# Nonlinear stability of entropy waves for the Euler equations

Wei Wang  School of Mathematical Sciences, Zhejiang University, Hangzhou,
People’s Republic of China<EMAIL_ADDRESS>,  Zhifei Zhang  School of
Mathematical Sciences, Peking University, Beijing, People’s Republic of China
<EMAIL_ADDRESS>and  Wenbin Zhao  School of Mathematical Sciences,
Peking University, Beijing, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

In this article, we consider a class of the contact discontinuity for the full
compressible Euler equations, namely the entropy wave, where the velocity is
continuous across the interface while the density and the entropy can have
jumps. The nonlinear stability of entropy waves is a longstanding open problem
in multi-dimensional hyperbolic conservation laws. The rigorous treatments are
challenging due to the characteristic discontinuity nature of the problem
(G.-Q. Chen and Y.-G. Wang in Nonlinear partial differential equations, Volume
7 of Abel Symp.(2012)). In this article, we discover that the Taylor sign
condition plays an essential role in the nonlinear stability of entropy waves.
By deriving the evolution equation of the interface in the Eulerian
coordinates, we relate the Taylor sign condition to the hyperbolicity of this
evolution equation, which reveals a stability condition of the entropy wave.
With the optimal regularity estimates of the interface, we can derive the a
priori estimates without loss of regularity.

###### Key words and phrases:

compressible Euler, free boundary problems, contact discontinuity, entropy
wave

###### 2010 Mathematics Subject Classification:

35Q31, 35Q35, 35R35, 76B03, 76N10

W. Wang is supported by NSF of China (Nos.11931010, 12271476). Z. Zhang is
supported by NSF of China (Nos. 12171010, 12288101). W. Zhao is supported by
China Postdoctoral Science Foundation (Nos.2020TQ001, 2021M690225).

##  1\. Introduction

###  1.1. Presentation of the problem

In this article, we consider the full compressible Euler equations:

$\begin{cases}\partial_{t}\rho+\nabla\cdot(\rho u)=0,\\\ \partial_{t}(\rho
u)+\nabla\cdot(\rho u\otimes u)+\nabla p=0,\\\
\partial_{t}(\rho\frac{|u|^{2}}{2}+\rho
e)+\nabla\cdot\Big{\\{}u(\rho\frac{|u|^{2}}{2}+\rho
e+p)\Big{\\}}=0,\end{cases}$ (1.1)

where $\rho,\,u,\,p=p(\rho,\,S),\,e=e(\rho,\,S)$ are the density, velocity,
pressure and internal energy with the entropy $S$, respectively. The pressure
$p=p(\rho,\,S)$ satisfies the following state equation

$p=p(\rho,\,S)=A\rho^{\gamma}\mathrm{e}^{S},$ (1.2)

with $A>0$ and $\gamma>1$ as constants.

For a piecewise smooth weak solution of (1.6) in a domain
$\Omega=\mathbb{T}^{2}\times(-1,\,1)$ with

$\Omega^{\pm}=\\{x=(\overline{x},\,x_{3})=(x_{1},\,x_{2},\,x_{3})\in\Omega:\,x_{3}\gtrless
f(t,\,\overline{x})\\},$

the Rankine–Hugoniot (RH) conditions are satisfied on the interface
$\Gamma_{f}=\\{(\overline{x},\,f(t,\,\overline{x}):\,\overline{x}\in\mathbb{T}^{2})\\}$:

$\begin{cases}\llbracket m_{N}\rrbracket=0,\\\ m_{N}\llbracket
u_{N}\rrbracket+|N|^{2}\llbracket p\rrbracket=0,\\\ m_{N}\llbracket
u_{\tau}\rrbracket=0,\\\
m_{N}\llbracket\frac{|u|^{2}}{2}+e\rrbracket+\llbracket
pu_{N}\rrbracket=0,\end{cases}$ (1.3)

where

$N=(-\partial_{1}f,\,-\partial_{2}f,\,1)^{\top},\quad\quad\tau_{1}=(1,\,0,\,\partial_{1}f)^{\top},\quad\quad\tau_{2}=(0,\,1,\,\partial_{2}f)^{\top},$
(1.4) $u_{N}=u\cdot N,\quad\quad
u_{\tau}=(u\cdot\tau_{1},\,u\cdot\tau_{2})^{\top},\quad\quad
m_{N}=\rho(u_{N}-\partial_{t}f).$ (1.5)

