅𝑢𝑞−𝜀𝐵⋅ℎ+𝜀𝜆𝐵⋅𝑢²+|ℎ−𝜆𝑢|²+1−𝜆²−𝜆²𝜀𝐵²|𝑢|². (62) If 1−𝜆²−𝜆²𝜀𝐵²>0, (63) that is to say, if 𝜆²<11+𝜀𝐵², (64) then the matrix 𝒜₀ is positive definite.

Lemma 7. There exists 𝜀₀>0 such that, for each 𝜀∈]0,𝜀₀], the choice 𝜆^±=𝜆^±≐[𝑣]×𝐵^∓𝐵⁺×𝐵⁻sign𝐵⁺₁𝐵⁻₂−𝐵⁺₂𝐵⁻₁ (65) satisfies (61) and hence makes the matrices 𝒜^±₀ positive definite.

Proof. Since [𝑣]×𝐵^∓𝐵⁺×𝐵⁻<1, (66) which follows from (22), we deduce that there exists 𝜀₀>0 such that, for each 𝜀∈[0,𝜀₀[, the inequality [𝑣]×𝐵^∓𝐵⁺×𝐵⁻≤11+𝜀𝐵^±2 (67) follows, so we conclude that (61) holds true for 𝜀 small enough.

Let us remark that 𝜆^± is a constant term independent of 𝜀; moreover, since 𝐵^±₃=0, this choice of 𝜆^± is the same as 𝜆^±=𝑣₁𝐵^∓₂−𝑣₂𝐵^∓₁𝐵⁺₁𝐵⁻₂−𝐵⁺₂𝐵⁻₁, (68) which will be useful in the following steps. Moreover, this choice is equivalent to 𝑣^′−𝜆𝐵^′=0. (69) In addition, we set Δ=𝐵⁺₁𝐵⁻₂−𝐵⁺₂𝐵⁻₁, (70) which is always different from zero thanks to (23).

4.3. Boundary Conditions in Algebraic Form

The boundary conditions can be recast so that the space derivatives of the front function 𝑓 do not appear explicitly.

Lemma 8. The boundary conditions (35) imply, on 𝜔_𝑇, the boundary conditions 𝑞𝑢=0,₃−𝜆ℎ₃=0 (71) and the front equation 𝜕_𝑡𝑓=𝑢^±₃+𝑆^±₊ℎ⁻₃−𝑆^±₋ℎ⁺₃, (72) with 𝑆^±₊=1Δ𝑣^±₁𝐵⁺₂−𝑣^±₂𝐵⁺₁,𝑆^±₋=1Δ𝑣^±₁𝐵⁻₂−𝑣^±₂𝐵⁻₁,Δ=𝐵⁺₁𝐵⁻₂−𝐵⁺₂𝐵⁻₁. (73)

Proof. Condition (41) can be recast as 𝐵⁺₁𝜕₁𝑓+𝐵⁺₂𝜕₂𝑓=ℎ⁺₃,𝐵⁻₁𝜕₁𝑓+𝐵⁻₂𝜕₂𝑓=ℎ⁻₃. (74) Since 𝐵Δ=det⁺¹𝐵⁺₂𝐵⁻₁𝐵⁻₂≠0 (75) by hypothesis, we can obtain 𝜕₁1𝑓=Δℎ⁺₃𝐵⁻₂−ℎ⁻₃𝐵⁺₂,𝜕₂1𝑓=−Δℎ⁺₃𝐵⁻₁−ℎ⁻₃𝐵⁺₁. (76) By substituting in the second equation of (35), we get 𝜕_𝑡𝑓=𝑢^±₃+𝑣^±₁𝐵⁺₂−𝑣^±₂𝐵⁺₁Δℎ⁻₃−𝑣^±₁𝐵⁻₂−𝑣^±₂𝐵⁻₁Δℎ⁺₃. (77)
Using the definition of 𝑆^±₊ and 𝑆^±₋, we deduce the conditions 𝜕_𝑡𝑓=𝑢^±₃+𝑆^±₊ℎ⁻₃−𝑆^±₋ℎ⁺₃,𝑢⁺₃+𝑆⁺₊ℎ⁻₃−𝑆⁺₋ℎ⁺₃=𝑢⁻₃+𝑆⁻₊ℎ⁻₃−𝑆⁻₋ℎ⁺₃. (78) The last condition can be recast as 𝑢₃=−𝑆₊ℎ⁻₃+𝑆₋ℎ⁺₃, (79) where 𝑆₊=𝑆⁺₊−𝑆⁻₊,𝑆₋=𝑆⁺₋−𝑆⁻₋. (80)
Recalling (68), we have 𝜆^±=𝑆_∓. (81) The previous condition becomes 𝑢₃−𝜆ℎ₃=0, (82) and the lemma is proved.

