# Nonlinear stability and existence of two-dimensional compressible current-
vortex sheets

Alessandro Morando  DICATAM, University of Brescia, Via Valotti 9, 25133
Brescia, Italy<EMAIL_ADDRESS>,  Paolo Secchi  DICATAM,
University of Brescia, Via Valotti 9, 25133 Brescia, Italy
<EMAIL_ADDRESS>,  Paola Trebeschi  DICATAM, University of Brescia, Via
Valotti 9, 25133 Brescia, Italy<EMAIL_ADDRESS>and  Difan Yuan
School of Mathematical Sciences, Beijing Normal University and Laboratory of
Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China.
<EMAIL_ADDRESS>
###### Abstract.

We are concerned with nonlinear stability and existence of two-dimensional
current-vortex sheets in ideal compressible magnetohydrodynamics. This is a
nonlinear hyperbolic initial-boundary value problem with characteristic free
boundary. It is well-known that current-vortex sheets may be at most weakly
(neutrally) stable due to the existence of surface waves solutions that yield
a loss of derivatives in the energy estimate of the solution with respect to
the source terms. We first identify a sufficient condition ensuring the weak
stability of the linearized current-vortex sheets problem. Under this
stability condition for the background state, we show that the linearized
problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a
loss of derivatives. Based on the weakly linear stability results, we then
establish the local-in-time existence and nonlinear stability of current-
vortex sheets by a suitable Nash-Moser iteration, provided the stability
condition is satisfied at each point of the initial discontinuity. This result
gives a new confirmation of the stabilizing effect of sufficiently strong
magnetic fields on Kelvin-Helmholtz instabilities.

###### Key words and phrases:

Vortex sheets, Ideal compressible Magnetohydrodynamics, Characteristic free
boundary, Nonlinear stability, Loss of derivatives

###### 2010 Mathematics Subject Classification:

76W05, 35L65, 76N10, 35Q35, 35R35, 76E17

##  1\. Introduction

In this paper, we are focusing on two-dimensional ideal compressible
magnetohydrodynamics (MHD)(See [15, 18, 19]):

$\begin{cases}\partial_{t}\rho+\mathrm{div}(\rho\mathbf{u})=0,\\\
\partial_{t}(\rho\mathbf{u})+\mathrm{div}(\rho\mathbf{u}\otimes\mathbf{u}-\mathbf{H}\otimes\mathbf{H})+\nabla(p+\frac{1}{2}|\mathbf{H}|^{2})=0,\\\
\partial_{t}\mathbf{H}-\nabla\times(\mathbf{u}\times\mathbf{H})=0,\\\
\partial_{t}(\rho e+\frac{1}{2}|\mathbf{H}|^{2})+\mathrm{div}((\rho
e+p)\mathbf{u}+\mathbf{H}\times(\mathbf{u}\times\mathbf{H}))=0,\\\
\end{cases}$ (1.1)

supplemented with

$\mathrm{div}\mathbf{H}=0$ (1.2)

on the initial data for the Cauchy problem. Here density $\rho,$ velocity
$\mathbf{u}=(u_{1},u_{2})$, magnetic field $\mathbf{H}=(H_{1},H_{2})$ and
pressure $p$ are unknown functions of time $t$ and spacial variables
$\mathbf{x}=(x_{1},x_{2})$. $p=p(\rho,S)$ stands for the pressure, $S$ is the
entropy, $e=E+\frac{1}{2}|\mathbf{u}|^{2}$, where $E=E(\rho,S)$ stands for the
internal energy. By using the state equation of gas, $\rho=\rho(p,S),$ and the
first principle of thermodynamics, we have that (1.1) is a closed system. We
write ${\bf{U}}={\bf{U}}(t,\mathbf{x})=(p,\mathbf{u},\mathbf{H},S)^{T},$ with
initial data ${\bf{U}}(0,\mathbf{x})={\bf{U}}_{0}(\mathbf{x}).$ By (1.2), we
can rewrite (1.1) into the following form

$\begin{cases}(\partial_{t}+\mathbf{u}\cdot\nabla)p+\rho
c^{2}\mathrm{div}\mathbf{u}=0,\\\
\rho(\partial_{t}+\mathbf{u}\cdot\nabla)\mathbf{u}-(\mathbf{H}\cdot\nabla)\mathbf{H}+\nabla
q=0,\\\
(\partial_{t}+\mathbf{u}\cdot\nabla)\mathbf{H}-(\mathbf{H}\cdot\nabla)\mathbf{u}+\mathbf{H}\mathrm{div}\mathbf{u}=0,\\\
(\partial_{t}+\mathbf{u}\cdot\nabla)S=0,\\\ \end{cases}$ (1.3)

where $q=p+\frac{1}{2}|\mathbf{H}|^{2}$ represents the total pressure, $c$
denotes the speed of sound and $\frac{1}{c^{2}}:=\frac{\partial\rho}{\partial
p}(\rho,S)=\rho_{p}(p,S)$. (1.3) can be written in the matrix form as

$A_{0}({\bf{U}})\partial_{t}{\bf{U}}+A_{1}({\bf{U}})\partial_{1}{\bf{U}}+A_{2}({\bf{U}})\partial_{2}{\bf{U}}=0,$
(1.4)

where

$A_{0}({\bf{U}}):=\mathrm{diag}\Big{\\{}\frac{1}{\rho
c^{2}},\rho,\rho,1,1,1\Big{\\}},$
$\begin{split}&A_{1}({\bf{U}}):=\begin{bmatrix}\frac{u_{1}}{\rho
c^{2}}&1&0&0&0&0\\\ 1&\rho u_{1}&0&0&H_{2}&0\\\ 0&0&\rho u_{1}&0&-H_{1}&0\\\
0&0&0&u_{1}&0&0\\\ 0&H_{2}&-H_{1}&0&u_{1}&0\\\ 0&0&0&0&0&u_{1}\\\
\end{bmatrix},\\\ &A_{2}({\bf{U}}):=\begin{bmatrix}\frac{u_{2}}{\rho
c^{2}}&0&1&0&0&0\\\ 0&\rho u_{2}&0&-H_{2}&0&0\\\ 1&0&\rho u_{2}&H_{1}&0&0\\\
0&-H_{2}&H_{1}&u_{2}&0&0\\\ 0&0&0&0&u_{2}&0\\\ 0&0&0&0&0&u_{2}\\\
\end{bmatrix}.\\\ \end{split}$ (1.5)

The quasilinear system (1.4) is symmetric hyperbolic if the state equation
$\rho=\rho(p,S)$ satisfies the hyperbolicity condition $A_{0}>0:$

$\rho(p,S)>0,\quad\rho_{p}(\rho,S)>0.$ (1.6)

We suppose that $\Gamma:=\bigcup\limits_{t\in[0,T]}\Gamma(t)$, where
$\Gamma(t):=\\{{\bf x}\in\mathbb{R}^{2}\,:\,\,x_{1}-\varphi(t,x_{2})=0\\}$, is
a smooth hypersurface in $[0,T]\times{\mathbb{R}}^{2}.$ The weak solutions of
(1.1) satisfy the Rankine-Hugoniot jump conditions on $\Gamma(t)$:

$\begin{cases}&[j]=0,\quad[H_{N}]=0,\\\ &j[u_{N}]+|N|^{2}[q]=0,\quad
j[u_{\tau}]=H_{N}[H_{\tau}],\\\ &H_{N}[u_{\tau}]=j[\frac{H_{\tau}}{\rho}],\\\
&j[e+\frac{1}{2}\frac{|\mathbf{H}|^{2}}{\rho}]+[qu_{N}-H_{N}(\mathbf{H}\cdot\mathbf{u})]=0.\end{cases}$
(1.7)

Here we denote $[\upsilon]=(\upsilon^{+}-\upsilon^{-})|_{\Gamma}$ the jump of
$\upsilon$, with $\upsilon^{\pm}:=\upsilon$ in
$\Omega^{\pm}(t)=\\{\pm(x_{1}-\varphi(t,x_{2}))>0\\},$ and
$j=\rho(u_{N}-\partial_{t}\varphi)$ is the mass transfer flux across the
discontinuity surface. We also denote the tangential and normal components of
velocity and magnetic fields
$u_{\tau}=\mathbf{u}\cdot\tau,H_{\tau}=\mathbf{H}\cdot\tau,$
$u_{N}=\mathbf{u}\cdot N,H_{N}=\mathbf{H}\cdot N,$ where
$N=(1,-\partial_{2}\varphi),\tau=(\partial_{2}\varphi,1).$

We are focusing on the current-vortex sheets solutions, which obey the
following additional conditions along the interface $\Gamma$:

$j^{\pm}|_{\Gamma}=0,\quad H^{\pm}_{N}|_{\Gamma}=0.$

Then, the Rankine-Hugoniot conditions reduce to the following boundary
conditions

$\partial_{t}\varphi=u^{\pm}_{N},\quad H^{\pm}_{N}=0,\quad[q]=0$ (1.8)

on $\Gamma(t).$

The tangential components of the velocity and the magnetic field may undergo
any jumps: $[u_{\tau}]\neq 0,\quad[H_{\tau}]\neq 0.$ The initial data are
given as follows:

$\mathbf{U}^{\pm}(0,\mathbf{x})=\mathbf{U}^{\pm}_{0}(\mathbf{x}),\quad\mathbf{x}\in\Omega^{\pm}(0),\quad\varphi(0,x_{2})=\varphi_{0}(x_{2}),\quad
x_{2}\in{\mathbb{R}}.$ (1.9)

It is obvious that there exist trivial vortex sheets (contact discontinuity)
solutions consisting of two constant states separated by a flat surface as
follows:

$\bar{\mathbf{U}}(\mathbf{x}):=\begin{cases}\bar{\mathbf{U}}^{+}:=(\bar{p}^{+},0,\bar{u}^{+}_{2},0,\bar{H}^{+}_{2},\bar{S}^{+}),&\text{
if }x_{1}>0,\\\
\bar{\mathbf{U}}^{-}:=(\bar{p}^{-},0,\bar{u}^{-}_{2},0,\bar{H}^{-}_{2},\bar{S}^{-}),&\text{
if }x_{1}<0,\\\ \end{cases}$ (1.10)

where on account of (1.8), we require that

$\bar{p}^{+}\neq\bar{p}^{-},\quad\bar{u}^{+}_{2}\neq\bar{u}^{-}_{2},\quad\bar{H}^{+}_{2}\neq\bar{H}^{-}_{2},\quad\bar{S}^{+}_{2}\neq\bar{S}^{-}_{2},\quad\bar{p}^{+}+\frac{1}{2}|\bar{H}^{+}_{2}|^{2}=\bar{p}^{-}+\frac{1}{2}|\bar{H}^{-}_{2}|^{2}.$

Compressible vortex sheets are fundamental waves in the study of entropy
solutions to multidimensional hyperbolic conservation laws, arising in many
important physical phenomena. For the two-dimensional compressible flows
governed by the Euler equations, Fejer and Miles [16, 22, 23], see also [12,
35], proved that vortex sheets are unstable when $M<\sqrt{2}$, while the
vortex sheets in three-dimensional Euler flows are always violently unstable
(the violent instability is the analogue of the Kelvin-Helmholtz instability
for incompressible fluids).

In their pioneer works [13, 14] Coulombel and Secchi proved the nonlinear
stability and existence of two-dimensional vortex sheets when the Mach number
$M>\sqrt{2}$. The linear and nonlinear stability of vortex sheets has also
been established in [25, 26] for two-dimensional nonisentropic Euler flows, in
[4] for two-dimensional relativistic fluids, in [41, 42] for three-dimensional
steady Euler flows.

For the three-dimensional compressible flows, various stabilizing effects on
vortex sheets have been considered. Taking into account the effect of magnetic
fields, Chen-Wang [5, 6] and Trakhinin [38, 39] proved that large non-parallel
magnetic fields stabilize the motion of three dimensional current-vortex
sheets. Elasticity can also provide stabilization of vortex sheets: Chen-Hu-
Wang [7, 8] and Chen-Hu-Wang-Wang-Yuan [9] successfully proved the linear
stability and nonlinear stability, respectively, of two-dimensional
compressible vortex sheets in elastodynamics by introducing the upper-
triangularization method. More recently, Chen-Huang-Wang-Yuan [10] confirmed
the stabilizing effect of elasticity also on three dimensional compressible
vortex sheets. Another stabilizing effect on vortex sheets is provided by
surface tension: for the three-dimensional compressible Euler flows, the local
existence and structural stability were proved in Stevens [36].

The analysis of three-dimensional current-vortex sheets in [5, 6, 38, 39] does
not cover the two dimensional case. In fact, such a case can be considered as
the case when the third component of magnetic field
$\mathbf{H}=(H_{1},H_{2},H_{3})$ is zero, i.e. $H_{3}=0,$ and therefore the
non-collinear stability conditions in [5, 38, 39] fail. As was shown in
Trakhinin [38], the case when either tangential magnetic fields are collinear
or one of them is zero corresponds to the transition to violent instability.

For the two-dimensional compressible flows, Wang and Yu [40] proved the linear
stability of rectilinear current-vortex sheets under suitable stability
conditions by the spectral analysis technique, through the computation of the
roots of the Lopatinski determinant and construction of a Kreiss symmetrizer.

In the present paper we investigate the nonlinear stability and existence of
two dimensional current-vortex sheets. From the mathematical point of view,
this is a nonlinear hyperbolic free boundary problem. Since the
Kreiss–Lopatinski condition does not hold uniformly, there is a loss of
tangential derivatives in the estimates of the solution. The free boundary is
characteristic, which yields a possible loss of regularity in the normal
direction to the boundary, and the loss of control of the traces of the
characteristic part of the solution.

Differently from [40], we consider the nonisentropic flows and for the
analysis of linear stability, instead of spectral analysis, we use a direct
energy estimate argument, adapting the dissipative symmetrizer technique
introduced by Chen-Wang [5, 6] and Trakhinin [38, 39] for the three-
dimensional problem. Moreover, in our linear stability result we study a
general case of $2D$ current-vortex sheets, while in [40] both states of the
background magnetic field are assumed to have the same strength, see the
subsequent Remark 5.2.

First, we introduce a secondary symmetrization of the system of equations by
multiplying by a suitable “secondary generalized Friedrichs symmetrizer” and
impose the hyperbolicity of the new system of equations. Then, we identify a
sufficient stability condition that makes the boundary conditions dissipative
for the new symmetrized system. The new stability condition on the boundary
takes the form

$\frac{c_{A}^{+}}{\sqrt{1+\left(\frac{c^{+}_{A}}{c^{+}}\right)^{2}}}+\frac{c_{A}^{-}}{\sqrt{1+\left(\frac{c^{-}_{A}}{c^{-}}\right)^{2}}}-|[\mathbf{u}\cdot\tau]|>0\,,$
(1.11)

where $c^{\pm}_{A}={|\mathbf{H}^{\pm}|}/{\sqrt{\rho^{\pm}}}$ stands for the
Alfvén speed. This condition indicates that larger magnetic fields than the
jump of tangential velocity play a stabilization effect; in some sense this
corresponds to the “subsonic” bubble in linear stability result of [40], see
Remark 5.2. Condition (1.11) is in agreement with the stability result found
for the three-dimensional current-vortex sheets of [5, 38, 39] that only holds
in the subsonic regime.

We observe that condition (1.11) has a strong similarity with the Syrovatskij
stability condition [3, 21, 37] that is necessary and sufficient for the
linear stability of the two-dimensional incompressible current-vortex sheets.
For our problem we show that condition (1.11) is sufficient for the linear
stability and optimal with respect to the specific dissipative symmetrizer
technique that we use in our proof. On the other hand, it is likely that
(1.11) is not necessary for the linear and nonlinear stability. Indeed, by
taking the incompressible limit as $c^{\pm}\to+\infty$ in (1.11), we formally
get the inequality

$|{\mathbf{H}}^{+}|/\sqrt{\bar{\rho}}+|{\mathbf{H}}^{-}|/\sqrt{\bar{\rho}}-|[{\mathbf{u}}\cdot\tau]|>0\,,$

where, because the incompressible flow has uniform density, $\rho^{\pm}$ have
been replaced by a constant $\bar{\rho}>0$. This inequality describes somehow
the “half” of the whole $2D$ neutral stability domain from [21]. Moreover, in
Wang-Yu [40] even a “supersonic” linear stability domain is found for the
studied case of particular piece-wise constant background states; the same
kind of “supersonic” region is therefore expected to appear for a general case
of $2D$ current-vortex sheet, see again Remark 5.2.

Under condition (1.11) for the background state, we show that the linearized
problem obeys an energy estimate in anisotropic weighted Sobolev spaces (see
[11, 28, 30, 31]) with a loss of derivatives. The energy estimate for the
linearized problem takes the form of a $\it{tame}$ estimate, since it exhibits
a fixed loss of derivatives from the basic state to the solutions. In order to
compensate the loss of derivatives, for the proof of the existence of the
solution to the nonlinear problem, we apply a modified Nash-Moser iteration
scheme. For an introduction to the Nash-Moser technique, refer to [2, 33].

Compared to [5, 39] for the three-dimensional problem, our existence result
shows a lower loss of regularity from the initial data to the solution. This
is mainly due to the use of finer Moser-type and imbedding estimates in
anisotropic Sobolev spaces.

The rest of the paper is organized as follows. In Section 2 we formulate the
nonlinear problem for the current-vortex sheets in a fixed domain. In Section
3 we introduce the definition of anisotropic Sobolev spaces and then state our
main Theorem 3.1. In Section 4 we linearize the nonlinear problem with respect
to the basic state. Then, we introduce dissipative Friedrichs symmetrizer for
two dimensional MHD equations. In Section 5 we study the well-posedness of the
linearized problems (4.20), (4.31) and determine the stability condition. In
Sections 6 and 7 we prove the tame estimate in anisotropic Sobolev spaces for
the linearized problems (4.31) and (4.20), respectively. In Section 8 we
formulate the compatibility conditions for the initial data and construct the
approximate solution. In Section 9 we introduce the Nash-Moser iteration
scheme and in Section 10 we prove the existence of the solution to the
nonlinear problem. In this section one important step is the new construction
of the modified state, notably the modified magnetic field, and the delicate
derivation of its estimates. In Appendix A we recall the trace theorem in
anisotropic Sobolev space $H^{m}_{\ast}$ and in Appendix B we give a detailed
proof of the well-posedness of the homogeneous linearized problem (4.31)
stated in Theorem 5.1.

##  2\. Reformulate Current-Vortex Sheets Problem in a Fixed Domain

Let us reformulate the current-vortex sheets problem into an equivalent one
posed in a fixed domain. Motivated by Métivier [20], we introduce the
functions

$\Phi^{\pm}(t,\mathbf{x}):=\Phi(t,\pm x_{1},x_{2})=\pm
x_{1}+\Psi^{\pm}(t,\mathbf{x}),\quad\Psi^{\pm}(t,\mathbf{x}):=\chi(\pm
x_{1})\varphi(t,x_{2}),$

where $\chi\in C^{\infty}_{0}({\mathbb{R}}),\quad\chi\equiv 1\text{ on
}[-1,1],$ and $||\chi^{\prime}||_{L^{\infty}({\mathbb{R}})}\leq\frac{1}{2}$.
Here, as in [20], we use the cut-off function $\chi$ to avoid assumptions
about compact support of the initial data in our subsequent nonlinear
existence Theorem 3.1. The unknowns ${\mathbf{U}}^{\pm}$ are smooth in
$\Omega^{\pm}(t)$ and can be replaced by the following

${\mathbf{U}}^{\pm}_{\natural}(t,\mathbf{x}):={\mathbf{U}}^{\pm}(t,\Phi(t,\pm
x_{1},x_{2}),x_{2})\,,$

after changing the variables, which are smooth in the fixed domain
$\Omega={\mathbb{R}}^{2}_{+}=\\{x_{1}>0,x_{2}\in{\mathbb{R}}\\}$. Dropping
$\natural$ in ${\mathbf{U}}^{\pm}_{\natural}$ for convenience, we reduce
(1.1), (1.8) and (1.9) into the following initial boundary value problem
(IBVP)

$\begin{cases}\mathbb{L}({\mathbf{U}}^{\pm},\Psi^{\pm})=0&\text{in
}[0,T]\times{\mathbb{R}}^{2}_{+},\\\
\mathbb{B}({\mathbf{U}}^{+},{\mathbf{U}}^{-},\varphi)=0&\text{on
}[0,T]\times\Gamma,\\\
{\mathbf{U}}^{\pm}(0,\mathbf{x})={\mathbf{U}}^{\pm}_{0}&\text{ in
}{\mathbb{R}}^{2}_{+},\\\ \varphi(0,x_{2})=\varphi_{0}&\text{ in
}{\mathbb{R}},\end{cases}$ (2.1)

where $\Gamma:=\\{x_{1}=0\\}\times{\mathbb{R}},$
$\mathbb{L}({\mathbf{U}}^{\pm},\Psi^{\pm})=L({\mathbf{U}}^{\pm},\Psi^{\pm}){\mathbf{U}}^{\pm}$,

$L({\mathbf{U}}^{\pm},\Psi^{\pm})=A_{0}({\mathbf{U}}^{\pm})\partial_{t}+\tilde{A}_{1}({\mathbf{U}}^{\pm},\Psi^{\pm})\partial_{1}+A_{2}({\mathbf{U}}^{\pm})\partial_{2}\,,$
(2.2)

with

$\tilde{A}_{1}({\mathbf{U}}^{\pm},\Psi^{\pm})=\frac{1}{\partial_{1}\Phi^{\pm}}\Big{(}A_{1}({\mathbf{U}}^{\pm})-A_{0}({\mathbf{U}}^{\pm})\partial_{t}\Psi^{\pm}-A_{2}({\mathbf{U}}^{\pm})\partial_{2}\Psi^{\pm}\Big{)},$
(2.3)

where $\partial_{1}\Phi^{\pm}=\pm 1+\partial_{1}\Psi^{\pm}.$ The boundary
condition in (2.1) takes the following form

$\partial_{t}\varphi=u^{\pm}_{N},\quad[q]=q^{+}-q^{-}=0\text{ on
}[0,T]\times\\{x_{1}=0\\}\times{\mathbb{R}},$

where $u^{\pm}_{N}=u^{\pm}_{1}-u^{\pm}_{2}\partial_{2}\varphi$.
The following Lemma 2.1 yields that the divergence constraints (1.2) and the
boundary conditions $H^{\pm}_{N}|_{x_{1}=0}=0$ on $\Gamma$ (that is not
included in (2.1)) can be regarded as the conditions on the initial data. The
proof follows by similar calculations as in [39, Proposition 1].

