| Russian Mathematical Surveys |
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On local well-posedness of problems with characteristic free boundary for hyperbolic systems of conservation laws Yu. L. Trakhinin Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences
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     1. IntroductionIn fluid mechanics many statements of free boundary problems, in one way or another, follow from the statement of the problem with Rankine–Hugoniot boundary conditions on a free surface on which the solutions of the system of differential conservation laws describing the flow of the continuous medium have a discontinuity. This is why we begin our survey with the description of precisely this most general problem. 1.1. Free boundary problem with Rankine–Hugoniot boundary conditionsWe consider the system of $m$ differential conservation laws
$$
\begin{equation}
\partial_t f^0_i(U )+\operatorname{div} f_i (U )=0,\qquad i=1,\dots,m,
\end{equation}
\tag{1}
$$
for the vector of unknows $U(t,x)=(u_1(t,x),\dots,u_m(t,x))^\top$, where $t$ is time, $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ are Cartesian coordinates, and $\partial_t:=\partial /\partial t$. The flux fuctions $f^0_i=f^0_i(U)$ and $f_i=(f_i^1,\dots,f_i^n)^\top=f_i(U)$ are assumed to be sufficiently smooth (for concrete systems they are usually infinitely smooth). Equations (1) are rewritten as the quasilinear system
$$
\begin{equation}
A_0(U )\,\partial_t U+\sum_{j=1}^nA_j(U)\,\partial_jU=0
\end{equation}
\tag{2}
$$
with matrices $A_{\alpha}=A_{\alpha}(U)=(\partial f^{\alpha}/\partial U)$, where $f^{\alpha}:=(f_1^{\alpha},\dots,f_m^{\alpha})^\top$, $\alpha=0,\dots,n$. Recall that system (2) is said to be symmetric hyperbolic [13], [14] if
$$
\begin{equation*}
A_{\alpha}=A_{\alpha}^\top\quad (\alpha =0,\dots,n),\qquad A_0>0
\end{equation*}
\notag
$$
for all $U\in G\subset\mathbb{R}^m$. Symmetric hyperbolic systems are hyperbolic in the sense of the following general definition of hyperbolic systems. Namely, the quasilinear system (2) with non-singular matrix $A_0$ is called hyperbolic if all the eigenvalues $\lambda_i$ ($i=1,\dots,m$) of the characteristic matrix are real for all ${\xi}=(\xi_1,\dots,\xi_n)^\top\in \mathbb{R}^n \setminus\{0\}$ and $U\in G\subset\mathbb{R}^m$ and this matrix is diagonalizable (with real diagonal elements $\lambda_i=\lambda_i(U,{\xi})$). In this case the subdomain $G$ of the state space is called the hyperbolicity domain. The local-in-time existence and uniqueness of smooth solutions of the Cauchy problem for a symmetric hyperbolic system was independently proved in [59] and [23]. Below we consider free boundary problems for symmetric hyperbolic systems. The general method of reducing a system of conservation laws to a symmetric form was proposed by Godunov [15], [16]. We do not discuss it here because it is enough for our goals to use an elementary symmetrization of the system of conservation laws which can be performed for the most famous models of continuum mechanics thanks to a suitable choice of the vector of unknowns $U$. The hyperbolicity condition $A_0(U_0)>0$ for the initial data $U\big|_{t=0}=U_0(x)$ is the main requirement guaranteeing the local-in-time existence and uniqueness of a smooth solution to the Cauchy problem for the symmetric system (2). If we are interested in the local existence and uniqueness of a piecewise smooth solution whose smooth parts are separated by a free surface of discontinuity, then the natural question of requirements for the initial data in the corresponding free boundary problem (apart from the hyperbolicity condition) arises. From both the physical and mathematical points of view such piecewise smooth solutions must be weak solutions to the system of conservation laws (1). A weak solution to (1) is defined in a standard way. Namely, the vector-function $U(t,x)\in L^2([0,T]\times \mathbb{R}^n)$ is called a weak solution of the system of conservation laws (1) if
$$
\begin{equation}
\int_{[0,T]\times\mathbb{R}^n}\biggl( f^0(U)\cdot\partial_t\psi + \sum_{j=1}^n f^j(U)\cdot \partial_j\psi\biggr)=0
\end{equation}
\tag{3}
$$
for all $\psi (t,x) =(\psi_1(t,x),\dots,\psi_m(t,x))^\top\in C_0^{\infty} ([0,T]\times \mathbb{R}^n)$. Let
$$
\begin{equation*}
\Sigma=\Sigma(t)=\{ x_1-\varphi (t,x')=0\},\qquad x':=(x_2,\dots,x_n)\in\mathbb{R}^{n-1},
\end{equation*}
\notag
$$
be a smooth hypersurface in $[0,T]\times\mathbb{R}^n$. We assume that $\Sigma (t)$ is a surface of strong discontinuity of the solution $U(t,x)$ of system (1) which is smooth on each side of the discontinuity $\Sigma$. In other words, not only the derivatives of $U$ but also the solution $U$ itself have a discontinuity on this surface. We leave outside the scope of our survey the question of the spontaneous formation of strong discontinuities (in particular, shock waves) in solutions of hyperbolic systems of conservation laws and refer the reader, for example, to [61]. Let
$$
\begin{equation*}
U=U^{\pm}\quad\text{in}\ \ \Omega^{\pm}(t)=\{\pm(x_1-\varphi (t,{x}'))>0\}.
\end{equation*}
\notag
$$
Multiplying system (1) scalarly by $\psi \in C_0^{\infty} ([0,T]\times \mathbb{R}^n)$, integrating the result by parts, and applying the Gauss–Ostrogradsky theorem we obtain
$$
\begin{equation}
\begin{aligned} \, &\int_{[0,T]\times\mathbb{R}^n}\biggl(f^0(U)\cdot\partial_t\psi+ \sum_{j=1}^n f^j(U)\cdot \partial_j\psi \biggr) \nonumber \\ &\qquad+\int_{\Sigma} \biggl\{\partial_t\varphi\,[f^0(U)]-[f^1(U)]+ \sum_{k=2}^{n}\partial_k\varphi\,[f^k(U)]\biggr\}\cdot \psi=0 \end{aligned}
\end{equation}
\tag{4}
$$
for all $\psi \in C_0^{\infty} ([0,T]\times \mathbb{R}^n)$. Here and below
$$
\begin{equation*}
[f^{\alpha}(U)]:=f^{\alpha}(U^+)\big|_{\Sigma}-f^{\alpha}(U^-)\big|_{\Sigma}
\end{equation*}
\notag
$$
is the usual notation for a jump. Then, in view of definition (3) of a weak solution and du Bois-Reymond’s lemma, it follows from (4) that the piecewise smooth solution ${U}$ of system (1) (or (2)) which undergoes a jump on $\Sigma(t)$ is a weak solution if and only if the Rankine–Hugoniot conditions hold at each point of $\Sigma$:
$$
\begin{equation}
\partial_t\varphi\,[f^0(U)]-[f^1(U)]+ \sum_{k=2}^{n}\partial_k\varphi\,[f^k(U)]=0\quad \text{on } \Sigma(t).
\end{equation}
\tag{5}
$$
It should be noted that the problem for system (2) in the domains $\Omega^{\pm}(t)$,
$$
\begin{equation}
A_0(U^\pm)\,\partial_t U^\pm + \sum_{j=1}^nA_j(U^\pm)\,\partial_jU^\pm=0\quad \text{in } \Omega^\pm(t),
\end{equation}
\tag{6}
$$
with boundary conditions (5) on the hypersurface $\Sigma(t)$, is a free boundary problem. Indeed, the function $\varphi(t,{x}')$ determining $\Sigma$ is one of the unknowns in problem (2), (5) with the corresponding initial data
$$
\begin{equation}
{U}^\pm\big|_{t=0}={U}^{\pm}_0({x})\quad\text{and}\quad \varphi\big|_{t=0}=\varphi_0(x').
\end{equation}
\tag{7}
$$
To prove the existence of a strong discontinuity for the system of conservation laws (1) we have to answer the following question: is there a smooth solution $(U^+,U^-,\varphi)$ of the free boundary problem (5), (6), at least locally in time? Moreover, we must also prove the uniqueness of this solution. A hypersurface in $[0,T]\times\mathbb{R}^n$ is said to be characteristic if it is a characteristic of the hyperbolic system (2). Then, at each point of this hypersurface its space-time normal $(\tau,\xi_1,\dots,\xi_n)^\top$ satisfies the equation
$$
\begin{equation*}
\det\biggl(\tau A_0(U)+\sum_{j=1}^n\xi_jA_j(U)\biggr) =0.
\end{equation*}
\notag
$$
That is, a surface of strong discontinuity $\Sigma(t)$ is characteristic if
$$
\begin{equation}
\det\biggl(\partial_t\varphi\,A_0(U^\pm)-A_1(U^\pm)+ \sum_{k=2}^n\partial_k\varphi\,A_n(U^\pm)\biggr)=0\quad\text{on } \Sigma(t).
\end{equation}
\tag{8}
$$
Following Lax [27], characteristic discontinuities are called contact, and a strong discontinuity is called a shock wave if at each point of this discontinuity the determinant in (8) does not vanish. However, it should be noted that from the physical point of view not all the contact discontinuities are contact surfaces (with non-zero mass transfer flux). In this survey we are interesting in problems with characteristic free boundary, in particular, problems for characteristic discontinuities. 1.2. Free boundary problem for the compressible Euler equationsThe system of Euler’s equations describing the motion of an ideal compressible fluid, in particular, a gas has the form of system (1) containing five conservation laws:
$$
\begin{equation}
\begin{cases} \partial_t\rho+\operatorname{div}(\rho v)=0, \\ \partial_t(\rho v)+\operatorname{div}(\rho v \otimes v)+\nabla p=0, \\ \partial_t(\rho E)+\operatorname{div}(\rho Ev+pv)=0, \end{cases}
\end{equation}
\tag{9}
$$
where $\rho$ is the density, $v=(v_1,v_2,v_3)^\top$ the velocity, $p=p(\rho,S)$ the pressure, $S$ the entropy, $E=\mathfrak{e}+|v|^2/2$ the total energy, and $\mathfrak{e}=\mathfrak{e}(\rho,S)$ the internal energy of the fluid. It follows from the thermodynamic identity
$$
\begin{equation}
{d}\mathfrak{e}=\frac{p}{\rho^2}\,{d}\rho+\vartheta\,dS
\end{equation}
\tag{10}
$$
(here $\vartheta$ is the temperature) that
$$
\begin{equation*}
p=\rho^2\,\frac{\partial\mathfrak{e}}{\partial\rho}(\rho,S) \quad\text{and}\quad \vartheta=\frac{\partial\mathfrak{e}}{\partial S}(\rho,S).
\end{equation*}
\notag
$$
Then (9) is a closed system, for example, for the vector of unknowns $U=(p,v,S)^\top$. Note that the additional law of entropy conservation is fulfilled on smooth solutions of system (9). Taking (11) into account, system (9) can be rewritten in the non-conservative form
$$
\begin{equation}
\frac{1}{\rho c^2}\,\frac{dp}{dt}+\operatorname{div}\, v =0,\qquad \rho\, \frac{dv}{dt}+\nabla p=0,\qquad \frac{dS}{dt}=0,
\end{equation}
\tag{12}
$$
where $c^2=\dfrac{\partial p}{\partial \rho}(\rho,S)$ is the square of the sound speed and $\dfrac{d}{dt}=\partial_t+v\cdot \nabla$. It is now more reasonable to consider $\rho=\rho(p,S)$ as a state equation. It is clear that (12) is a symmetric system in the form (2) (for $n=3$) for the chosen unknown $U=(p,v,S)^\top$. The symmetric matrices $A_{\alpha}$ can easily be written down, and the diagonal matrix $A_0$ is positive definite under the natural physical assumptions
$$
\begin{equation}
\rho (p,S) >0\quad\text{and}\quad \frac{\partial \rho}{\partial p}(p,S)>0,
\end{equation}
\tag{13}
$$
which are the hyperbolicity conditions for the symmetric system (12). Note that the hyperbolicity conditions in the sense of the assumption $A_0>0$ can differ for different symmetric forms of the system. For example, for Godunov’s symmetrization of Euler’s equations [15] these conditions, in combination with inequalities (13), contain the convexity requirements for the state equation $\mathfrak{e}=\mathfrak{e}(V,S)$, where $V=1/\rho$ (see [18]). The Rankine–Hugoniot jump conditions for the conservation laws (9) can be written as
$$
\begin{equation}
[\mathfrak{m}]=0,\quad \mathfrak{m}[v_{N}]+|N|^2[p]=0,\quad \mathfrak{m}[v_{\tau}]=0,\quad \mathfrak{m}[ E]+[pv_{N}]=0,
\end{equation}
\tag{14}
$$
where $\mathfrak{m}=\mathfrak{m}^{\pm}\big|_{\Sigma}$ is the mass transfer flux through the discontinuity, $N=(1,-\partial_2\varphi, -\partial_3\varphi)^\top$ the normal to $\Sigma(t)$, and we have
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{m}^{\pm}=\rho^{\pm} (v_{N}^{\pm}-\partial_t\varphi),\quad v_{N}^{\pm}=v^\pm\cdot N,\quad v_{\tau}^{\pm}=(v_{\tau_1}^{\pm},v_{\tau_2}^{\pm})^\top, \\ v_{\tau_i}^{\pm}= v^{\pm}\cdot\tau_i,\quad i=1,2,\quad {\tau}_1=(\partial_2\varphi,1,0)^\top,\quad\text{and}\quad {\tau}_2=(\partial_3\varphi,0,1)^\top. \end{gathered}
\end{equation*}
\notag
$$
It can easily be verified that if $\mathfrak{m}=0$, then condition (8) holds, that is, the hypersurface $\Sigma(t)$ is characteristic. Such a discontinuity is called tangential [25] (or a vortex sheet). If $\mathfrak{m}\ne 0$ and $[\rho ]\ne 0$, then we have a shock wave (if $\mathfrak{m}\ne 0$ and $[\rho]=0$, then the solution is continuous). It follows from (14) that the boundary conditions on a vortex sheet read [25]
$$
\begin{equation}
[p]=0\quad\text{and}\quad \partial_t\varphi=v_{N}^\pm\quad \text{on } \Sigma(t).