4.4. Equivalence of Systems

Obviously, if (𝑈^𝜀,𝑓^𝜀) is a solution to system (49) with boundary conditions (35), then it also satisfies system (59) with conditions (71) (we are assuming the same initial conditions); the converse holds true as well.

Proposition 9. If   𝑈^𝜀 is a solution to system (59) with boundary conditions (71), then one can find a unique 𝑓^𝜀, so that (𝑈^𝜀,𝑓^𝜀) is a solution to system (49) with boundary conditions (35) and the same initial conditions.

Proof. First, we prove that (49) is satisfied. Since (59) holds, then we have (51), (52), and (53). If we substitute (54) = (47) +  𝜀𝐵(45) into (52), we get (46) =𝜆(47), and substituting this relation in (53), we deduce 𝐼₃−𝜆²𝐼₃−𝜆²𝜀²𝐵⊗𝐵(47)=0. (83) Since det𝐼₃−𝜆²𝐼₃−𝜆²𝜀²𝐵⊗𝐵=1−𝜆²²1−𝜆²−𝜆²𝜀²𝐵²≠0 (84) thanks to (61), then we get (47) =0 and consequently (46) =𝜆(47) =0.
From (51), in order to prove (45), it is now sufficient to establish that divℎ^𝜀=0, since we already know that (46) =0. With this aim, we substitute (51) into (47) and get 𝜕_𝑡ℎ^𝜀+(𝑣⋅∇)ℎ^𝜀−𝜆𝐵divℎ^𝜀+𝐵div𝑢^𝜀−𝐵⋅∇𝑢^𝜀=𝐹₂. (85) Taking the divergence of the above equation, exploiting the assumption div𝐹₂=0 and the fact that 𝑣, 𝐵, 𝜆 are piecewise constant, and simplifying, we deduce that 𝜕_𝑡+𝑣−𝜆𝐵⋅∇divℎ^𝜀=0. (86) Since divℎ^𝜀=0 in {𝑡=0}×Ω^± by assumption on the initial data, we conclude that divℎ^𝜀=0 in 𝑄^±_𝑇, so that (45) =0 from (51). Then (45), (46), and (47) are satisfied, which means that (49) holds.
Now, we have to prove (35). Condition [𝑞^𝜀]=0 on 𝜔_𝑇 obviously holds, since we have the same condition in (71). We need to prove that there exists 𝑓^𝜀 so that 𝜕_𝑡𝑓^𝜀+𝑣^′±⋅∇^′𝑓^𝜀=𝑢₃^𝜀,± on 𝜔_𝑇. Given 𝑢₃^𝜀,± and ℎ₃^𝜀,±, let us define 𝑓^± as the solutions of 𝜕_𝑡𝑓^±+𝑣^′±⋅∇^′𝑓^±=𝑢₃^𝜀,±in𝜔_𝑇,𝑓^±_𝑡=0=𝑓₀onΓ. (87) Thus, for the completion of the proof of (35), it is sufficient to show that 𝑓⁺=𝑓⁻ in 𝜔_𝑇. Proceeding as in the proof of Proposition 4, we first deduce ℎ₃^𝜀,±=(𝐵^′±⋅∇^′)𝑓^± in 𝜔_𝑇. Let us compute 𝑢₃^𝜀,±−𝜆^±ℎ₃^𝜀,±=𝜕_𝑡𝑓^±+𝑣^′±⋅∇^′𝑓^±−𝜆^±𝐵^′±⋅∇^′𝑓^±. (88) Since [𝑢^𝜀₃−𝜆ℎ^𝜀₃]=0 by (71), we obtain 𝜕_𝑡+𝑣^′+−𝜆⁺𝐵^′+⋅∇^′𝑓⁺=𝜕_𝑡+𝑣^′−−𝜆⁻𝐵^′−⋅∇^′𝑓⁻. (89) Thanks to (69), we have 𝑤^′≐𝑣^′+−𝜆⁺𝐵^′+=𝑣^′−−𝜆⁻𝐵^′−, and hence, 𝜕_𝑡+𝑤^′⋅∇^′𝑓⁺=𝜕_𝑡+𝑤^′⋅∇^′𝑓⁻in𝜔_𝑇,𝑓^±_𝑡=0=𝑓₀onΓ. (90) Thus, we can take 𝑓^𝜀≐𝑓⁺=𝑓⁻ in 𝜔_𝑇, and (35) holds.