######  Lemma 2.1.

Let the initial data in (2.1) satisfy the following

$\mathrm{div}\mathbf{h}^{\pm}=0,\quad\mbox{in}\,\,\,\mathbb{R}^{2}_{+}$ (2.4)

and the boundary conditions

$H^{\pm}_{N}|_{x_{1}=0}=0,\quad\mbox{on}\,\,\,\\{x_{1}=0\\}\times\mathbb{R}\,,$
(2.5)

where

$\mathbf{h}^{\pm}=(H^{\pm}_{n},H^{\pm}_{2}\partial_{1}\Phi^{\pm}),\quad
H^{\pm}_{n}=H^{\pm}_{1}-H^{\pm}_{2}\partial_{2}\Psi^{\pm},\quad
H^{\pm}_{N}|_{x_{1}=0}=H^{\pm}_{n}|_{x_{1}=0}.$

If the IBVP (2.1) admits a solution $({\mathbf{U}}^{\pm},\varphi),$ then this
solution satisfies (2.4) and (2.5) for all $t\in[0,T].$

##  3\. Properties of Function Spaces and Main Theorem

In this section, we first introduce some notations, then anisotropic Sobolev
spaces are defined. At the end, we are ready to state the main result of this
paper.

###  3.1. Notations

Let us denote $\Omega_{T}:=(-\infty,T)\times\Omega$ and
$\Gamma_{T}:=(-\infty,T)\times\Gamma$ for $T>0.$ We write
$\partial_{t}=\frac{\partial}{\partial
t},\partial_{i}=\frac{\partial}{\partial
x_{i}},i=1,2,\nabla_{t,\mathbf{x}}=(\partial_{t},\nabla).$
$D^{\alpha}_{\ast}:=\partial^{\alpha_{0}}_{t}(\sigma\partial_{1})^{\alpha_{1}}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}}_{1},\alpha:=(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}),\quad|\alpha|=\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}.$
Here $\sigma$ is an increasing smooth function, which satisfies
$\sigma(x_{1})=x_{1}$ for $0\leq x_{1}\leq\frac{1}{2}$ and $\sigma(x_{1})=1$
for $x_{1}\geq 1.$ The symbol $A\lesssim B$ represents that $A\leq CB$ holds
uniformly for some universal positive constant $C.$

###  3.2. Anisotropic Sobolev spaces

For any integer $m\in\mathbb{N}$ and the interval $I\subseteq{\mathbb{R}},$
function spaces $H^{m}_{\ast}(\Omega)$ and $H^{m}_{\ast}(I\times\Omega)$ are
defined by

$H^{m}_{\ast}(\Omega):=\\{u\in L^{2}(\Omega):D^{\alpha}_{\ast}u\in
L^{2}(\Omega)\text{ for }\langle\alpha\rangle:=|\alpha|+\alpha_{3}\leq
m,\text{ with }\alpha_{0}=0\\},$ $H^{m}_{\ast}(I\times\Omega):=\\{u\in
L^{2}(I\times\Omega):D^{\alpha}_{\ast}u\in L^{2}(I\times\Omega)\text{ for
}\langle\alpha\rangle:=|\alpha|+\alpha_{3}\leq m\\},$

and equipped with the norm $||\cdot||_{H^{m}_{\ast}(\Omega)}$ and
$||\cdot||_{H^{m}_{\ast}(I\times\Omega)}$ respectively, where

$||u||^{2}_{H^{m}_{\ast}(\Omega)}:=\sum_{\langle\alpha\rangle\leq
m,\alpha_{0}=0}||D^{\alpha}_{\ast}u||^{2}_{L^{2}(\Omega)},$ (3.1)
$||u||^{2}_{H^{m}_{\ast}(I\times\Omega)}:=\sum_{\langle\alpha\rangle\leq
m}||D^{\alpha}_{\ast}u||^{2}_{L^{2}(I\times\Omega)}.$ (3.2)

Define the norm

$|||u(t)|||^{2}_{m,\ast}:=\sum^{m}_{j=0}||\partial^{j}_{t}u(t)||^{2}_{H^{m-j}_{\ast}(\Omega)}\,.$
(3.3)

We also write $||\cdot||_{m,\ast,t}:=||u||_{H^{m}_{\ast}(\Omega_{t})}$ for
convenience. Then, from (3.2), we have

$||u||^{2}_{m,\ast,t}=\int^{t}_{-\infty}|||u(s)|||^{2}_{m,\ast}ds\,.$ (3.4)

###  3.3. Moser-type calculus inequalities

Now, we introduce two lemmata, which will be useful in the proof of tame
estimates in $H^{m}_{\ast}(\Omega_{T})$ for the problem (4.20) when $m$ is
large enough. We first introduce the Moser-type calculus inequalities in
$H^{m},$ see [2, Propositions 2.1.2 and 2.2].

######  Lemma 3.1.

Let $m\in\mathbb{N}_{+}.$ Let $\mathcal{O}$ be an open subset of
${\mathbb{R}}^{n}$ with Lipschitz boundary. Assume that $F\in C^{\infty}$ in a
neighbourhood of the origin with $F(0)=0$ and that $u,v\in
H^{m}(\mathcal{O})\cap L^{\infty}(\mathcal{O})$. Then,

$||\partial^{\alpha}u\,\partial^{\beta}v||_{L^{2}}\lesssim||u||_{H^{m}}||v||_{L^{\infty}}+||u||_{L^{\infty}}||v||_{H^{m}},$
$||F(u)||_{H^{m}}\leq C||u||_{H^{m}},$

for all $\alpha,\beta\in\mathbb{N}^{n}$ with $|\alpha|+|\beta|\leq m$ and
where $C$ depends only on $F$ and $\|u\|_{L^{\infty}}$.

Next, we introduce the Moser-type calculus inequalities for $H^{m}_{\ast}$.
Let us define the space

$W^{1,\infty}_{\ast}(\Omega_{T}):=\\{u\in
L^{\infty}(\Omega_{T})\,:\,\,\,D^{\alpha}_{\ast}u\in
L^{\infty}(\Omega_{T})\,,\,\,\langle\alpha\rangle\leq 1\\}\,,$

equipped with the natural norm

$||u||_{W^{1,\infty}_{\ast}(\Omega_{T})}:=\sum_{\langle\alpha\rangle\leq
1}||D^{\alpha}_{\ast}u||_{L^{\infty}(\Omega_{T})}\,,$ (3.5)

where $D^{\alpha}_{\ast}$ and $\langle\alpha\rangle$ are defined in Section
3.1.

######  Lemma 3.2 ([24, 39]).

Let $m\in\mathbb{N}_{+}.$ Assume that $F$ is a $C^{\infty}-$function and
$u,v\in H^{m}_{\ast}(\Omega_{T})\cap W^{1,\infty}(\Omega_{T})$. Then, there
hold

$||D^{\alpha}_{\ast}uD^{\beta}_{\ast}v||_{L^{2}(\Omega_{T})}\lesssim||u||_{m,\ast,T}||v||_{W^{1,\infty}_{\ast}(\Omega_{T})}+||v||_{m,\ast,T}||u||_{W^{1,\infty}_{\ast}(\Omega_{T})},$
(3.6)
$||uv||_{m,\ast,T}\lesssim||u||_{m,\ast,T}||v||_{W^{1,\infty}_{\ast}(\Omega_{T})}+||v||_{m,\ast,T}||u||_{W^{1,\infty}_{\ast}(\Omega_{T})},$
(3.7)

for any multi-index $\alpha,\beta\in\mathbb{N}^{4}$ with
$\langle\alpha\rangle+\langle\beta\rangle\leq m.$ Let $M_{\ast}$ be a positive
constant such that

$||u||_{W^{1,\infty}_{\ast}(\Omega_{T})}\leq M_{\ast}\,.$

Moreover, if we assume that $F(0)=0$, then there holds

$||F(u)||_{m,\ast,T}\leq C(M_{\ast})||u||_{m,\ast,T}\,.$ (3.8)

For the proof of (3.6) and (3.7), one can check [24]. For (3.8) one can check
[39].

###  3.4. Embedding and trace theorem

Now, we introduce the Sobolev embedding theorem for
$H^{m}_{\ast}(\Omega_{T})$.

######  Lemma 3.3 ([24]).

The following inequalities hold:

$||u||_{L^{\infty}(\Omega_{T})}\lesssim||u||_{3,\ast,T},\quad||u||_{W^{1,\infty}_{\ast}(\Omega_{T})}\lesssim||u||_{4,\ast,T},$
(3.9)
$||u||_{W^{1,\infty}(\Omega_{T})}\lesssim||u||_{5,\ast,T},\quad||u||_{W^{2,\infty}_{\ast}(\Omega_{T})}\lesssim||u||_{6,\ast,T},$
(3.10)

where $||u||_{W^{1,\infty}_{\ast}(\Omega_{T})}$ is defined by (3.5) and

$||u||_{W^{2,\infty}_{\ast}(\Omega_{T})}:=\sum_{\langle\alpha\rangle\leq
1}||D^{\alpha}_{\ast}u||_{W^{1,\infty}(\Omega_{T})}.$

###### Proof.

Using [24, Theorem B.4], we obtain the first inequality in (3.9) in
$\Omega_{T}\subseteq{\mathbb{R}}^{3}.$ Then, the second one in (3.9) can be
obtained by definition. Observing that

$||u||_{W^{1,\infty}(\Omega_{T})}\lesssim\sum_{\langle\alpha\rangle\leq
2}||D^{\alpha}_{\ast}u||_{L^{\infty}(\Omega_{T})},$

we can obtain the first inequality in (3.10) from the first inequality in
(3.9). Similarly the second one in (3.10) can be obtained by definition. ∎

For higher order energy estimate, we also need to use the following trace
theorem by Ohno, Shizuta, Yanagisawa [27] for the anisotropic Sobolev spaces
$H^{m}_{\ast},$ see also Appendix A.

######  Lemma 3.4 ([27]).

Let $m\geq 1$ be an integer. Then, the following arguments hold:
(i) If $u\in H^{m+1}_{\ast}(\Omega_{T}),$ then its trace $u|_{x_{1}=0}$
belongs to $H^{m}(\Gamma_{T}),$ and it satisfies

$||u|_{x_{1}=0}||_{H^{m}(\Gamma_{T})}\lesssim||u||_{m+1,\ast,T}.$

(ii) There exists a continuous lifting operator $\mathcal{R}_{T}:$

$H^{m}(\Gamma_{T})\rightarrow H^{m+1}_{\ast}(\Omega_{T}),$

such that $(\mathcal{R}_{T}u)|_{x_{1}=0}=u\text{ and
}||\mathcal{R}_{T}u||_{m+1,\ast,T}\lesssim||u||_{H^{m}({\Gamma_{T}})}.$

###  3.5. Main Theorem

######  Theorem 3.1.

Let $m\in\mathbb{N}$ and $m\geq 15,$ and let $\bar{\mathbf{U}}^{\pm}$ be
defined in (1.10). Suppose that the initial data in (2.1) satisfy

$({\mathbf{U}}^{\pm}_{0}-\bar{\mathbf{U}}^{\pm},\varphi_{0})\in
H^{m+11.5}({\mathbb{R}}^{2}_{+})\times H^{m+11.5}({\mathbb{R}}),$

and also satisfy the hyperbolicity condition (1.6), the divergence-free
constraint (2.4) for all $\mathbf{x}\in{\mathbb{R}}^{2}_{+}.$ Let the initial
data at $x_{1}=0$ satisfy the stability condition (1.11). The hyperbolicity
condition (1.6) and the stability condition (1.11) have to be satisfied
uniformly in the sense of (4.2) and (4.4), for suitable $k>0$. Assume that the
initial data are compatible up to order m+10 in the sense of Definition 8.1
and satisfy the boundary constraints (2.5). Assume also that

$\|\varphi_{0}\|_{L^{\infty}(\mathbb{R})}<\frac{1}{2}.$

Then, there exists a sufficiently short time $T>0$ such that problem (2.1) has
a unique solution on the time interval $[0,T]$ satisfying

$({\mathbf{U}}^{\pm}-\bar{\mathbf{U}}^{\pm},\varphi)\in
H^{m}_{\ast}([0,T]\times{\mathbb{R}}^{2}_{+})\times
H^{m}([0,T]\times{\mathbb{R}}).$

######  Remark 3.1.

Since the initial data ${\bf U}_{0}^{\pm}$ have the form ${\bf
U}_{0}^{\pm}=\overline{{\bf U}}^{\pm}+\tilde{{\bf U}}_{0}^{\pm}$, with
$\tilde{{\bf U}}_{0}^{\pm}\in H^{m}(\mathbb{R}^{2}_{+})$ vanishing at infinity
(as $|{\bf x}|\to+\infty$), the hyperbolicity and stability conditions
satisfied in the sense of (4.2), (4.4) (see Remark 4.1) yield that the same
conditions hold for the constant states (1.10).

######  Remark 3.2.

We note that Theorem 3.1 implies corresponding results for the original free
boundary problem (1.1), (1.8) and (1.9), posed in the moving domain, because
Lemma 2.1 and the relation $\partial_{1}\Phi^{+}\geq\frac{1}{2}$ and
$\partial_{1}\Phi^{-}\leq-\frac{1}{2}$ hold in
$[0,T]\times{\mathbb{R}}^{2}_{+}$ for sufficiently small $T>0.$

######  Remark 3.3.

Compared to [5, 39], in Theorem 3.1 there is less loss of regularity from the
initial data to the solution. This is mainly due to the use of finer Moser-
type and imbedding estimates in anisotropic Sobolev spaces.

######  Remark 3.4.

The analysis in this paper could also be applied to prove the nonlinear
stability and existence of two dimensional relativistic current-vortex sheets,
see [4, 17].

##  4\. Linearized Problem

###  4.1. The basic state

Let the basic state

$(\hat{{\mathbf{U}}}^{\pm}(t,\mathbf{x}),\hat{\varphi}(t,x_{2}))$ (4.1)

be a given vector-valued and sufficiently smooth function, where
$\hat{{\mathbf{U}}}^{\pm}=(\hat{p}^{\pm},\hat{\mathbf{u}}^{\pm},\hat{\mathbf{H}}^{\pm},\hat{S}^{\pm})^{T}$
are defined in $\Omega_{T}.$ We assume that we shall linearize the problem
(2.1) around the basic state (4.1), which satisfy the hyperbolicity condition
(1.6) in $\Omega_{T}:$

$\rho(\hat{p}^{\pm},\hat{S}^{\pm})\geq
k>0,\quad\rho_{p}(\hat{p}^{\pm},\hat{S}^{\pm})\geq k>0\,,$ (4.2)

the Rankine-Hugoniot jump condition

$\partial_{t}\hat{\varphi}=\hat{u}^{\pm}_{N}|_{x_{1}=0}\,,$ (4.3)

where
$\hat{u}^{\pm}_{N}=\hat{u}^{\pm}_{1}-\hat{u}^{\pm}_{2}\partial_{2}\hat{\varphi}$
and the stability condition on the boundary:

$\hat{a}^{+}|\hat{H}^{+}_{2}|+\hat{a}^{-}|\hat{H}^{-}_{2}|-|[\hat{u}_{2}]|\geq
k>0\,,$ (4.4)

for a suitable constant $k$, where
$\hat{c}^{\pm}_{A}=\frac{|\hat{\mathbf{H}}^{\pm}|}{\sqrt{\hat{\rho}^{\pm}}}$
denotes the Alfvén speed and

$\hat{a}^{\pm}:=\sqrt{\frac{1}{\hat{\rho}^{\pm}(1+\frac{(\hat{c}^{\pm}_{A})^{2}}{(\hat{c}^{\pm})^{2}})}}\,.$
(4.5)

######  Remark 4.1.

The stability condition (1.11) can be written in uniform form as (4.4) by
making use of (4.3) and the following boundary constraint in (4.10).

######  Remark 4.2.

Let us observe that, differently from [39], in the Rankine-Hugoniot conditions
we dot not require that the basic state satisfies $[\hat{q}]=0$. It seems that
this condition could not be helpful to simplify the boundary quadratic form
which appear from the application of the energy method to the linearized
problem.

######  Remark 4.3.

Estimate (4.4) implies in particular that at least one among $\hat{H}^{+}_{2}$
and $\hat{H}^{-}_{2}$ must be nonzero along the boundary.

######  Remark 4.4.

The presence of the positive constant $k$ in the hyperbolicity and stability
assumptions (4.2), (4.4) is needed in order to ensure the boundedness of all
coefficients, nonlinearly depending on the basic state
$\hat{\mathbf{U}}^{\pm}$, $\hat{\varphi}$, appearing in the arguments of the
energy method developed in the sequel.

Let us assume

$\begin{split}&\hat{{\mathbf{U}}}^{\pm}\in
W^{3,\infty}(\Omega_{T}),\quad\hat{\varphi}\in W^{4,\infty}(\Gamma_{T}),\\\
&||\hat{{\mathbf{U}}}^{\pm}||_{W^{3,\infty}(\Omega_{T})}+||\hat{\varphi}||_{W^{4,\infty}(\Gamma_{T})}\leq
K,\end{split}$ (4.6)

where $K>0$ is a constant. Moreover, without loss of generality, we assume

$||\hat{\varphi}||_{L^{\infty}(\Gamma_{T})}<\frac{1}{2}.$ (4.7)

This implies

$\partial_{1}\hat{\Phi}^{+}\geq\frac{1}{2},\quad\partial_{1}\hat{\Phi}^{-}\leq-\frac{1}{2},$

with

$\hat{\Phi}^{\pm}(t,\mathbf{x}):=\pm
x_{1}+\hat{\Psi}^{\pm}(t,\mathbf{x}),\quad\hat{\Psi}^{\pm}(t,\mathbf{x}):=\chi(\pm
x_{1})\hat{\varphi}(t,x_{2}).$

We also assume the following nonlinear constraints on the background states:

$\partial_{t}\hat{\mathbf{H}}^{\pm}+\frac{1}{\partial_{1}\hat{\Phi}^{\pm}}\Big{(}(\hat{\mathbf{w}}^{\pm}\cdot\nabla)\hat{\mathbf{H}}^{\pm}-(\hat{\mathbf{h}}\cdot\nabla)\hat{\mathbf{u}}^{\pm}+\hat{\mathbf{H}}^{\pm}\mathrm{div}{\hat{\mathrm{\mathbf{v}}}^{\pm}}\Big{)}=0,$
(4.8)

where

$\begin{split}&\hat{\mathbf{v}}^{\pm}=(\hat{u}^{\pm}_{n},\hat{u}^{\pm}_{2}\partial_{1}\hat{\Phi}^{\pm}),\quad\hat{u}^{\pm}_{n}=\hat{u}^{\pm}_{1}-\hat{u}^{\pm}_{2}\partial_{2}\hat{\Psi}^{\pm},\\\
&\hat{\mathbf{h}}^{\pm}=(\hat{H}^{\pm}_{n},\hat{H}^{\pm}_{2}\partial_{1}\hat{\Phi}^{\pm}),\quad\hat{H}^{\pm}_{n}=\hat{H}^{\pm}_{1}-\hat{H}^{\pm}_{2}\partial_{2}\hat{\Psi}^{\pm},\\\
&\hat{\mathbf{w}}^{\pm}=\hat{\mathbf{v}}^{\pm}-(\partial_{t}\hat{\Psi}^{\pm},0)\,.\end{split}$
(4.9)

Then, we can obtain that the constraints (2.4) and (2.5), that is

$\mathrm{div}\hat{\mathbf{h}}^{\pm}=0,\quad\hat{H}^{\pm}_{N}|_{x_{1}=0}=0,$
(4.10)

hold for all times $t>0$ if they hold initially (see Appendix A in [39]),
where
$\hat{H}^{\pm}_{N}=\hat{H}^{\pm}_{1}-\hat{H}^{\pm}_{2}\partial_{2}\hat{\varphi}.$

###  4.2. The linearized equations

The linearized equations for (2.1) around the basic state (4.1) can be defined
as follows:

$\mathbb{L^{\prime}}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})(\delta{\mathbf{U}}^{\pm},\delta\Psi^{\pm}):=\frac{d}{d\varepsilon}\mathbb{L}({\mathbf{U}}^{\pm}_{\varepsilon},\Psi^{\pm}_{\varepsilon})|_{\varepsilon=0}=\mathbf{f}^{\pm}\text{
in }\Omega_{T},$
$\mathbb{B^{\prime}}(\hat{{\mathbf{U}}}^{+},\hat{{\mathbf{U}}}^{-},\hat{\varphi})(\delta{\mathbf{U}}^{+},\delta{\mathbf{U}}^{-},\delta\varphi)=\frac{d}{d\varepsilon}\mathbb{B}({\mathbf{U}}^{+}_{\varepsilon},{\mathbf{U}}^{-}_{\varepsilon},\varphi_{\varepsilon})|_{\varepsilon=0}=\mathbf{g}\text{
on }\Gamma_{T},$

where
${\mathbf{U}}^{\pm}_{\varepsilon}=\hat{{\mathbf{U}}}^{\pm}+\varepsilon\delta{\mathbf{U}}^{\pm},\quad\varphi_{\varepsilon}=\hat{\varphi}+\varepsilon\delta\varphi$
and

$\Psi^{\pm}_{\varepsilon}(t,\mathbf{x}):=\chi(\pm
x_{1})\varphi_{\varepsilon}(t,x_{2}),\quad\Phi^{\pm}_{\varepsilon}(t,\mathbf{x}):=\pm
x_{1}+\Psi^{\pm}_{\varepsilon}(t,\mathbf{x}),\quad\delta\Psi^{\pm}(t,\mathbf{x}):=\chi(\pm
x_{1})\delta\varphi(t,x_{2}).$

In the following argument, we shall drop $\delta$ for simplicity. The
linearized operators have the following form:

$\begin{split}&\mathbb{L}^{\prime}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})({\mathbf{U}}^{\pm},\Psi^{\pm})\\\
&=L(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm}){\mathbf{U}}^{\pm}+C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm}){\mathbf{U}}^{\pm}-\\{L(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\Psi^{\pm}\\}\frac{\partial_{1}\hat{{\mathbf{U}}}^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}},\end{split}$
(4.11)
$\mathbb{B}^{\prime}(\hat{{\mathbf{U}}}^{+},\hat{{\mathbf{U}}}^{-},\hat{\varphi})({\mathbf{U}}^{+},{\mathbf{U}}^{-},\varphi)=\left[\begin{array}[]{c}\partial_{t}\varphi+\hat{u}^{+}_{2}\partial_{2}\varphi-u^{+}_{N}\\\
\partial_{t}\varphi+\hat{u}^{-}_{2}\partial_{2}\varphi-u^{-}_{N}\\\
q^{+}-q^{-}\end{array}\right],$ (4.12)

where the operator $L(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})$ is defined
in (2.2),
$q^{\pm}=p^{\pm}+\hat{\mathbf{H}}^{\pm}\cdot\mathbf{H}^{\pm},u^{\pm}_{N}=u^{\pm}_{1}-u^{\pm}_{2}\partial_{2}\hat{\varphi},$
and the matrix $C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})$ is defined as
follows:

$\begin{split}C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})Y=(Y,\nabla_{y}A_{0}(\hat{{\mathbf{U}}}^{\pm}))\partial_{t}\hat{{\mathbf{U}}}^{\pm}+(Y,\nabla_{y}\tilde{A}_{1}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm}))\partial_{1}\hat{{\mathbf{U}}}^{\pm}+(Y,\nabla_{y}A_{2}(\hat{{\mathbf{U}}}^{\pm}))\partial_{2}\hat{{\mathbf{U}}}^{\pm}\,,\end{split}$
$(Y,\nabla_{y}A(\hat{{\mathbf{U}}}^{\pm}))=\sum_{i=1}^{6}y_{i}(\frac{\partial
A(Y)}{\partial y_{i}}\Big{|}_{Y=\hat{{\mathbf{U}}}^{\pm}}),\quad
Y=(y_{1},\cdots,y_{6}).$

We introduce the Alinhac’s good unknown [1]

$\dot{{\mathbf{U}}}^{\pm}:={\mathbf{U}}^{\pm}-\frac{\partial_{1}\hat{{\mathbf{U}}}^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}}\Psi^{\pm}.$
(4.13)

In terms of (4.13), the linearized interior equations have the following form

$L(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}+C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}-\frac{\Psi^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}}\partial_{1}\\{\mathbb{L}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\\}=\mathbf{f^{\pm}}.$
(4.14)

Since the zero order terms in $\Psi^{\pm}$ can be regarded as the error terms
in the Nash-Moser iteration, we drop these terms and consider the effective
linear operators:

$\begin{split}\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}&:=L(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}+C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}\\\
&=A_{0}(\hat{{\mathbf{U}}}^{\pm})\partial_{t}\dot{{\mathbf{U}}}^{\pm}+\tilde{A}_{1}(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\partial_{1}\dot{{\mathbf{U}}}^{\pm}+A_{2}(\hat{{\mathbf{U}}}^{\pm})\partial_{2}\dot{{\mathbf{U}}}^{\pm}+C(\hat{{\mathbf{U}}}^{\pm},\hat{\Psi}^{\pm})\dot{{\mathbf{U}}}^{\pm}.\end{split}$
(4.15)

Concerning the boundary differential operator $\mathbb{B}^{\prime}$, we
rewrite (4.12) in terms of the Alinhac’s good unknowns (4.13) to get

$\mathbb{B}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\varphi})(\dot{{\mathbf{U}}},\varphi)=\left[\begin{array}[]{c}\partial_{t}\varphi+\hat{u}^{+}_{2}\partial_{2}\varphi-\dot{u}^{+}_{N}-\varphi\partial_{1}\hat{u}^{+}_{N}\\\
\partial_{t}\varphi+\hat{u}^{-}_{2}\partial_{2}\varphi-\dot{u}^{-}_{N}+\varphi\partial_{1}\hat{u}^{-}_{N}\\\
\dot{q}^{+}-\dot{q}^{-}+\varphi(\partial_{1}\hat{q}^{+}+\partial_{1}\hat{q}^{-})\end{array}\right],$
(4.16)

where
$\dot{u}^{\pm}_{N}=\dot{u}^{\pm}_{1}-\dot{u}^{\pm}_{2}\partial_{2}\hat{\varphi}$
(recall also that $\partial_{1}\hat{\Phi}^{\pm}|_{x_{1}=0}=\pm 1$).

######  Remark 4.5.

Notice that, due to the fact that we have transformed the domains
$\Omega^{\pm}(t)$ into the same half-space $\mathbb{R}^{2}_{+}$, the jump on
the boundary of a normal derivative of a function $a=a(t,{\bf x})$ is defined
as follows

$[\partial_{1}a]:=\partial_{1}a^{+}_{|x_{1}=0}+\partial_{1}a^{-}_{|x_{1}=0}.$
(4.17)

This is why the jump of the total pressure $q$ in the last row of (4.16)
reduces under Alinhac’s change of unknowns to

$\dot{q}^{+}-\dot{q}^{-}+\varphi(\partial_{1}\hat{q}^{+}+\partial_{1}\hat{q}^{-})\,.$

In the following, according to (4.17), we will set

$[\partial_{1}\hat{q}]:=\partial_{1}\hat{q}^{+}|_{x_{1}=0}+\partial_{1}\hat{q}^{-}|_{x_{1}=0}\,.$
(4.18)

Denote the operator:

$\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\dot{{\mathbf{U}}}:=\left[\begin{array}[]{c}\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}}^{+},\hat{\Psi}^{+})\dot{{\mathbf{U}}}^{+}\\\
\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}}^{-},\hat{\Psi}^{-})\dot{{\mathbf{U}}}^{-}\end{array}\right]\,.$
(4.19)

Now, we are focusing on the following linear problem for
$(\dot{{\mathbf{U}}}^{\pm},\varphi):$

$\begin{cases}\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\dot{{\mathbf{U}}}=\mathbf{f}&\text{in
}\Omega_{T},\\\
\mathbb{B}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\varphi})(\dot{{\mathbf{U}}},\varphi)=\mathbf{g}&\text{on
}\Gamma_{T},\\\ (\dot{{\mathbf{U}}},\varphi)=0&\text{for }t<0,\\\ \end{cases}$
(4.20)

where
$\mathbf{f}=(\mathbf{f}^{+},\mathbf{f}^{-})=(f^{+}_{1},\cdots,f^{+}_{6},f^{-}_{1},\cdots,f^{-}_{6}),$
and $\mathbf{g}=(g^{+}_{1},g^{-}_{1},g_{2})$ vanish in the past. In order to
prove the well-posedness of the linearized problem (4.20), we can state the
following Lemma 4.1, that can be proved as in [39, Proposition 2].

######  Lemma 4.1.

Let the basic state (4.1) satisfy the assumption (4.8) and (4.10). Then, the
solutions of the problem (4.20) satisfy

$\mathrm{div}\dot{\mathbf{h}}^{\pm}=r^{\pm}\text{ in }\Omega_{T},$ (4.21)
$\hat{H}_{2}^{\pm}\partial_{2}\varphi-\dot{H}^{\pm}_{N}\mp\varphi\partial_{1}\hat{H}^{\pm}_{N}=g^{\pm}_{3}\\\
\text{ on }\Gamma_{T}.$ (4.22)

Here

$\dot{\mathbf{h}}^{\pm}=(\dot{H}^{\pm}_{n},\dot{H}^{\pm}_{2}\partial_{1}\hat{\Phi}^{\pm}),\quad\dot{H}^{\pm}_{n}=\dot{H}^{\pm}_{1}-\dot{H}^{\pm}_{2}\partial_{2}\hat{\Psi}^{\pm},\quad\dot{H}^{\pm}_{N}|_{x_{1}=0}=\dot{H}^{\pm}_{n}|_{x_{1}=0},$

where $r^{\pm}=r^{\pm}(t,\mathbf{x}),\quad g^{\pm}_{3}=g^{\pm}_{3}(t,x_{2}),$
which vanish in the past, are determined by the source terms and the basic
state as solutions to the linear equations

$\partial_{t}R^{\pm}+\frac{1}{\partial_{1}\hat{\Phi}^{\pm}}\Big{\\{}(\hat{\mathbf{w}}^{\pm}\cdot\nabla
R^{\pm})+R^{\pm}\mathrm{div}\hat{\mathbf{v}}^{\pm}\Big{\\}}=\mathcal{F}^{\pm}\text{
in }\Omega_{T},$ (4.23)
$\partial_{t}g^{\pm}_{3}+\hat{u}^{\pm}_{2}\partial_{2}g^{\pm}_{3}+\partial_{2}\hat{u}^{\pm}_{2}g^{\pm}_{3}=\mathcal{G}^{\pm}\text{
on }\Gamma_{T},$ (4.24)

where
$R^{\pm}=\frac{r^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}},\quad\mathcal{F}^{\pm}=\frac{\mathrm{div}\mathbf{f}^{\pm}_{h}}{\partial_{1}\hat{\Phi}^{\pm}},\quad\mathbf{f}^{\pm}_{h}=(f^{\pm}_{n},f^{\pm}_{5}),\quad
f^{\pm}_{n}=f^{\pm}_{4}-f^{\pm}_{5}\partial_{2}\hat{\Psi}^{\pm}$,
$\mathcal{G}^{\pm}=\\{\partial_{2}(\hat{H}^{\pm}_{2}g^{\pm}_{1})-f^{\pm}_{n}\\}|_{x_{1}=0}$,
with $\hat{\mathbf{w}}^{\pm}$ and $\hat{\mathbf{v}}^{\pm}$ defined in (4.9).

###  4.3. Reduction to homogeneous boundary conditions

In this section, we reduce the inhomogeneous boundary condition in (4.20) to
the homogeneous one. We follow the same ideas in Trakhinin [39], that for
reader’s convenience, we recall here. Suppose there exists a solution
$(\dot{{\mathbf{U}}},\varphi)\in H^{s}_{\ast}(\Omega_{T})\times
H^{s}({\Gamma_{T}})$ to problem (4.20), with a given $s\in\mathbb{N}.$ We
define a vector-valued function

$\tilde{{\mathbf{U}}}=(\tilde{p}^{+},\tilde{\mathbf{u}}^{+},\tilde{\mathbf{H}}^{+},\tilde{S}^{+},\tilde{p}^{-},\tilde{\mathbf{u}}^{-},\tilde{\mathbf{H}}^{-},\tilde{S}^{-})\in
H^{s+2}_{\ast}(\Omega_{T})$

that vanishes in the past and such that it is a “suitable lifting” of the
boundary data $({\mathbf{g}},g^{+}_{3},g^{-}_{3})\in H^{s+1}(\Gamma_{T})$. We
choose $\tilde{{\mathbf{U}}}$ such that on the boundary $\Gamma_{T}$, it
satisfies the boundary conditions in (4.20) with $\varphi=0$, i.e.

$\mathbb{B}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\varphi})(\tilde{{\mathbf{U}}},0)=\mathbf{g},$

and both conditions (4.22) and (4.24) with $\varphi=0$ (where $\tilde{q}$,
$\tilde{u}^{\pm}_{N}$, $\tilde{H}_{N}^{\pm}$ are defined similarly to
$\dot{q}$, $\dot{u}_{N}^{\pm}$ and $\dot{H}_{N}^{\pm}$). More explicitly we
require

$\tilde{u}^{\pm}_{N}|_{x_{1}=0}=-g^{\pm}_{1},$
$\tilde{q}^{+}|_{x_{1}=0}-\tilde{q}^{-}|_{x_{1}=0}=g_{2},$
$\tilde{H}^{\pm}_{N}|_{x_{1}=0}=-g^{\pm}_{3}$

with $g^{\pm}_{3}$ solution of (4.24). Then, we define $\tilde{q}^{\pm}$,
$\tilde{u}_{n}^{\pm}$ and $\tilde{H}_{n}^{\pm}$ in the interior domain
$\Omega_{T}$ by using the lifting operator (that exists thanks to the trace
theorem in anisotropic Sobolev spaces $H^{s}_{*}$, see [27] and Appendix A)

$\mathcal{R}_{T}:H^{s+1}(\Gamma_{T})\rightarrow H^{s+2}_{*}(\Omega_{T})$

which gives

$\tilde{q}^{\pm}=\mathcal{R}_{T}(\tilde{q}^{\pm}|_{x_{1}=0}),\quad\tilde{u}^{\pm}_{n}=\mathcal{R}_{T}(\tilde{u}^{\pm}_{N}|_{x_{1}=0}),\quad\tilde{H}^{\pm}_{n}=\mathcal{R}_{T}(\tilde{H}^{\pm}_{N}|_{x_{1}=0}).$

Let also $\tilde{u}_{1}^{\pm}$ and $\tilde{H}^{\pm}_{1}$ be such that

$\tilde{u}_{n}^{\pm}=\tilde{u}_{1}^{\pm}-\tilde{u}_{2}^{\pm}\partial_{2}\hat{\Psi}^{\pm},\quad\tilde{H}_{n}^{\pm}=\tilde{H}_{1}^{\pm}-\tilde{H}_{2}^{\pm}\partial_{2}\hat{\Psi}^{\pm},$

where $\tilde{u}_{2}^{\pm}$ is arbitrary and can be taken for instance as
zero, and where we can define $\tilde{H}^{\pm}_{2}$ in such a way that it
satisfies equation (4.23) for
$R^{\pm}=\frac{\text{div}\tilde{\mathbf{h}}^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}}$,
where
$\tilde{\mathbf{h}}^{\pm}=(\tilde{H}_{n}^{\pm},\tilde{H}_{2}^{\pm}\partial_{1}\hat{\Phi}^{\pm})$
(this is possible since we have a freedom in the choice of “characteristic
unknown” $\tilde{H}^{\pm}_{2}$ ). The last components $\tilde{S}^{\pm}$ of
$\tilde{\mathbf{U}}$ can again be taken as zero. To sum up, the vector
$\tilde{\mathbf{U}}$ is defined as

$\tilde{{\mathbf{U}}}=(\tilde{p}^{+},\tilde{u}^{+}_{n},0,\tilde{H}^{+}_{n}+\tilde{H}^{+}_{2}\partial_{2}\hat{\Psi}^{+},\tilde{H}^{+}_{2},0,\tilde{p}^{-},\tilde{u}^{-}_{n},0,\tilde{H}^{-}_{n}+\tilde{H}^{-}_{2}\partial_{2}\hat{\Psi}^{-},\tilde{H}^{-}_{2},0)\,,$

where $\tilde{H}^{\pm}_{2}$ satisfies equation (4.23) for
$R^{\pm}=\frac{\text{div}\tilde{\mathbf{h}}^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}}$
and
$\tilde{\mathbf{h}}^{\pm}=(\tilde{H}_{n}^{\pm},\tilde{H}_{2}^{\pm}\partial_{1}\hat{\Phi}^{\pm})$.
We define
$\dot{{\mathbf{U}}}^{\natural}=\dot{{\mathbf{U}}}-\tilde{{\mathbf{U}}},$ then
$\dot{{\mathbf{U}}}^{\natural}$ satisfies

$\begin{cases}\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\dot{{\mathbf{U}}}^{\natural}=\mathbf{F}=\mathbf{f}-\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\tilde{{\mathbf{U}}}&\text{
in }\Omega_{T},\\\
\mathbb{B}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})(\dot{{\mathbf{U}}}^{\natural},\varphi)=0&\text{
on }\Gamma_{T}.\end{cases}$ (4.25)

Here $\mathbf{F}=(F^{+}_{1},\cdots,F^{+}_{6},F^{-}_{1},\cdots,F^{-}_{6}).$
Moreover, in view of equations (4.23) for
$R^{\pm}=\frac{\mathrm{div}\tilde{\mathbf{h}}^{\pm}}{\partial_{1}\hat{\Phi}^{\pm}},$
condition (4.22) for $\tilde{\mathbf{U}}$ with $\varphi=0,$ and (4.24), we
have, from (4.25), that (4.23) and (4.24) are satisfied for
$R^{\pm}=\frac{\mathrm{div}\dot{\mathbf{h}}^{\natural\pm}}{\partial_{1}\hat{\Phi}^{\pm}},$
$g^{\pm}_{3}=(\hat{H}^{\pm}_{2}\partial_{2}\varphi-\dot{H}^{\natural\pm}_{N}\mp\varphi\partial_{1}\hat{H}^{\pm}_{N})|_{x_{1}=0}$
with right-hand sides $\mathcal{F}^{\pm}=\mathcal{G}^{\pm}=0$. Here
$\dot{\mathbf{h}}^{\natural\pm}$ and $\dot{H}^{\natural\pm}_{N}$ are defined
similarly to $\dot{\mathbf{h}}^{\pm}$ and $\dot{H}^{\pm}_{N}.$ Notice that
$\dot{{\mathbf{U}}}^{\natural}=0,\text{ for }t<0.$ Hence, the conditions

$\mathrm{div}\dot{\mathbf{h}}^{\natural\pm}=0,\quad(\hat{H}^{\pm}_{2}\partial_{2}\varphi-\dot{H}^{\natural\pm}_{N}{\mp}\varphi\partial_{1}\hat{H}^{\pm}_{N})|_{x_{1}=0}=0$
(4.26)

hold for $t<0$. Then, by standard method of characteristic curves, we get that
equations (4.26) are satisfied for all $t\in(-\infty,T].$ Notice that from

$||\partial_{1}(\cdot)||_{s,\ast,T}\leq||\cdot||_{s+2,\ast,T},$

we have that

$||\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\tilde{{\mathbf{U}}}||_{s,\ast,T}\leq
C||\tilde{{\mathbf{U}}}||_{s+2,\ast,T}.$ (4.27)

By the definition of ${\mathbf{F}}$, using (4.27) and the definition of
$\tilde{\mathbf{U}}$ as a lifting of the boundary data
$({\mathbf{g}},g_{3}^{+},g_{3}^{-})$, we obtain that

$||\mathbf{F}||_{s,\ast,T}\leq
C\Big{(}||\mathbf{f}||_{s,\ast,T}+||\mathbf{g}||_{H^{s+1}(\Gamma_{T})}+||(g_{3}^{+},g_{3}^{-})||_{H^{s+1}(\Gamma_{T})}\Big{)}.$
(4.28)

Using (4.24) and the trace Theorem in the anisotropic Sobolev spaces to
estimate

$\|f_{n}|_{x_{1}=0}\|_{H^{s+1}(\Gamma_{T})}\leq
C\|\mathbf{f}\|_{s+2,\ast,T}\,,$

we get

$\begin{split}||(g^{+}_{3},g^{-}_{3})||_{H^{s+1}(\Gamma_{T})}\leq
C||(\mathcal{G}^{+},\mathcal{G}^{-})||_{H^{s+1}(\Gamma_{T})}\leq
C\Big{(}||\mathbf{f}||_{s+2,\ast,T}+||(g^{+}_{1},g^{-}_{1})||_{H^{s+2}(\Gamma_{T})}\Big{)}.\end{split}$
(4.29)

Then, from (4.28) and (4.29), we derive

$||\mathbf{F}||_{s,\ast,T}\leq
C\Big{(}||\mathbf{f}||_{s+2,\ast,T}+||\mathbf{g}||_{H^{s+2}(\Gamma_{T})}\Big{)}.$
(4.30)

We obtain that $\dot{{\mathbf{U}}}^{\natural}$ solves the following problem:

$\begin{cases}\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\dot{{\mathbf{U}}^{\natural}}=\mathbf{F}&\text{
in }\Omega_{T},\\\
\mathbb{B}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\varphi})(\dot{{\mathbf{U}}^{\natural}},\varphi)=0&\text{
on }\Gamma_{T},\\\ (\dot{{\mathbf{U}}^{\natural}},\varphi)=0&\text{ for
}t<0,\end{cases}$ (4.31)

where the source term $\mathbf{F}$ satisfies the estimate (4.30) and solutions
of (4.31) satisfy the constraints

$\mathrm{div}\dot{\mathbf{h}^{\natural}}^{\pm}=0\text{ in }\Omega_{T},$ (4.32)
$\hat{H}^{\pm}_{2}\partial_{2}\varphi-\dot{H}^{\natural\,\pm}_{N}{\mp}\varphi\partial_{1}\hat{H}^{\pm}_{N}=0\text{
on }\Gamma_{T}.$ (4.33)

We have thus proved the following result.

######  Lemma 4.2.

Let problem (4.31) be well-posed and its unique solution
$(\dot{{\mathbf{U}}^{\natural}},\varphi)$ belongs to
$H^{s}_{\ast}(\Omega_{T})\times H^{s}({\Gamma_{T}})$ for $\mathbf{F}\in
H^{s}_{\ast}(\Omega_{T}),$ where $s\in\mathbb{N}$ is a given number. Then
problem (4.20) is well-posed, namely it admits a unique solution
$(\dot{\mathbf{U}},\varphi)\in H^{s}_{\ast}(\Omega_{T})\times
H^{s}({\Gamma_{T}})$ for data $(\mathbf{f},\mathbf{g})\in
H^{s+2}_{\ast}(\Omega_{T})\times H^{s+2}(\Gamma_{T}).$

######  Remark 4.6.

Let us observe that loss of regularity from the data to the solution in the
inhomogeneous problem (4.20) is due to the introduction of lifting function
$\tilde{{\mathbf{U}}}$.