There are two kinds of characteristic discontinuities on which $\llbracket
p\rrbracket=\llbracket u_{N}\rrbracket=m_{N}=0$ (see [4, 5, 7, 13]):

  * •

Vortex sheets:

$\llbracket u_{\tau}\rrbracket\neq 0;$

  * •

Entropy waves:

$\llbracket u_{\tau}\rrbracket=0,\quad\quad\llbracket\rho\rrbracket\neq
0,\quad\quad\llbracket S\rrbracket\neq 0.$

In this article, we focus on the entropy waves. The system (1.1) can be
rewritten as a symmetric hyperbolic system of $(p^{\pm},\,u^{\pm},\,S^{\pm})$
in $\Omega^{\pm}$ respectively:

$\begin{cases}\frac{1}{\gamma p^{\pm}}D_{t}^{\pm}p^{\pm}+\nabla\cdot
u^{\pm}=0,\\\ \rho^{\pm}D_{t}^{\pm}u^{\pm}+\nabla p^{\pm}=0,\\\
D_{t}^{\pm}S^{\pm}=0,\end{cases}$ (1.6)

where $D_{t}^{\pm}=\partial_{t}+u^{\pm}\cdot\nabla$. Since $\llbracket
u\rrbracket=0$, we shall just use $D_{t}=\partial_{t}+u\cdot\nabla$ to denote
the material derivative. The density $\rho^{\pm}$ can be recovered from (1.2)
by

$\rho^{\pm}=A^{-\frac{1}{\gamma}}(p^{\pm})^{\frac{1}{\gamma}}\mathrm{e}^{-\frac{S^{\pm}}{\gamma}}.$
(1.7)

On the interface $\Gamma_{f}$, the RH conditions across $\Gamma_{f}$ are

$\llbracket p\rrbracket=p^{+}-p^{-}=0,\quad\quad\llbracket
u\rrbracket=u^{+}-u^{-}=0,\quad\quad\text{on $\Gamma_{f}$}.$ (1.8)

Meanwhile, the density and the entropy can have jumps across $\Gamma_{f}$. We
shall assume that

$\llbracket\rho\rrbracket=\rho^{+}-\rho^{-}\neq 0,\quad\quad\llbracket
S\rrbracket=S^{+}-S^{-}\neq 0,\quad\quad\text{on $\Gamma_{f}$}.$ (1.9)

The evolution of the interface $\Gamma_{f}$ is given by

$\partial_{t}f=u^{\pm}\cdot N.$ (1.10)

We shall also use $N^{\pm}=\mp N$ to indicate outer normal directions of
$\Gamma_{f}$ in $\Omega^{\pm}$ respectively.

On the fixed upper and lower boundaries
$\Gamma^{\pm}=\mathbb{T}^{2}\times\\{\pm 1\\}$, there holds that

$u^{\pm}\cdot n^{\pm}\big{|}_{\Gamma^{\pm}}=0,$ (1.11)

with $n^{\pm}=(0,\,0,\,\pm 1)^{\top}$.

In this article, we shall prove the a priori estimates of the problem
(1.6)–(1.11) without loss of regularity under the Taylor sign condition

$\llbracket\nabla_{N}p\rrbracket=\nabla_{N}p^{+}-\nabla_{N}p^{-}=-\nabla_{N^{+}}p^{+}-\nabla_{N^{-}}p^{-}>0.$
(1.12)

More precise statement of the main result will be presented in Section 3. In
Appendix A, we also show that the a priori estimates of the problem hold true
for the incompressible Euler equations.

###  1.2. History and related works

There are three fundamental waves in the multi-dimensional hyperbolic
conservation laws: shock waves, rarefaction waves and contact discontinuities
(including vortex sheets and entropy waves). The interested reader is referred
to [4, 13] for detailed discussion. The nonlinear stability of shock waves and
rarefaction waves were proved in [23, 24] and [2, 3] respectively. As for
contact discontinuities, they are characteristic discontinuities and usually
subject to the Kelvin–Helmholtz instability and the Rayleigh-Taylor
instability (see [14, 16]). If $\llbracket u\rrbracket\neq 0$, the contact
discontinuity is also called the vortex sheet. The 3D vortex sheets are
violently unstable while the 2D vortex sheets are weakly stable under the
supersonic condition (see [9, 15, 26]). The nonlinear stability of the 2D
vortex sheets were proved in [11, 12] (see also [30, 31]).