4.5. Existence of the Approximate Solution

Now, consider the system (59) with the boundary conditions (71) and the initial condition 𝑈^𝜀,±(0)=𝑈^±₀≐(0,𝑢^±₀,ℎ^±₀)^𝑇. This system can be written as ℒ𝑈^𝜀=ℱin[0,𝑇]×Ω,ℳ𝑈^𝜀=0[]𝑈in0,𝑇×Γ,_𝜀(0)=𝑈₀inΩ, (91) where we have set 𝑈^𝜀=(𝑈^𝜀,+,𝑈^𝜀,−)^𝑇 and similarly for the other quantities; moreover, we set ‖𝑈^𝜀‖_Ω=‖𝑈^𝜀,+‖_Ω⁺+‖𝑈^𝜀,−‖_Ω⁻ and so on. Notice that ℳ𝑈^𝜀=0 corresponds to the boundary conditions (71).

Let us note that, since 𝑞^±₀≐𝑞^𝜀,±|_𝑡=0≡0, the condition on Γ given by [𝑞₀]=0 is automatically satisfied. Moreover, if we denote by (𝜕_𝑡𝑞^±,𝜕_𝑡𝑢^±,𝜕_𝑡ℎ^±)|_𝑡=0 the homonymous quantities in (91) evaluated at time 𝑡=0 and calculated in terms of the initial data, from the equations and the hypotheses on 𝑞^±₀, 𝑢^±₀, ℎ^±₀, we get the condition (𝜕_𝑡𝑞^±,𝜕_𝑡𝑢^±,𝜕_𝑡ℎ^±)|_𝑡=0∈𝐿²(Ω^±). Let us note that the hypothesis 𝑢^±₀,ℎ^±₀ in 𝐻¹(Ω^±), rather than simply in 𝐻¹_∗(Ω^±), is needed exactly here. We have the following result.

Proposition 10. There exists 𝜀₀>0 so that, for each 𝜀∈]0,𝜀₀], there exists a unique solution 𝑈^𝜀∈𝒞⁰([0,𝑇];𝐻¹_∗(Ω))∩𝒞¹([0,𝑇];𝐿²(Ω)) to problem (91); moreover, one has |||𝑈^𝜀(𝑡)|||_1,∗;Ω≤𝐶_𝜀e^𝐶𝜀𝑡|||𝑈^𝜀₀|||_1,∗;Ω+[ℱ]_{1,∗,𝑡;Ω} (92) for each 𝑡∈[0,𝑇] and a suitable 𝐶_𝜀>0.