###  4.4. New Friedrichs Symmetrizer for 2D MHD equations

Motivated by the idea of Trakhinin [39], in the following we will make use of
a new symmetric form of the MHD system. This symmetric form is the result of
the application of a “secondary generalized Friedrichs symmetrizer”
$\mathbb{S}=(S({\mathbf{U}}),\mathcal{T}({\mathbf{U}}))$ to system (1.4):

$\begin{split}&\mathcal{S}({\mathbf{U}})A_{0}({\mathbf{U}})\partial_{t}{\mathbf{U}}+\mathcal{S}({\mathbf{U}})A_{1}({\mathbf{U}})\partial_{1}{\mathbf{U}}+\mathcal{S}({\mathbf{U}})A_{2}({\mathbf{U}})\partial_{2}{\mathbf{U}}+\mathcal{T}({\mathbf{U}})\mathrm{div}\mathbf{H}\\\
&\quad:=B_{0}({\mathbf{U}})\partial_{t}{\mathbf{U}}+B_{1}({\mathbf{U}})\partial_{1}{\mathbf{U}}+B_{2}({\mathbf{U}})\partial_{2}{\mathbf{U}}=0.\end{split}$
(4.34)

The Friedrichs symmetrizer can be written as (see [38])

$\mathcal{S}({\mathbf{U}})=\left[\begin{array}[]{cccccc}1&\frac{\lambda
H_{1}}{\rho c^{2}}&\frac{\lambda H_{2}}{\rho c^{2}}&0&0&0\\\ \lambda
H_{1}\rho&1&0&-\rho\lambda&0&0\\\ \lambda H_{2}\rho&0&1&0&-\rho\lambda&0\\\
0&-\lambda&0&1&0&0\\\ 0&0&-\lambda&0&1&0\\\
0&0&0&0&0&1\end{array}\right],\quad\mathcal{T}({\mathbf{U}})=-\lambda\left[\begin{array}[]{c}1\\\
0\\\ 0\\\ H_{1}\\\ H_{2}\\\ 0\end{array}\right],$ (4.35)
$B_{0}=\mathcal{S}A_{0}=\left[\begin{array}[]{cccccc}\frac{1}{\rho
c^{2}}&\frac{\lambda H_{1}}{c^{2}}&\frac{\lambda H_{2}}{c^{2}}&0&0&0\\\
\frac{\lambda H_{1}}{c^{2}}&\rho&0&-\rho\lambda&0&0\\\ \frac{\lambda
H_{2}}{c^{2}}&0&\rho&0&-\rho\lambda&0\\\ 0&-\rho\lambda&0&1&0&0\\\
0&0&-\rho\lambda&0&1&0\\\ 0&0&0&0&0&1\end{array}\right],$ (4.36)

where $\lambda=\lambda(\mathbf{U})$ is an arbitrary function.
In order to make system (4.34) symmetric hyperbolic, we need $B_{0}>0,$ i.e.,

$\rho\lambda^{2}<\frac{1}{1+\frac{c^{2}_{A}}{c^{2}}}.$ (4.37)

Condition (4.37) ensures the equivalence of system (1.4) and (4.34) on smooth
solutions if $\lambda({\mathbf{U}})$ is a smooth function of ${\mathbf{U}}$
(see Trakhinin [38] and [39]).
Now let us apply the new symmetrization to the homogeneous linearized problem
(4.31). From now on we drop the index $\natural$ from the unknown
$\dot{\mathbf{U}}^{\natural}$ of system (4.31).
Multiplying (4.31) on the left by $\mathcal{S}(\hat{{\mathbf{U}}})$ and adding
to the result the vector

$\left[\begin{array}[]{c}\frac{\mathrm{div}\dot{\mathbf{h}}^{+}}{\partial_{1}\hat{\Phi}^{+}}\mathcal{T}(\hat{{\mathbf{U}}}^{+})\\\
\frac{\mathrm{div}\dot{\mathbf{h}}^{-}}{\partial_{1}\hat{\Phi}^{-}}\mathcal{T}(\hat{{\mathbf{U}}}^{-})\\\
\end{array}\right],$ (4.38)

we obtain

$B_{0}(\hat{{\mathbf{U}}})\partial_{t}\dot{{\mathbf{U}}}+\tilde{B}_{1}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{1}\dot{{\mathbf{U}}}+B_{2}(\hat{{\mathbf{U}}})\partial_{2}\dot{{\mathbf{U}}}+\tilde{C}(\hat{{\mathbf{U}}},\hat{\Psi})\dot{{\mathbf{U}}}=\tilde{\mathbf{F}}(\hat{{\mathbf{U}}}),$
(4.39)

where
$\tilde{C}(\hat{{\mathbf{U}}},\hat{\Psi})=\mathcal{S}(\hat{{\mathbf{U}}})C(\hat{{\mathbf{U}}},\hat{\Psi}),\quad\tilde{\mathbf{F}}(\hat{{\mathbf{U}}})=\mathcal{S}(\hat{{\mathbf{U}}}){\mathbf{F}}$,

$S(\hat{{\mathbf{U}}})=\mathrm{diag}(\mathcal{S}(\hat{{\mathbf{U}}}^{+}),S(\hat{{\mathbf{U}}}^{-})),\quad
B_{\alpha}(\hat{{\mathbf{U}}})=\mathrm{diag}(B_{\alpha}(\hat{{\mathbf{U}}}^{+}),B_{\alpha}(\hat{{\mathbf{U}}}^{-})),\,\,\,\alpha=0,2\,,$
$C(\hat{{\mathbf{U}}},\hat{\Psi})=\mathrm{diag}(C(\hat{{\mathbf{U}}}^{+},\hat{\Psi}^{+}),C(\hat{{\mathbf{U}}}^{-},\hat{\Psi}^{-})),\quad\tilde{B}_{1}(\hat{{\mathbf{U}}},\hat{\Psi})=\mathrm{diag}(\tilde{B}_{1}(\hat{{\mathbf{U}}}^{+},\hat{\Psi}^{+}),\tilde{B}_{1}(\hat{{\mathbf{U}}}^{-},\hat{\Psi}^{-})),$
$\tilde{B}_{1}({\mathbf{U}}^{\pm},\Psi^{\pm})=\frac{1}{\partial_{1}\Phi^{\pm}}\Big{(}B_{1}({\mathbf{U}}^{\pm})-B_{0}({\mathbf{U}}^{\pm})\partial_{t}\Psi^{\pm}-B_{2}({\mathbf{U}}^{\pm})\partial_{2}\Psi^{\pm}\Big{)}.$

##  5\. Stability condition and well-posedness of the linearized problem

In this section, let us introduce the new unknown
$\mathbf{V}=(\mathbf{V}^{+},\mathbf{V}^{-}),$ where

$\mathbf{V}^{\pm}=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{u}^{\pm}_{2},\dot{H}^{\pm}_{n},\dot{H}^{\pm}_{2},\dot{S}^{\pm}).$

Rewriting system (4.31) in terms of $\mathbf{V}$ shows in clear way that the
boundary matrix of the resulting system has constant rank at the boundary (see
(5.6)), i.e. the system is symmetric hyperbolic with characteristic boundary
of constant multiplicity (in the sense of Rauch [29]). Indeed, we obtain that

$\dot{{\mathbf{U}}}=J{\mathbf{V}}\,,$ (5.1)

where $J=\mathrm{diag}(J^{+},J^{-}),$

$J^{\pm}=\left[\begin{array}[]{cccccc}1&0&0&-\hat{H}^{\pm}_{1}&-\hat{H}^{\pm}_{\tau}&0\\\
0&1&\partial_{2}\hat{\Psi}^{\pm}&0&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&1&\partial_{2}\hat{\Psi}^{\pm}&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\end{array}\right],$ (5.2)

with
$\hat{H}^{\pm}_{\tau}=\hat{H}\cdot\tau^{\pm},\quad\tau^{\pm}=(\partial_{2}\hat{\Psi}^{\pm},1).$
In terms of $\mathbf{V}$, systems (4.31) and (4.39) can be equivalently
rewritten as:

$\mathcal{A}_{0}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{t}{\mathbf{V}}+\mathcal{A}_{1}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{1}{\mathbf{V}}+\mathcal{A}_{2}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{2}{\mathbf{V}}+\mathcal{A}_{3}(\hat{{\mathbf{U}}},\hat{\Psi}){\mathbf{V}}=\mathcal{F}(\hat{{\mathbf{U}}},\hat{\Psi}),$
(5.3)
$\mathcal{B}_{0}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{t}{\mathbf{V}}+\mathcal{B}_{1}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{1}{\mathbf{V}}+\mathcal{B}_{2}(\hat{{\mathbf{U}}},\hat{\Psi})\partial_{2}{\mathbf{V}}+\mathcal{B}_{3}(\hat{{\mathbf{U}}},\hat{\Psi}){\mathbf{V}}=\tilde{\mathcal{F}}(\hat{{\mathbf{U}}},\hat{\Psi}),$
(5.4)

where
$\mathcal{A}_{\alpha}=J^{T}A_{\alpha}J,\quad\mathcal{B}_{\alpha}=J^{T}B_{\alpha}J,\quad\alpha=0,2,$

$\mathcal{A}_{1}=J^{T}\tilde{A}_{1}J,\quad\mathcal{B}_{1}=J^{T}\tilde{B}_{1}J,\quad\mathcal{F}=J^{T}\mathbf{F},\quad\tilde{\mathcal{F}}=J^{T}\tilde{\mathbf{F}},$
$\mathcal{A}_{3}=J^{T}\Big{(}CJ+A_{0}\partial_{t}J+\tilde{A}_{1}\partial_{1}J+A_{2}\partial_{2}J\Big{)},$
$\mathcal{B}_{3}=J^{T}\Big{(}CJ+B_{0}\partial_{t}J+\tilde{B}_{1}\partial_{1}J+B_{2}\partial_{2}J\Big{)}.$

In view of constraints (4.3) and (4.10) on the basic state $\hat{\mathbf{U}}$,
the boundary matrix in (5.3) has the following form

$\mathcal{A}_{1}=\mathcal{A}+\mathcal{A}_{(0)},\quad\mathcal{A}=\mathrm{diag}\Big{(}\frac{1}{\partial_{1}\hat{\Phi}^{+}}\mathcal{E}_{12},\frac{1}{\partial_{1}\hat{\Phi}^{-}}\mathcal{E}_{12}\Big{)},\quad\mathcal{A}_{(0)}|_{x_{1}=0}=0.$
(5.5)

Here, $\mathcal{E}_{ij}$ is a $6\times 6$ symmetric matrix, in which $(ij)$th
and $(ji)$th elements are 1 and the remaining elements are 0. The explicit
form of $\mathcal{A}_{(0)}$ is of no interest and it is only important that
the all non zero elements of $\mathcal{A}_{(0)}$ are multiplied either by the
function $\hat{u}_{n}-\partial_{t}\hat{\Psi}$ or by the function
$\hat{H}_{n}$. Therefore, the boundary matrix

$\mathcal{A}_{1}|_{x_{1}=0}=\mathrm{diag}(\mathcal{E}_{12},-\mathcal{E}_{12}).$
(5.6)

This is a matrix of constant rank 4 and has two positive and two negative
eigenvalues.
Concerning system (5.4), note that $\mathcal{B}_{0}>0$ because of the
hyperbolicity condition (4.37) satisfied for the basic state, hence the
symmetric system (5.4) is hyperbolic. Moreover, considering the boundary
matrix $\mathcal{B}_{1}$ in (5.4), we will only need its explicit form on the
boundary:

$\mathcal{B}_{1}|_{x_{1}=0}=\mathrm{diag}\Big{(}\mathcal{B}(\hat{{\mathbf{U}}}^{+}|_{x_{1}=0}),-\mathcal{B}(\hat{{\mathbf{U}}}^{-}|_{x_{1}=0})\Big{)},\quad\mathcal{B}(\hat{\mathbf{U}}^{\pm})=\mathcal{E}_{12}-\lambda(\hat{\mathbf{U}}^{\pm})\mathcal{E}_{14},$

which gives

$(\mathcal{B}_{1}{\mathbf{V}},{\mathbf{V}})|_{x_{1}=0}=2[\dot{q}(\dot{u}_{N}-\hat{\lambda}\dot{H}_{N})],$
(5.7)

with $\hat{\lambda}:=\lambda(\hat{\mathbf{U}})$.
Using (4.31) and (4.33) (see also (4.16)) for $\dot{u}^{\pm}_{N}$ and
$\dot{H}^{\pm}_{N},$ (recall that we have dropped the index $\natural$ from
the unknown) and the boundary constraint
$[\dot{q}]=-\varphi[\partial_{1}\hat{q}]$ (recall the definition of
$[\partial_{1}\hat{q}]$ in (4.18)) we obtain that

$(\mathcal{B}_{1}{\mathbf{V}}\cdot{\mathbf{V}})=2[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}]\dot{q}^{+}\partial_{2}\varphi+\mbox{l.o.t}\,,\quad\mbox{on}\,\,\,\\{x_{1}=0\\}\,,$
(5.8)

with

$\begin{split}\mbox{l.o.t.}:=&-2[\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{1}\hat{H}_{N}]\dot{q}^{+}\varphi-2[\partial_{1}\hat{q}]\varphi\partial_{t}\varphi-2[\partial_{1}\hat{q}](\hat{u}^{-}_{2}-\hat{\lambda}\hat{H}^{-}_{2})\varphi\partial_{2}\varphi\\\
&-2[\partial_{1}\hat{q}](\partial_{1}\hat{u}^{-}_{N}-\hat{\lambda}^{-}\partial_{1}\hat{H}^{-}_{N})\varphi^{2}\,.\end{split}$
(5.9)

We use “l.o.t.” to mean boundary terms that can be manipulated in the energy
estimate by passing to volume integral and using integration by parts, so that
they do not give any trouble in the derivation of the energy estimate, see
Appendix B.

Let us now make a suitable choice of the function
$\hat{\lambda}=\lambda(\hat{\mathbf{U}})$. We first take $\hat{\lambda}$ as
follows, see [39],

$\hat{\lambda}^{\pm}:=\lambda(\hat{{\mathbf{U}}}^{\pm}):=\eta(x_{1})\lambda^{\pm}(t,x_{2}),$
(5.10)

with $\eta(x_{1})$ a smooth monotone decreasing function satisfying
$\eta(0)=1$ and $\eta(x_{1})=0,$ for $x_{1}>\varepsilon$, with $\varepsilon>0$
sufficiently small. The functions $\lambda^{\pm}$ will be chosen below.

######  Remark 5.1.

The motivation of the definition of $\hat{\lambda}$ by using the cut-off
function $\eta=\eta(x_{1})$ is that it guarantees that the hyperbolicity
condition (4.37), that trivially holds for $\hat{\lambda}=0$ remains true for
the basic state in a small neighborhood $\\{0<x_{1}<\varepsilon\\}$ of the
boundary, thanks to the continuity of the basic state. Hence, by definition of
$\eta$, the hyperbolicity condition still holds in the whole domain
$\Omega_{T}$, see [39] for more details.

The functions $\lambda^{\pm}(t,x_{2})$ are chosen in this way: if the jump
$[\hat{u}_{2}](t,x_{2})=0,$ we define $\lambda^{\pm}(t,x_{2})=0$; otherwise,
if the jump $[\hat{u}_{2}](t,x_{2})\neq 0,$ we choose $\lambda^{+}(t,x_{2})$
and $\lambda^{-}(t,x_{2})$ satisfying the following relation

$[\hat{u}_{2}-\lambda\hat{H}_{2}]=0.$

The following Lemma ensures the existence of such functions
$\lambda^{+}(t,x_{2})$, $\lambda^{-}(t,x_{2})$ satisfying the above equation
and the hyperbolicity assumption (4.37) written for the basic state, see
(5.12).

######  Lemma 5.1.

For all $(t,x_{2})\in\Gamma_{T}$, there exist $\lambda^{\pm}(t,x_{2})$
satisfying

$[\hat{u}_{2}-\lambda\hat{H}_{2}]=0,$ (5.11)

and the hyperbolicity conditions

$|{\lambda}^{\pm}|<\hat{a}^{\pm}$ (5.12)

if and only if the basic states $\hat{\mathbf{U}}^{\pm}$ obey the stability
estimate

$|[\hat{u}_{2}]|<|\hat{H}^{+}_{2}|\hat{a}^{+}+|\hat{H}^{-}_{2}|\hat{a}^{-}\,,$
(5.13)

where $\hat{a}^{\pm}$ are defined in (4.5). In particular, under (5.13) we can
set

$\begin{split}\lambda^{+}=sgn(\hat{H}^{+}_{2})\frac{\hat{a}^{+}[\hat{u}_{2}]}{\hat{a}^{+}|\hat{H}^{+}_{2}|+\hat{a}^{-}|\hat{H}^{-}_{2}|},\\\
\lambda^{-}=-sgn(\hat{H}^{-}_{2})\frac{\hat{a}^{-}[\hat{u}_{2}]}{\hat{a}^{+}|\hat{H}^{+}_{2}|+\hat{a}^{-}|\hat{H}^{-}_{2}|}.\end{split}$
(5.14)

###### Proof.

For shortness in the proof we drop the hat and the subscript $2$ on the
variables, moreover we do not write the considered variables are restricted
along the boundary $\\{x_{1}=0\\}$ that is we write

$u=\hat{u}^{\pm}_{2}|_{x_{1}=0},\quad
H^{\pm}=\hat{H}^{\pm}_{2}|_{x_{1}=0},\quad\text{etc.}\,.$

Let us first note that equation (5.11) can be restated as

$[u]=\lambda^{+}H^{+}-\lambda^{-}H^{-}.$ (5.15)

Let us consider the simple case $H^{+}\neq 0$ and $H^{-}=0$. Then, from
equation (5.15) we get at once

$\lambda^{+}=\frac{[u]}{H^{+}}$

and plugging the above into the constraint in (5.12) for $\lambda^{+}$ gives

$|[u]|<a^{+}|H^{+}|\,,$ (5.16)

that is (5.13). In this case $\lambda^{-}$ can be whatever function satisfying
the corresponding constraint (5.12)

$|\lambda^{-}|<a^{-}$

(for instance $\lambda^{-}\equiv 0$); it does not yield any condition on the
background state in addition to (5.16). Of course the symmetric case $H^{+}=0$
and $H^{-}\neq 0$ is similar with

$\lambda^{-}=-\frac{[u]}{H^{-}},\quad\lambda^{+}\equiv 0$

and the condition

$|[u]|<a^{-}|H^{-}|$ (5.17)

on the background state, which is again (5.13).
Let us consider now the case $H^{+}\neq 0$ and $H^{-}\neq 0$. From (5.15), we
write $\lambda^{+}$ as a known function of $\lambda^{-}$ as

$\lambda^{+}=\frac{[u]+\lambda^{-}H^{-}}{H^{+}}$ (5.18)

and plug the above into $|\lambda^{+}|<a^{+}$ to get, after simple
manipulations,

$-a^{+}-\frac{[u]}{H^{+}}<\frac{H^{-}}{H^{+}}\lambda^{-}<a^{+}-\frac{[u]}{H^{+}}.$
(5.19)

We have that $\lambda^{-}$ must obey simultaneously the two constraints (5.19)
and

$-a^{-}<\lambda^{-}<a^{-}.$ (5.20)

Let assume now that $\frac{H^{-}}{H^{+}}>0$. Then equation (5.19) becomes

$b_{2}<\lambda^{-}<b_{1},$ (5.21)

where

$b_{1}=\frac{H^{+}}{H^{-}}\left(a^{+}-\frac{[u]}{H^{+}}\right)\quad\mbox{and}\quad
b_{2}=\frac{H^{+}}{H^{-}}\left(-a^{+}-\frac{[u]}{H^{+}}\right).$ (5.22)

A necessary condition for (5.20) and (5.21) to hold simultaneously is that

$b_{1}>-a^{-}\quad\text{and}\quad a^{-}>b_{2}.$ (5.23)

After some calculations, in view of the definition (5.22), the above becomes
equivalent to

$-\left(\frac{H^{+}}{H^{-}}a^{+}+a^{-}\right)<\frac{[u]}{H_{-}}<\frac{H^{+}}{H^{-}}a^{+}+a^{-}.$

Assuming $H^{-}<0$ (hence $H^{+}<0$) the above is equivalent to

$|[u]|<-\left(H^{+}a^{+}+H^{-}a^{-}\right)=|H^{+}|a^{+}+|H^{-}|a^{-},$

which is the estimate (5.13). Similar calculation in the case $H^{-}>0$ and
$H^{+}>0$ still yield to (5.13).

To sum up if $\frac{H^{-}}{H^{+}}>0$ we get that condition (5.13) is at least
necessary in order to find $\lambda^{-}$ satisfying (5.20) and (5.21), that is

$\max\\{-a^{-},b_{2}\\}<\lambda^{-}<\min\\{a^{-},b_{1}\\}.$

If such $\lambda^{-}$ actually exists, then $\lambda^{+}$ will be defined by
(5.18).

From (5.19), arguing in the same way as above when $\frac{H^{-}}{H^{+}}<0$ we
obtain once again that (5.13) is at least necessary for the existence of such
a $\lambda^{-}$ satisfying both (5.19) and (5.20), that in this case are
equivalent to

$\max\\{-a^{-},b_{1}\\}<\lambda^{-}<\min\\{a^{-},b_{2}\\}\,,$

where $b_{1}$ and $b_{2}$ are defined in (5.22).