\end{equation}
\tag{15}
$$
Now let the domain on one side of the vortex sheet, for example, $\Omega^-(t)$, be the vacuum region. Then to find $U:=U^+$ and $\varphi$ in the domain $\Omega(t):=\Omega^+(t)$ we obtain the classical free boundary problem for Euler’s equations (12) with the boundary conditions
$$
\begin{equation}
p=0\quad\text{and}\quad \partial_t\varphi=v_{N}\quad \text{on } \Sigma(t)
\end{equation}
\tag{16}
$$
and the initial data
$$
\begin{equation}
{U}\big|_{t=0}={U}_0({x})\quad\text{and}\quad \varphi\big|_{t=0}=\varphi_0(x').
\end{equation}
\tag{17}
$$
The free boundary problem (12), (16), (17) is formulated for the unbounded domain $\Omega (t)=\{x_1>\varphi(t,x')\}$ whose free boundary $\Sigma(t)$ has the form of a graph $x_1=\varphi(t,x')$. The impermeability condition in (16) can be rewritten as
$$
\begin{equation*}
\frac{d\eta}{dt}=0,\quad\text{where}\ \ \eta(t,x)=x_1-\varphi(t,x').
\end{equation*}
\notag
$$
For the free boundary problem in a bounded domain $\Omega(t)$ this condition has exactly this form with an arbitrary smooth function $\eta(t,x)$. 1.3. Free boundary problems for current-vortex sheets and magnetohydrodynamic contact discontinuitiesThe system of magnetohydrodynamics (MHD) describing the motion of a compressible inviscid perfectly conducting fluid (for example, a plasma) in a magnetic field contains eight conservation laws [26]:
$$
\begin{equation}
\begin{cases} \partial_t \rho +\operatorname{div} (\rho v )=0, \\ \partial_t(\rho v)+\operatorname{div}(\rho v \otimes v -H \otimes H )+\nabla q =0, \\ \partial_t H-\nabla\times(v\times H)=0, \\ \partial_t\biggl(\rho E+\dfrac{1}{2}|H|^2\biggr)+ \operatorname{div}\bigl(\rho Ev+pv+H\times(v\times H)\bigr)=0, \end{cases}
\end{equation}
\tag{18}
$$
where $H=(H_1,H_2,H_3)^\top$ is the magnetic field, $q=p+|H|^2/2$ is the total pressure, and the rest of notation is the same as for system (9). For system (18) the divergence constraint on the initial data must be satisfied. It is easily seen that it holds for all $t>0$ if it is true at $t=0$. In view of the thermodynamic identity (10) system (18) is closed, for example, for the vector of unknowns $U=(p,v,H,S)^\top$. As for Euler’s equations (9), the entropy conservation law (11) holds on smooth solutions of system (18). Taking into account (11) and (19), system (18) can be rewritten in the non-conservative form as
$$
\begin{equation}
\begin{cases} \dfrac{1}{\rho c^2}\,\dfrac{d p}{dt} +\operatorname{div}{v}=0,\quad \rho\, \dfrac{dv}{dt}-({H}\cdot\nabla ){H}+{\nabla}q =0 , \\ \dfrac{dH}{dt}-(H \cdot\nabla ){v}+{H}\,\operatorname{div}\,{v}=0,\quad \dfrac{dS}{dt}=0, \vphantom{\Biggr\}} \end{cases}
\end{equation}
\tag{20}
$$
where, as in (12), $c$ and $d/dt$ are the sound speed and material derivative, respectively. System (20) is a symmetric system in the form (2). The symmetric matrices $A_{\alpha}$ can easily be written down; in particular,
$$
\begin{equation*}
A_0=\operatorname{diag}\biggl(\frac{1}{\rho c^2}\,,\rho I_3,I_4\biggr),
\end{equation*}
\notag
$$
where $I_k$ is the identity matrix of order $k$. The hyperbolicity condition $A_0 >0$ holds if the state equation $\rho =\rho(p,S)$ satisfies inequalities (13). The Rankine–Hugoniot conditions for system (18) read
$$
\begin{equation}
\begin{cases} [\mathfrak{m}]=0,\quad [H_{N}]=0,\quad \mathfrak{m}[v_{N}] + |N|^2[q]=0,\quad \mathfrak{m}[{v}_{\tau}]=H_{N}[{H}_{\tau}], \\ H_{N}[{v}_{\tau}]=\mathfrak{m}\biggl[\dfrac{{H}_{\tau}}{\rho}\biggr],\quad \mathfrak{m}\biggl[E+\dfrac{1}{2}\,\dfrac{|{H}|^2}{\rho}\biggr]+ [qv_{N}-H_{N}({H} \cdot {v})]=0, \vphantom{\Biggr\}} \end{cases}
\end{equation}
\tag{21}
$$
where
$$
\begin{equation*}
\begin{gathered} \, H_{N}^{\pm}=H\cdot N,\qquad H_{N}\big|_{\Sigma}:=H_{N}^{\pm}\big|_{\Sigma}, \\ H_{\tau}=(H_{\tau_1},H_{\tau_2})^\top,\quad\text{and}\quad H_{\tau_i}= H\cdot\tau_i,\quad i=1,2, \end{gathered}
\end{equation*}
\notag
$$
and the rest of notation is the same as in (14). If $\mathfrak{m}=0$, then two types of strong discontinuities are possible [26]: tangential discontinuities (for $H_N=0$; they are also current-vortex sheets) and contact discontinuities (for $H_N\ne 0$). It is easily verified that both of these discontinuities are characteristic because condition (8) holds for them. Here we are going to consider exactly these discontinuities. Note that if $\mathfrak{m}\ne 0$ and $[\rho ] =0$, then the discontinuity is also characteristic. It is called rotational [26] (or Alfvén), and for $\mathfrak{m}\ne 0$ and $[\rho] \ne 0$ we have a shock wave. It follows from (21) that the boundary conditions on a current-vortex sheet read [26]
$$
\begin{equation}
[q]=0,\quad \partial_t\varphi=v_{N}^\pm,\quad H_{N}^+=0,\quad\text{and}\quad H_{N}^-=0\quad\text{on }\Sigma(t).
\end{equation}
\tag{22}
$$
At the same time, it can easily be shown that the boundary conditions [26]
$$
\begin{equation}
[p]=0,\quad [v]=0,\quad [H]=0,\quad\text{and}\quad \partial_t\varphi=v_{N}^+\quad \text{on }\Sigma (t)
\end{equation}
\tag{23}
$$
hold on a contact discontinuity, whereas the density and entropy can make arbitrary jumps on its surface. The corresponding free boundary problems for these discontinuities have the form (6), (7) with boundary conditions (22) or (23). 1.4. Plasma-vacuum interface problemConsidering formally the problem for a current-vortex sheet, we assume that the domain $\Omega^-(t)$ on one side of the discontinuity is the vacuum region. Then for $U:=U^+$ and $\varphi$ we obtain a free boundary problem for the quasilinear system (20) in the domain $\Omega(t):=\Omega^+(t)$ with boundary conditions
$$
\begin{equation}
q=0,\quad \partial_t\varphi=v_{N},\quad\text{and}\quad H_{N}=0\quad \text{on }\Sigma (t)
\end{equation}
\tag{24}
$$
and the initial data (17). In this case, if the MHD equations are used to describe the plasma flow, then such a problem is usually called the plasma–vacuum interface problem. However, the most complete statement of this problem includes the description of the magnetic field ${h}=(h_1,h_2,h_3)^\top$ and the electric field ${e}=(e_1,e_2,e_3)^\top$ in vacuum, which must satisfy the equations of pre-Maxwell dynamics
$$
\begin{equation}
\nabla\times {h} =0, \qquad \operatorname{div}{h} =0,
\end{equation}
\tag{25a}
$$
$$
\begin{equation}
\nabla\times {e} =-\partial_t h, \qquad \operatorname{div} {e} =0,
\end{equation}
\tag{25b}
$$
where the speed of light is set to be equal to one. These equations follow from Maxwell’s equations in vacuum by neglecting the displacement current $\partial_t e$, which is natural because it is assumed to be sufficiently small in the derivation of the non-relativistic MHD system (we also refer the reader to [3], [4], [31], [32], and [52], where problem statements in relativistic and non-relativistic MHD with displacement current in vacuum are considered). In this case the vacuum electric field in (25) is a secondary variable, which can be computed from the magnetic field $h$. That is, in the vacuum region $\Omega^-(t)$ we have the elliptic system (25a) for $h$, which is equivalently rewritten as
$$
\begin{equation}
\sum_{j=1}^{3}A_j^-\partial_j {h}=0\quad \text{in } \Omega^-(t),
\end{equation}
\tag{26}
$$
where
$$
\begin{equation*}
A_1^-:=\begin{pmatrix} 0 & 0 &\!\hphantom{-}0 \\ 0 & 0 &\!-1 \\ 0 & 1 &\!\hphantom{-}0 \\ 1 & 0 &\!\hphantom{-}0 \end{pmatrix}, \quad A_2^-:=\begin{pmatrix} \hphantom{-}0 & 0 &1 \\ \hphantom{-}0 & 0 &0 \\ -1 & 0 &0 \\ \hphantom{-}0 & 1 &0 \end{pmatrix},\quad\text{and} \quad A_3^-:=\begin{pmatrix} 0 & \!-1 &0 \\ 1 & \!\hphantom{-}0 &0 \\ 0 & \!\hphantom{-}0 &0 \\ 0 & \!\hphantom{-}0 &1 \end{pmatrix}.
\end{equation*}
\notag
$$
On the infinitely conducting free boundary $\Sigma (t)$ we set the boundary condition $h_{N}\big|_{\Sigma}=0$ for the elliptic system (26), where $h_{N} = h\cdot N$. The first condition in (24) is replaced by the jump condition
$$
\begin{equation*}
[q]:=\biggl(q-\frac{1}{2}|h|^2\biggr)\bigg|_{\Sigma}=0
\end{equation*}
\notag
$$
for the total pressure. However, since the domain $\Omega^-(t)=\{x_1<\varphi(t,x')\}$ is simply connected, the only solution of the elliptic system (26) with boundary condition $h_{N}\big|_{\Sigma}=0$ is $h=0$. In this case we arrive automatically at the free boundary problem (20), (24) in the domain $\Omega (t)=\{x_1>\varphi(t,x')\}$. The vacuum region can be a non-simply connected domain in more general situations, when the free boundary is a compact hypersurface $F(t,x)=0$. It seems that the difficulties in the proof of the local-in-time well-posedness of the problem (under suitable assumptions on the initial data) are technical. However, they have not yet been overcome formally and this question is still open. On the other hand the elliptic problem in a simply connected domain has a non-trivial solution if the plasma-vacuum system is not isolated from the outside world owing to the presence of a prescribed surface current $\boldsymbol{j}_{\rm c}$ on the outer fixed boundary (see [19]). Following [44], below we formulate the free boundary problem for this case. We assume that the space domain occupied by plasma and vacuum is given by with boundaries and let
$$
\begin{equation}
\Omega^{\pm}(t)=\{x\in \Omega\colon \pm(x_1-\varphi(t,x'))>0 \}
\end{equation}
\tag{29}
$$
be the plasma and vacuum regions, respectively. Here $\mathbb{T}^2$ denotes the 2-torus, that is, for technical simplicity we consider periodic (with respect to $x_2$ and $x_3$) boundary conditions. The boundary conditions are set not only on the free boundary but also on the fixed top and bottom boundaries $\Sigma^{\pm}$:
$$
\begin{equation}
q-\frac{1}{2}|{h}|^2 =0, \quad \partial_t \varphi =v_{N} \qquad \text{on } \Sigma(t),
\end{equation}
\tag{31a}
$$
$$
\begin{equation}
H_{N} =0, \quad {h}_{N} =0 \qquad \text{on } \Sigma(t),
\end{equation}
\tag{31b}
$$
$$
\begin{equation}
{h}\times \mathbf{e}_1 =\boldsymbol{j}_{\rm c} \qquad\text{on } \Sigma^-,
\end{equation}
\tag{31d}
$$
where $\mathbf{e}_j:=(\delta_{1j},\delta_{2j},\delta_{3j})^\top$, $j=1,2,3$, and $\delta_{ij}$ is the Kronecker symbol. The boundary conditions (31c) on the fixed outer boundary of plasma are standard perfectly conducting wall boundary conditions. The prescribed surface current $\boldsymbol{j}_{\rm c}$ on the fixed outer boundary of the vacuum forces oscillations onto the plasma–vacuum system. For example, in laboratory plasmas this external excitation can be caused by a system of coils (see [19] for a detailed discussion of the boundary condition (31d)). The plasma–vacuum interface problem is thus the problem for the hyperbolic system (20) for $U=(p,v,H,S)^\top$ in $\Omega^+(t)$ and the elliptic system (26) for $h$ in $\Omega^-(t)$ with boundary conditions (31) and initial data in the form (17). 2. Equivalent fixed boundary problem and its linearizationFor reasons of presentation, all the above problems will be considered in the domain (27) equal to the union of the movable domains (29) from both sides of the free boundary (30). In particular, for current-vortex sheets and MHD contact discontinuities we set the standard perfectly conducting wall boundary conditions on the fixed outer boundaries (28) of the movable domains $\Omega^\pm (t)$:
$$
\begin{equation}
v_1^\pm=0,\quad H_1^\pm =0 \quad \text{on } \Sigma^\pm.