Proof. The boundary matrix of system (91) is given by 𝒜_𝜈=𝒜⁺_𝜈𝑂₇𝑂₇𝒜⁻_𝜈, (93) where 𝒜^±_𝜈=∓(𝒜^±₃+(1/𝜀)ℰ_1,4−(𝜆^±/𝜀)ℰ_1,7). The matrix 𝒜_𝜈 is singular with constant rank (indeed, 𝒜^±₃ is the null matrix) and, moreover, kerℳ is maximally nonnegative for 𝒜_𝜈, that is to say, 𝒜_𝜈𝑈^𝜀⋅𝑈^𝜀≥0 (94) for each 𝑈^𝜀∈kerℳ (nonnegativity) and such a property does not hold in any other vector space properly containing kerℳ (maximality). Indeed, we have 𝒜_𝜈Γ𝑈^𝜀⋅𝑈^𝜀=−𝒜⁺_𝜈Γ𝑈^𝜀,+⋅𝑈^𝜀,++𝒜⁻_𝜈Γ𝑈^𝜀,−⋅𝑈^𝜀,−2=−𝜀𝑞^𝜀𝑢₃^𝜀,+−𝜆⁺ℎ₃^𝜀,++𝑞^𝜀,−𝑢^𝜀₃−𝜆ℎ^𝜀₃=0 (95) thanks to [𝑞^𝜀]=0 and [𝑢^𝜀₃−𝜆ℎ^𝜀₃]=0 on kerℳ. Moreover, let us observe that both 𝒜⁺_𝜈 and 𝒜⁻_𝜈 have exactly one negative eigenvalue, so we have the same number of conditions defining kerℳ and negative eigenvalues, that is, 2; this implies maximality.
Thus, we can apply the results in Lax and Phillips [12] (existence in 𝐿², see as well Friedrichs [20] for the noncharacteristic case) and Secchi [13, 14] (regularity in 𝐻^𝑚_∗), and deduce the existence of a unique global solution 𝑈^𝜀∈𝒞⁰([0,𝑇];𝐻¹_∗(Ω))∩𝒞¹([0,𝑇];𝐿²(Ω)) to problem (91) such that (92) holds for each 𝑡∈[0,𝑇], provided 𝜀>0 is sufficiently small.

Remark 11. Let us note that, in the previous result, the smallness of 𝜀 is necessary in order to have 𝒜^±₀ positive definite.
We need to have a number of boundary conditions in (71), that is, defining kerℳ, equal to the number of negative eigenvalues of 𝒜_𝜈, that is to say, 2, in order to have maximality.

Corollary 12. There exists 𝜀₀>0, so that, for each 𝜀∈]0,𝜀₀], there exists a unique solution (𝑈^𝜀,𝑓^𝜀) to problem (49), (35), and (36) with 𝑞^𝜀,±|_𝑡=0≡0 such that 𝑈^𝜀∈𝒞⁰([0,𝑇];𝐻¹_∗(Ω))∩𝒞¹([0,𝑇];𝐿²(Ω)), (𝑞,𝑢₃,ℎ₃)^𝜀,±|_𝜔𝑇∈𝒞⁰([0,𝑇];𝐻^1/2(Γ)) and 𝑓^𝜀∈𝒞⁰([0,𝑇];𝐻^1/2(Γ)); moreover, one has |||𝑈^𝜀(𝑡)|||_1,∗;Ω+𝑓^𝜀(𝑡)_𝐻1/2(Γ)≤𝐶_𝜀e^𝐶𝜀𝑡|||𝑈^𝜀₀|||_1,∗;Ω+[ℱ]_{1,∗,𝑡;Ω}+𝑓_0𝐻1/2(Γ) (96) for each 𝑡∈[0,𝑇] and a suitable 𝐶_𝜀>0.