If $\llbracket u\rrbracket=0$ where the velocity is continuous across the
interface, the contact discontinuity is also called the entropy wave. The
normal modes analysis shows that the entropy wave is only weakly stable (see
[5]). However, the nonlinear stability of entropy waves in multi-dimensional
situations is a longstanding open problem (see [7]). There even lacks
stability conditions addressing this problem. As stated in [7], “it would be
interesting to analyze entropy waves to explore new phenomena and features of
these waves in two-dimensions and even higher dimensions.”

In this article, we discover that the Taylor sign condition is essential to
the nonlinear stability of entropy waves and prove the a priori estimates of
the problem without loss of regularity. More precisely:

  * •

We shall derive the evolution equation of the interface and study the problem
in the Eulerian coordinates. This approach was first used in [33] to
investigate the stability of the incompressible current-vortex sheets. It was
generalized by the authors to the one-phase compressible Euler equations in
[38], where it is vacuum in $\Omega^{+}$. The entropy wave can be seen as a
two-phase problem, where we have two fluids $(p^{\pm},\,u^{\pm},\,S^{\pm})$ in
$\Omega^{\pm}$ respectively.

  * •

With the evolution equation of the interface, we discover that the Taylor sign
condition is a natural stability condition since it is equivalent to the
hyperbolicity of this evolution equation. Then we can derive the optimal
regularity estimates of the interface. This enables us to investigate the
quantities inside $\Omega^{\pm}$ in a simpler way. We do not need to make a
change of coordinates or use the Alinhac’s good unknowns. The a priori
estimates can be derived without loss of regularity. This is important since
loss of regularity is a common phenomenon for characteristic discontinuities
(see [4]).

  * •

The Taylor sign condition is commonly used when treating the free boundary
problems. However, for the two-phase compressible flow without gravity or
other forces, it is a strong requirement that the Taylor sign condition (1.12)
holds at each point on the interface. On the other hand, the violation of the
Taylor sign condition will lead to the Rayleigh-Taylor instability (see [16]).

  * •

In order for the piecewise smooth solution to be the entropy wave defined in
(1.8)–(1.9), there need some high order compatibility conditions (3.14) on the
interface. As discussed in Remark 3.1, the violation of the compatibility
conditions could transform the entropy wave to a vortex sheet and lead to the
Kelvin–Helmholtz instability. This suggests that the entropy wave is a really
special class of contact discontinuities of the compressible Euler equations.
See also the discussion of viscous contact discontinuities in [17, 18].

  * •

When considering the incompressible Euler equations, the Taylor sign condition
also indicates the nonlinear stability of the interface as long as $\llbracket
u\rrbracket=0$ where the Kevin-Helmholtz instability is ruled out. In this
case, we are able to give a construction of the solution.

There is also a huge literature investigating other stabilizing effects on the
interface. When taking magnetic fields under consideration, there are more
types of characteristic discontinuities (see [5, 19]). If the magnetic fields
are parallel to the interface, the characteristic discontinuities are called
current-vortex sheets. The stability of the current-vortex sheets were
investigated in [6, 34, 36, 41] for the compressible case and in [10, 21, 27,
33, 35, 42] for the incompressible case. If the magnetic fields are continuous
across the interface and not parallel to the interface, the characteristic
discontinuities are called MHD contact discontinuities. In [28, 29] Morando et
al proved the nonlinear stability assuming that the Taylor sign condition
holds. See also [37] for the case with surface tension. Recently, Wang and Xin
in [39] managed to prove the nonlinear stability without the assumption of the
Taylor sign condition. They used the Lagrangian coordinates and verified that
the normal component of the magnetic field can stabilize the interface (see
also [40] for the incompressible case with surface tension).

The rest of the article is organized as follows. After laying out some
preliminaries in Section 2, we shall derive the evolution equation of the
interface and present the main result in Section 3. We prove some basic
estimates in Section 4. The evolution of the interface is estimated in Section
5. The estimates of the pressure, the velocity and the entropy are derived in
Section 6. In Appendix A, we give a discussion of corresponding problem in the
incompressible Euler equations. In Appendix B, some results on the elliptic
systems are presented. We also list some analytic tools in Appendix C.