\end{equation}
\tag{32}
$$
We require the fulfilment of these conditions on $\Sigma^+$ also for problem (20), (24), (17) (for $U:=U^+$). Likewise, for the free boundary problem (12), (16), (17), on the fixed boundary $\Sigma^+$ we set the impermeability condition Again, for reasons of presentation we suppose that $U^-\equiv 0$ in $\Omega^-(t)$ (and also $U^+:=U$) for those problems which are formulated only in the domain $\Omega^+(t)$, and $U^-:=h$ for problem (20), (26), (31), (17). To reduce the problem with free boundary $\Sigma(t)$ to the one with fixed boundary we perform the change of variables $U^\pm_{\sharp}(t,x):=U^\pm(t,\Phi(t,x),x')$, where
$$
\begin{equation}
\Phi(t,x):=x_1+\Psi(t,x)=x_1+\chi( x_1)\varphi(t,x'),
\end{equation}
\tag{34}
$$
and $\chi\in C^{\infty}_0(-1,1)$ is the cut-off function that satisfies $\|\chi'\|_{L^{\infty}(\mathbb{R})} < 2$ and is equal to 1 in a small neighborhood of the origin (in [6] and [44] a different choice of the function $\Phi(t,x)$ was used, which resulted in a not too important increase by 1/2 of the regularity of the function $\varphi$). The non-degeneracy of the change of variables, $\partial_1\Phi >0$, is guaranteed for $\|\varphi\|_{L^{\infty}([0,T]\times\mathbb{T}^2)}< 1/2$. This inequality holds for a short time if, without loss of generality, we consider the initial data satisfying
$$
\begin{equation}
\|\varphi _0\|_{L^{\infty}(\mathbb{T}^2)} < \frac{1}{4}\,.
\end{equation}
\tag{35}
$$
The change of variables (34) reduces our problem to that with fixed boundary:
$$
\begin{equation}
\mathbb{L}_+(U^+,\Phi) :=L_+(U^+,\Phi)U^+ =0 \qquad \text{in}\ \ \Omega^+:=(0,1)\times \mathbb{T}^2,
\end{equation}
\tag{36a}
$$
$$
\begin{equation}
\mathbb{L}_-(U^-,\Phi) :=L_-(U^-,\Phi)U^- =0 \qquad \text{in}\ \ \Omega^-:=(-1,0)\times \mathbb{T}^2,
\end{equation}
\tag{36b}
$$
$$
\begin{equation}
\mathbb{B}(U^+,U^-,\varphi)=0 \qquad \text{on}\ \ \Sigma^l\times \Sigma^+\times \Sigma^-,
\end{equation}
\tag{36c}
$$
$$
\begin{equation}
U^\pm\big|_{t=0}=U_0^\pm,\qquad \varphi\big|_{t=0}=\varphi_0,
\end{equation}
\tag{36d}
$$
where we have dropped the subscript ‘$\sharp$’ for convenience. Here $\Sigma:=\{0\}\times \mathbb{T}^2$,
$$
\begin{equation}
L_\pm(U,\Phi) :=A_0(U^\pm)\partial_t+\widetilde{A}_1(U^\pm,\Phi)\partial_1+ A_2(U^\pm)\partial_2+A_3(U^\pm)\partial_3,
\end{equation}
\tag{37}
$$
$$
\begin{equation}
\widetilde{A}_1(U^\pm,\Phi) := \frac{1}{\partial_1\Phi}\big(A_1(U^\pm)-\partial_t\Phi A_0(U^\pm)- \partial_2\Phi A_2(U^\pm)-\partial_3\Phi A_3(U^\pm)\big),
\end{equation}
\tag{38}
$$
and $l$ is the number of boundary conditions on the ‘straightened’ free boundary $\Sigma$. For example, for problem (20), (26), (31), (17) we have $l=3$ and
$$
\begin{equation}
\mathbb{B}(U^+,U^-,\varphi)=\mathbb{B}(U,h,\varphi):= \begin{pmatrix} \partial_t \varphi -v_{N} \\ q-\dfrac{1}{2}|{h}|^2 \\ {h}_{N} \\ v_1 \\ {h}\times \mathbf{e}_1 \end{pmatrix}
\end{equation}
\tag{39}
$$
(the equalities $H_{N}\big|_{\Sigma}=0$ and $H_1\big|_{\Sigma^-}=0$ are not included in the boundary conditions (36c) because, as shown in [49], they are restrictions on the initial data; the boundary conditions $H_{N}^\pm\big|_{\Sigma}=0$ in (22) have the same property). Moreover, in (36c) we employ the notation $\Sigma^l\times \Sigma^+\times \Sigma^-$ to denote that the first $l$ boundary conditions in (36c) are set on $\Sigma$, whereas the last two boundary conditions are set on $\Sigma^+$ and $\Sigma^-$, respectively. We note that if (36b) is used to denote the result of the change of variables (34) in the elliptic system (26), then in (37) and (38) we have $A_0(U^-)=0$, and the $A_j(U^-):=A_j$ ($j=1,2,3$) are the constant matrices from (26), in particular,
$$
\begin{equation*}
\widetilde{A}_1(U^-,\Phi) := \widetilde{A}_1(\Phi)= \frac{A_1-\partial_2\Phi A_2-\partial_3\Phi A_3}{\partial_1\Phi}\,.
\end{equation*}
\notag
$$
Finally, we note that for problem (20), (26), (31), (17) we must write $(0,0,0,0,\boldsymbol{j}_{\rm c})^\top$ instead of the zero vector on the right-hand side of conditions (36c), and in (36d) we do not need to prescribe the initial data $U^-\big|_{t=0}=U_0^-$ for the ‘elliptic’ unknown $U^-=h$. If we are going to prove the existence and uniqueness of the solution $(U^+,U^-,\varphi )$ to problem (36) using a fixed-point argument, then we must consider the corresponding linearized problem. As is known, for a successful application of this argument we have to derive a priori estimates for the linearized problem without loss of derivatives from the initial data and the source terms. At the same time, if a loss of derivatives takes place in a priori estimates, then one can sometimes apply the Nash–Moser method (see § 6). This method also involves linearization. That is, anyway we should perform the linearization of problem (36). Let
$$
\begin{equation*}
\Omega_T^{\pm}:=(-\infty,T)\times \Omega^{\pm}\quad\text{and}\quad \Sigma_T:=(-\infty,T)\times \Sigma.
\end{equation*}
\notag
$$
The triple will be called the basic state, where $\mathring{U}^+=\mathring{U}^+(t,x)$ and $\mathring{U}^-=\mathring{U}^-(t,x)$ are fixed sufficiently smooth vector functions defined in $\Omega_{T}^+$ and $\Omega_{T}^-$, respectively, and $\mathring{\varphi}=\mathring{\varphi}(t,x')$ is a fixed sufficiently smooth function on $\Sigma_T$. The basic state (40) can be, in particular, an arbitrary solution to problem (36) (whose existence is yet to be proved). The functions $\mathring{U}^\pm$ and $\mathring{\varphi}$ should be bounded in the norms of some Sobolev spaces. We do not write here the corresponding inequalities because they depend on the concrete problem. Moreover, the basic state should satisfy certain requirements, in particular, the hyperbolicity condition $A_0(\mathring{U}^\pm )>0$. These requirements usually include part of the boundary conditions (36c) so that the boundary matrices $\widetilde{A}_1(U^\pm,\Phi)\big|_{\Sigma}$ calculated on the basic state have the same structure as these matrices calculated on the solution of problem (36). For example, for problem (20), (26), (31), (17) the basic state should satisfy all boundary conditions in (31) except the first condition in (31a). The operators for problem (36) linearized about the basic state (40) read
$$
\begin{equation*}
\begin{aligned} \, \mathbb{L}_{\pm}'\bigl(\mathring{U}^{\pm},\mathring{\Phi}\bigr)(V^{\pm},\Psi) &:=\biggl.\frac{\mathrm{d}}{\mathrm{d}\theta}\mathbb{L}_{\pm} \bigl(\mathring{U}^{\pm}+{\theta}V^{\pm}, \mathring{\Phi}+{\theta}\Psi\bigr)\bigg|_{\theta=0}, \\ \mathbb{B}'\bigl(\mathring{U}^+,\mathring{U}^-,\mathring{\varphi}\bigr) (V^+,V^-,\psi)&:=\frac{\mathrm{d}}{\mathrm{d}\theta} \mathbb{B}(\mathring{U}^{+}+{\theta}V^+,\mathring{U}^{-}+{\theta}V^-, \mathring{\varphi}+{\theta}\psi)\bigg|_{\theta=0}, \end{aligned}
\end{equation*}
\notag
$$
where $\Psi(t,x):=\chi(x_1)\psi(t,x')$ and $\mathring{\Phi}(t,x):=x_1+\chi(x_1)\mathring{\varphi}(t,x')$. We can easily compute their explicit form. In particular,
$$
\begin{equation}
\mathbb{L}_{\pm}'\bigl(\mathring{U}^{\pm},\mathring{\Phi}\bigr)(V^{\pm},\Psi)= \mathbb{L}'_{\mathrm{e}\pm}\bigl(\mathring{U}^{\pm},\mathring{\Phi}\bigr)V^{\pm}- \frac{L_{\pm}(\mathring{U}^{\pm}, \mathring{\Phi})\Psi} {\partial_1 \mathring{\Phi}}\,\partial_1\mathring{U}^{\pm},
\end{equation}
\tag{41}
$$
where
$$
\begin{equation*}
\mathbb{L}'_{\mathrm{e}\pm}(U,\Phi)Y:=L_{\pm}(U,\Phi)Y+\mathcal{C}_{\pm}( U,\Phi)Y
\end{equation*}
\notag
$$
for some $Y=(y_1,\dots,y_m)^\top$ and the matrices $\mathcal{C}_{\pm}(U,\Phi)$ are defined as follows:
$$
\begin{equation*}
\mathcal{C}_{\pm}({U},{\Phi})Y:=\sum_{k=1}^{N}y_k \biggl(\frac{\partial A^{\pm}_0}{\partial {u_k}}({U})\,\partial_t{U}+ \frac{\partial\widetilde{A}^{\pm}_1}{\partial{u_k}}(U,\Phi)\,\partial_1{U}+ \sum_{i=2,3}\frac{\partial A^{\pm}_i}{\partial{u_k}}(U)\,\partial_i{U}\biggr)
\end{equation*}
\notag
$$
(in fact, the concrete form of these matrices is not of interest because lower-order terms play no role in the well-posedness of the problem). The differential operator in (41) has the first order in $\Phi$, which presents a potential difficulty for deriving a priori estimates for linearized problems. To overcome this difficulty we pass to Alinhac’s good unknown [1]
$$
\begin{equation}
\begin{aligned} \, \dot{V}^{\pm}:=V^{\pm}- \frac{\Psi}{\partial_1 \mathring{\Phi}}\,\partial_1\mathring{U}^{\pm}. \end{aligned}
\end{equation}
\tag{42}
$$
In terms of (42) the operator in (41) becomes
$$
\begin{equation}
\mathbb{L}_{\pm}'\big(\mathring{U}^{\pm},\mathring{\Phi}\big)(V^{\pm},\Psi)= \mathbb{L}'_{\mathrm{e}\pm}\big(\mathring{U}^{\pm},\mathring{\Phi}\big)\dot{V}^{\pm}+ \frac{\Psi}{\partial_1\mathring{\Phi}}\,\partial_1\mathbb{L}_{\pm} (\mathring{U}^{\pm},\mathring{\Phi}).
\end{equation}
\tag{43}
$$
Regarding linearized boundary conditions, for example, for the plasma–vacuum interface problem, the linearization of the boundary operator in (39) and the passage to the good unknown yield
$$
\begin{equation}
\begin{aligned} \, \nonumber &\mathbb{B}'\bigl(\mathring{U}^+,\mathring{U}^-,\mathring{\varphi}\bigr) (V^+,V^-,\psi) \\ &\qquad=\mathbb{B}'_{\mathrm{e}}(\mathring{U}^+, \mathring{U}^-, \mathring{\varphi}) (\dot{V}^+,\dot{V}^-,\psi)=\begin{pmatrix} (\partial_t + \mathring{v}' \cdot \mathrm{D}_{x'}+\mathring{b}_1) \psi- \dot{v}\cdot\mathring{N} \\ \dot{p}+\mathring{H}\cdot\dot{H}-\mathring{h}\cdot\dot{h}+\mathring{b}_2\psi \\ \dot{{h}}\cdot \mathring{N}-\mathrm{D}_{x'}\cdot(\mathring{h}' \psi) \\ \dot{v}_1 \\ \dot{h}\times \mathbf{e}_1 \end{pmatrix}, \end{aligned}
\end{equation}
\tag{44}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathring{U}^+:=\mathring{U}=(\mathring{p},\mathring{v},\mathring{H}, \mathring{S})^\top,\quad \mathring{U}^-:=\mathring{h}, \\ \dot{V}^+:=(\dot{p},\dot{v},\dot{H},\dot{S})^\top,\quad \dot{V}^-:=\dot{h},\quad \mathring{v}'=(\mathring{v}_2,\mathring{v}_3)^\top,\quad \mathring{b}_1:=-\partial_1 \mathring{v}\cdot \mathring{N}, \\ \mathring{N}=(1,-\partial_2\mathring{\varphi}, -\partial_3\mathring{\varphi})^\top,\quad \mathring{b}_2:=\partial_1\mathring{p}+\mathring{H}\cdot\partial_1\mathring{H} -\mathring{{h}}\cdot \partial_1\mathring{{h}},\quad\text{and}\quad \mathrm{D}_{x'}:=(\partial_2,\partial_3)^\top. \end{gathered}
\end{equation*}
\notag
$$
The linearization of problem (36) results in the following linear initial-boundary value problem with variable coefficients:
$$
\begin{equation}
\mathbb{L}'_{\mathrm{e}\pm}(\mathring{U}^\pm, \mathring{\Phi}) \dot{V}^\pm =f^\pm \quad \text{in}\ \ \Omega^\pm_T,
\end{equation}
\tag{45a}
$$
$$
\begin{equation}
\mathbb{B}'_{\mathrm{e}}(\mathring{U}^+, \mathring{U}^-, \mathring{\varphi}) (\dot{V}^+,\dot{V}^-,\psi)=g \quad\text{on}\ \ \Sigma_T^l\times \Sigma_T^+\times \Sigma_{T}^-,
\end{equation}
\tag{45b}
$$
where $\Sigma_T^{\pm}:=(-\infty,T)\times \Sigma^{\pm}$, the source terms $f^{\pm}(t,x)$ and $g(t,x')$ are supposed to be prescribed, and the notation $\Sigma_T^l\times \Sigma_T^+\times \Sigma_{T}^-$ is analogous to that in (36c). Moreover, we consider the zero initial data because the case of non-zero initial data can be deferred to nonlinear analysis, namely, to the construction of a so-called approximate solution (see, for example, [50]). The source terms $f^{\pm}(t,x)$ and $g(t,x')$ are assumed to vanish in the past. Note also that on the left-hand sides of systems (45a) we keep only the so-called effective operators, whereas the last terms in (43) (zero-order terms in $\psi$) are dropped. In the subsequent nonlinear analysis these terms are considered as additional errors of the Nash–Moser iterations.