Proof. Problem (91) with 𝑓^𝜀|_𝑡=0=𝑓₀ on Γ is a different way to say that (59), (71), and (36) with 𝑞^𝜀,±|_𝑡=0≡0 are satisfied. Using Propositions 9 and 10, we get the result for 𝑈^𝜀. Let us note that, since 𝑢^𝜀,±∈𝒞⁰([0,𝑇];𝐻¹_∗(Ω)) and the normal component 𝑢₃^𝜀,± at the boundary is noncharacteristic, then the trace 𝑢₃^𝜀,±|_𝜔𝑇 is in 𝒞⁰([0,𝑇];𝐻^1/2(Γ)), as shown in Secchi [13, 14] and Shizuta [21]. In particular, 𝜕₃𝑢^𝜀₃ can be estimated using the other terms appearing in (45), so that we have 𝑢^𝜀₃(𝑡)_𝐻1(Ω)≤𝐶|||𝑈^𝜀(𝑡)|||_1,∗;Ω. (97)
The same holds true for 𝑞^𝜀 as well, for the same reason (𝜕₃𝑞^𝜀 is obtained from (46)), and for 𝜕₃ℎ^𝜀₃, since divℎ^𝜀=0 is bounded (this follows from (54) proceeding similarly as in Proposition 4; note that, taking the divergence, the terms containing 𝑢^𝜀 cancel together). Thus, from the transport equation (87), we find 𝑓^𝜀∈𝒞⁰([0,𝑇];𝐻^1/2(Γ)) and 𝑓^𝜀(𝑡)_𝐻1/2(Γ)≤𝑓_0𝐻1/2(Γ)+𝑡𝑢₃^𝜀,+_{𝐿∞(0,𝑡;𝐻1/2(Γ))}≤𝑓_0𝐻1/2(Γ)+𝐶𝑡𝑢₃^𝜀,+_{𝐿∞(0,𝑡;𝐻1(Ω+))}≤𝑓_0𝐻1/2(Γ)+𝐶_𝜀𝑒^𝐶𝜀𝑡_±|||𝑈^±₀|||_1,∗;Ω^±+𝐹^±_{1,∗,𝑡;Ω±}. (98) This concludes the proof.

4.6. A Priori Energy Estimate Uniform in 𝜀

Since estimate (96) depends on 𝜀, we cannot use it when passing to the limit as 𝜀→0. We can deduce the following a priori estimate for (𝑈^𝜀,±,𝑓^𝜀), which is an 𝐿²-estimate uniform in 𝜀, useful to pass to the limit as 𝜀→0. We recall that the choice 𝜆=𝜆 makes 𝒜^±₀ positive definite, and hence, if we fix 𝜀₀>0 sufficiently small, there exists a suitable real number 𝛼₀∈]0,1], so that 𝒜^±₀≥𝛼₀𝐼₇ for each 𝜀∈]0,𝜀₀].

Proposition 13. There exists 𝜀₀>0, such that, for each 𝜀∈]0,𝜀₀], the a priori estimate _±|||𝑈^𝜀,±(𝑡)|||²_1,∗;Ω^±+𝑢₃,ℎ₃^𝜀,±(𝑡)²_𝐻1Ω±+𝑓^𝜀(𝑡)²_𝐻1/2(Γ)≤𝐶𝛼₀𝑒^{𝐶𝑡/𝛼0}_±|||𝑈^±₀|||²_1,∗;Ω^±+𝐹^±2_{1,∗,𝑡;Ω±}+𝑓₀²_𝐻1/2(Γ) (99) holds for each 𝑡∈[0,𝑇] and for a suitable 𝐶>0 independent of 𝜀, where 𝑈^𝜀 is the solution to problem (49), (35), and (36) and 𝑓^𝜀 satisfies (87) with initial condition 𝑓^𝜀|_𝑡=0=𝑓₀.