To complete the proof, it remains to show that condition (5.13) is also
sufficient for the existence of functions $\lambda^{\pm}$ satisfying (5.11)
and (5.12).
We already proved it in the simplest cases $H^{+}\neq 0$ and $H^{-}=0$ or
viceversa. Now let us do the same in the case $H^{+}\neq 0$ and $H^{-}\neq 0$.
So let us assume that the background state satisfies condition (5.13). If we
assume for a while that $\lambda^{\pm}$ exist and satisfy (5.11) and (5.12),
we derive that

$|[u]|\leq|\lambda^{+}||H^{+}|+|\lambda^{-}||H^{-}|<a^{+}|H^{+}|+a^{-}|H^{-}|\,,$

that is

$\frac{|[u]|}{a^{+}|H^{+}|+a^{-}|H^{-}|}<1\,.$ (5.24)

The last inequality (5.24) suggests to take $\lambda^{\pm}$ such that

$|\lambda^{\pm}|=\frac{a^{\pm}|[u]|}{a^{+}|H^{+}|+a^{-}|H^{-}|}\,.$ (5.25)

Indeed from (5.24) it immediately follows that $\lambda^{\pm}$ as above
satisfy the constraint (5.12). Formula (5.25) defines $\lambda^{\pm}$ up to
the sign. One can directly check that taking

$\lambda^{+}=sgn(H^{+})\frac{a^{+}[u]}{a^{+}|H^{+}|+a^{-}|H^{-}|}\quad\mbox{and}\quad\lambda^{-}=-sgn(H^{-})\frac{a^{-}[u]}{a^{+}|H^{+}|+a^{-}|H^{-}|}$
(5.26)

makes the equation (5.11) to be satisfied. The above definitions of
$\lambda^{\pm}$ are just equal to (5.14). This ends the proof. ∎

######  Remark 5.2.

In order to make a comparison between our stability condition (5.13) and the
subsonic part of the stability condition found in [40], let us consider the
case of background piece-wise constant state

$\hat{\mathbf{U}}^{\pm}:=(\hat{\rho}^{\pm},0,\hat{u}^{\pm}_{2},0,\hat{H}_{2}^{\pm},\hat{S}^{\pm})^{T}\,.$
(5.27)

Then the right-hand side of estimate (5.13) can be restated in terms of sound
speeds $\hat{c}^{\pm}$ and the Alfvén speeds
$\hat{c}^{\pm}_{A}=\frac{|\hat{\mathbf{H}}^{\pm}|}{\sqrt{\rho^{\pm}}}=\frac{|\hat{H}^{\pm}_{2}|}{\sqrt{\rho^{\pm}}}$
as follows

$\hat{a}^{+}|\hat{H}_{2}^{+}|+\hat{a}^{-}|\hat{H}_{2}^{-}|=\frac{\hat{c}^{+}\hat{c}^{+}_{A}}{((\hat{c}^{+})^{2}+(\hat{c}^{+}_{A})^{2})^{1/2}}+\frac{\hat{c}^{-}\hat{c}^{-}_{A}}{((\hat{c}^{-})^{2}+(\hat{c}^{-}_{A})^{2})^{1/2}}$

and (5.13) becomes

$|[\hat{u}_{2}]|<\frac{\hat{c}^{+}\hat{c}^{+}_{A}}{((\hat{c}^{+})^{2}+(\hat{c}^{+}_{A})^{2})^{1/2}}+\frac{\hat{c}^{-}\hat{c}^{-}_{A}}{((\hat{c}^{-})^{2}+(\hat{c}^{-}_{A})^{2})^{1/2}}\,.$
(5.28)

In their two dimensional linear stability analysis [40], Wang and Yu are able
to perform a complete normal modes analysis of the linearized problem for an
isentropic flow, when the planar piece-wise constant basic state (5.27)
(without $\hat{S}^{\pm}$) satisfies suitable technical restrictions.
Precisely, the constant components of $\hat{{\mathbf{U}}}^{\pm}$ are required
to satisfy, besides the Rankine–Hugoniot conditions

$\hat{u}_{2}^{+}+\hat{u}_{2}^{-}=0,\quad\hat{u}_{2}^{+}>0,\quad\hat{p}^{+}+\frac{(\hat{H}^{+}_{2})^{2}}{2}=\hat{p}^{-}+\frac{(\hat{H}^{+}_{2})^{2}}{2}\,,$
(5.29)

the following additional assumptions

$|\hat{H}_{2}^{+}|=|\hat{H}_{2}^{-}|=:|\hat{H}_{2}|,\quad\hat{c}>\hat{c}_{A}=\sqrt{\frac{|\hat{H}_{2}|^{2}}{\hat{\rho}}}$
(5.30)

(here the notation of [40] is adapted to our current setting). Notice in
particular that, since the flow is isentropic, the assumption
$|\hat{H}_{2}^{+}|=|\hat{H}_{2}^{-}|$ and the last Rankine–Hugoniot condition
in (5.29) imply $\hat{\rho}^{-}=\hat{\rho}^{+}=:\hat{\rho}$ so that the sound
speed and the Alfvén speed take the same value
$\hat{c}=c(\hat{\rho}):=\sqrt{p^{\prime}(\hat{\rho})}$ and
$\hat{c}_{A}:=\sqrt{\frac{|\hat{H}_{2}|^{2}}{\hat{\rho}}}$ on both
$\pm$-states of the flow.
The linear stability conditions by Wang-Yu [40] read as follows:

$(\hat{u}_{2}^{+})^{2}<\hat{c}^{2}-\hat{c}^{2}\sqrt{\frac{\hat{c}^{2}-\hat{c}^{2}_{A}}{\hat{c}^{2}+\hat{c}^{2}_{A}}}\qquad\text{or}\qquad(\hat{u}^{+}_{2})^{2}>\hat{c}^{2}+\hat{c}^{2}\sqrt{\frac{\hat{c}^{2}-\hat{c}^{2}_{A}}{\hat{c}^{2}+\hat{c}^{2}_{A}}}\,.$
(5.31)

The left inequality in (5.31) identifies a “subsonic” weak stability region,
becoming empty in the absence of a magnetic field (namely for compressible
Euler equations), whereas the right inequality corresponds to the “supersonic”
weak stability region of $2D$ compressible vortex sheets (to which it reduces,
when formally setting $\hat{c}_{A}=0$). With respect to (5.31), our stability
condition in (5.28) provides only a subsonic stability domain. For a planar
isentropic background state obeying (5.30), condition (5.28) reduces to

$(\hat{u}_{2}^{+})^{2}<\frac{\hat{c}^{2}_{A}\hat{c}^{2}}{(\hat{c})^{2}+(\hat{c}_{A})^{2}}.$
(5.32)

To compare the left inequality in (5.31) and (5.32), one can easily check that

$\frac{\hat{c}^{2}_{A}\hat{c}^{2}}{(\hat{c})^{2}+(\hat{c}_{A})^{2}}<\hat{c}^{2}-\hat{c}^{2}\sqrt{\frac{\hat{c}^{2}-\hat{c}_{A}^{2}}{\hat{c}^{2}+\hat{c}_{A}^{2}}}\,,$

so that our subsonic condition (5.32) is more restrictive than the one in
[40].
Let us also note that we are not able, by our approach, to obtain a
counterpart of the supersonic condition in [40] for isentropic flows in the
constant coefficients case (that is the right inequality in (5.31)). On the
other hand, our stability condition applies also to nonisentropic flows and to
general (non piece-wise constant) background states.

We are now in the position to state the well-posedness of the “homogeneous”
linearized problem (4.31).

######  Theorem 5.1.

Let all assumptions (4.2)-(4.10) be satisfied for the basic state (4.1) (note
that (4.4) implies the stability condition (5.13)). Then, for all
$\mathbf{F}\in H^{1}_{\ast}(\Omega_{T})$ that vanish in the past, problem
(4.31) has a unique solution $(\dot{{\mathbf{U}}}^{\natural},\varphi)\in
H^{1}_{\ast}(\Omega_{T})\times H^{1}(\Gamma_{T}).$ The solution satisfies the
following a priori estimate

$||\dot{{\mathbf{U}}}^{\natural}||_{1,\ast,T}+||\varphi||_{H^{1}(\Gamma_{T})}\leq
C||\mathbf{F}||_{1,\ast,T},$ (5.33)

where $C=C(K,T)>0$ is a constant independent of the data $\mathbf{F}$.

The proof of Theorem 5.1 is given in Appendix B.

As a consequence of Lemma 4.2 we have the following well-posedness result for
the nonhomogeneous problem (4.20).

######  Theorem 5.2.

Let all assumptions of Theorem 5.1 be satisfied for the basic state (4.1).
Then, for all $(\mathbf{f},\mathbf{g})\in H^{3}_{\ast}(\Omega_{T})\times
H^{3}(\Gamma_{T})$ that vanish in the past, problem (4.20) has a unique
solution $(\dot{{\mathbf{U}}},\varphi)\in H^{1}_{\ast}(\Omega_{T})\times
H^{1}(\Gamma_{T})$. The solution satisfies the following a priori estimate

$||\dot{{\mathbf{U}}}||_{1,\ast,T}+||\varphi||_{H^{1}(\Gamma_{T})}\leq
C\Big{(}||\mathbf{f}||_{3,\ast,T}+||\mathbf{g}||_{H^{3}(\Gamma_{T})}\Big{)},$
(5.34)

where $C=C(K,T)>0$ is a constant independent of the data
$\mathbf{f},\mathbf{g}.$

##  6\. Higher-order energy estimates for the homogeneous problem (4.31)

In order to get an a priori tame estimate in $H^{s}_{\ast}$ for solution of
problem (4.20) with $s$ sufficiently large, as a preliminary step we derive a
tame estimate for the homogenous problem (4.31), that we state in the
following theorem.

For shortness, in all this section we write
$\dot{{\mathbf{U}}}^{\natural}=\dot{{\mathbf{U}}}$.

######  Theorem 6.1.

Let $T>0$ and $s$ be positive integer, $s\geq 6.$ Assume that the basic state
$(\breve{\mathbf{U}}^{\pm},\hat{\varphi})\in H^{s+4}_{\ast}(\Omega_{T})\times
H^{s+5}(\Gamma_{T}),$

$||\breve{\mathbf{U}}^{\pm}||_{9,\ast,T}+||\hat{\varphi}||_{H^{10}(\Gamma_{T})}\leq\hat{K},\text{
where }\breve{\mathbf{U}}^{\pm}:=\hat{\mathbf{U}}^{\pm}-\bar{\mathbf{U}}.$
(6.1)

Assume that $\mathbf{F}\in H^{s}_{\ast}(\Omega_{T})$ vanishes in the past.
Then, there exists a positive constant $K_{0},$ that does not depend on $s$
and $T,$ and there exists a constant $C(K_{0})>0$ such that if $\hat{K}\leq
K_{0},$ then there exists a unique solution $(\dot{{\mathbf{U}}},\varphi)\in
H^{s}_{\ast}(\Omega_{T})\times H^{s}(\Gamma_{T})$ to homogeneous problem
(4.31) satisfying the following estimate:

$\begin{split}||\dot{{\mathbf{U}}}||_{s,\ast,T}+||\varphi||_{H^{s}(\Gamma_{T})}\leq
C(K_{0})\Big{(}||\mathbf{F}||_{s,\ast,T}+||\mathbf{F}||_{6,\ast,T}||\hat{W}||_{s+4,\ast,T}\Big{)}\,,\end{split}$
(6.2)

for $T$ small enough, where $\hat{W}=(\breve{\mathbf{U}},\nabla_{t,\bf
x}\hat{\Psi}).$

To prove the above theorem we need to obtain several higher-order energy
estimates.

###  6.1. Estimate of the normal derivative of the “non-characteristic”
unknown

In this section we will prove the estimate of the normal derivative of the
“non-characteristic” unknown

${\mathbf{V}}_{n}=({\mathbf{V}}^{+}_{n},{\mathbf{V}}^{-}_{n}),\quad{\mathbf{V}}^{\pm}_{n}=(\dot{q}^{\pm},\dot{u}^{\pm}_{n},\dot{H}^{\pm}_{n}).$

Let $s$ be positive integer, we need to estimate
$\partial_{1}{\mathbf{V}}_{n}$ in $H^{s-1}_{\ast},$ obtained from (5.3) and
the divergence constraint (4.32) as follows

$\partial_{1}{\mathbf{V}}^{+}_{n}=\left[\begin{array}[]{c}\mathcal{K}_{1}\\\
\mathcal{K}_{2}\\\
-\partial_{2}(\dot{H}^{+}_{2}\partial_{1}\hat{\Phi}^{+})\end{array}\right],\quad\partial_{1}{\mathbf{V}}^{-}_{n}=\left[\begin{array}[]{c}\mathcal{K}_{7}\\\
\mathcal{K}_{8}\\\
-\partial_{2}(\dot{H}^{-}_{2}\partial_{1}\hat{\Phi}^{-})\end{array}\right]\,.$
(6.3)

Here $\mathcal{K}_{i}\in{\mathbb{R}}$ is the $i-$th scalar component of the
vector

$\mathcal{K}:=\tilde{\mathcal{A}}\Big{(}\mathcal{F}-\mathcal{A}_{0}\partial_{t}{\mathbf{V}}-\mathcal{A}_{2}\partial_{2}{\mathbf{V}}-\mathcal{A}_{3}{\mathbf{V}}-\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}}\Big{)},$
(6.4)

where
$\tilde{\mathcal{A}}=\mathrm{diag}\Big{(}(\partial_{1}\hat{\Phi}^{+})\mathcal{E}_{12},(\partial_{1}\hat{\Phi}^{-})\mathcal{E}_{12}\Big{)},$
$\partial_{1}\hat{\Phi}^{\pm}=\pm 1+\partial_{1}\hat{\Psi}^{\pm},$
$\mathcal{F}=J^{T}\mathbf{F}.$

Now, we are ready to prove the following Lemma, which is needed in the proof
of weighted normal derivatives and non-weighted tangential derivatives, see
Section 6.2 and Section 6.4.

######  Lemma 6.1.

The following estimate

$||\partial_{1}{\mathbf{V}}_{n}||^{2}_{s-1,\ast,t}\leq C(K)\mathcal{M}(t),$
(6.5)

with

$\begin{split}\mathcal{M}(t)&=||({\mathbf{V}},\mathbf{F})||^{2}_{s,\ast,t}+(||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}+||\mathbf{F}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})})||\hat{W}||^{2}_{s+2,\ast,t}\\\
\end{split}$ (6.6)

holds for problem (4.31) for all $t\leq T.$

###### Proof.

Using Moser-type calculus inequalities (3.7) and (3.8), we can estimate the
right-hand side of (6.4):

$\begin{split}||\tilde{\mathcal{A}}\mathcal{F}||^{2}_{s-1,\ast,t}\leq
C(K)\Big{(}||\mathbf{F}||^{2}_{s-1,\ast,t}+||\mathbf{F}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s-1,\ast,t}\Big{)}.\end{split}$
(6.7)

For $j=0,2,$

$\begin{split}||\tilde{\mathcal{A}}\mathcal{A}_{j}\partial_{j}{\mathbf{V}}||^{2}_{s-1,\ast,t}&\lesssim||\tilde{\mathcal{A}}\mathcal{A}_{j}{\mathbf{V}}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)},\end{split}$
(6.8)
$\begin{split}||\tilde{\mathcal{A}}\mathcal{A}_{3}{\mathbf{V}}||^{2}_{s-1,\ast,t}\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s-1,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s+1,\ast,t}\Big{)},\end{split}$
(6.9)

where $C(K)$ stands for positive constants that depends on $K$ and we used
that $\tilde{\mathcal{A}}J^{T}$, $\tilde{\mathcal{A}}\mathcal{A}_{j}$ for
$j=0,2$ and $\tilde{\mathcal{A}}\mathcal{A}_{3}$ are all nonlinear smooth
functions of the basic state $\hat{W}$.

Now, we estimate the last term in (6.4), where for simplicity we denote
$\mathcal{A}=\tilde{\mathcal{A}}\mathcal{A}_{(0)}$. We get

$\begin{split}||\mathcal{A}\partial_{1}{\mathbf{V}}||_{s-1,\ast,t}&=||\frac{\mathcal{A}}{\sigma}\sigma\partial_{1}{\mathbf{V}}||_{s-1,\ast,t}\\\
&\leq||\frac{\mathcal{A}}{\sigma}||_{W^{1,\infty}_{\ast}(\Omega_{t})}||\sigma\partial_{1}{\mathbf{V}}||_{s-1,\ast,t}+||\frac{\mathcal{A}}{\sigma}||_{s-1,\ast,t}||\sigma\partial_{1}{\mathbf{V}}||_{W^{1,\infty}_{\ast}(\Omega_{t})}\\\
&\leq||\mathcal{A}||_{W^{2,\infty}_{\ast}(\Omega_{t})}||{\mathbf{V}}||_{s,\ast,t}+||\mathcal{A}||_{s+1,\ast,t}||{\mathbf{V}}||_{W^{2,\infty}_{\ast}(\Omega_{t})}\\\
&\leq||\hat{W}||_{W^{2,\infty}_{\ast}(\Omega_{t})}||{\mathbf{V}}||_{s,\ast,t}+||\hat{W}||_{s+1,\ast,t}||{\mathbf{V}}||_{W^{2,\infty}_{\ast}(\Omega_{t})}\,,\end{split}$
(6.10)

where we used that $\mathcal{A}|_{x_{1}=0}=0$, thus the $L^{\infty}-$norm of
$\mathcal{A}/\sigma$ can be estimated by the $L^{\infty}$-norms of
$\mathcal{A}$ and $\partial_{1}\mathcal{A}$, see [24, Lemma B.9], [32].

For the rest of the term in (6.3), using (3.7) and (3.8), we obtain that

$\begin{split}||\partial_{2}(\dot{H}^{+}_{2}\partial_{1}\hat{\Phi}^{+})||^{2}_{s-1,\ast,t}&\leq||\dot{H}^{+}_{2}\partial_{1}\hat{\Phi}^{+}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}.\end{split}$
(6.11)

The second term in (6.3) can be controlled similarly. Summarizing all the
estimates (6.7)-(6.11) for the terms in (6.3) and (6.4), Lemma 6.1 is
concluded. ∎

We also need the following estimate, still for $\partial_{1}{\mathbf{V}}_{n}$,
which is also essential in the proof of non-weighted tangential derivatives,
see (6.47) in Section 6.4.

######  Lemma 6.2.

The estimate

$|||\partial_{1}{\mathbf{V}}_{n}(t)|||^{2}_{s-1,\ast}\leq
C(K)\Big{(}|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+\mathcal{M}(t)\Big{)}$ (6.12)

holds for problem (4.31) for all $t\leq T$, where $s$ is a positive integer
and $\mathcal{M}(t)$ is defined in (6.6).

###### Proof.

Denote the differential operator
$D^{\alpha}_{\ast}=\partial^{\alpha_{0}}_{t}(\sigma\partial_{1})^{\alpha_{1}}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}}_{1}$,
$\langle\alpha\rangle:=|\alpha|+\alpha_{3}.$ We estimate the right-hand side
of (6.4). Using the elementary estimate

$|||u(t)|||^{2}_{s-1,\ast}\lesssim||u||^{2}_{s,\ast,t},$ (6.13)

we get

$\begin{split}|||(\tilde{\mathcal{A}}\mathcal{F})(t)|||^{2}_{s-1,\ast}&\lesssim||\tilde{\mathcal{A}}J^{T}\mathbf{F}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||\mathbf{F}||^{2}_{s,\ast,t}+||\mathbf{F}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}\\\
&\leq C(K)\mathcal{M}(t).\end{split}$ (6.14)

The second term of (6.4) can be controlled as follows:

$\begin{split}|||\tilde{\mathcal{A}}\mathcal{A}_{0}&\partial_{t}{\mathbf{V}}(t)|||^{2}_{s-1,\ast}\lesssim\sum\limits_{\langle\alpha\rangle\leq
s-1}||\tilde{\mathcal{A}}\mathcal{A}_{0}D^{\alpha}_{\ast}\partial_{t}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\\\
&+\sum\limits_{\langle\alpha\rangle\leq
s-1}||D_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{0})D^{\alpha-1}_{\ast}\partial_{t}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\sum\limits_{\langle\alpha\rangle=2}||D^{\alpha}_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{0})\partial_{t}{\mathbf{V}}(t)||^{2}_{s-3,\ast}\\\
&=\Sigma_{1}^{\prime}+\Sigma^{\prime}_{2}+\Sigma_{3}^{\prime}\,,\end{split}$

with $D_{\ast}$ denoting any tangential derivative in $t$ or ${\bf x}$ of
order one and $D^{\alpha-1}_{\ast}$ denoting the derivative of order
$\langle\alpha\rangle-1$ obtained “subtracting” $D_{\ast}$ from
$D_{\ast}^{\alpha}$.
We estimate separately each $\Sigma^{\prime}_{i}$, $i=1,2,3$ as follows:

$\Sigma_{1}^{\prime}\lesssim||\hat{W}||^{2}_{L^{\infty}(\Omega_{t})}|||\mathbf{V}(t)|||^{2}_{s,\ast}\leq
C(K)|||\mathbf{V}(t)|||^{2}_{s,\ast}\,;$
$\Sigma_{2}^{\prime}\lesssim||\hat{W}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}|||\mathbf{V}(t)|||^{2}_{s-1,\ast}\leq
C(K)|||\mathbf{V}(t)|||^{2}_{s,\ast}\,;$
$\begin{split}\Sigma_{3}^{\prime}&\lesssim\sum\limits_{\langle\alpha\rangle=2}||D^{\alpha}_{\ast}(\tilde{A}\mathcal{A}_{0})\partial_{t}\mathbf{V}|||^{2}_{s-2,\ast,t}\\\
&\lesssim\sum\limits_{\langle\alpha\rangle=2}\left(||D^{\alpha}_{\ast}(\tilde{A}\mathcal{A}_{0})||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\partial_{t}\mathbf{V}||^{2}_{s-2,\ast,t}+||D^{\alpha}_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{0})||^{2}_{s-2,\ast,t}||\partial_{t}\mathbf{V}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}\right)\\\
&\leq
C(K)||\mathbf{V}||^{2}_{s-1,\ast,t}+||\hat{W}||^{2}_{s,\ast,t}||\mathbf{V}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}\,.\end{split}$

Adding the above inequalities, we get

$|||\tilde{\mathcal{A}}\mathcal{A}_{0}\partial_{t}{\mathbf{V}}(t)|||^{2}_{s-1,\ast}\leq
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+C(K)||\mathbf{V}||^{2}_{s-1,\ast,t}+||\hat{W}||^{2}_{s,\ast,t}||\mathbf{V}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}\,.$
(6.15)

Similarly, the third term can be estimated by

$|||\tilde{\mathcal{A}}\mathcal{A}_{2}\partial_{2}{\mathbf{V}}(t)|||^{2}_{s-1,\ast}\leq
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+C(K)||\mathbf{V}||^{2}_{s-1,\ast,t}+||\hat{W}||^{2}_{s,\ast,t}||\mathbf{V}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}.$
(6.16)

Using Moser-type calculus inequalities (3.7) and (3.8), we can estimate

$\begin{split}|||(\tilde{\mathcal{A}}\mathcal{A}_{3}\mathbf{V})(t)|||^{2}_{s-1,\ast}&\lesssim||\tilde{\mathcal{A}}\mathcal{A}_{3}\mathbf{V}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||\mathbf{V}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}\\\
&\leq C(K)\mathcal{M}(t).\end{split}$ (6.17)

Now, we estimate the last term
$\tilde{\mathcal{A}}\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}}:$

$|||(\tilde{\mathcal{A}}\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}})(t)|||^{2}_{s-1,\ast}\lesssim\Sigma_{1}+\Sigma_{2}+\Sigma_{3}\,,$

where:

$\begin{split}\Sigma_{1}&=\sum_{\langle\alpha\rangle\leq
s-1}||\tilde{\mathcal{A}}\mathcal{A}_{(0)}D^{\alpha}_{\ast}\partial_{1}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\\\
&=\sum_{\langle\alpha\rangle\leq
s-1}||\frac{\tilde{\mathcal{A}}\mathcal{A}_{(0)}}{\sigma}\sigma
D^{\alpha}_{\ast}\partial_{1}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\\\
&\lesssim||\tilde{\mathcal{A}}\mathcal{A}_{(0)}||^{2}_{W^{1,\infty}(\Omega_{t})}|||\mathbf{V}(t)|||^{2}_{s,\ast}\lesssim||\hat{W}||_{W^{1,\infty}(\Omega_{t})}^{2}|||\mathbf{V}(t)|||^{2}_{s,\ast}\,;\end{split}$
$\begin{split}\Sigma_{2}&=\sum\limits_{\langle\alpha\rangle\leq
s-1}||\frac{D_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{(0)})}{\sigma}\sigma
D^{\alpha-1}_{\ast}\partial_{1}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\\\
&\lesssim||D_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{(0)})||^{2}_{W^{1,\infty}(\Omega_{t})}|||\mathbf{V}(t)|||^{2}_{s-1,\ast}\lesssim||\hat{W}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}|||\mathbf{V}(t)|||^{2}_{s-1,\ast}\,;\end{split}$
$\begin{split}\Sigma_{3}&=\sum\limits_{\langle\alpha\rangle=2}|||D^{\alpha}_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{(0)})\partial_{1}\mathbf{V}(t)|||^{2}_{s-3,\ast}\lesssim\sum\limits_{\langle\alpha\rangle=2}|||D^{\alpha}_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{(0)})\partial_{1}\mathbf{V}|||^{2}_{s-2,\ast,t}\\\
&\lesssim||\hat{W}||^{2}_{W^{3,\infty}(\Omega_{t})}||\mathbf{V}||^{2}_{s,\ast,t}+||\hat{W}||^{2}_{s,\ast,t}||\mathbf{V}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})},\end{split}$

and where we exploit again the vanishing of
$\tilde{\mathcal{A}}\mathcal{A}_{(0)}$ and
$D_{\ast}(\tilde{\mathcal{A}}\mathcal{A}_{(0)})$ along the boundary
$\\{x_{1}=0\\}$ in the estimates of $\Sigma_{1}$ and $\Sigma_{2}$, see [24,
Lemma B.9], [32].
Adding the estimates of $\Sigma_{i}$, $i=1,2,3$ above, we get

$|||(\tilde{\mathcal{A}}\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}})(t)|||^{2}_{s-1,\ast}\leq
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+C(K)||\mathbf{V}||^{2}_{s,\ast,t}+||\hat{W}||^{2}_{s,\ast,t}||\mathbf{V}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}\,.$
(6.18)

For the rest of the term in (6.3), using (3.7) and (3.8), we obtain that

$\begin{split}|||\partial_{2}(\dot{H}^{+}_{2}\partial_{1}\hat{\Phi}^{+})(t)|||^{2}_{s-1,\ast,t}&\leq||\dot{H}^{+}_{2}\partial_{1}\hat{\Phi}^{+}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}\\\
&\leq C(K)\mathcal{M}(t).\end{split}$ (6.19)

The second term in (6.3) can be controlled similarly. Summarizing from
(6.14)-(6.19), we conclude that (6.12) holds. ∎

The following lemma gives the estimate of the normal derivative of the
entropy, which is treated differently from the other components of the vector
${\mathbf{V}}$.