3. Non-ellipticity of the symbol of the free boundary and the Rayleigh–Taylor sign conditionConsider the free boundary problem (12), (16), (17) (completed by the rigid wall boundary condition (33)). First let the basic state be a constant vector $(\mathring{U},0)$ satisfying the second condition in (16), that is, $\mathring{v}_1=0$. Then for this problem the linear boundary conditions (45b) read
$$
\begin{equation}
\dot{v}_1= (\partial_t + \mathring{v}' \cdot \mathrm{D}_{x'}) \psi,\qquad \dot{p}=0 \quad\text{on } \Sigma_T,\qquad \dot{v}_1=0 \quad\text{on } \Sigma_T^+,
\end{equation}
\tag{47}
$$
and the boundary matrix $\widetilde{A}_1(\mathring{U},\mathring{\Phi})$ has the form
$$
\begin{equation}
\widetilde{A}_1(\mathring{U},\mathring{\Phi})=\mathring{B}=\begin{pmatrix} 0 & 1 &0 &0 & 0 \\ 1 & 0 &0 &0 & 0 \\ 0 & 0 &0 &0 & 0 \\ 0 & 0 &0 &0 & 0 \\ 0 & 0 &0 &0 & 0 \end{pmatrix}.
\end{equation}
\tag{48}
$$
For simplicity we consider here the homogeneous boundary conditions, that is, in (45b) the source term satisfies $g\equiv 0$. By the standard arguments of the energy method (see, for instance, [2], [14], [17]), for the linearized problem for the symmetric hyperbolic acoustics system1 with source term $f:=f^+$ and boundary conditions (47) we easily deduce the a priori estimate
$$
\begin{equation}
\|\dot{V}\|_{L^2(\Omega^+_T)}\lesssim \|f\|_{L^2(\Omega^+_T)},
\end{equation}
\tag{49}
$$
since the boundary conditions (47) are dissipative [2], [14], [17] because
$$
\begin{equation}
\big({A}_1(\mathring{U})\dot{V}\cdot\dot{V}\big)=2\dot{p}\dot{v_1}=0 \quad \text{on } \Sigma_T\text{ and on } \Sigma_T^+.
\end{equation}
\tag{50}
$$
Here and below the notation $A\lesssim_{a_1,\dots,a_m} B$ means that $A \leqslant C(a_1,\dots,a_m)B$ for the fixed parameters $a_1,\dots,a_m$, where $C(\,\cdot\,)$ is a positive constant depending on the values in curly brackets. We just write $\lesssim$ (as in the a priori estimate (49) for the constant coefficient problem) if the dependence of this constant on parameters is not important. Note that the standard shift of solution [38] to the vector function satisfying conditions (47) for $g\ne 0$ enables one to derive from estimate (49) the a priori estimate
$$
\begin{equation}
\|\dot{V}\|_{L^2(\Omega^+_T)}\lesssim \|f\|_{L^2(\Omega^+_T)} + \|g\|_{H^{1/2}(\Sigma_T)\times H^{1/2}(\Sigma_T^+)}
\end{equation}
\tag{51}
$$
for the linear problem with inhomogeneous boundary conditions. As we can see, since the boundary conditions are merely dissipative but not strictly dissipative [2], [17], that is, the quadratic form is not positive or negative definite (depending on the direction of the outer normal to the boundary), in the a priori estimate (51) we have loss of derivatives from the source term $g$. The presence of the a priori estimate (49) shows that an Hadamard-type ill- posedness example cannot be constructed for our problem. In other words, the boundary conditions (47) satisfy the Kreiss–Lopatinski condition [24] for any coefficients of the problem, that is, for any vector $\mathring{U}$. In many cases (but not in ours) the Kreiss–Lopatinski condition enables one to obtain an a priori estimate also for the linear problem with variable coefficients without additional restrictions on these coefficients, as well as to derive an estimate for the function $\psi$ describing the perturbation of the free boundary. For dissipative boundary conditions this is achieved by prolonging the problem up to the first-order tangential derivatives, passing from the boundary integral to a volume integral in the energy inequality and integrating by parts (see § 5). The peculiarity of the boundary conditions (16) is that for the application of such arguments we have to make an additional assumption on the coefficients of the linearized problem. The point is that the boundary conditions (16) are such that the symbol of free boundary is not elliptic. The concept of the ellipticity of the boundary symbol (see, for example, [7]) has come from paradifferential calculus. In fact, the non-ellipticity of the boundary symbol means that the boundary conditions (16) are not uniquely solved for the gradient $(\partial_t\varphi,\partial_2\varphi,\partial_3\varphi)^\top$, for whose components we have only the second equation in (16). For the problem linearized about the basic state (46) satisfying (33) and the second condition in (16) the homogeneous boundary conditions read
$$
\begin{equation}
\begin{alignedat}{3} \dot{v}\cdot\mathring{N}&=(\partial_t+\mathring{v}' \cdot \mathrm{D}_{x'}+ \mathring{b}_1) \psi,&\quad \dot{p}+\mathring{b}_2 \psi&=0 &&\quad\text{on } \Sigma_T, \\ &&\dot{v}_1&=0&&\quad\text{on } \Sigma_T^+, \end{alignedat}
\end{equation}
\tag{52}
$$
where $\mathring{b}_2:=\partial_1\mathring{p}$. In this case the boundary matrix
$$
\begin{equation*}
B(\mathring{U},\mathring{\Phi}):=J(\mathring{\Phi})^\top \widetilde{A}_1(\mathring{U},\mathring{\Phi})J(\mathring{\Phi})
\end{equation*}
\notag
$$
of the linear symmetric hyperbolic system for the vector of unknowns
$$
\begin{equation}
W=(\dot{p},\dot{v}\cdot\mathring{N},\dot{v}_2,\dot{v}_3,\dot{S})^\top= \bigl(J(\mathring{\Phi})\bigr)^{-1}\dot{V}
\end{equation}
\tag{53}
$$
computed on the boundaries $\Sigma_T$ and $\Sigma_T^+$ coincides with the constant matrix in (48), where the matrix $J(\mathring{\Phi})$ can easily be written down. In view of the boundary conditions (52) the quadratic form with matrix $B(\mathring{U},\mathring{\Phi})$ is zero on the boundary $\Sigma_T^+$, and on the ‘straightened’ free boundary we have the relations
$$
\begin{equation*}
\begin{aligned} \, Q&=(\widetilde{A}_1(\mathring{U},\mathring{\Phi})\dot{V}\cdot \dot{V})\big|_{\Sigma_T}= (B(\mathring{U},\mathring{\Phi})W\cdot W)\big|_{\Sigma_T} \\ &=2\dot{p}(\dot{v}\cdot\mathring{N})\big|_{\Sigma_T}= -2\mathring{b}_2\psi(\partial_t+\mathring{v}' \cdot \mathrm{D}_{x'}+ \mathring{b}_1)\psi\big|_{\Sigma_T}. \end{aligned}
\end{equation*}
\notag
$$
In the general case the quadratic form $Q$ is sign indefinite and we are not able to close the a priori estimate in $L^2$. However, since
$$
\begin{equation}
\begin{aligned} \, \nonumber -Q\big|_{\Sigma_T}&= \partial_t\bigl(\mathring{b}_2\big|_{\Sigma_T}\psi^2\bigr)+ \partial_2\bigl((\widehat{v}_2\mathring{b}_2)\big|_{\Sigma_T}\psi^2\bigr)+ \partial_3\bigl((\widehat{v}_3\mathring{b}_2)\big|_{\Sigma_T}\psi^2\bigr) \\ &\qquad-\bigl(\partial_t\mathring{b}_2+ \partial_2(\widehat{v}_2\mathring{b}_2)+ \partial_3(\widehat{v}_3\mathring{b}_2)+ 2\mathring{b}_1\mathring{b}_2\bigr)\big|_{\Sigma_T}\psi^2, \end{aligned}
\end{equation}
\tag{54}
$$
it follows from the energy inequality
$$
\begin{equation*}
\int_{\Omega^+}A_0\dot{V}\cdot\dot{V}-\int_{\Sigma_t}Q\lesssim_{K} \|f\|^2_{L^2(\Omega_T^+)}+\|\dot{V}\|^2_{L^2(\Omega_t^+)}
\end{equation*}
\notag
$$
that
$$
\begin{equation}
\int_{\Omega^+}A_0\dot{V}\cdot\dot{V}+ \int_{\Sigma}\mathring{b}_2\psi^2 \lesssim_{K}\| f\|^2_{L^2(\Omega_T^+)} +\| \dot{V}\|^2_{L^2(\Omega_t^+)}+ \| \psi\|^2_{L^2(\Sigma_t)},
\end{equation}
\tag{55}
$$
where the positive constant $K$ bounds the $W^2_{\infty}$-norms of the coefficients of the problem, that is, the components of the basic state $(\mathring{U},\mathring{\varphi})$. Then the physical condition $\mathring{b}_2\big|_{\Sigma}>0$ arises naturally. This is the condition for the unperturbed pressure $\mathring{p}$, where $\varepsilon$ is a fixed constant. Condition (56) is nothing else than the famous Rayleigh–Taylor sign condition $(\partial p /\partial N )|_{\Sigma (t)}\leqslant -\varepsilon < 0$ written in the ‘straightened variables’. If $\mathring{U}$ satisfies (56) and the hyperbolicity conditions (13) (that is, the inequality $A_0(\mathring{U})\big|_{\Omega_T^+}>0$), then applying Gronwall’s lemma, from (55) we deduce the a priori estimate
$$
\begin{equation}
\|\dot{V}\|_{L^2(\Omega^+_T)}+\|\psi\|_{L^2(\Sigma_T)}\lesssim_K \|f\|_{L^2(\Omega^+_T)}.
\end{equation}
\tag{57}
$$
Note that by assuming that the hyperbolicity conditions (13) hold up to the free boundary we exclude the case of compressible gas, when $p\big|_{\Sigma (t)}=0$ implies that $\rho\big|_{\Sigma (t)}=0$. That is, it is supposed that the compressible medium is a fluid but not a gas. The local well-posedness of the free boundary problem for the case of a compressible gas ($\rho\big|_{\Sigma (t)}=0$) was proved in [10], [22], and [29], provided that the physical vacuum condition $(\partial c^2/\partial N)\big|_{\Sigma(t)}\leqslant -\varepsilon <0$ holds. The existence of solutions to the linearized problem was proved in [50] by the duality argument. In particular, it includes deriving an $L^2$ a priori estimate for the dual problem which is analogous to estimate (51). With the help of estimate (51) we also prove the uniqueness of the solution to the original nonlinear problem. The existence of solutions of the free boundary problem (12), (16), (17) or, more precisely, of its equivalent reformulation (36a), (36c), (36d) was proved in [50] by Nash–Moser iterations. Below we state the theorem from [50] on the local existence and uniqueness of smooth solutions to this problem in a form adapted to the additional boundary conditions considered here, namely, the rigid wall boundary condition (33) and the periodic (in tangential directions) boundary conditions. Theorem 3.1. Let $m\in\mathbb{N}$ and $m\geqslant 6$. Assume that the initial data (36d) for $(U_0,\varphi_0)\in H^{m+7}(\Omega^+)\times H^{m+7}(\Sigma)$ satisfy the hyperbolicity conditions (13), the Rayleigh–Taylor sign condition (56), condition (35), and the corresponding compatibility conditions up to order $m+7$ (see [50]). Then there exists sufficiently small time $T > 0$ such that problem (36a), (36c), (36d) has a unique solution $(U,\varphi )\in H^m([0,T]\times\Omega^+)\times H^m ([0,T]\times\Sigma)$. The result analogous to that of Theorem 3.1 was obtained in [28] for the isentropic Euler equations by passing to the Lagrangian coordinates under the assumption that the initial flow domain is diffeomorphic to a ball. As in [50], the existence of solutions to the nonlinear problem was proved in [28] by the Nash–Moser method, which implies a loss of derivatives from the initial data. One has to apply the Nash–Moser method because of loss of derivatives in the a priori estimates for the linearized problem. This is a consequence of the violation of the uniform Kreiss–Lopatinski condition [24] for this problem. A priori estimates for the linearized problem are necessary for the proof of the convergence of the Nash–Moser iterations. Sometimes it is possible to prove the existence of solutions to the nonlinear problem without linearizing it, which enables one to avoid the loss of derivatives from the initial data of this problem. For instance, recently the results of [28] and [50] were eignificantly improved in [30] (true, in the isentropic case so far) in the following sense: local well-posedness was proved with no loss of derivatives from the initial data. This was done by passing to Lagrangian coordinates and using the method of tangential smoothing [9]. This method involves considering an approximate (regularized) problem [30] and deducing for it a priori estimates uniform with respect to the parameter of regularization. It is worth noting that the Rayleigh–Taylor sign condition is not only sufficient but also necessary for the local well-posedness of the problem, since, as shown in [20], the original nonlinear free boundary problem is ill-posed if this condition fails. Finally, we note that, as in [28], we neglect gravitation in problem (36a), (36c), (36d) which may seem odd when Rayleigh–Taylor instability can occur. However, taking the effect of gravity into account in Euler’s equations is connected with lower-order (non-differential) terms, which play a crucial role in the stability of solutions but play no role in the local well-posedness of the problem and can thus be neglected. Taking the effect of gravity into account in [30] and [50] was only caused by the fact that the flow domain is unbounded (owing to the corresponding ‘gravitational’ lower-order terms in the second equation in (12), there is no contradiction between inequality (56) and the fact that the velocity $v$ in $H^m$ vanishes at infinity.) Another example of a problem with non-elliptic symbol of free boundary is problem (6), (7), (23) for the MHD contact discontinuity. We do not discuss here the derivation of an a priori estimate for the corresponding linearized problem and only note that the Rayleigh–Taylor sign condition
$$
\begin{equation}
[\partial_1{p}]\big|_{\Sigma}\geqslant \varepsilon >0
\end{equation}
\tag{58}
$$
written for the jump of the normal derivative of the unperturbed pressure $\mathring{p}$ arises naturally in the derivation of this estimate in [34]. In this case the arguments towards the proof of the a priori estimate in [34] are not as simple as in (54) and (55). This is connected with the presence of additional terms in the boundary integral, which depend on the magnetic field. In [34] the basic a priori estimate for the linearized problem could only be closed in $H^1$ and for the 2D problem ($x=(x_1,x_2)$). Namely, for the 2D version of problem (45) written for the MHD contact discontinuity in the planar case $v=(v_1,v_2)$, $H=(H_1,H_2)$ the a priori estimate
$$
\begin{equation}
\sum_{\pm}\|\dot{V}^\pm \|_{H^{1}(\Omega_T^\pm)}+\|\psi\|_{H^1(\Sigma_T)} \lesssim_{K,\varepsilon,T}\sum_{\pm}\|f^\pm \|_{H^{1}(\Omega_T^\pm)}+ \|g\|_{H^{3/2}(\Sigma_T)}
\end{equation}
\tag{59}
$$
was obtained, where
$$
\begin{equation*}
\|\mathring{U}^+\|_{W^2_{\infty}(\Omega_T^+)}+ \|\mathring{U}^-\|_{W^2_{\infty}(\Omega_T^-)} +\|\mathring{\varphi}\|_{W^3_{\infty}(\Sigma_T)} \leqslant K<\infty.