##  2\. Preliminaries

In this section, we recall some preliminary results on the harmonic
coordinates and the Dirichlet-Neumann (DN) operators from [33, 38]. The
notations and basic properties of paradifferential operators are included in
Appendix C.

###  2.1. Harmonic coordinates

Given a smooth function $f_{\ast}=f_{\ast}(\overline{x})$, we define a
reference domain

$\Omega_{\ast}^{\pm}=\\{x\in\Omega:\,x_{3}\gtrless
f_{\ast}(\overline{x})\\},\quad\quad\Gamma_{*}=\\{(\overline{x},\,f_{*}(\overline{x})):\,\overline{x}\in\mathbb{T}^{2}\\}.$

We shall consider the free boundary problem that lies in a neighborhood of the
reference domain $\Omega_{\ast}$. To this end, we define

$\mathcal{N}(\delta,\,\kappa)=\\{f\in
H^{\kappa}(\mathbb{T}^{2}):\,\big{\|}f-f_{\ast}\big{\|}_{H^{\kappa}(\mathbb{T}^{2})}\leq\delta\\}.$

For a function $f\in\mathcal{N}(\delta,\,\kappa)$, set

$\Omega_{f}^{\pm}=\\{x\in\Omega:\,x_{3}\gtrless
f(\overline{x})\\},\quad\quad\Gamma_{f}=\\{(\overline{x},\,f(\overline{x})):\,\overline{x}\in\mathbb{T}^{2}\\}.$

Then we can introduce the harmonic coordinates. Define
$\Phi_{f}^{\pm}:\,\Omega_{\ast}^{\pm}\to\Omega_{f}^{\pm}$ by the harmonic
extension:

$\begin{cases}\Delta\Phi_{f}^{\pm}=0,&\text{in $\Omega_{\ast}^{\pm}$},\\\
\Phi_{f}^{\pm}(\overline{x},\,f_{\ast}(\overline{x}))=(\overline{x},\,f(\overline{x})),&\text{on
$\Gamma_{\ast}$},\\\ \Phi_{f}^{\pm}(\overline{x},\,\pm 1)=(\overline{x},\,\pm
1),&\text{on $\Gamma^{\pm}$}.\end{cases}$ (2.1)

Given $f_{\ast}$, there exists $\delta_{0}>0$ such that $\Phi_{f}^{\pm}$ is
bijective when $\delta\in[0,\,\delta_{0}]$. Thus we can also define the
inverse map $(\Phi_{f}^{\pm})^{-1}:\,\Omega_{f}^{\pm}\to\Omega_{\ast}^{\pm}$
such that

$\Phi_{f}^{\pm}\circ(\Phi_{f}^{\pm})^{-1}=\mathrm{Id},\quad\quad(\Phi_{f}^{\pm})^{-1}\circ\Phi_{f}^{\pm}=\mathrm{Id}.$

Let us list some basic inequalities about harmonic coordinates without proof.

######  Lemma 2.1.

Assume that $f\in\mathcal{N}(\delta_{0},\,\kappa)$ with $\kappa\geq 4$. Then
there exists a constant
$C=C(\delta_{0},\,\big{\|}f_{\ast}\big{\|}_{H^{\kappa}(\mathbb{T}^{2})})$ such
that

  1. (1)

If $u\in H^{s}(\Omega_{f}^{\pm})$ with $s\in[0,\,\kappa]$, then

$\big{\|}u\circ\Phi_{f}^{\pm}\big{\|}_{H^{s}(\Omega_{\ast}^{\pm})}\leq
C\big{\|}u\big{\|}_{H^{s}(\Omega_{f}^{\pm})}.$

  2. (2)

If $u\in H^{s}(\Omega_{\ast}^{\pm})$ with $s\in[0,\,\kappa]$, then

$\big{\|}u\circ(\Phi_{f}^{\pm})^{-1}\big{\|}_{H^{s}(\Omega_{f}^{\pm})}\leq
C\big{\|}u\big{\|}_{H^{s}(\Omega_{\ast}^{\pm})}.$

  3. (3)