######  Lemma 6.3.

The estimate

$|||S^{\pm}(t)|||^{2}_{s,\ast}\leq C(K)\mathcal{M}(t)$ (6.20)

holds for problem (4.31) for all $t\leq T$, where $s$ is a positive integer
and $\mathcal{M}(t)$ is defined in (6.6).

###### Proof.

The linearized equation for the entropy $S^{\pm}$ is an evolution-like
equation because the coefficient of the normal derivative of the entropy
vanishes on the boundary; this yields that no boundary conditions are needed
to be coupled to the equation in order to derive an apriori energy estimate.
So, to estimate $S^{\pm}$, we just handle the equation of $S^{\pm}$ alone by
the standard energy method tools. The details of the proof are similar to
those of the following Lemma 6.4. ∎

Now, we derive weighted derivative estimates.

###  6.2. Estimate of weighted derivatives

Since the differential operators $(\sigma\partial_{1})^{\alpha_{1}}$ and
$\sigma^{\alpha_{1}}\partial^{\alpha_{1}}_{1}$ are equivalent, see [28], in
the following we discuss the term $D^{\alpha}_{\ast}{\mathbf{V}}$, with
$\alpha_{1}>0$ and $\langle\alpha\rangle=|\alpha|+\alpha_{3}\leq s$, in its
equivalent form
$\sigma^{\alpha_{1}}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}\mathbf{V}$,
where
$D^{\alpha^{\prime}}_{t,x}:=\partial^{\alpha_{0}}_{t}\partial^{\alpha_{1}}_{1}\partial^{\alpha_{2}}_{2}$
($\alpha^{\prime}=(\alpha_{0},\alpha_{1},\alpha_{2})$).

######  Lemma 6.4.

The following estimate holds for (4.31) for all $t\leq T:$

$\sum_{\langle\alpha\rangle\leq
s,\alpha_{1}>0}||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\Omega)}\leq
C(K)\mathcal{M}(t),$ (6.21)

where $s$ is a positive integer and $\mathcal{M}(t)$ is defined in (6.6).

###### Proof.

It is obvious that when $\alpha_{1}>0,$
$D^{\alpha}_{\ast}{\mathbf{V}}|_{x_{1}=0}=0.$ Note that

$\sigma^{\alpha_{1}}\partial^{m+1}_{1}=\partial_{1}(\sigma^{\alpha_{1}}\partial^{m}_{1})-\alpha_{1}\sigma^{\prime}\sigma^{\alpha_{1}-1}\partial^{m}_{1},$

for some nonnegative integer $m.$ Applying $D^{\alpha}_{\ast}$ to (5.3) and
using the standard energy method, we obtain that

$\int_{{\mathbb{R}}^{2}_{+}}(\mathcal{A}_{0}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})(t)d\mathbf{x}=\int_{\Omega_{t}}\Big{(}((\partial_{t}\mathcal{A}_{0}+\partial_{1}\mathcal{A}_{1}+\partial_{2}\mathcal{A}_{2})D^{\alpha}_{\ast}{\mathbf{V}}+2\mathcal{R})\cdot
D^{\alpha}_{\ast}{\mathbf{V}}\Big{)}d\mathbf{x}d\tau,$

where

$\quad\mathcal{R}=D^{\alpha}_{\ast}\mathcal{F}+\mathcal{R}_{0}+\mathcal{R}_{1},\quad\mathcal{R}_{0}=\alpha_{1}\sigma^{\prime}\mathcal{A}_{1}\sigma^{\alpha_{1}-1}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}{\mathbf{V}},$
$\mathcal{R}_{1}=-[D^{\alpha}_{\ast},\mathcal{A}_{0}]\partial_{t}{\mathbf{V}}-\sum^{2}_{j=1}[D^{\alpha}_{\ast},\mathcal{A}_{j}]\partial_{j}{\mathbf{V}}-D^{\alpha}_{\ast}(\mathcal{A}_{3}{\mathbf{V}})$

(the brackets $[\cdot,\cdot]$ here denotes the commutator between the
operators). Notice that $\mathcal{A}_{0}$ is positive definite, then

$\int_{{\mathbb{R}}^{2}_{+}}(\mathcal{A}_{0}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})(t)d\mathbf{x}\geq
c_{0}||D^{\alpha}_{\ast}\mathbf{V}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})},$

where $c_{0}$ depends on the number $k$ in (4.2) and (4.4). Hence, we obtain
that

$||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}({\mathbb{R}}^{2}_{+})}\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\mathcal{R}||^{2}_{L^{2}(\Omega_{t})}\Big{)}.$
(6.22)

Now, we estimate $||\mathcal{R}||^{2}_{L^{2}(\Omega_{t})}$ in (6.22). Recall
that
$\mathcal{R}=D^{\alpha}_{\ast}\mathcal{F}+\mathcal{R}_{0}+\mathcal{R}_{1}.$
Using Moser-type calculus inequalities (3.7) and (3.8), we can prove that

$\begin{split}||D^{\alpha}_{\ast}\mathcal{F}||^{2}_{L^{2}(\Omega_{t})}&\lesssim||J^{T}\mathbf{F}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||\mathbf{F}||^{2}_{s,\ast,t}+||\mathbf{F}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}.\end{split}$
(6.23)

Using the decomposition of boundary matrix $\mathcal{A}_{1}$ in (5.5),
$\mathcal{A}_{1}=\mathcal{A}+\mathcal{A}_{(0)}$, following arguments similar
to those used in Lemma 6.1, we obtain that

$\begin{split}||\mathcal{R}_{0}||^{2}_{L^{2}(\Omega_{t})}&\lesssim||\mathcal{A}\sigma^{\alpha_{1}-1}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}+||\frac{\mathcal{A}_{(0)}}{\sigma}\sigma^{\alpha_{1}}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}\\\
&\lesssim
C(K)\left(||\sigma^{\alpha_{1}-1}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}{\mathbf{V}}_{n}||^{2}_{L^{2}(\Omega_{t})}+||\sigma^{\alpha_{1}}D^{\alpha^{\prime}}_{t,x}\partial^{\alpha_{3}}_{1}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}\right)\\\
&\leq
C(K)\Big{(}||\partial_{1}{\mathbf{V}}_{n}||^{2}_{s-1,\ast,t}+||{\mathbf{V}}||^{2}_{s,\ast,t}\Big{)}\\\
&\leq C(K)\mathcal{M}(t),\end{split}$ (6.24)

recall here above that the matrix $\mathcal{A}$ acts only on the
noncharacteristic part ${\mathbf{V}}_{n}$ of the unknown $\mathbf{V}$.
Now, we estimate the commutators in $\mathcal{R}_{1}:$ For $j=0,2,$ we obtain
that

$\begin{split}||[D^{\alpha}_{\ast},\mathcal{A}_{j}]\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}\leq&\sum_{0<\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{j}D^{\alpha-\beta}_{\ast}\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}\,.\end{split}$
(6.25)

For $\langle\beta\rangle=1$, we get

$\sum_{\langle\beta\rangle=1\,,\,\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{j}D^{\alpha-\beta}_{\ast}\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}\leq
C(K)||{\mathbf{V}}||^{2}_{s,\ast,t}\,.$ (6.26)

For $\langle\beta\rangle\geq 2,$ we obtain that

$\begin{split}\sum_{\langle\beta\rangle\geq
2\,,\,\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{j}D^{\alpha-\beta}_{\ast}\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}&\lesssim\sum_{\langle\beta^{\prime}\rangle=2,\beta^{\prime}\leq\beta\leq\alpha}||D^{\beta-\beta^{\prime}}_{\ast}(D^{\beta^{\prime}}_{\ast}\mathcal{A}_{j})D^{\alpha-\beta}_{\ast}(\partial_{j}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s-1,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}\,.\end{split}$
(6.27)

For $j=1,$ we need to be very careful. We have

$\begin{split}||[D^{\alpha}_{\ast},\mathcal{A}_{1}]\partial_{1}{\mathbf{V}}||^{2}_{L^{2}(\Omega_{t})}&\lesssim\sum_{0<\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{1}D^{\alpha-\beta}_{\ast}(\partial_{1}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}\,.\end{split}$
(6.28)

For $\langle\beta\rangle=1$ we get $\beta_{3}=0.$ Therefore, from (5.5) it
follows that $D^{\beta}_{\ast}\mathcal{A}_{1}|_{x_{1}=0}=0$ and we use Moser-
type calculus inequalities (3.7), (3.8) to obtain

$\sum_{\langle\beta\rangle=1\,,\,\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{1}D^{\alpha-\beta}_{\ast}(\partial_{1}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}\leq
C(K)||{\mathbf{V}}||^{2}_{s,\ast,t}\,.$ (6.29)

For $\langle\beta\rangle\geq 2$, we have

$\begin{split}&\sum_{\langle\beta\rangle\geq
2\,,\,\beta\leq\alpha}||D^{\beta}_{\ast}\mathcal{A}_{1}D^{\alpha-\beta}_{\ast}(\partial_{1}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}\\\
&\lesssim\sum_{\langle\beta^{\prime}\rangle=2,\beta^{\prime}\leq\beta\leq\alpha}||D^{\beta-\beta^{\prime}}_{\ast}(D^{\beta^{\prime}}_{\ast}\mathcal{A}_{1})D^{\alpha-\beta}_{\ast}(\partial_{1}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}.\end{split}$
(6.30)

Using Moser-type calculus inequalities (3.7), (3.8), we can prove that

$\begin{split}||D^{\alpha}_{\ast}(\mathcal{A}_{3}{\mathbf{V}})||^{2}_{L^{2}(\Omega_{t})}&\lesssim||\mathcal{A}_{3}{\mathbf{V}}||^{2}_{s,\ast,t}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}.\end{split}$
(6.31)

Notice that summing up (6.25)-(6.31) gives

$||\mathcal{R}_{1}||^{2}_{L^{2}(\Omega_{t})}\leq C(K)\mathcal{M}(t).$ (6.32)

Hence, using (6.23), (6.24) and (6.32), we obtain (6.21). Lemma 6.4 is
concluded. ∎

Now, we are going to estimate the non-weighted normal derivatives.

###  6.3. Estimate of non-weighted normal derivatives

Now we perform the differential operator $D^{\alpha}_{\ast}$, in the case
$\alpha_{1}=0,\alpha_{3}\geq 1$, that is
$D^{\alpha}_{\ast}=\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}}_{1}$
with $\langle\alpha\rangle\leq s$. Now, we are ready to prove the following
Lemma 6.5.

######  Lemma 6.5.

The estimate

$\sum_{\langle\alpha\rangle\leq s,\alpha_{1}=0,\alpha_{3}\geq
1}||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\leq
C(K)\mathcal{M}(t)$ (6.33)

holds for problem (4.31) for all $t\leq T,$ where $s$ is a positive integer,
$\mathcal{M}(t)$ is given in (6.6).

###### Proof.

Similar as in Lemma 6.4, applying the operator $D^{\alpha}_{\ast}$ on (5.3)
and using the standard energy method, we obtain that

$\begin{split}\int_{{\mathbb{R}}^{2}_{+}}(\mathcal{A}_{0}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})(t)d\mathbf{x}&=\int_{\Omega_{t}}\Big{(}((\partial_{t}\mathcal{A}_{0}+\partial_{1}\mathcal{A}_{1}+\partial_{2}\mathcal{A}_{2})D^{\alpha}_{\ast}{\mathbf{V}}+2\mathcal{R})\cdot
D^{\alpha}_{\ast}{\mathbf{V}}\Big{)}d\mathbf{x}d\tau\\\
&\quad+\int_{\Gamma_{t}}(\mathcal{A}_{1}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})|_{x_{1}=0}dx_{2}d\tau\,,\end{split}$

where $\mathcal{R}$ is defined as in the proof of Lemma 6.4. Thus

$||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\mathcal{R}||^{2}_{L^{2}(\Omega_{t})}+\tilde{\mathcal{M}}(t)\Big{)},$

where

$\tilde{\mathcal{M}}(t)=\Big{|}\int_{\Gamma_{t}}(\mathcal{A}_{1}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})|_{x_{1}=0}dx_{2}d\tau\Big{|}=\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{q}D^{\alpha}_{\ast}\dot{u}_{n})|_{x_{1}=0}dx_{2}d\tau\Big{|}.$

When $\alpha_{1}=0,$ by definition, $\mathcal{R}_{0}=0.$ Hence, we obtain the
estimate

$||\mathcal{R}||^{2}_{L^{2}(\Omega_{t})}\leq C(K)\mathcal{M}(t).$

This yields that

$||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}({\mathbb{R}}^{2}_{+})}\leq
C(K)(\mathcal{M}(t)+\tilde{\mathcal{M}}(t)).$ (6.34)

Using (6.3) and (6.4), we obtain that

$\tilde{\mathcal{M}}(t)\lesssim||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}\mathcal{K}||^{2}_{L^{2}(\Gamma_{t})}\,,$

where $\mathcal{K}$ is defined in (6.4). Moreover, recalling that
$\tilde{\mathcal{A}}|_{x_{1}=0}=\mathrm{diag}\Big{(}\mathcal{E}_{12},-\mathcal{E}_{12}\Big{)}$
because $\partial_{1}\hat{\Phi}^{\pm}|_{x_{1}=0}=\pm 1,$ we notice that
$||\tilde{\mathcal{A}}|_{x_{1}=0}||_{L^{\infty}(\Gamma_{t})}=1.$ Using
$\mathcal{A}_{(0)}|_{x_{1}=0}=0$ we obtain that

$\tilde{\mathcal{M}}(t)\leq
C\Big{(}\Sigma_{1}(t)+\Sigma_{2}(t)+\Sigma_{3}(t)+||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}(\mathcal{A}_{3}{\mathbf{V}})||^{2}_{L^{2}(\Gamma_{t})}+||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}\mathcal{F}||^{2}_{L^{2}(\Gamma_{t})}\Big{)},$
(6.35)

where

$\Sigma_{1}(t)=\sum_{j=0,2}||\mathcal{A}_{j}\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Gamma_{t})},$
$\Sigma_{2}(t)=\sum_{j=0,2}\sum_{\langle\beta^{\prime}\rangle\geq
1,\beta^{\prime}+\beta^{\prime\prime}=(\alpha_{0},\alpha_{2},\alpha_{3}-1)}||D^{\beta^{\prime}}_{\ast}\mathcal{A}_{j}D^{\beta^{\prime\prime}}_{\ast}\partial_{j}{\mathbf{V}}||^{2}_{L^{2}(\Gamma_{t})};$

notice that in spite of the notation here $D^{\beta^{\prime}}_{\ast}$ and
$D^{\beta^{\prime\prime}}_{\ast}$ dot not involve any weighted derivative
$\sigma\partial_{1}$ (see the beginning of this section); notice also that
$\beta^{\prime}+\beta^{\prime\prime}=(\alpha_{0},\alpha_{2},\alpha_{3}-1)$
implies that
$\langle\beta^{\prime}\rangle+\langle\beta^{\prime\prime}\rangle\leq s-2$;

$\Sigma_{3}(t)=\sum_{\alpha_{3}^{\prime}+\alpha_{3}^{\prime\prime}\leq\alpha_{3},\alpha_{3}^{\prime},\alpha_{3}^{\prime\prime}\geq
1}||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}(\partial^{\alpha_{3}^{\prime}}_{1}\mathcal{A}_{(0)}\partial^{\alpha_{3}^{\prime\prime}}_{1}{\mathbf{V}})||^{2}_{L^{2}(\Gamma_{t})}.$

Now, we estimate $\Sigma_{2}(t)$ and $\Sigma_{3}(t)$, which can be controlled
by using Lemma 3.4 (i) and Moser-type calculus inequalities (3.7) and (3.8):

$\begin{split}\Sigma_{2}(t)&\lesssim\sum_{j=0,2}\sum_{\langle\beta^{\prime}\rangle=1}\Big{(}||D^{\beta^{\prime}}_{\ast}\mathcal{A}_{j}\partial_{j}{\mathbf{V}}||^{2}_{s-3,\ast,t}+||\partial_{1}(D^{\beta^{\prime}}_{\ast}\mathcal{A}_{j}\partial_{j}{\mathbf{V}})||^{2}_{s-3,\ast,t}\Big{)}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}.\end{split}$
(6.36)

In the above estimate, it is noted that
$\langle\beta^{\prime\prime}\rangle\leq s-3$.

$\begin{split}\Sigma_{3}(t)&\lesssim||\partial_{1}\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}}||^{2}_{s-4,\ast,t}+||\partial_{1}(\partial_{1}\mathcal{A}_{(0)}\partial_{1}{\mathbf{V}})||^{2}_{s-4,\ast,t}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{t})}||\hat{W}||^{2}_{s,\ast,t}\Big{)}.\end{split}$
(6.37)

For the term $\Sigma_{1}(t),$ passing to the volume integral then using
Leibniz’s rule, we get (for shortness, in the sequel we denote
$D^{\alpha}:=\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}$ whereas
$D^{\alpha}_{\ast}:=\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}}_{1}$):

$\begin{split}\Sigma_{1}(t)&=-\sum_{j=0,2}\int_{\Omega_{t}}\partial_{1}|\mathcal{A}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}|^{2}d\mathbf{x}d\tau\\\
&=-2\sum_{j=0,2}\int_{\Omega_{t}}\Big{(}(\mathcal{A}^{2}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}\partial_{j}{\mathbf{V}})\\\
&\quad+(\mathcal{A}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}\cdot\partial_{1}\mathcal{A}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}})\Big{)}d\mathbf{x}d\tau\,.\end{split}$

Then integration by parts with respect to $\partial_{j}$ (for $j=0,2)$ gives

$\Sigma_{1}(t)=\tilde{\Sigma}_{1}(t)+J_{0}(t)\,,$

where

$\begin{split}\tilde{\Sigma}_{1}(t)&=2\sum_{j=0,2}\int_{\Omega_{t}}\Big{(}\mathcal{A}^{2}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial^{2}_{j}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}}\Big{)}\\\
&\quad+\Big{(}(\mathcal{A}_{j}\partial_{j}\mathcal{A}_{j}+\partial_{j}\mathcal{A}_{j}\mathcal{A}_{j})D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}}\Big{)}\\\
&\quad-\Big{(}\mathcal{A}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}\cdot\partial_{1}\mathcal{A}_{j}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{j}{\mathbf{V}}\Big{)}d\mathbf{x}d\tau,\end{split}$
$J_{0}(t)=-2\int_{\mathbb{R}^{2}_{+}}\Big{(}\mathcal{A}^{2}_{0}D^{\alpha}\partial^{\alpha_{3}-1}_{1}\partial_{t}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}}\Big{)}(t)d\mathbf{x}.$

Since $|\alpha|+2+2(\alpha_{3}-1)\leq s$ we obtain that

$\tilde{\Sigma}_{1}(t)\leq C(K)||{\mathbf{V}}||^{2}_{s,\ast,t}.$ (6.38)

Using Young’s inequality, we obtain that

$J_{0}(t)\leq
C(K)\Big{(}\varepsilon||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\frac{1}{\varepsilon}|||{\mathbf{V}}(t)||||^{2}_{s-1,\ast}\Big{)},$
(6.39)

for small $\varepsilon.$ Therefore, we conclude from (6.38), (6.39) and
elementary inequalities (6.13) that

$\Sigma_{1}(t)\leq
C(K)\Big{(}\varepsilon||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\frac{1}{\varepsilon}||{\mathbf{V}}||^{2}_{s,\ast,t}\Big{)}.$
(6.40)

The last two terms in (6.35) can be estimated by using Lemma 3.4 (i), Moser-
type calculus inequalities (3.7) and (3.8):

$||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}(\mathcal{A}_{3}{\mathbf{V}})||^{2}_{L^{2}(\Gamma_{t})}+||\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}\partial^{\alpha_{3}-1}_{1}\mathcal{F}||^{2}_{L^{2}(\Gamma_{t})}\leq
C(K)\mathcal{M}(t).$ (6.41)

Summarizing (6.34)-(6.37) and using (6.40) and (6.41), taking $\varepsilon$
sufficiently small, we get the estimate (6.33). Therefore, Lemma 6.5 is
proved. ∎

###  6.4. Estimate of non-weighted tangential derivatives

Now, we are going to obtain estimates of non-weighted tangential derivatives,
i.e. $\alpha_{1}=\alpha_{3}=0$, that is
$D^{\alpha}_{\ast}=\partial^{\alpha_{0}}_{t}\partial^{\alpha_{2}}_{2}$ with
$|\alpha|\leq s$. This is the most important case because we shall use the
boundary conditions. This gives the loss of two additional derivatives that
imply that in final tame estimate we will have the “$s+4,\ast,t$” loss of
derivatives from the coefficients, see Theorem 6.1. This loss is caused by the
presence of zero order terms in $\varphi$ (see (4.16)).