\end{equation*}
\notag
$$
In estimate (59) we have a loss of derivatives from $g$. This is why the local existence of solutions to the original nonlinear problem (for the 2D case) was proved in [35] by the Nash–Moser method, provided that the Rayleigh–Taylor sign condition (58) holds on the initial contact discontinuity (condition (58) is the Rayleigh–Taylor sign condition $[\partial p/\partial N]\big|_{\Sigma(t)}\leqslant -\varepsilon <0$ written in ‘straightened’ variables). We note, however, that recently the local well-posedness of the free boundary problem for the MHD contact discontinuity was proved in [60] for the general 3D case and without assuming the Rayleigh–Taylor sign condition at the first moment of time. This result was obtained by passing to Lagrangian coordinates and a suitable viscous regularization. It can seem odd that local well-posedness was proved in [60] without the assumption that the Rayleigh–Taylor sign condition holds, whereas in [34] this physical condition appears naturally in the analysis of the linearized problem. At the same time, we should keep in mind that the existence of solutions to the nonlinear problem was proved in [60] without considering its linearization. But, if our aim is the study of the stability of solutions to the free boundary problem with respect to small perturbations, then the proof of the well-posedness of the linearized problem must precede this study. The result in [34], where the Rayleigh–Taylor sign condition is required for the unperturbed contact discontinuity, is thus in agreement with linear and nonlinear Rayleigh–Taylor instability, which can occur for MHD contact discontinuities and is detected in astrophysical calculations (see, for instance, [11]). It is connected with the Rayleigh–Taylor instability fingers observed along the boundaries of various nebulae. 4. Loss of control of normal derivatives and anisotropic weighted Sobolev spacesAs already noted above, for the linearized free boundary problem for Euler’s equations the boundary matrix computed on the straightened free boundary coincides with (48). Since $\det B(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T}=0$, the boundary $\Sigma_T$ is characteristic. This implies that $\partial_1W$, which is the derivative in the direction normal to the boundary (we call it just the normal derivative below), cannot be expressed from the linearized Euler equations in terms of the tangential derivatives $\partial_tW$, $\partial_2W$, and $\partial_3W$. By differentiating the linear problem with respect to $t$, $x_2$, and $x_3$, for the tangential derivatives we obtain an augmented system analogous to the original system for the unknown $W$. For this augmented problem we can apply the same arguments towards the proof of the $L^2$ a priori estimate as for the original problem. However, we are not able to close the estimate for the unknown $Y=(W,\partial_tW,\partial_2W,\partial_3W)$ because of the presence of terms like $\partial_\ell B(\mathring{U},\mathring{\Phi})\,\partial_1W$, where $\ell=0,2,3$, and $\partial_0:=\partial_t$, in the augmented problem. Thus, in view of the loss of control of normal derivatives, in the general case we cannot derive a priori $H^m$-estimates from the $L^2$-estimate. However, despite the fact that the boundary is characteristic, we can do this for the linearized free boundary problem for Euler’s equations. First, thanks to the structure of the boundary matrix we can include the $H^m$-norm of the non-characteristic unknown $(\dot{p},\dot{v}\cdot\mathring{N})$ in the energy inequality for $Y$ (see (53)). Second, the missing normal derivatives of the characteristic unknowns $\dot{v}_2$, $\dot{v}_3$, and $\dot{S}$ are estimated by using the last equation for $\dot{S}$ and the equations for the linearized vorticity $\xi =\nabla \tilde{v}$, where $\tilde{v}=(\dot{v}_1,\dot{v}_2\cdot \mathring{\tau}_2,\dot{v}_3\cdot\mathring{\tau}_3)$. It should be noted here that the differential operators in the equations for $\dot{S}$ and $\xi$ (without taking lower-order terms into account) read
$$
\begin{equation*}
\partial_t + \frac{1}{\partial_1\mathring{\Phi}} \bigl((\mathring{v}\cdot\mathring{N}-\partial_t\mathring{\Phi})\partial_1+ \mathring{v}_2\partial_2+\mathring{v}_3\partial_3\bigr).
\end{equation*}
\notag
$$
But these equations do not need boundary conditions on $\Sigma_T$ and $\Sigma_T^+$ because of the assumptions on the basic state
$$
\begin{equation*}
(\mathring{v}\cdot\mathring{N}-\partial_t\mathring{\Phi})\big|_{\Sigma_T}=0 \quad \text{and} \quad \big(\mathring{v}\cdot\mathring{N}- \partial_t\mathring{\Phi}\big)\big|_{\Sigma_T^+}= \mathring{v_1}\big|_{\Sigma_T^+}=0.
\end{equation*}
\notag
$$
Using the compensation of the loss of control on normal derivatives, the following theorem on the a priori estimate of solutions to the linear problem in usual Sobolev spaces was proved. Theorem 4.1. Let $T>0$, $m\in \mathbb{N}$, and $m\geqslant 3$. Assume that the basic state $(\mathring{U} ,\mathring{\varphi})\in H^{m+3}(\Omega_T^+ )\times H^{m+3}(\Sigma_T)$ satisfies all assumptions above (in particular, the Rayleigh–Taylor sign condition (56)) and A priori estimates like (61) are called tame estimates. They play a crucial role in the proof of the convergence of the Nash–Moser iterations with the help of which one obtains the existence of solutions to the nonlinear problem (see Theorem 3.1). The problem for the MHD contact discontinuity is another example of the problem with characteristic free boundary for which one can compensate the loss of control of normal derivatives. As we can see, the basic a priori estimate (59) for solutions of the linearized problem was derived in [34] in the norms of usual Sobolev spaces. In [35] a suitable tame estimate similar to (61) (to be precise, tame estimates in $H^m$ for $m\geqslant 3$) was deduced. However, in the general case of problem with characteristic boundary it is impossible to compensate the loss of control of normal derivatives. For example, it was shown in [36] that while the linearized problem for the MHD system (18) in a bounded domain with fixed boundary and the standard perfectly conducting wall boundary conditions is well posed in $H^1$, it is not well posed in $H^m$ for $m\geqslant 2$. At the same time, this problem is an example of a problem for a linear symmetric hyperbolic system with characteristic boundary whose well-posedness was proved in [40] in the anisotropic weighted Sobolev spaces $H^m_*$ for $m\geqslant 1$. The spaces $H^m_*$ were first introduced in [5] (see also [33] and [41], and the references there). We define them here for the domains $I\times\Omega^{\pm}$, where $I\subset \mathbb{R}$ (in our case $I=[0,T]$ or $I=(-\infty,T)$). Let
$$
\begin{equation}
\begin{aligned} \, \mathrm{D}_*^{\alpha}:=\partial_t^{\alpha_0}(\sigma \partial_1)^{\alpha_1} \partial_2^{\alpha_2} \partial_3^{\alpha_3} \partial_1^{\alpha_{4}} \quad\text{for}\ \ \alpha:=(\alpha_0,\dots,\alpha_{4})\in\mathbb{N}^{5}, \end{aligned}
\end{equation}
\tag{62}
$$
where $\sigma=\sigma(x_1):=x_1(1-x_1)(1+x_1)$. The anisotropic weighted Sobolev space $H_*^{m}(I\times \Omega^{\pm})$ for $m\in\mathbb{N}$ is defined by
$$
\begin{equation*}
H_*^m(I\times\Omega^{\pm}):=\{u\in L^2(I\times\Omega^{\pm}): \mathrm{D}_*^{\alpha} u\in L^2(I\times\Omega^{\pm})\ \text{for}\ \langle \alpha \rangle\leqslant m\},
\end{equation*}
\notag
$$
where $\langle\alpha\rangle:=\displaystyle\sum_{i=0}^3\alpha_i+2\alpha_{4}$. This space is equipped with the norm
$$
\begin{equation*}
\|\cdot\|_{H^m_*(I\times \Omega^{\pm})}^2:=\|{u}\|_{m,*,t}^2:= \sum_{\langle \alpha\rangle\leqslant m} \|\mathrm{D}_*^{\alpha}(\cdot)\|_{L^2(I \times\Omega^{\pm})}^2.
\end{equation*}
\notag
$$
By definition we have the obvious embeddings
$$
\begin{equation*}
H^m(I\times \Omega^{\pm})\hookrightarrow H_*^m(I\times \Omega^{\pm}) \hookrightarrow H^{\lfloor m/2\rfloor}(I\times \Omega^{\pm}),
\end{equation*}
\notag
$$
where $\lfloor m \rfloor$ is the floor function of $m$ that maps $m$ to the greatest integer not exceeding $m$. Roughly speaking, in comparison to the usual space $H^m$ we loose half of the $x_1$-derivatives in the $H^m_*$ norm. Let us now consider the plasma–vacuum interface problem (20), (24), (17) completed by the boundary conditions on the fixed wall $\Sigma^+$. For the corresponding linearized problem rewritten for the unknown
$$
\begin{equation*}
W=(\dot{q},\dot{v}\cdot\mathring{N},\dot{v}_2,\dot{v}_3,\dot{H},\dot{S})^\top
\end{equation*}
\notag
$$
($\dot{q}=\dot{p} +\mathring{H}\cdot\dot{H}$) the boundary matrix $B(\mathring{U},\mathring{\Phi})$ calculated on the boundaries $\Sigma_T$ and $\Sigma_T^+$ has the same structure as the constant matrix (48) (this boundary matrix differs from (48) only by the presence of the additional sixth, seventh, and eighth lines and columns of zeros). Therefore, the arguments towards the proof of an a priori $L^2$-estimate are analogous to those in (54) and (55). Then the modified Rayleigh–Taylor sign condition for the unperturbed total pressure $\mathring{q}$ arises naturally, which gives us the a priori estimate (57) for the linearized problem. To derive from (57) an estimate in higher norms we must consider an augmented problem for tangential derivatives of the unknown $W$. As noted above, the linear problem prolonged up to first-order derivatives has dangerous lower-order terms like $\partial_\ell B(\mathring{U},\mathring{\Phi})\,\partial_1W$ ($\ell=0,2,3$). Since
$$
\begin{equation*}
B(\mathring{U},\mathring{\Phi})=\mathring{B}+ B_0(\mathring{U},\mathring{\Phi}),
\end{equation*}
\notag
$$
where $\mathring{B}:=B(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T}= B(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T^+}$ is a constant matrix (cf. (48)) and
$$
\begin{equation*}
B_0(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T}= B_0(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T^+}=0,
\end{equation*}
\notag
$$
it follows that
$$
\begin{equation*}
\partial_\ell B(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T}= \partial_\ell B(\mathring{U},\mathring{\Phi})\big|_{\Sigma_T^+}=0.
\end{equation*}
\notag
$$
We can prove (see [37] and [41]) that for any matrix $A$ vanishing on the boundaries, in particular, for $\partial_\ell B(\mathring{U},\mathring{\Phi})$, one can deduce the estimate
$$
\begin{equation*}
\|A\partial_1W\|_{L^2(\Omega_T^+)} \lesssim \|A\|_{W^1_{\infty}(\Omega_T^+)}\|\sigma \partial_1W\|_{L^2(\Omega_T^+)} \lesssim\|A\|_{H^s_*(\Omega_T^+)}\|\sigma \partial_1W\|_{L^2(\Omega_T^+)},
\end{equation*}
\notag
$$
where $s=2\lfloor n/2\rfloor+4$, and $\sigma=\sigma (x_1)$ is a weight function from the definition (62). The spaces $H^m_*(\Omega_T^+)$ are thus natural spaces in which we can close the a priori estimate in higher norms (for the linearization of problem (20), (24), (32), (17), just as for the linear problem in [36], it is impossible to compensate the loss of control of normal derivatives). The following a priori estimate in $H^m_*(\Omega_T^+)$ was obtained in [55] for the linearized plasma-vacuum interface problem (20), (24), (17), (32):
$$
\begin{equation}
\begin{aligned} \, &\|\dot{V}\|_{m,*,T} +\|\psi\|_{H^{m}(\Sigma_T)} \lesssim_{K_0} \|f\|_{m,*,T} +\|g\|_{H^{m+1}(\Sigma_T)} \nonumber \\ &\qquad+(\|f\|_{6,*,T}+\|g\|_{H^{7}(\Sigma_T)}) (\|\mathring{U}\|_{{m+4},*,T}+\|\mathring{\varphi}\|_{H^{m+4} (\Sigma_T)}), \end{aligned}
\end{equation}
\tag{64}
$$
where $m\in\mathbb{N}$, $m\geqslant 6$, and the basic state $(\mathring{U} ,\mathring{\varphi})\in H^{m+4}_* \times H^{m+4}(\Sigma_T)$ satisfies the corresponding assumptions from [55], in particular, the Rayleigh–Taylor sign condition (63). The existence of solutions to the original nonlinear problem (20), (24), (17), (32) was proved in [55] by Nash–Moser iterations, whose convergence was shown just with the help of estimate (64). Note that, owing to the obvious embedding $H_*^m([0,T]\times \Omega^+) \hookrightarrow H^{\lfloor m/2\rfloor}([0,T]\times\Omega^+)$ mentioned above, the local existence and uniqueness theorem for this problem was formulated in [55] in usual Sobolev spaces, namely,
$$
\begin{equation*}
(U,\varphi)\in H^{{\lfloor (m-9)/2\rfloor}}([0,T]\times\Omega^+)\times H^{m-{9}}([0,T]\times\Sigma),
\end{equation*}
\notag
$$
provided that the initial data $(U_0,\varphi_0)$ belong to $H^{m+3/2}(\Omega^+)\times H^{m+2}(\Sigma)$ for an integer $m\geqslant 20$ and satisfy the Rayleigh–Taylor sign condition (63) (as well as condition (35), the hyperbolicity conditions, and the corresponding compatibility conditions [55]).