If $u,\,v\in H^{s}(\Omega_{f}^{\pm})$ with $s\in[2,\,\kappa]$, then

$\big{\|}uv\big{\|}_{H^{s}(\Omega_{f}^{\pm})}\leq
C\big{\|}u\big{\|}_{H^{s}(\Omega_{f}^{\pm})}\big{\|}v\big{\|}_{H^{s}(\Omega_{f}^{\pm})}.$

###  2.2. The Dirichlet-Neumann operator

For a smooth enough function $g=g(\overline{x})$ on
$\Gamma_{f}=\\{(\overline{x},\,f(\overline{x})):\,\overline{x}\in\mathbb{T}^{2}\\}$,
denote the harmonic extension of $g$ to $\Omega_{f}^{\pm}$ by
$\mathcal{H}_{f}^{\pm}g$, that is,

$\begin{cases}\Delta\mathcal{H}_{f}^{\pm}g=0,&\text{in $\Omega_{f}^{\pm}$},\\\
(\mathcal{H}_{f}^{\pm}g)(\overline{x},\,f(\overline{x}))=g(\overline{x}),&\text{on
$\Gamma_{f}$},\\\ (\mathcal{H}_{f}^{\pm}g)(\overline{x},\,\pm 1)=0,&\text{on
$\Gamma^{\pm}$}.\end{cases}$ (2.2)

Here we use the Dirichlet boundary condition on the bottom $\Gamma^{\pm}$
instead of the Neumann boundary condition as in the usual case. This
modification is useful in the energy estimates in the following sections.

For a smooth enough function $g=g(\overline{x})$ on
$\Gamma_{f}=\\{(\overline{x},\,f(\overline{x})):\,\overline{x}\in\mathbb{T}^{2}\\}$,
define

$\mathcal{G}_{f}^{\pm}g=N^{\pm}\cdot\nabla\mathcal{H}_{f}^{\pm}g\big{|}_{\Gamma_{f}}=\mp
N\cdot\nabla\mathcal{H}_{f}^{\pm}g\big{|}_{\Gamma_{f}},$ (2.3)

where $N=(-\partial_{1}f,\,-\partial_{2}f,\,1)^{\top}$ is the scaled normal
vector on the surface $\Gamma_{f}$. However, all the regularity properties of
the Dirichlet-Neumann operator will be kept in spite of the modification, as
discussed in the appendix of [38]. The same arguments in [20] yield the
following basic properties of the DN operator.

######  Lemma 2.2.

Let $f\in\mathcal{N}(\delta_{0},\,\kappa)$ with $\kappa\geq 4$. Then there
exists a constant
$C=C(\delta_{0},\,\big{\|}f\big{\|}_{H^{\kappa}(\mathbb{T}^{2})})$ such that

  1. (1)

$\mathcal{G}_{f}^{\pm}$ is self-adjoint:

$(\mathcal{G}_{f}^{\pm}\phi,\,\psi)=(\phi,\,\mathcal{G}_{f}^{\pm}\psi),\quad\quad\forall\phi,\,\psi\in
H^{\frac{1}{2}}(\mathbb{T}^{2}).$

  2. (2)

$\mathcal{G}_{f}^{\pm}$ is positive:

$(\mathcal{G}_{f}^{\pm}\phi,\,\phi)\geq
C\big{\|}\phi\big{\|}_{\dot{H}^{\frac{1}{2}}(\mathbb{T}^{2})}.$

By the appendix of [33], we also have the following paralinearization of the
DN operators.

######  Lemma 2.3.

Assume that $f\in H^{\kappa}(\mathbb{T}^{2})$ with $\kappa\geq 4$. Then the DN
operators $\mathcal{G}_{f}^{\pm}$ can be decomposed as

$\mathcal{G}_{f}^{\pm}=T_{\lambda}+R_{f}^{\pm},$ (2.4)

where the symbol of the leading term is

$\lambda(x,\,\xi)=\sqrt{(1+|\nabla f|^{2})|\xi|^{2}-(\nabla f\cdot\xi)^{2}}$
(2.5)

and the remainder terms $R_{f}^{\pm}$ satisfy that

$\big{\|}R_{f}^{\pm}g\big{\|}_{H^{s}}\leq
C\big{(}\big{\|}f\big{\|}_{H^{\kappa}}\big{)}\big{\|}g\big{\|}_{H^{s}},\quad\quad\forall
s\in[1/2,\,\kappa-1].$ (2.6)