######  Lemma 6.6.

The following estimate holds for (4.31) for all $t\leq T:$

$\begin{split}\sum_{|\alpha|\leq
s\,,\,\alpha_{1}=\alpha_{3}=0}||D^{\alpha}_{\ast}&{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\leq\varepsilon
C(K)\Big{(}|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+|||\varphi(t)|||^{2}_{H^{s-1}({\mathbb{R}})}\Big{)}\\\
&+\frac{C(K)}{\varepsilon}\Big{(}\mathcal{M}(t)+||\varphi||^{2}_{H^{s-1}(\Gamma_{t})}+||\varphi||^{2}_{W^{2,\infty}_{\ast}(\Gamma_{t})}||\hat{W}||^{2}_{s+4,\ast,t}\Big{)},\end{split}$
(6.42)

where $s$ is positive integer, $\varepsilon$ is a positive constant and
$\mathcal{M}(t)$ is defined in (6.6).

###### Proof.

We only need to estimate the highest-order tangential derivatives with
$|\alpha|=s,$ since the lower order terms can be controlled by definition of
the anisotropic Sobolev norm, through the following estimate:

$\sum_{|\alpha|\leq
s-1}||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\leq
C||{\mathbf{V}}||^{2}_{s,\ast,t}.$ (6.43)

Therefore, applying same argument as in Lemmata 6.4 and 6.5, we obtain that

$||D^{\alpha}_{\ast}{\mathbf{V}}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}\leq
C(K)(\mathcal{M}(t)+\mathcal{J}(t)),$ (6.44)

for $\alpha=(\alpha_{0},\alpha_{2}),$ with $|\alpha|=s,$ where

$\mathcal{J}(t)=\Big{|}\int_{\Gamma_{t}}(\mathcal{B}_{1}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})|_{x_{1}=0}dx_{2}d\tau\Big{|}.$

Taking into account the boundary conditions (4.31) and (4.33), and also the
important dissipative structure (5.8), the explicit quadratic boundary term in
the integral can be rewritten in a suitable form: let us denote

$c^{\pm}=\sum_{|\alpha^{\prime}|+|\alpha^{\prime\prime}|=s,|\alpha^{\prime}|\geq
1}D^{\alpha^{\prime}}_{\ast}\hat{\lambda}^{\pm}D^{\alpha^{\prime\prime}}_{\ast}\dot{H}^{\pm}_{n},\quad\dot{w}^{\pm}=\dot{u}^{\pm}_{n}-\hat{\lambda}^{\pm}\dot{H}^{\pm}_{n}\,,$

where

$[\partial_{1}\hat{q}]=(\partial_{1}\hat{q}^{+}+\partial_{1}\hat{q}^{-})|_{x_{1}=0}\,,\,\,\dot{w}^{\pm}|_{x_{1}=0}=(\dot{u}^{\pm}_{N}-\hat{\lambda}^{\pm}\dot{H}^{\pm}_{N})|_{x_{1}=0}$

and $\hat{\lambda}^{\pm}$ were defined in (5.10), see also Lemma 5.1.
The boundary quadratic form becomes:

$\begin{split}(\mathcal{B}_{1}D^{\alpha}_{\ast}{\mathbf{V}}\cdot
D^{\alpha}_{\ast}{\mathbf{V}})|_{x_{1}=0}&=2[(D^{\alpha}_{\ast}\dot{u}_{N}-\hat{\lambda}D^{\alpha}_{\ast}\dot{H}_{N})D^{\alpha}_{\ast}\dot{q}]=2[cD^{\alpha}_{\ast}\dot{q}]+2[D^{\alpha}_{\ast}\dot{w}D^{\alpha}_{\ast}\dot{q}]\\\
&=2[cD^{\alpha}_{\ast}\dot{q}]+2D^{\alpha}_{\ast}\dot{w}^{-}|_{x_{1}=0}[D^{\alpha}_{\ast}\dot{q}]+2D^{\alpha}_{\ast}\dot{q}^{+}|_{x_{1}=0}[D^{\alpha}_{\ast}\dot{w}]\\\
&=2D^{\alpha}_{\ast}\dot{q}^{+}|_{x_{1}=0}D^{\alpha}_{\ast}(\partial_{2}\varphi[\hat{u}_{2}-\hat{\lambda}\hat{H}_{2}])+\text{l.o.t},\end{split}$
(6.45)

where lower order term l.o.t can be expressed by

$2[cD^{\alpha}_{\ast}\dot{q}]-2D^{\alpha}_{\ast}\dot{w}^{-}|_{x_{1}=0}D^{\alpha}_{\ast}([\partial_{1}\hat{q}]\varphi)-2D^{\alpha}_{\ast}\dot{q}^{+}|_{x_{1}=0}D^{\alpha}_{\ast}([\partial_{1}\hat{u}_{N}-\hat{\lambda}\partial_{1}\hat{H}_{N}]\varphi).$

Since the boundary conditions are dissipative, it is noted that first term on
the right-hand side of (6.45) vanish by Lemma 5.1. The boundary terms can be
estimated separately:

$\mathcal{J}(t)\leq\sum_{\pm}\sum^{4}_{i=1}\mathcal{J}^{\pm}_{i}(t),$ (6.46)

where

$\mathcal{J}^{\pm}_{1}(t)=\Big{|}\int_{\Gamma_{t}}(c^{\pm}D^{\alpha}_{\ast}\dot{q}^{\pm})|_{x_{1}=0}d{x_{2}}d\tau\Big{|},$
$\mathcal{J}^{\pm}_{2}(t)=\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{w}^{-}D^{\alpha}_{\ast}(\varphi\partial_{1}\hat{q}^{\pm}))|_{x_{1}=0}d{x_{2}}d\tau\Big{|},$
$\mathcal{J}^{\pm}_{3}(t)=\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{q}^{+}D^{\alpha}_{\ast}(\varphi\partial_{1}\hat{u}^{\pm}_{n}))|_{x_{1}=0}d{x_{2}}d\tau\Big{|},$
$\mathcal{J}^{\pm}_{4}(t)=\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{q}^{+}D^{\alpha}_{\ast}(\varphi\hat{\lambda}^{\pm}\partial_{1}\hat{H}^{\pm}_{n}))|_{x_{1}=0}d{x_{2}}d\tau\Big{|}.$

Recall that $|\alpha|=s\geq 2.$ We denote
$D^{\alpha}_{\ast}=\partial_{l}D^{\gamma}$,
$D^{\gamma}:=\partial^{\gamma_{0}}_{t}\partial^{\gamma_{2}}_{2}$ for
$\gamma=(\gamma_{0},\gamma_{2}),|\gamma|=s-1\geq 1$, where

$l=\begin{cases}2,&\text{ if }\alpha_{0}\neq s,\\\ 0,&\text{ if
}\alpha_{0}=s.\end{cases}$

In the following estimate of $\mathcal{J}^{+}_{1}(t)$, we separate the
analysis into two cases.

Case A: When $\alpha_{0}=s$, then
$l=0,D^{\alpha}_{\ast}=\partial_{t}D^{\gamma}.$ Using the normal derivative
estimates (6.5), (6.12) for non-characteristic variables, the elementary
inequality (6.13) and integration by parts, we obtain that

$\begin{split}\mathcal{J}^{+}_{1}(t)&=\Big{|}\int_{\Omega_{t}}(\partial_{t}c^{+}\partial_{1}D^{\gamma}\dot{q}^{+}-\partial_{1}c^{+}D^{\alpha}_{\ast}\dot{q}^{+})d\mathbf{x}d\tau-\int_{\mathbb{R}^{2}_{+}}c^{+}\partial_{1}D^{\gamma}\dot{q}^{+}d\mathbf{x}\Big{|}\\\
&\leq
C\Big{(}||\dot{q}^{+}||^{2}_{s,\ast,t}+||\partial_{1}\dot{q}^{+}||^{2}_{s-1,\ast,t}+||\partial_{t}c^{+}||^{2}_{L^{2}(\Omega_{t})}\\\
&\quad+||\partial_{1}c^{+}||^{2}_{L^{2}(\Omega_{t})}+\frac{1}{\varepsilon}||c^{+}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\varepsilon||\partial_{1}\dot{q}^{+}||^{2}_{H^{s-1}_{\ast}(\mathbb{R}^{2}_{+})}\Big{)}\\\
&\leq
C\Big{(}||\dot{q}^{+}||^{2}_{s,\ast,t}+||\partial_{1}\dot{q}^{+}||^{2}_{s-1,\ast,t}+||\partial_{t}c^{+}||^{2}_{L^{2}(\Omega_{t})}\\\
&\quad+||\partial_{1}c^{+}||^{2}_{L^{2}(\Omega_{t})}+\frac{1}{\varepsilon}||c^{+}(t)||^{2}_{L^{2}(\mathbb{R}^{2}_{+})}+\varepsilon|||\partial_{1}\dot{q}^{+}(t)|||^{2}_{s-1,\ast}\Big{)}\\\
&\leq\varepsilon
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+\frac{C(K)}{\varepsilon}\mathcal{M}(t).\end{split}$
(6.47)

Here, $\varepsilon$ is an arbitrary fixed constant. Similar argument also
holds for $\mathcal{J}^{-}_{1}(t).$ Therefore, we obtain that

$\mathcal{J}^{+}_{1}(t)+\mathcal{J}^{-}_{1}(t)\leq\varepsilon
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+\frac{C(K)}{\varepsilon}\mathcal{M}(t).$

Case B: When $\alpha_{0}\neq s$, then
$l=2,D^{\alpha}_{\ast}=\partial_{2}D^{\gamma}.$ Using integration by parts, we
could deduce that

$|\mathcal{J}^{\pm}_{1}(t)|\leq C(K)\mathcal{M}(t).$ (6.48)

Next, we estimate $\mathcal{J}^{+}_{3}$ and separate this term into two parts

$\mathcal{J}^{+}_{3}\leq\mathcal{J}_{0}(t)+\Sigma_{4}(t),$

where

$\mathcal{J}_{0}(t)=\Big{|}\int_{\Gamma_{t}}(\varphi
D^{\alpha}_{\ast}\dot{q}^{+}D^{\alpha}_{\ast}\partial_{1}\hat{u}^{+}_{n})|_{x_{1}=0}dx_{2}d\tau\Big{|},$
$\Sigma_{4}(t)=\sum_{|\alpha^{\prime}|+|\alpha^{\prime\prime}|=s,|\alpha^{\prime}|\geq
1}\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{q}^{+}D^{\alpha^{\prime}}_{\ast}\varphi
D^{\alpha^{\prime\prime}}_{\ast}\partial_{1}\hat{u}^{+}_{n})\Big{|}_{x_{1}=0}dx_{2}d\tau\Big{|}.$

For the term $\mathcal{J}_{0}(t),$ after integrating by parts, we obtain that

$\begin{split}\mathcal{J}_{0}(t)&=\Big{|}\int_{\Omega_{t}}(\partial_{1}\partial_{l}D^{\alpha}_{\ast}\hat{u}^{+}_{n})\varphi\partial_{1}D^{\gamma}\dot{q}^{+}-(\partial^{2}_{1}D^{\alpha}_{\ast}\hat{u}^{+}_{n})\varphi
D^{\alpha}_{\ast}\dot{q}^{+}\\\
&\quad+\partial_{1}D^{\alpha}_{\ast}\hat{u}^{+}_{n}\partial_{l}\varphi\partial_{1}D^{\gamma}\dot{q}^{+}d\mathbf{x}d\tau-\frac{l-2}{2}\int_{\mathbb{R}^{2}_{+}}(\partial_{1}D^{\alpha}_{\ast}\hat{u}^{+}_{n})\varphi\partial_{1}D^{\gamma}\dot{q}^{+}d\mathbf{x}\Big{|}.\end{split}$

We note that to estimate $\partial^{2}_{1}D^{\alpha}_{\ast}\hat{u}^{+}_{n}$,
we need $\hat{W}$ with regularity $s+4.$ For the term $\mathcal{J}_{0}(t),$
using Moser-type calculus inequalities (3.7), (3.8), (6.5) and (6.12) we get

$\mathcal{J}_{0}(t)\leq\varepsilon
C(K)|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+\frac{C(K)}{\varepsilon}\Big{(}\mathcal{M}(t)+||\varphi||^{2}_{W^{1,\infty}{(\Gamma_{t})}}||\hat{W}||^{2}_{s+4,\ast,t}\Big{)}.$

Now, we start to estimate $\Sigma_{4}.$ It is noted that for
$|\alpha^{\prime}|\geq 1,$ we can isolate one tangential derivative in the
differential operator:

$D^{\alpha^{\prime}}_{\ast}\varphi=D^{\gamma^{\prime}}(\partial_{j}\varphi),\quad|\gamma^{\prime}|\leq
s-1,\quad j=0\text{ or }j=2.$

Using (B.55) and (B.42), we obtain that

$\nabla_{t,x_{2}}\varphi=\mathbf{H}(\hat{{\mathbf{U}}},\hat{\varphi}){\mathbf{V}}_{n}|_{x_{1}=0}+\mathbf{G}(\hat{{\mathbf{U}}},\hat{\varphi})\varphi.$
(6.49)

Here
$\mathbf{H}(\hat{{\mathbf{U}}},\hat{\varphi}),\mathbf{G}(\hat{{\mathbf{U}}},\hat{\varphi})$
depend on
$\hat{{\mathbf{U}}}|_{x_{1}=0},\partial_{1}\hat{{\mathbf{U}}}|_{x_{1}=0}$ and
second order derivatives of $\hat{\varphi}.$

Using (6.49), we can write the derivatives of $\varphi$ by using the non-
characteristic unknown ${\mathbf{V}}_{n}$. We insert these derivatives
$D^{\alpha^{\prime}}_{\ast}\varphi=D^{\gamma^{\prime}}(\cdots)$, with
$|\gamma^{\prime}|\leq s-1$, into $\Sigma_{4}$ to obtain

$\Sigma_{4}(t)=\sum_{|\alpha^{\prime}|+|\alpha^{\prime\prime}|=s,|\alpha^{\prime}|\geq
1}\Big{|}\int_{\Gamma_{t}}(D^{\alpha}_{\ast}\dot{q}^{+}D^{\gamma^{\prime}}(\mathbf{H}{\mathbf{V}}_{n}+\mathbf{G}\varphi)D^{\alpha^{\prime\prime}}_{\ast}\partial_{1}\hat{u}^{+}_{n})\Big{|}_{x_{1}=0}dx_{2}d\tau\Big{|}\,,$

that can be controlled by the right-hand side of (6.42). Indeed, by passing to
the volume integral on $\Omega_{t}$, we note that the highest order of
regularity for
$D^{\alpha^{\prime\prime}}_{\ast}\partial_{1}^{2}\hat{u}^{+}_{n}$ is $s+3$
because $|\gamma^{\prime}|+|\alpha^{\prime\prime}|=s-1$ (since
$|\alpha^{\prime}|=|\gamma^{\prime}|+1$ and
$|\alpha^{\prime}|+|\alpha^{\prime\prime}|=s$). Similar estimates also hold
for $\mathcal{J}^{-}_{3},\mathcal{J}^{\pm}_{2},\mathcal{J}^{\pm}_{4}.$ Using
(6.43), (6.44), (6.46), (6.47), (6.48), we obtain the estimate (6.42).
Therefore, Lemma 6.6 is concluded. ∎

###  6.5. Estimate of front

Now, we are going to estimate the front $\varphi(t)$ in
$H^{s-1}({\mathbb{R}})$ and its tangential derivatives
$\nabla_{t,x_{2}}\varphi$ in $H^{s-1}(\Gamma_{t}).$

######  Lemma 6.7.

Given the solution $\varphi$ of (4.31), for all $t\leq T$ and positive integer
$s,$ the following estimate holds:

$\begin{split}&|||\varphi(t)|||^{2}_{H^{s-1}({\mathbb{R}})}+||\nabla_{t,x_{2}}\varphi||^{2}_{H^{s-1}(\Gamma_{t})}\\\
&\quad\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\varphi||^{2}_{H^{s-1}(\Gamma_{t})}+(||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}+||\varphi||^{2}_{W^{1,\infty}_{\ast}(\Gamma_{t})})||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}.\end{split}$
(6.50)

###### Proof.

Applying tangential derivatives on the first boundary conditions in (4.31), we
obtain that

$\begin{split}|||\varphi(t)|||^{2}_{H^{s-1}({\mathbb{R}})}&\leq
C(K)\Big{(}||\dot{u}_{n}|_{x_{1}=0}||^{2}_{H^{s-1}(\Gamma_{t})}+||\varphi||^{2}_{H^{s-1}(\Gamma_{t})}+\sum_{|\alpha|\leq
s-1}||\mathcal{G}_{\alpha}||^{2}_{L^{2}(\Gamma_{t})}\Big{)},\end{split}$
(6.51)

where

$\mathcal{G}_{\alpha}=D^{\alpha}_{\ast}(\varphi\partial_{1}\hat{u}^{+}_{N})-([D^{\alpha}_{\ast},\hat{u}^{+}_{2}]\partial_{2}\varphi+\frac{1}{2}\partial_{2}\hat{u}^{+}_{2}D^{\alpha}_{\ast}\varphi).$

For the first term on the right-hand side of (6.51), using trace Theorem in
Lemma 3.4, we obtain that

$||\dot{u}_{n}|_{x_{1}=0}||^{2}_{H^{s-1}(\Gamma_{t})}\lesssim||{\mathbf{V}}||^{2}_{s,\ast,t}.$
(6.52)

For the third term on the right-hand side of (6.51), when $|\alpha|\leq s-1,$
we obtain that

$\begin{split}||\mathcal{G}_{\alpha}||^{2}_{L^{2}(\Gamma_{t})}&\lesssim||\varphi\partial_{1}\hat{u}^{+}_{N}||^{2}_{H^{s-1}(\Gamma_{t})}+||[D^{\alpha}_{\ast},\hat{u}^{+}_{2}]\partial_{2}\varphi||^{2}_{L^{2}(\Gamma_{t})}+||\partial_{2}\hat{u}^{+}_{2}D^{\alpha}_{\ast}\varphi||^{2}_{L^{2}(\Gamma_{t})}\\\
&\leq
C(K)\Big{(}||\varphi||^{2}_{H^{s-1}(\Gamma_{t})}+||\varphi||^{2}_{W^{1,\infty}(\Gamma_{t})}||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}.\end{split}$
(6.53)

Hence, we obtain the estimate of the first term in the left-hand side of
(6.50).

For the estimate of the second term in the left-hand side of (6.50), we use
the relation (6.49),
$||\hat{\varphi}||_{H^{s}(\Gamma_{t})}\lesssim||\hat{\Psi}||_{s,\ast,t}\leq||\hat{\varphi}||_{H^{s}(\Gamma_{t})},$
and Moser-type calculus inequalities (3.7) and (3.8) to obtain that

$\begin{split}&||\nabla_{t,x_{2}}\varphi||^{2}_{H^{s-1}(\Gamma_{t})}\leq||\mathbf{H}(\hat{\mathbf{U}},\hat{\varphi}){\mathbf{V}}_{n}|_{x_{1}=0}||^{2}_{H^{s-1}(\Gamma_{t})}+||\mathbf{G}(\hat{\mathbf{U}},\hat{\varphi})\varphi||^{2}_{H^{s-1}(\Gamma_{t})}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,t}+||\varphi||^{2}_{H^{s-1}(\Gamma_{t})}+(||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{t})}+||\varphi||^{2}_{W^{1,\infty}(\Gamma_{t})})||\hat{W}||^{2}_{s+2,\ast,t}\Big{)}.\end{split}$

Therefore, Lemma 6.7 is concluded. ∎

Collecting all the previously established higher order estimates, we can prove
the following lemma.