5. Non-dissipative boundary conditions and secondary symmetrization of system of conservations laws5.1. Secondary symmetrization of the system of magnetohydrodynamicsIn the case of constant coefficients the linearization of the free boundary problem (12), (16), (17), (33) has dissipative boundary conditions (see (50)). This means that the Kreiss–Lopatinski condition is fulfilled. The possible ill-posedness of the linearized problem appears on the level of variable coefficients and is connected with the violation of the Rayleigh–Taylor sign condition (56). However, in the general case, on the one hand the Kreiss–Lopatinski condition can fail and, on the other hand, even if it holds in some domain of parameters of the linearized problem with constant coefficients, the boundary conditions do not have to be dissipative. At the same time the non-dissipativity of the boundary conditions does not mean that the linear problem is ill posed. Either it is indeed ill posed, or the energy method for deriving a priori estimates does not work in its pure form. In the case when the boundary conditions are not dissipative, one can sometimes adapt the energy method for deriving an a priori estimate for a concrete problem. For shock waves, the problem with boundary conditions on the gas dynamical shock is an example of such a problem. For the corresponding linearized problem Blokhin [2] succeeded in making the boundary conditions strictly dissipative in the whole domain of fulfillment of the uniform Kreiss–Lopatinski condition [24] by a successful construction of a problem prolonged up to second-order derivatives. Regarding problems with characteristic free boundaries, we discuss here examples of problems for which the difficulty connected with the non-dissipativity of the boundary conditions can be overcome by means of the so-called secondary symmetrization of the original quasilinear hyperbolic system. Secondary symmetrization was originally proposed in [48] for the equations of ideal compressible MHD. The MHD system written in the non-conservative form (20) is already symmetric, and it is hyperbolic under conditions (13). The idea of secondary symmetrization proposed in [48] consists in multiplying system (20) from the left by a non-singular matrix such that the resulting system is also symmetric. The hyperbolicity conditions for the new system turn out to be more restrictive than conditions (13). In this sense secondary symmetrization is useless for the Cauchy problem for the MHD system. However, for the linearized problem for current-vortex sheets it enables one to make the boundary conditions dissipative (under certain conditions for the basic state $(\mathring{U}^+,\mathring{U}^-,\mathring{\varphi})$). A natural question arising here is how to find the non-singular matrix mentioned above such that after multiplying by it the MHD system remains symmetric. Consider system (20) written in the symmetric form (2) and linearized about its constant solution $\mathring{U}=(\mathring{p},\mathring{v},\mathring{H},\mathring{S})^\top$:
$$
\begin{equation}
A_0(\mathring{U})\,\partial_t U+\sum_{j=1}^nA_j(\mathring{U})\,\partial_jU=0.
\end{equation}
\tag{65}
$$
The magnetoacoustics system (65) has obviously the following conserved integral
$$
\begin{equation*}
I(t)=\int_{\mathbb{R}^3}A_0(\mathring{U})U\cdot U = \int_{\mathbb{R}^3}\biggl(\frac{1}{\mathring{\rho}\mathring{c}^2}p^2+ \mathring{\rho}|v |^2 +|H |^2 +S^2\biggr),
\end{equation*}
\notag
$$
that is, where $\mathring{\rho} =\rho (\mathring{p},\mathring{S})$ and $1/\mathring{c}^2=(\partial\rho /\partial p)(\mathring{p},\mathring{S})$. On the other hand, taking the divergence constraint (19) into account we can show that there exists the additional conserved integral
$$
\begin{equation*}
J(t)=\int_{\mathbb{R}^3}\biggl(\frac{1}{\mathring{c}^2}(\mathring{H}\cdot v) p-\mathring{\rho}v \cdot H\biggr).
\end{equation*}
\notag
$$
This integral is a counterpart of the cross-helicity integral which is conserved for the system of an ideal incompressible fluid (unlike the case of a compressible fluid under consideration, the conservation of the cross-helicity integral takes place even for the original quasilinear system of ideal incompressible MHD). A linear combination of the integrals $I(t)$ and $J(t)$ yields
$$
\begin{equation*}
\frac{d}{dt}\bigl(I(t)+2\mathring{\lambda}J(t)\bigr)=0
\end{equation*}
\notag
$$
or
$$
\begin{equation*}
\frac{d}{dt}\int_{\mathbb{R}^3}{B}_0(\mathring{U})U\cdot U=0,
\end{equation*}
\notag
$$
where the symmetric matrix $B_0={B}_0({U})$ reads
$$
\begin{equation*}
B_0= \begin{pmatrix} \dfrac{1}{\rho c^2} & \dfrac{\lambda }{c^2}H^\top & 0 &0 \\ \dfrac{\lambda}{c^2}H & \rho I_3 & -\rho\lambda I_3 & 0 \\ 0 &-\rho\lambda I_3 & I_3 &0 \\ 0 & 0 & 0 &1 \end{pmatrix},
\end{equation*}
\notag
$$
and $\mathring{\lambda}$ is an arbitrary constant. Moreover, we can suppose that $\mathring{\lambda}=\lambda (\mathring{U})$, where $\lambda =\lambda (U)$ is an arbitrary function. Then we can easily find a matrix $\mathcal{S}=\mathcal{S}(U)$ such that $B_0=\mathcal{S}A_0$:
$$
\begin{equation*}
{\mathcal S}=\begin{pmatrix} 1 & \dfrac{\lambda }{\rho c^2}H^\top & 0 &0 \\ \lambda \rho H & I_3 & -\rho\lambda I_3 & 0 \\ 0 &-\lambda I_3 & I_3 &0 \\ 0 & 0 & 0 &1 \end{pmatrix}.
\end{equation*}
\notag
$$
Multiplying the original quasilinear symmetric MHD system (2) from the left by the non-singular matrix $\mathcal{S}(U)$ we obtain a system which is not symmetric yet. However, using the divergence constraint (19) we can ‘correct’ it so that the resulting system becomes symmetric:
$$
\begin{equation}
\begin{aligned} \, \nonumber &{\mathcal S}(U )A_0(U )\,\partial_t{U}+ \sum_{j=1}^{3}{\mathcal S}(U )A_j(U )\,\partial_j{U} + {R}(U )\operatorname{div}{H} \\ &\qquad=B_0(U)\,\partial_t{U}+\sum_{j=1}^{3}B_j(U)\,\partial_j{U}=0, \end{aligned}
\end{equation}
\tag{66}
$$
where the vector $R=R(U)$ and the symmetric matrices $B_j=B_j(U)$ are easily calculated:
$$
\begin{equation*}
\begin{gathered} \, R=-\lambda\,(1 ,0,0,0,H_1,H_2,H_3,0)^{\textsf{T}}, \\ B_j=\begin{pmatrix} \dfrac{h_j}{\rho c^2} & \mathbf{e}_j^\top+\dfrac{\lambda v_j}{c^2}H^\top & -\lambda \mathbf{e}_j^\top & 0 \\ \mathbf{e}_j+\dfrac{\lambda v_j}{c^2}H \; & \rho h_jI_3 & \mathbf{e}_j\otimes H +m_jI_3 & 0 \vphantom{\biggl\}^2} \\ -\lambda \mathbf{e}_j & H \otimes\mathbf{e}_j +m_jI_3 & h_jI_3 -\lambda (\mathbf{e}_j\otimes H + H \otimes\mathbf{e}_j) & 0 \\ 0& 0 & 0 & v_j \end{pmatrix}, \\ h_j=v_j+\lambda H_j,\quad m_j = -\rho\lambda v_j -H_j, \end{gathered}
\end{equation*}
\notag
$$
the $\mathbf{e}_j=(\delta_{1j},\delta_{2j},\delta_{3j})^\top$ are unit vectors ($j=1,2,3$), and $\delta_{ij}$ is the Kronecker symbol. It is clear that the symmetric matrices (2) and (66) coincide for $\lambda =0$. The symmetric system (66) is hyperbolic if $B_0>0$, which is true under inequalities (13) and the additional condition where $c_{\rm A}=|{H} |/\sqrt{\rho}$ is the Alfvén speed. As noted above, the hyperbolicity conditions (66) are thus more restrictive than conditions (13).Let us now discuss how the secondary symmetrization of the MHD system is used in [48] for the free boundary problem (6), (7), (22), (32) (in [48] the problem for the current-vortex sheet was actually formulated in the whole of $\mathbb{R}^3$, but here for uniformity of presentation, we consider it in the domain (27) by completing it by conditions (32)). Consider the corresponding constant-coefficient linearized problem. It is a problem for the magnetoacoustics systems (65) written on the left- and right-hand sides of the unperturbed planar discontinuity $x_1=0$. It has the form of problem (45) under the assumption that the basic state (40) is a constant vector $(\mathring{U}^+,\mathring{U}^-,0)$ satisfying the boundary conditions (22):
$$
\begin{equation}
A_0(\mathring{U}^\pm)\,\partial_t \dot{V}^\pm + \sum_{j=1}^nA_j(\mathring{U}^\pm)\,\partial_j\dot{V}^\pm =f^\pm \quad \text{in} \ \ \Omega^\pm_T,
\end{equation}
\tag{68a}
$$
$$
\begin{equation}
\dot{v}_1^\pm = (\partial_t + \mathring{v}'_\pm \cdot \mathrm{D}_{x'}) \psi,\quad \dot{q}^+=\dot{q}^- \text{on } \Sigma_T,
\end{equation}
\tag{68b}
$$
where $\mathring{v}'_{\pm}=(\mathring{v}_2^{\pm},\mathring{v}_3^{\pm})^\top$ and the hyperbolicity conditions are assumed to hold for
$$
\begin{equation*}
\mathring{U}^\pm = (\mathring{p}^\pm , \mathring{v}^\pm , \mathring{H}^\pm,\mathring{S}^\pm )^\top ,
\end{equation*}
\notag
$$
where $\mathring{v}^\pm =(0, \mathring{v}_2^{\pm},\mathring{v}_3^{\pm})^\top$ and $\mathring{H}^\pm =(0,\mathring{H}_2^{\pm},\mathring{H}_3^{\pm})^\top$. For simplicity we consider here the homogeneous boundary conditions. The conditions
$$
\begin{equation}
\dot{H}_1^\pm =\mathring{H}_2^\pm\partial_2\psi + \mathring{H}_3^\pm\partial_3\psi \quad \text{on } \Sigma_T
\end{equation}
\tag{69a}
$$
and are not included into (68b) and (68c) because they are restrictions to the initial data for $f^\pm =0$ (see [48]). However, if $f^\pm \ne 0$, then the original boundary conditions for the linearized problem are inhomogeneous and, as in (68d), the initial data are zero, then conditions (69) can be fulfilled automatically (see [49]) while reducing the problem to the homogeneous boundary conditions. The same takes place also for the condition $\operatorname{div}\dot{H}^\pm =0$ in $\Omega^\pm_T$. The boundary conditions (68c) on the rigid walls $\Sigma_T^\pm$ are dissipative, and we have no troubles with them. As regards the boundary conditions (68b) on the unperturbed planar discontinuity, taking them into account we see that the quadratic form
$$
\begin{equation*}
Q = [{A}_1(\mathring{U})\dot{V}\cdot\dot{V}]= 2\dot{q}^+\big|_{\Sigma_T}[\dot{v}_1]= 2\dot{q}^+\big|_{\Sigma_T}[\mathring{v}']\cdot \mathrm{D}_{x'}\psi
\end{equation*}
\notag
$$
is sign indefinite (here and below we exploit the usual notation for a jump, in particular, $[\dot{v}_1]:=\dot{v}_1^+\big|_{\Sigma_T}- \dot{v}_1^-\big|_{\Sigma_T}$). It follows from the energy inequality
$$
\begin{equation}
\sum_{\pm}\int_{\Omega^\pm}A_0(\mathring{U}^\pm)\dot{V}^\pm\cdot \dot{V}^\pm- \int_{\Sigma_t}Q \lesssim \sum_{\pm} \bigl(\| f^\pm\|^2_{L^2(\Omega_T^\pm)} + \| \dot{V}^\pm\|^2_{L^2(\Omega_t^\pm)}\bigr),
\end{equation}
\tag{70}
$$
which is valid for problem (68), that we cannot deduce an a priori estimate in $L^2$. At the same time, if instead of the magnetoacoustics equations we use the linear analogue of the secondary symmetrization (66), that is, the systems
$$
\begin{equation*}
B_0(\mathring{U}^\pm)\,\partial_t \dot{V}^\pm + \sum_{j=1}^nB_j(\mathring{U}^\pm)\,\partial_j\dot{V}^\pm = {\mathcal S}( \mathring{U}^\pm )f^\pm \quad \text{in } \Omega^\pm_T,
\end{equation*}
\notag
$$
which can easily be shown to be equivalent to (68a) under the inequalities
$$
\begin{equation}
\mathring{\rho}^\pm(\mathring{\lambda}^\pm)^2< \frac{1}{1+(\mathring{c}_{\rm A}^\pm )^2/\mathring{c}_\pm^2}
\end{equation}
\tag{71}
$$
(cf. (67)), then in the boundary integral of the analogue of the energy inequality (70), instead of the quadratic form $Q$ we have the form
$$
\begin{equation}
\mathcal{Q}=[{B}_1(\mathring{U})\dot{V}\cdot\dot{V}]= 2\dot{q}^+\big|_{\Sigma_T}[\dot{v}_1-\mathring{\lambda}\dot{H}_1]= 2\dot{q}^+\big|_{\Sigma_T}[\mathring{v}'-\mathring{\lambda}\mathring{H}'] \cdot\mathrm{D}_{x'}\psi,
\end{equation}
\tag{72}
$$
where
$$
\begin{equation*}
\mathring{c}_{\rm A}^\pm = \frac{|\mathring{H}^\pm |}{\sqrt{\mathring{\rho}^\pm}}\,,\quad \frac{1}{\mathring{c}_\pm^2}= \frac{\partial \rho}{\partial p}(\mathring{p}^\pm ,\mathring{S}^\pm ),\quad \mathring{\lambda}^\pm =\lambda (\mathring{U}^\pm),\quad\text{and}\quad \mathring{H}'_{\pm}=(\mathring{H}_2^{\pm},\mathring{H}_3^{\pm})^\top.