Proof. Let us take the scalar product of (59), which is equivalent to (49), with 𝑈^𝜀,± and integrate over 𝑄^±_𝑡 (recall that 𝜆^±=𝜆^±); integrating by parts the terms with spatial derivatives of 𝑈^𝜀,±, exploiting the fact that the coefficient matrices are constant, and recalling that the boundary matrix is given by 𝒜^±_𝜈=∓(𝒜^±₃+(1/𝜀)ℰ_1,4−(𝜆^±/𝜀)ℰ_1,7)=∓((1/𝜀)ℰ_1,4−(𝜆^±/𝜀)ℰ_1,7), since 𝒜^±₃=𝑂₇, we get _𝑄±𝑡𝒜^±₀𝜕_𝑡𝑈^𝜀,±⋅𝑈^𝜀,±∓1𝑑𝑥𝑑𝑡2𝜀_𝜔𝑡ℰ_1,4−𝜆^±ℰ_1,7𝑈^𝜀,±⋅𝑈^𝜀,±𝑑𝑥^′=1𝑑𝑡2_𝑄^±_𝑡³_𝑗=1𝜕_𝑗𝒜^±_𝑗+1𝜀ℰ_1,𝑗+1−𝜆^±𝜀ℰ_1,𝑗+4𝑈^𝜀,±⋅𝑈^𝜀,±+𝑑𝑥𝑑𝑡_𝑄±𝑡ℱ^±⋅𝑈^𝜀,±=𝑑𝑥𝑑𝑡_𝑄±𝑡ℱ^±⋅𝑈^𝜀,±𝑑𝑥𝑑𝑡, (100) where sum over ± is assumed. Let us note that, since the basic state is piecewise constant, then 𝒜^±_𝑗,ℰ_ℎ,𝑘 and 𝜆^± are piecewise constant as well; thus, ³_𝑗=1𝜕_𝑗𝒜^±_𝑗+1𝜀ℰ_1,𝑗+1−𝜆^±𝜀ℰ_1,𝑗+4=0 (101) in 𝑄^±_𝑡. Moreover, _±∓12𝜀_𝜔𝑡ℰ_1,4−𝜆^±ℰ_1,7𝑈^𝜀,±⋅𝑈^𝜀,±𝑑𝑥^′1𝑑𝑡=−𝜀_𝜔𝑡𝑞^𝜀𝑢₃^𝜀,+−𝜆⁺ℎ₃^𝜀,++𝑞^𝜀,−𝑢^𝜀₃−𝜆ℎ^𝜀₃𝑑𝑥^′𝑑𝑡=0, (102) thanks to the boundary conditions [𝑞^𝜀]=0 and [𝑢^𝜀₃−𝜆ℎ^𝜀₃]=0, that hold in 𝜔_𝑡.
We deduce the equality _±𝑄±𝑡𝒜^±₀𝜕_𝑡𝑈^𝜀,±⋅𝑈^𝜀,±𝑑𝑥𝑑𝜏=_±𝑄±𝑡ℱ^±⋅𝑈^𝜀,±𝑑𝑥𝑑𝜏. (103) For the left-hand side, integrating by parts in time and exploiting the symmetry of 𝒜^±₀, we have _Ω±^𝑡₀𝒜^±₀𝜕_𝑡𝑈^𝜀,±⋅𝑈^𝜀,±1𝑑𝜏𝑑𝑥=2_Ω^±𝒜^±₀𝑈^𝜀,±(𝑡)⋅𝑈^𝜀,±(𝑡)−1𝑑𝑥2_Ω^±𝒜^±₀𝑈^±₀⋅𝑈^±₀𝑑𝑥. (104) By applying the Cauchy-Schwarz inequality in the right-hand side, we obtain _±Ω±𝒜^±₀𝑈^𝜀,±(𝑡)⋅𝑈^𝜀,±(𝑡)≤𝑑𝑥_𝑡0±𝑈^𝜀,±(𝜏)²_𝐿2Ω±𝑑𝜏+_±Ω±𝒜^±₀𝑈^±₀⋅𝑈^±₀𝑑𝑥+_±^𝑡₀ℱ^±(𝜏)²_𝐿2Ω±𝑑𝜏, (105) and hence, by exploiting 𝒜^±₀≥𝛼₀𝐼₇, _±𝑈^𝜀,±(𝑡)²_𝐿2Ω±≤1𝛼₀^𝑡_0±𝑈^𝜀,±(𝜏)²_𝐿2Ω±+1𝑑𝜏𝛼_0±𝐶𝑈^±₀²_𝐿2Ω±+^𝑡₀ℱ^±(𝜏)²_𝐿2Ω±𝑑𝜏. (106) Through a standard application of Grönwall’s lemma, we deduce that _±𝑈^𝜀,±(𝑡)