######  Lemma 6.8.

The solution of the homogeneous problem (4.31) satisfies the following a
priori estimate

$||\dot{{\mathbf{U}}}||^{2}_{s,\ast,T}+||\varphi||^{2}_{H^{s}(\Gamma_{T})}\leq
C(K)Te^{C(K)T}\mathcal{N}(T)$ (6.54)

for positive integer $s,$ where

$\begin{split}\mathcal{N}(T)&=||\mathbf{F}||^{2}_{s,\ast,T}+(||\dot{{\mathbf{U}}}||^{2}_{W^{2,\infty}_{\ast}(\Omega_{T})}+||\varphi||^{2}_{W^{2,\infty}(\Gamma_{T})}+||\mathbf{F}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})})||\hat{W}||^{2}_{s+4,\ast,T}.\end{split}$

###### Proof.

Combining the estimates (6.21), (6.33), (6.42) choosing $\varepsilon$ small
enough and (6.50), we obtain that

$\mathcal{I}(t)\leq
C(K)\Big{(}\mathcal{N}(T)+\int^{t}_{0}\mathcal{I}(\tau)d\tau\Big{)},$ (6.55)

where

$\mathcal{I}(t)=|||{\mathbf{V}}(t)|||^{2}_{s,\ast}+|||\varphi(t)|||^{2}_{H^{s-1}({\mathbb{R}})}.$

Notice that

$\mathcal{I}(0)=0.$

Then, using Grönwall’s lemma to (6.55), we obtain

$\mathcal{I}(t)\leq C(K)e^{C(K)T}\mathcal{N}(T),\text{ for }0\leq t\leq T.$

Integrating (6.55) with respect to $t\in(-\infty,T],$ we get

$||{\mathbf{V}}||^{2}_{s,\ast,T}+||\varphi||^{2}_{H^{s-1}(\Gamma_{T})}\leq
C(K)Te^{C(K)T}\mathcal{N}(T).$ (6.56)

Notice that $\dot{{\mathbf{U}}}=J{\mathbf{V}}.$ Then, using the decomposition
$J=J(\hat{W})=I+J_{0}(\hat{W}),$ where $J_{0}$ satisfies $J_{0}(0)=0,$ we
apply the Moser-type calculus inequalities (3.7) and (3.8) and derive

$\begin{split}||\dot{{\mathbf{U}}}||^{2}_{s,\ast,T}&=||{\mathbf{V}}+J_{0}{\mathbf{V}}||^{2}_{s,\ast,T}\\\
&\leq
C(K)\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,T}+||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})}||\hat{W}||^{2}_{s,\ast,T}\Big{)}\\\
&\leq
C(K)||{\mathbf{V}}||^{2}_{s,\ast,T}+TC(K)||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})}||\hat{W}||^{2}_{s+1,\ast,T},\end{split}$
(6.57)

and, using (6.50) with $t=T$,

$\begin{split}&||\nabla_{t,x_{2}}\varphi||^{2}_{H^{s-1}(\Gamma_{T})}\\\
&\quad\lesssim\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,T}+||\varphi||^{2}_{H^{s-1}(\Gamma_{T})}+(||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})}+||\varphi||^{2}_{W^{1,\infty}(\Gamma_{T})})||\hat{W}||^{2}_{s+2,\ast,T}\Big{)}\\\
&\quad\lesssim\Big{(}||{\mathbf{V}}||^{2}_{s,\ast,T}+||\varphi||^{2}_{H^{s-1}(\Gamma_{T})}\\\
&\qquad+TC(K)(||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})}+||\varphi||^{2}_{W^{1,\infty}(\Gamma_{T})})||\hat{W}||^{2}_{s+3,\ast,T}\Big{)}\,.\end{split}$
(6.58)

In the proof of last inequalities in (6.57), (6.58), we have used the
following relation by applying Sobolev imbedding:

$\begin{split}||\hat{W}||^{2}_{s,\ast,T}&=\sum^{s}_{j=0}\int^{T}_{0}||\partial^{j}_{t}\hat{W}(t)||^{2}_{s-j,\ast}dt\leq\sum^{s}_{j=0}T\max_{t\in[0,T]}||\partial^{j}_{t}\hat{W}(t)||^{2}_{s-j,\ast}\\\
&\lesssim
T\sum^{s}_{j=0}\Big{\\{}\int^{T}_{0}||\partial^{j}_{t}\hat{W}(t)||^{2}_{s-j,\ast}dt+\int^{T}_{0}||\partial^{j+1}_{t}\hat{W}(t)||^{2}_{s-j,\ast}dt\Big{\\}}\\\
&\lesssim
T\int^{T}_{0}|||\partial^{j}_{t}\hat{W}(t)|||^{2}_{s+1-j,\ast}dt\lesssim
T||\hat{W}||^{2}_{s+1,\ast,T}.\end{split}$

Using (6.56) and (6.57), we obtain

$||\dot{{\mathbf{U}}}||^{2}_{s,\ast,T}+||\varphi||^{2}_{H^{s-1}(\Gamma_{T})}\leq
C(K)Te^{C(K)T}\mathcal{N}(T).$ (6.59)

Similar to (6.57), we can get

$||{\mathbf{V}}||^{2}_{s,\ast,T}\leq
C(K)||\dot{{\mathbf{U}}}||^{2}_{s,\ast,T}+TC(K)||\dot{{\mathbf{U}}}||^{2}_{W^{1,\infty}_{\ast}(\Omega_{T})}||\hat{W}||^{2}_{s+1,\ast,T}.$
(6.60)

Adding (6.58) and (6.59), and using (6.60), we conclude Lemma 6.8. ∎

Using (6.54), we are ready to prove the tame estimate for the homogeneous
problem (4.31).

Proof of Theorem 6.1: Using Lemma 6.8, the Sobolev inequalities (3.9), (3.10)
for $s\geq 6,$ we obtain

$\begin{split}||\dot{{\mathbf{U}}}||_{s,\ast,T}+||\varphi||_{H^{s}(\Gamma_{T})}&\leq
C(K)T^{\frac{1}{2}}e^{C(K)T}\Big{(}||\mathbf{F}||_{s,\ast,T}+(||\dot{{\mathbf{U}}}||_{6,\ast,T}\\\
&\quad+||\varphi||_{H^{6}(\Gamma_{T})}+||\mathbf{F}||_{4,\ast,T})||\hat{W}||_{s+4,\ast,T}\Big{)}.\end{split}$
(6.61)

Taking $T$ sufficiently small and $s=6,$ using (6.1), we obtain that

$||\dot{{\mathbf{U}}}||_{6,\ast,T}+||\varphi||_{H^{6}(\Gamma_{T})}\leq
C(K_{0})||\mathbf{F}||_{6,\ast,T}.$ (6.62)

Hence, (6.61) and (6.62) implies (6.2). The existence and uniqueness of the
solution comes from Theorem 5.1. The proof of Theorem 6.1 is complete.

##  7\. Higher order energy estimate for problem (4.20)

Now, we are ready to obtain an a priori tame estimate in $H^{s}_{\ast}$ for
the nonhomogeneous problem (4.20).

######  Theorem 7.1.

Let $T>0$ and $s$ be an integer, $s\geq 6.$ Assume that the basic state
$(\hat{{\mathbf{U}}},\hat{\varphi})$ satisfies (4.2)-(4.10), and
$(\breve{\mathbf{U}}^{\pm},\hat{\varphi})\in H^{s+4}_{\ast}(\Omega_{T})\times
H^{s+5}(\Gamma_{T})$ satisfies (6.1). Assume that $\mathbf{f}\in
H^{s+2}_{\ast}(\Omega_{T})$, $\mathbf{g}\in H^{s+2}(\Gamma_{T})$ vanish in the
past. Then, there exists a positive constant $K_{0},$ that does not depend on
$s$ and $T,$ and there exists a constant $C(K_{0})>0$ such that if
$\hat{K}\leq K_{0}$, then there exists a unique solution
$(\dot{{\mathbf{U}}},\varphi)\in H^{s}_{\ast}(\Omega_{T})\times
H^{s}({\Gamma_{T}})$ to the problem (4.20) that allows the tame estimate

$\begin{split}&||\dot{{\mathbf{U}}}||_{s,\ast,T}+||\varphi||_{H^{s}(\Gamma_{T})}\\\
&\quad\leq
C(K_{0})\Big{(}||\mathbf{f}||_{s+2,\ast,T}+||\mathbf{g}||_{H^{s+2}(\Gamma_{T})}+(||\mathbf{f}||_{8,\ast,T}+||\mathbf{g}||_{H^{8}(\Gamma_{T})})||\hat{W}||_{s+4,\ast,T}\Big{)},\end{split}$
(7.1)

for $T$ small enough, where $\hat{W}=(\breve{\mathbf{U}},\nabla_{t,\bf
x}\hat{\Psi}).$

######  Remark 7.1.

The lower regularity in (6.1) and low norms in (6.2) and (7.1), for both the
even and odd case, differ from Trakhinin [39], see Theorem 3 and Theorem 4,
due to finer Sobolev imbeddings (3.9), (3.10).

###### Proof.

Using the Moser-type calculus inequalities (3.7) and (3.8), we obtain a
refined version of estimate (4.27) in tame form:

$||\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\tilde{{\mathbf{U}}}||_{s,\ast,T}\leq
C(K)\Big{(}||\tilde{{\mathbf{U}}}||_{s+2,\ast,T}+||\tilde{{\mathbf{U}}}||_{W^{2,\infty}_{\ast}(\Omega_{T})}||\hat{\mathbf{U}},\hat{\Psi}||_{s+2,\ast,T}\Big{)}.$

Then, using Sobolev embedding inequalities (3.10), we get

$||\mathbb{L}^{\prime}_{e}(\hat{{\mathbf{U}}},\hat{\Psi})\tilde{{\mathbf{U}}}||_{s,\ast,T}\leq
C(K)\Big{(}||\tilde{{\mathbf{U}}}||_{s+2,\ast,T}+||\tilde{{\mathbf{U}}}||_{6,\ast,T}||\hat{\mathbf{U}},\hat{\Psi}||_{s+2,\ast,T}\Big{)}.$

Using the above estimate, (4.30) and recalling the definition of
$\tilde{\mathbf{U}}$, see Section 4.3, it holds that

$\begin{split}||\mathbf{F}||_{s,\ast,T}\leq
C(K)\Big{(}||\mathbf{f}||_{s+2,\ast,T}+||\mathbf{g}||_{H^{s+2}(\Gamma_{T})}+(||\mathbf{f}||_{8,\ast,T}+||\mathbf{g}||_{H^{8}(\Gamma_{T})})||\hat{\mathbf{U}},\hat{\Psi}||_{s+2,\ast,T}\Big{)}.\end{split}$
(7.2)

Using the assumption (6.1) and (7.2) with $s=6$, we get

$||\mathbf{F}||_{6,\ast,T}\leq
C(K_{0})\Big{(}{||\mathbf{f}||_{8,\ast,T}+||\mathbf{g}||_{H^{8}(\Gamma_{T})}}\Big{)}.$
(7.3)

Combining the estimates (6.61), (7.2) and (7.3), we obtain the tame estimate
(7.1). ∎

##  8\. Construction of Approximate Solutions

Suppose the initial data

$({\mathbf{U}}^{\pm}_{0},\varphi_{0})=(\bar{\mathbf{U}}^{\pm}+\tilde{\mathbf{U}}^{\pm}_{0},\varphi_{0})=(p^{\pm}_{0},u^{\pm}_{1,0},u^{\pm}_{2,0},H^{\pm}_{1,0},H^{\pm}_{2,0},S^{\pm}_{0},\varphi_{0})$

satisfy the stability condition (4.4) and restriction (2.5) at $x_{1}=0$ for
$x_{2}\in{\mathbb{R}}$. Since ${H}^{+}_{2,0}\neq 0$ or ${H}^{-}_{2,0}\neq 0$
at ${x_{1}=0}$, see also Remark 4.3, from (2.5) we can solve
$\partial_{2}\varphi$ as follows (we drop the sub-index 0 for simplicity):

$\partial_{2}\varphi=\mu({\mathbf{U}})|_{x_{1}=0}=\frac{H^{+}_{1}H^{+}_{2}+H^{-}_{1}H^{-}_{2}}{(H^{+}_{2})^{2}+(H^{-}_{2})^{2}}\Big{|}_{x_{1}=0}\,,$
(8.1)

where ${\mathbf{U}}:=({\mathbf{U}}^{+},{\mathbf{U}}^{-}).$ Then, using the
boundary condition (4.3), we have

$\partial_{t}\varphi=\eta({\mathbf{U}})|_{x_{1}=0},$ (8.2)

with

$\eta({\mathbf{U}})=u^{+}_{1}-u^{+}_{2}\mu({\mathbf{U}}).$

By using the hyperbolicity condition (1.6), we can write the system in (2.1):

$\partial_{t}{\mathbf{U}}=-(A_{0}({\mathbf{U}}))^{-1}\Big{(}\tilde{A}_{1}({\mathbf{U}},\Psi)\partial_{1}{\mathbf{U}}+A_{2}({\mathbf{U}})\partial_{2}{\mathbf{U}}\Big{)},$
(8.3)

where $\Psi:=(\Psi^{+},\Psi^{-})$, and the matrices
$A_{0},A_{2},\tilde{A}_{1}$ are defined by (1.5) and (2.3). The traces

${\mathbf{U}}_{j}=(p^{+}_{j},u^{+}_{1,j},u^{+}_{2,j},H^{+}_{1,j},H^{+}_{2,j},S^{+}_{j},p^{-}_{j},u^{-}_{1,j},u^{-}_{2,j},H^{-}_{1,j},H^{-}_{2,j},S^{-}_{j})=\partial^{j}_{t}{\mathbf{U}}|_{t=0}$

and

$\varphi_{j}=\partial^{j}_{t}\varphi|_{t=0},\quad j\geq 1$

can be defined step by step by applying operator $\partial^{j-1}_{t}$ to (8.2)
and (8.3), for $j\geq 1$ and evaluating $\partial^{j}_{t}{\mathbf{U}}$ and
$\partial^{j}_{t}\varphi$ at $t=0$ in terms of the initial data. Notice that

$\Psi^{\pm}_{j}=\partial^{j}_{t}\Psi^{\pm}|_{t=0}=\chi(\pm x_{1})\varphi_{j}.$

Define the zero-th order compatibility condition:

$[u_{1,0}]-[u_{2,0}]\partial_{2}\varphi_{0}=0,\quad[p_{0}+\frac{|H_{0}|^{2}}{2}]=0.$
(8.4)

Taking (8.1), (8.2) evaluated at $t=0,$ and using (8.4), we obtain that

$\varphi_{1}=u^{\pm}_{1,0}-u^{\pm}_{2,0}\partial_{2}\varphi_{0}\Big{|}_{x_{1}=0}.$
(8.5)

Denote $(H^{\pm}_{N})_{j}=\partial^{j}_{t}H^{\pm}_{N}\Big{|}_{t=0}.$ Using
(8.3) and (8.5), taking $t=0,$ we obtain that

$(H^{\pm}_{N})_{1}=-\Big{(}u^{\pm}_{2,0}\partial_{2}(H^{\pm}_{N})_{0}+\partial_{2}u^{\pm}_{2,0}(H^{\pm}_{N})_{0}\Big{|}_{x_{1}=0}\Big{)}.$

Therefore, $(H^{\pm}_{N})_{0}|_{x_{1}=0}=0$ implies
$(H^{\pm}_{N})_{1}|_{x_{1}=0}=0.$ Once we have defined
${\mathbf{U}}_{1},\varphi_{1},$ we can deduce ${\mathbf{U}}_{2},\varphi_{2}$
and so on. Moreover, at each step, we can prove that

$(H^{\pm}_{N})_{j}|_{x_{1}=0}=0,\quad j\geq 2,$

provided that ${\mathbf{U}}_{j}$, $\varphi_{j}$ satisfy the compatibility
condition (see Definition 8.1).
The following Lemma 8.1 is necessary for the approximate solutions; we refer
to [14] and [20, Lemma 4.2.1]. Differently from [39], we take the initial data
in the standard Sobolev spaces.

######  Lemma 8.1.

Let $\mu\in\mathbb{N},\mu\geq 3,$
$\tilde{{\mathbf{U}}}_{0}:=\mathbf{U}_{0}-\bar{\mathbf{U}}\in
H^{\mu+1.5}(\mathbb{R}^{2}_{+})$ and $\varphi_{0}\in
H^{\mu+1.5}({\mathbb{R}})$. Then, we can determine
$\tilde{{\mathbf{U}}}_{j}\in H^{\mu+1.5-j}(\mathbb{R}^{2}_{+})$ and
$\varphi_{j}\in H^{\mu+1.5-j}({\mathbb{R}})$ by induction and set
$\mathbf{U}_{j}=\tilde{\mathbf{U}}_{j}+\bar{\mathbf{U}},$ for
$j=1,\cdots,\mu$. Besides we prove

$\sum^{\mu}_{j=0}(||\tilde{{\mathbf{U}}}_{j}||_{H^{\mu+1.5-j}(\mathbb{R}^{2}_{+})}+||\varphi_{j}||_{H^{\mu+1.5-j}({\mathbb{R}})})\leq
C(M_{0}),$ (8.6)

where $C>0$ depends only on $\mu$,
$||\tilde{{\mathbf{U}}}_{0}||_{W^{1,\infty}(\mathbb{R}^{2}_{+})}$ and
$||\varphi_{0}||_{W^{1,\infty}({\mathbb{R}})}$, and

$M_{0}:=||\tilde{{\mathbf{U}}}_{0}||_{{H^{\mu+1.5}}(\mathbb{R}^{2}_{+})}+||\varphi_{0}||_{H^{\mu+1.5}({\mathbb{R}})}\,.$
(8.7)

######  Definition 8.1.

Let $\mu\in\mathbb{N},\mu\geq 3$. The initial data
$(\tilde{{\mathbf{U}}}_{0},\varphi_{0})\in
H^{\mu+1.5}(\mathbb{R}^{2}_{+})\times H^{\mu+1.5}({\mathbb{R}})$ are defined
to be compatible up to order $\mu$ if $(\tilde{{\mathbf{U}}}_{j},\varphi_{j})$
satisfy (8.4) for $j=0$ and

$\sum^{j}_{l=0}([u_{1,j-l}]-[u_{2,j-l}]\partial_{2}\varphi_{l})=0,\quad[p_{j}]+\sum^{j-1}_{l=0}C_{l,j-1}[(H_{l},H_{j-l})]=0,\quad\mbox{on}\,\,\\{x_{1}=0\\}\,,$

for $j=1,\cdots,\mu$, where $C_{l,j-1}$ are suitable constants.

To use the tame estimate for the proof of convergence of the Nash–Moser
iteration, we should reduce our nonlinear problem to that whose solution
vanishes in the past. This is achieved by the construction of the so-called
“approximate solution” that allows to “absorb” the initial data into the
interior equation. The “approximate solution” is in the sense of Taylor’s
series at $t=0$.
Below, we will use the notation

$\mathbb{L}({\mathbf{U}},\Psi):=\left[\begin{array}[]{c}\mathbb{L}({\mathbf{U}}^{+},\Psi^{+})\\\
\mathbb{L}({\mathbf{U}}^{-},\Psi^{-})\end{array}\right].\quad$

######  Lemma 8.2.

Let $\mu\in\mathbb{N},\mu\geq 3$ and let $\delta>0$. Suppose the initial data
$(\tilde{{\mathbf{U}}}_{0},\varphi_{0})\in
H^{\mu+1.5}(\mathbb{R}^{2}_{+})\times H^{\mu+1.5}({\mathbb{R}})$ are
compatible up to order $\mu$ and satisfy the assumptions (1.6), (2.4), (2.5),
(4.4). Then, there exist $T>0$ and $({\tilde{\mathbf{U}}}^{a},\varphi^{a})\in
H^{\mu+2}(\Omega_{T})\times H^{\mu+2}(\Gamma_{T})$ such that

$\partial^{j}_{t}\mathbb{L}({\mathbf{U}}^{a},\Psi^{a})|_{t=0}=0\text{ in
}\Omega,\text{ for }j\in\\{0,\cdots,\mu-1\\},$ (8.8)

where

$\mathbf{U}^{a}:=\tilde{\mathbf{U}}^{a}+\overline{\mathbf{U}}\,,\quad\Psi^{a\,\pm}=\chi(\pm
x_{1})\varphi^{a}\,.$

We call $(\mathbf{U}^{a},\varphi^{a})$ the approximate solution to problem
(4.20). Moreover the approximate solution satisfies the following estimate

$||\tilde{{\mathbf{U}}}^{a}||_{H^{\mu+2}(\Omega_{T})}+||\varphi^{a}||_{H^{\mu+2}(\Gamma_{T})}<\delta\,,$
(8.9)

the stability conditions (4.4) on $\Gamma_{T}$, the hyperbolicity condition
(4.2) on $\Omega_{T}$.

###### Proof.

Let us first denote $\Phi^{a\,\pm}=\pm x_{1}+\Psi^{a\,\pm}$,
$\tilde{\mathbf{U}}^{a}=(\tilde{\mathbf{U}}^{a+},\tilde{\mathbf{U}}^{a-})^{T}$,
$\tilde{p}^{a}=(\tilde{p}^{a+},\tilde{p}^{a-})^{T}$,
$\tilde{u}^{a}=(\tilde{u}^{a+},\tilde{u}^{a-})^{T}$,
$\tilde{H}^{a}=(\tilde{H}^{a+},\tilde{H}^{a-})^{T}$,
$\tilde{S}^{a}=(\tilde{S}^{a+},\tilde{S}^{a-})^{T}$. Consider
$\tilde{\mathbf{U}}^{a}\in
H^{\mu+2}({\mathbb{R}}\times\mathbb{R}^{2}_{+}),\varphi^{a}\in
H^{\mu+2}({\mathbb{R}}^{2})$, such that

$\partial^{j}_{t}\tilde{\mathbf{U}}^{a}|_{t=0}=\tilde{\mathbf{U}}_{j}\in
H^{\mu-j+2}(\mathbb{R}^{2}_{+}),\text{ for }j=0,\cdots,\mu,$
$\partial^{j}_{t}\varphi^{a}|_{t=0}=\varphi_{j}\in
H^{\mu-j+2}({\mathbb{R}}),\quad\text{ for }j=0,\cdots,\mu,$

where $\tilde{\mathbf{U}}_{j}$ and $\varphi_{j}$ are given by Lemma 8.1. Since
$(\tilde{\mathbf{U}}^{a},\varphi^{a})$ satisfies the hyperbolicity condition
(4.2) and the stability condition (4.4) at $t=0$, by continuity
$(\tilde{\mathbf{U}}^{a},\varphi^{a})$ satisfy (4.4) at $x_{1}=0$ and (4.2)
for small times. By multiplication of $(\tilde{\mathbf{U}}^{a},\varphi^{a})$
by a cut-off function in time supported on $[-T,T]$ we can assume that (4.2),
(4.4) hold for all times (in this regard, recall Remark 3.1). Given any
$\delta>0$, by taking $T>0$ sufficiently small, we can assume that
$\tilde{\mathbf{U}}^{a}$, $\varphi^{a}$ are small in the sense of (8.9). ∎

######  Remark 8.1.

Let us remark that we do not require any constraint (that is interior
equations or boundary condition) to be satisfied by the approximate solution
constructed above. This allows us to the use of cut-off argument making the
hyperbolicity condition (4.2) and the stability condition (4.4) to be
satisfied globally in time, without any trouble.

######  Remark 8.2.

In the sequel, in the proof of the main Theorem 3.1, estimate (8.9) will be
used with $\mu=m+10$, being $m$ an integer as in the statement of that
theorem.

We assume that

$||\varphi_{0}||_{L^{\infty}({\mathbb{R}})}<\frac{1}{2},$

then we fix $T>0$ sufficiently small so that
$||\varphi^{a}||_{L^{\infty}([0,T]\times{\mathbb{R}})}\leq\frac{1}{2}.$ Hence,
we get

$\partial_{1}\Phi^{a+}\geq\frac{1}{2},\quad\partial_{1}\Phi^{a-}\leq-\frac{1}{2}$

(recall that $||\chi^{\prime}||_{L^{\infty}({\mathbb{R}})}\leq 1/2$, see
Section 2).