\end{equation*}
\notag
$$
Recall that the constants $\mathring{\lambda}^\pm$ are entirely in our hands. The only thing is that they must satisfy the hyperbolicity conditions (71). Assume that the constant vectors $\mathring{H}^+$ and $\mathring{H}^-$ satisfy the non-collinearity condition
$$
\begin{equation}
\mathring{H}^+_2\mathring{H}^-_3-\mathring{H}^+_3\mathring{H}^-_2 \ne 0,
\end{equation}
\tag{73}
$$
which, by taking the equality $H_{N}^\pm\big|_{\Sigma}=0$ into account (see (22)), in the case of variable coefficients and the initial data of the original nonlinear problem is written as
$$
\begin{equation}
\bigl|({H}^+\times {H}^-)\big|_{\Sigma}\bigr| \geqslant \epsilon > 0,
\end{equation}
\tag{74}
$$
where $\epsilon$ is a fixed constant. Under the non-collinearity condition (73) we have the representation
$$
\begin{equation}
[\mathring{v}']=\mathring{a}^+\mathring{H}'_+-\mathring{a}^-\mathring{H}'_-,
\end{equation}
\tag{75}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathring{a}^{\pm}=-\frac{|[\mathring{v}']| \sin\mathring{\varphi}^{\mp}}{|\mathring{H}'_\pm| \sin (\mathring{\varphi}^+ -\mathring{\varphi}^-)}\,,\quad \mathring{\varphi}^\pm=\varphi^\pm (\mathring{U}^+,\mathring{U}^-), \\ \varphi^{\pm}=\varphi^{\pm}(U^+,U^-),\quad\text{and}\quad \cos\varphi^{\pm}=\frac{([{v}'],{H}'_\pm)}{ |[{v}']|\,|{H}'_\pm|}\,. \end{gathered}
\end{equation*}
\notag
$$
Choosing $\mathring{\lambda}^\pm =\mathring{a}^{\pm}$, it follows from (72) and (75) that $\mathcal{Q}=0$, so that the boundary conditions are dissipative for this choice. Then the counterpart of the energy inequality (70) for secondary symmetrization implies the a priori $L^2$-estimate, provided that the hyperbolicity conditions (71) hold, which for $\mathring{\lambda}^\pm$ as chosen above read where
$$
\begin{equation*}
\begin{gathered} \, G(U^+,U^-)=|\!\sin({\varphi}^+-{\varphi}^-)| \min\biggl\{\frac{{\gamma}^+}{|\!\sin{{\varphi}^-}|}\,, \frac{{\gamma}^-}{|\!\sin{{\varphi}^+}|}\biggr\}-|[{v}']|, \\ {\gamma}^{\pm}=\gamma ({U}^\pm ), \quad\text{and}\quad \gamma=\gamma(U)=\frac{{c}_{\rm A}c}{\sqrt{c^2+{c}_{\rm A}^2}}\,. \end{gathered}
\end{equation*}
\notag
$$
Condition (76), found in [48], has independent significance, since in astrophysics it can be treated as a sufficient condition for the macroscopic stability of the heliopause [39]. In [49] the local existence and uniqueness theorem in $H^m_*$ was proved for the free boundary problem (6), (7), (22), (32), provided that conditions (73) and (76) hold at each point on the initial surface of the current-vortex sheet, more precisely, if the initial data (7) satisfy the non-collinearity condition (74), together with the stability condition where $\varepsilon$ is a fixed constant.Note that the non-collinearity condition (74) for the basic state (40) is important not only for making the boundary conditions of the linearized problem dissipative but also for closing the a priori estimate in the case of variable coefficients. Namely, if this condition holds, then the boundary symbol is elliptic. Indeed, using (74), from the last two conditions in (22) we find $\partial_2\varphi$ and $\partial_3\varphi$ in terms of the traces $H_1^\pm\big|_{\Sigma}$, and then we find $\partial_t\varphi$ from the second condition in (22). The ellipticity of the symbol of free boundary enables one to express, based on the boundary conditions (45b), the gradient $(\partial_t\psi,\mathrm{D}_{x'}\psi )$ in terms of the traces $(\dot{H}^\pm \cdot\mathring{N})\big|_{\Sigma_T}$, $(\dot{v}^+\cdot\mathring{N})\big|_{\Sigma_T}$ and the function $\psi$ itself. Unlike the constant coefficient problem, the a priori estimate for the linear variable coefficient problem cannot be closed in $L^2$ because of the presence of ‘lower-order’ terms like $\operatorname{coeff}(\dot{v}^+\cdot\mathring{N})\psi$, $\operatorname{coeff}(\dot{H}^\pm \cdot\mathring{N})\psi$, and so on, in the boundary integral, where $\operatorname{coeff}$ is a generic coefficient depending on the basic state. However, instead of these terms we obtain terms like $\operatorname{coeff} \partial_k(\dot{v}^+\cdot\mathring{N})\,\partial_k\psi$ ($k=2,3$), $\operatorname{coeff} \partial_t(\dot{v}^+\cdot\mathring{N})\,\partial_t\psi$, and so on while prolonging the problem up to first-order tangential derivatives. Then using that the boundary symbol is elliptic we can reduce the terms like $\operatorname{coeff} \partial_2\dot{v}^+_N\partial_2\psi$ to terms like $\operatorname{coeff} \dot{H}_N^\pm\partial_2\dot{v}^+_N$, and so on, where $\dot{v}^+_N=\dot{v}^+\cdot\mathring{N}$ and $\dot{H}_N^\pm=\dot{H}^\pm \cdot\mathring{N}$. For example, the boundary integral of $\operatorname{coeff} \dot{H}_N^+\partial_2\dot{v}^+_N$ appearing in the energy inequality for the problem prolonged up to first-order tangential derivatives is no longer an obstacle to closing the a priori estimate in $H^1$ (more precisely, in $H^1_*$) since it can be absorbed in the right-hand side of this inequality by reducing to a volume integral and integrating by parts:
$$
\begin{equation*}
\begin{aligned} \, \int_{\Sigma_t}\operatorname{coeff}\dot{H}_N^+\,\partial_2\dot{v}^+_N&= -\int_{\Omega^+_t}\partial_1(\operatorname{coeff}\dot{H}_N^+\, \partial_2\dot{v}^+_N) \\ &=\int_{\Omega^+_t}\bigl(\operatorname{coeff}\partial_2\dot{H}_N^+\, \partial_1\dot{v}^+_N+\operatorname{coeff}\partial_1\dot{H}_N^+\, \partial_2\dot{v}^+_N + \operatorname{coeff}\dot{H}_N^+\, \partial_1\dot{v}^+_N\bigr) \end{aligned}
\end{equation*}
\notag
$$
(recall that $\operatorname{coeff}$ is the general notation for a coefficient; in particular, the coefficients of terms in the last integral are different). Let us now formulate the local existence and uniqueness theorem proved in [49] for compressible current-vortex sheets. Theorem 5.1. Let $m\in\mathbb{N}$ and $m\geqslant 12$. Assume that the initial data (36d), where Note that the question of finding not only a sufficient but also a necessary condition for the well-posedness of the problem for compressible current-vortex sheets remains open. Necessary conditions can be found only by means of spectral analysis (a test for the Kreiss–Lopatinski condition), which seems to be a technically difficult problem at the moment (see the discussion in [48]). Regarding incompressible current-vortex sheets, for them a necessary and sufficient condition of well-posedness, which is in fact a condition for the linear stability of a planar discontinuity which holds at each point on the surface of initial discontinuity, can be found by means of spectral analysis. It was found relatively long time ago by Syrovatsky [47]. The local well-posedness of the nonlinear problem was recently proved in [46], provided that this condition is satisfied at the first moment of time. 5.2. Secondary symmetrization of the vacuum Maxwell equationsConsider now another example of secondary symmetrization, which enables one to achieve the dissipativity of the boundary conditions of the linearized problem. The local well-posedness of the plasma–vacuum interface problem (20), (26), (31), (17) was established in [44] in the case when the initial data satisfy the non-collinearity condition (74). However, the boundary conditions (45b) (with boundary operator (44)) of the linearized problem are not dissipative under the non-collinearity condition (74) (for $H^+=\mathring{H}$ and $H^-=\mathring{h}$). This difficulty was overcome in [43] by means of a hyperbolic $\varepsilon$-regularization of the elliptic system for $\dot{h}$, secondary symmetrization of the regularized system, and the subsequent limit transition as $\varepsilon \to 0$. For technical simplicity we consider the case of constant coefficients of the linearized problem, that is, the case when for the basic state $\mathring{U}$ and $\mathring{h}$ are constant vectors,and $\mathring{\varphi}=0$. We also assume that, following [43], the linear problem has already been reduced to the homogeneous system in vacuum and homogeneous boundary conditions. Then the hyperbolic regularization proposed in [43] consists in considering, instead of the original elliptic system
$$
\begin{equation*}
\nabla\times {h}=0,\quad \operatorname{div} {h}=0 \quad \text{in } \Omega^-_T
\end{equation*}
\notag
$$
(here and below points over unknowns are dropped), its hyperbolic $\varepsilon$- regularization in $\Omega^-_T$
$$
\begin{equation}
\varepsilon\,\partial_t{h}+\nabla\times {e}=0,\quad \varepsilon\,\partial_t{e}-\nabla\times {h}=0,
\end{equation}
\tag{78a}
$$
$$
\begin{equation}
\operatorname{div} {h}=0 ,\quad \operatorname{div} {e}=0,
\end{equation}
\tag{78b}
$$
where $\varepsilon >0$ is a small parameter of regularization and ${e}=({e}_1,{e}_2,{e}_3)^\top$ is a new vector unknown, which is in a certain sense an ‘artificial’ electric field in vacuum for Maxwell’s equations (78) (recall that the real electric field in vacuum is a secondary unknown in the framework of the original non-relativistic statement of pre-Maxwell dynamics (25)). Moreover, the regularized boundary conditions on $\Sigma_T$ contain two additional conditions for $e$:
$$
\begin{equation}
{v}_1= (\partial_t + \mathring{v}' \cdot \mathrm{D}_{x'}) \psi,
\end{equation}
\tag{79a}
$$
$$
\begin{equation}
{p}+\mathring{H}\cdot {H}=\mathring{h}\cdot {h}- \varepsilon \mathring{e}\cdot {e},
\end{equation}
\tag{79b}
$$
$$
\begin{equation}
{h}_1 =\mathring{h}_2\,\partial_2\psi+\mathring{h}_3\,\partial_3\psi,
\end{equation}
\tag{79c}
$$
$$
\begin{equation}
{e}_2 =\varepsilon\mathring{h}_3\,\partial_t\psi - \varepsilon\mathring{e}_1\,\partial_2\psi,
\end{equation}
\tag{79d}
$$
$$
\begin{equation}
{e}_3 =\varepsilon\mathring{h}_2\,\partial_t\psi + \varepsilon\mathring{e}_1\,\partial_3\psi,
\end{equation}
\tag{79e}
$$
where $\mathring{e}=(\mathring{e}_1,\mathring{e}_2,\mathring{e}_3)^\top$, and the $\mathring{e}_j$ are some fixed constants (in the case of variable coefficients they are functions) defined after the secondary symmetrization of Maxwell’s system (78) so as to make the boundary conditions (79) dissipative. The linear system in the domain $\Omega^+_T$ remains the same, that is, it is the linearized MHD system, namely, for the case of constant coefficients it is the magnetoacoustics system (65) for $U= (p,v,H,S)^\top$ with source term $f^+$, where $\mathring{U}=(\mathring{p}, 0,\mathring{v}_2,\mathring{v}_3,0,\mathring{H}_2,\mathring{H}_3, \mathring{S})^\top$. Note that for the regularized problem equations (78b) and the boundary condition (79c) are automatically satisfied (for zero initial data). System (78a) for $W=(h,e)^\top$, written as is symmetric hyperbolic, where
$$
\begin{equation*}
\begin{gathered} \, B_j=\varepsilon^{-1}\begin{pmatrix} 0 & b_j \\ b_j^\top & 0 \end{pmatrix}, \\ b_1=\begin{pmatrix} 0 & 0 &\! \hphantom{-}0 \\ 0 & 0 &\! -1 \\ 0 & 1 &\! \hphantom{-}0 \end{pmatrix},\quad b_2=\begin{pmatrix} \hphantom{-} 0 & 0 & 1 \\ \hphantom{-} 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix},\quad \text{and}\quad b_3=\begin{pmatrix} 0 &\! -1 & 0 \\ 1 &\! \hphantom{-}0 & 0 \\ 0 &\! \hphantom{-}0 & 0 \end{pmatrix}. \end{gathered}
\end{equation*}
\notag
$$
The idea of hyperbolic regularization is inspired by the physical formulation of the relativistic plasma–vacuum interface problem whose well-posedness was studied in [52]. In fact, (78) is Maxwell’s system in vacuum, and $\varepsilon$ is a dimensionless quantity which is equal to the ratio of the characteristic speed of the problem and the speed of light in vacuum, and it is really negligible in the non-relativistic setting. Let us now discuss the secondary symmetrization of the hyperbolic system (80) proposed in [52]. Consider this system in the whole of $\mathbb{R}^3$. It is clear that On the other hand, using the divergence constraints (78b), for system (78a) we find several additional conserved integrals:
$$
\begin{equation*}
\begin{gathered} \, \frac{d}{{d}t}\int_{\mathbb{R}^3}(h_2e_3-h_3e_2)=0, \\ \frac{d}{{d}t}\int_{\mathbb{R}^3}(h_3e_1-h_1e_3)=0,\qquad \frac{d}{{d}t}\int_{\mathbb{R}^3}(h_1e_2-h_2e_1)=0. \end{gathered}
\end{equation*}
\notag
$$
Then
$$
\begin{equation}
\frac{d}{{d}t}\int_{\mathbb{R}^3}\big(|W|^2 +\varepsilon\nu_1(h_2e_3-h_3e_2) +\varepsilon\nu_2(h_3e_1-h_1e_3) +\varepsilon\nu_3(h_1e_2-h_2e_1)\big) =0,
\end{equation}
\tag{81}
$$
where the $\nu_i$ ($i=1,2,3$) are arbitrary constants. The energy identity (81) is rewritten as
$$
\begin{equation*}
\frac{d}{{d}t}\int_{\mathbb{R}^3} \mathcal{B}_0W\cdot W =0,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\mathcal{B}_0=\begin{pmatrix} 1 & 0 & 0& 0 & \varepsilon\nu_3 & -\varepsilon\nu_2 \\ 0 & 1 & 0& -\varepsilon\nu_3 & 0 & \varepsilon\nu_1 \\ 0 & 0 & 1& \varepsilon\nu_2 & -\varepsilon\nu_1 & 0 \\ 0 & -\varepsilon\nu_3 & \varepsilon\nu_2& 1 & 0 & 0 \\ \varepsilon\nu_3 & 0 & -\varepsilon\nu_1& 0 & 1 & 0 \\ -\varepsilon\nu_2 & \varepsilon\nu_1 & 0& 0 & 0 & 1 \end{pmatrix}.
\end{equation*}
\notag
$$
By virtue of the divergence constraints (78b) it follows from system (80) that
$$
\begin{equation}
\begin{aligned} \, &\mathcal{B}_0\partial_tW +\sum_{j=1}^3\mathcal{B}_0B_j\,\partial_jW + R_1\operatorname{div} h+R_2\operatorname{div} e \nonumber \\ &\qquad=\mathcal{B}_0\,\partial_tW +\sum_{j=1}^3\mathcal{B}_j\,\partial_jW=0, \end{aligned}
\end{equation}
\tag{82}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathcal{B}_1=\begin{pmatrix} \nu_1 & \nu_2 & \nu_3& 0 & 0 & 0 \\ \nu_2 & -\nu_1 & 0& 0 & 0 & -\varepsilon^{-1} \\ \nu_3 & 0 & -\nu_1& 0 & \varepsilon^{-1} & 0 \\ 0 & 0 & 0& \nu_1 & \nu_2 & \nu_3 \\ 0 & 0 & \varepsilon^{-1}& \nu_2 & -\nu_1 & 0 \\ 0 & -\varepsilon^{-1} & 0& \nu_3 & 0 & -\nu_1 \end{pmatrix}, \\ \mathcal{B}_2=\begin{pmatrix} -\nu_2 & \nu_1 & 0& 0 & 0 & \varepsilon^{-1} \\ \nu_1 & \nu_2 & \nu_3& 0 & 0 & 0 \\ 0 & \nu_3 & -\nu_2& -\varepsilon^{-1} & 0 & 0 \\ 0 & 0 & -\varepsilon^{-1}& -\nu_2 & \nu_1 & 0 \\ 0 & 0 & 0& \nu_1 & \nu_2 & \nu_3 \\ \varepsilon^{-1} & 0 & 0& 0 & \nu_3 & -\nu_2 \end{pmatrix}, \\ \mathcal{B}_3=\begin{pmatrix} -\nu_3 & 0 & \nu_1& 0 & -\varepsilon^{-1} & 0 \\ 0 & -\nu_3 & \nu_2& \varepsilon^{-1} & 0 & 0 \\ \nu_1 & \nu_2 & \nu_3& 0 & 0 & 0 \\ 0 & \varepsilon^{-1} & 0& -\nu_3 & 0 & \nu_1 \\ -\varepsilon^{-1} & 0 & 0& 0 & -\nu_3 & \nu_2 \\ 0 & 0 & 0& \nu_1 & \nu_2 & \nu_3 \end{pmatrix}, \\ R_1=\begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \\ 0 \\ 0 \\ 0 \end{pmatrix},\quad\text{and}\quad R_2=\begin{pmatrix} 0 \\ 0 \\ 0 \\ \nu_1 \\ \nu_2 \\ \nu_3 \end{pmatrix}. \end{gathered}
\end{equation*}
\notag
$$
The symmetric matrix (82) is hyperbolic if $\mathcal{B}_0>0$, that is, which is true for the given vector $\nu=(\nu_1,\nu_2,\nu_3)^\top$ and small $\varepsilon$. Note that, for the linearized variable coefficient problem, $\nu$ in (82) can be an arbitrary vector function $\nu=\nu(t,x)$. In the energy inequality for systems (65) (with $f^+$ on the right-hand side) and (82) we must compute the quadratic form
$$
\begin{equation*}
Q=-(A_1(\mathring{U})U\cdot U)\big|_{\Sigma_T}+ (\mathcal{B}_1W\cdot W)\big|_{\Sigma_T}.
\end{equation*}
\notag
$$
For the choice
$$
\begin{equation*}
\nu = \mathring{v} = (0,\mathring{v}_2,\mathring{v}_3)^\top
\end{equation*}
\notag
$$
made in [43] (recall once again that here, unlike [43], we consider the case of constant coefficients for simplicity) we have
$$
\begin{equation*}
Q=\bigl(-qv_1+\varepsilon^{-1}(h_3e_2-h_2e_3)+(\mathring{v}_2h_2+ \mathring{v}_3h_3)h_1+(\mathring{v}_2e_2+ \mathring{v}_3e_3)e_1\bigr)\big|_{\Sigma_T}.
\end{equation*}
\notag
$$
Then, choosing
$$
\begin{equation*}
\mathring{e} =(\mathring{v}_3\mathring{h}_2- \mathring{v}_2\mathring{h}_3,0,0)^\top
\end{equation*}
\notag
$$
and using the boundary conditions (79) we obtain $Q=0$. The boundary conditions for the secondary symmetrization are thus dissipative, and we can easily derive an a priori $L^2$-estimate. Moreover, since the matrix $\mathcal{B}_0$ is bounded as $\varepsilon \to 0$, this estimate is uniform in $\varepsilon$. Also in the case of variable coefficients, the a priori estimate derived in [43] is uniform in $\varepsilon$, but we can close it only in $H^1$. The non-collinearity condition (74) for $\mathring{H}$ and $\mathring{h}$ plays the crucial role in closing this estimate. We must pay special attention to proving the existence of solutions to the regularized problem. We do not discuss here this question, and we just refer the reader to [43]. Then, since the a priori estimate is uniform, we can easily pass to the limit as $\varepsilon \to 0$ and establish the existence of solutions to the original linear problem. An alternative proof of the existence of solutions to the linear problem (without the hyperbolic regularization of the elliptic problem for the perturbation of the magnetic field in vacuum) was recently given in [45]. Applying the results of [43] to the linearized problem, a local existence and uniqueness theorem for the original nonlinear plasma-vacuum interface problem (20), (26), (31), (17) was proved [44] by the Nash–Moser method. Here we formulate this theorem in the form given in [45]. Theorem 5.2. Assume that $\boldsymbol{j}_{\rm c}\in H^{m+3/2}([0,T_0]\times \Sigma^-)$ for some $T_0>0$ and $m\in\mathbb{N}$ with $m\geqslant 20$. Assume further that $(U_0,\varphi_0)\in H^{m+3/2}(\Omega^+)\times H^{m+2}(\mathbb{T}^{2})$ satisfy (35), the divergence constraint (19), the first condition in (31b) and second condition in (31c) (written in straightened variables: see (34)), the hyperbolicity conditions (13), the non-collinearity condition (74) (for $H^+={H}_0$ and $H^-={h}_0$), and the compatibility conditions up to order $m$ (see [43] and [45]). Then there exists small time $T > 0$ such that problem (36) (corresponding to the original statement (20), (26), (31), (17)) has a unique solution Note that the question of establishing the local well-posedness of the plasma- vacuum interface problem (20), (26), (31), (17) when the non-collinearity condition is violated and the alternative condition
$$
\begin{equation}
[\partial_1{q}]\big|_{\Sigma}\geqslant \varepsilon >0
\end{equation}
\tag{84}
$$
for the total pressure $q$ (cf. (58)) is fulfilled at the first moment of time, which was stated in [51], remains open even in the case when instead of (20) we consider the system of ideal incompressible MHD. In [53], by constructing an Hadamard-type ill-posedness example for the linearized frozen coefficient problem2 it was shown that the simultaneous failure of both conditions (84) and (74) leads to Rayleigh–Taylor instability. In [53] the natural conjecture was formulated that problem (20), (26), (31), (17) is locally well posed if and only if at each point on the initial interface either condition (84) or condition (74) is satisfied. The proof of this conjecture is a more challenging problem than the open problem of local well-posedness under the generalized Rayleigh–Taylor sign condition (84). Returning to the question of the secondary symmetrization of a quasilinear symmetric hyperbolic system, we note the obvious fact that if the corresponding linear constant coefficient system has conserved integrals different from the standard conserved integral equivalent to the square of the $L^2(\mathbb{R}^n)$ norm of the solution, then secondary symmetrization of the original quasilinear system exists. However, apart from the two examples of such systems given above, which are the quasilinear MHD system and the linear system of Maxwell’s equations in vacuum, we know only of two other physical examples of symmetric hyperbolic systems having secondary symmetrization. These are the systems of relativistic MHD and shallow water MHD, whose secondary symmetrizations were found in [12] and [54] respectively. These secondary symmetrizations were used in [12] and [54] to study the well-posedness of problems for current-vortex sheets in these MHD models. 6. Concluding remarksIf one uses the linearization method to prove the existence of solutions, while for the linearized problem we are able to obtain only a priori estimates with a loss of derivatives from the coefficients and source terms, then perhaps the only known possibility to prove the existence of solutions to the original nonlinear problem is the use of the Nash–Moser method. For example, Theorems 3.1, 5.1, and 5.2 formulated above were proved by Nash–Moser iterations. As already noted in § 3, in [30] the authors managed to avoid the loss of smoothness of solutions relative to the smoothness of the initial data that takes place in Theorem 3.1 in [50], as well as in a similar theorem proved in [28] for the isentropic Euler equations. That is, even if the linearized problem is only weakly well posed in the sense that the uniform Kreiss–Lopatinski condition is violated [24], which implies a loss of derivatives in a priori estimates for solutions to this problem, this does not necessarily mean a loss of smoothness in the original nonlinear problem. On the other hand there are many examples of problems in which the existence of solutions was proved with a loss of smoothness, but it is still unclear whether this is an essential property of the problem or such a loss of smoothness can be avoided by using a different method of the proof. This, for example, concerns the free boundary problems for which Theorems 5.1 and 5.2 formulated above were proved, that is, problems (6), (7), (22), (32) for current-vortex sheets and the plasma–vacuum interface problem (20), (26), (31), (17). Regarding the Nash–Moser method, its detailed description, and references to the original works of the authors of this method can be found in [21] and [42]. Briefly speaking, the idea of the method consists in solving a nonlinear equation ${\mathcal F}(u )=0$ by using the iteration scheme
$$
\begin{equation*}
{\mathcal F}'(S_{\theta_n}u_n)(u_{n+1}-u_{n} )=-{\mathcal F}(u_n),
\end{equation*}
\notag
$$
where ${\mathcal F}'$ is the linearization (first variation) of the functional ${\mathcal F}$, and $S_{\theta_n}$ is a sequence of smoothing operators such that $S_{\theta_n}\to I$ as $n\to \infty$. This is the classical Newton scheme if $S_{\theta_n}= I$. The application of the smoothing operator at each step of the Nash–Moser scheme aims to compensate for the loss of derivatives not only from the source terms (as, for example, in estimate (51)) but also from the coefficients of the linearized problem, that is, from the basic state (40). The errors of the classical Nash–Moser scheme are the quadratic error of Newton’s scheme and the substitution error caused by the application of the smoothing operators $S_{\theta_n}$. For all free boundary problems under discussion in this survey, the Nash–Moser method is not completely standard, because we have to take into account the additional error caused by the introduction of an intermediate (modified) state $u_{n+1/2}$ satisfying some nonlinear constraints. The main constraint is usually the requirement of the fulfillment of part of the boundary conditions by the coefficients of the linear problem at each step of the iteration scheme. For example, for the free boundary problem (12), (16), (17), (33) the modified state must satisfy the second condition in (16) and the boundary condition (33). This is necessary in order that the boundary matrix of the linear system have the same structure on the boundaries as the boundary matrix of the original nonlinear problem. Finally, we note that another additional error of the iteration scheme is caused by dropping the last term on the left-hand side of (43) (this term equals zero if the basic state satisfies the original nonlinear problem, and therefore the iteration scheme converges indeed to the solution, then this error of the scheme also tends to zero). The basic a priori estimates like (51) are (59) are not sufficient for the proof of the convergence of the Nash–Moser iterations. We must deduce a more delicate a priori estimate in higher norms of Sobolev spaces, which takes into account the number of ‘lost’ derivatives not only from the source terms but also from the coefficients, that is, from the basic state (40). As already noted above, such estimates are called tame estimates, and their main properties are linearity in higher norms and a fixed loss of derivatives, that is, it is necessary that the number of ‘lost’ derivatives remains the same for each index $m$ of the Sobolev space (see estimates (61) and (64)). All the examples of free boundary problems considered for hyperbolic systems of conservation laws can also be formulated with modified boundary conditions which take into account the influence of the surface tension on the evolution of the boundary. Surface tension usually plays a stabilizing role. In particular, in the case of non-zero surface tension, conditions (56), (58), (63) and condition (74) (for the plasma-vacuum interface problem) become unnecessary for the local well-posedness of the corresponding problems. In this survey we do not discuss the proof of existence and uniqueness theorems for these problems and just refer the reader to the original works [8] and [56]–[58] (see also [45]). 
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