Furthermore, there holds that

$\big{\|}\mathcal{G}_{f}^{\pm}g\big{\|}_{H^{s-1}}\leq
C\big{(}\big{\|}f\big{\|}_{H^{\kappa}}\big{)}\big{\|}g\big{\|}_{H^{s}},\quad\quad\forall
s\in[1/2,\,\kappa].$ (2.7)

###  2.3. Notations

When there is no ambiguity, we shall use
$u(x)=\mathbf{1}_{\Omega^{+}}(x)u^{+}(x)+\mathbf{1}_{\Omega^{-}}(x)u^{-}(x)$
and
$\big{\|}u\big{\|}_{H^{s}(\Omega)}=\big{\|}u^{+}\big{\|}_{H^{s}(\Omega^{+})}+\big{\|}u^{-}\big{\|}_{H^{s}(\Omega^{-})}$
to simplify notations.

Since $u$ is continuous across $\Gamma_{f}$ from (1.8), we shall just use
$D_{t}=\partial_{t}+u\cdot\nabla$ as the material derivative. The tangential
derivatives in $\Omega^{\pm}$ are

$\displaystyle\overline{\partial}_{t}=\partial_{t}+\mathcal{H}_{f}^{\pm}(\partial_{t}f)\partial_{3},\quad\quad\overline{\partial}_{j}=\partial_{j}+\mathcal{H}_{f}^{\pm}(\partial_{j}f)\partial_{3}\quad(j=1,\,2).$
(2.8)

By (1.10), it is direct to verify that

$D_{t}=\overline{\partial}_{t}+u_{1}\overline{\partial}_{1}+u_{2}\overline{\partial}_{2},\quad\quad\text{on
$\Gamma_{f}$}.$ (2.9)

By the definition of the harmonic extension (2.2), the derivatives
$\overline{\partial}=(\overline{\partial}_{1},\,\overline{\partial}_{2})$ are
tangential to both $\Gamma_{f}$ and $\Gamma^{\pm}$. We denote by

$\Lambda=\langle\nabla\rangle=(1-\Delta)^{1/2},\quad\quad\Upsilon=\langle\overline{\partial}\rangle=(1+|\overline{\partial}|^{2})^{1/2}$
(2.10)

when treating high order derivatives.

The minuscule indices $i,\,j,\,k$ are in $\\{1,\,2\\}$ and the capital indices
$J,\,K$ are in $\\{1,\,2,\,3\\}$. We shall use the Einstein summation
convention, that is, a repeated index in a term means summation of terms over
the index. To simply the arguments, we also omit all the binomial coefficients
$\binom{l}{m}=\frac{l!}{m!(l-m)!}$.

In the following, we shall use $\mathcal{H}^{\pm}=\mathcal{H}_{f}^{\pm}$ and
$\mathcal{G}^{\pm}=\mathcal{G}_{f}^{\pm}$ if there is no confusion of the
function $f$.

##  3\. Reformulation and main result

###  3.1. Evolution of interface $f$

By (2.9), we can rewrite the evolution equation of $f$ in (1.10) as

$D_{t}f=u_{3}^{\pm}.$ (3.1)

By taking another derivative $D_{t}$ to both sides of (3.1), we have

$D_{t}^{2}f=D_{t}u^{\pm}_{3}.$ (3.2)

For $i=1,\,2$, there holds that

$\displaystyle D_{t}^{2}\overline{\partial}_{i}f=$
$\displaystyle\overline{\partial}_{i}D_{t}u_{3}^{\pm}+[D_{t}^{2},\,\overline{\partial}_{i}]f$
(3.3) $\displaystyle=$
$\displaystyle\overline{\partial}_{i}D_{t}u_{3}^{\pm}+[D_{t},\,\overline{\partial}_{i}]D_{t}f+D_{t}[D_{t},\,\overline{\partial}_{i}]f$
$\displaystyle=$
$\displaystyle\overline{\partial}_{i}D_{t}u_{3}^{\pm}-\overline{\partial}_{i}u_{j}^{\pm}\overline{\partial}_{j}D_{t}f-D_{t}(\overline{\partial}_{i}u_{j}^{\pm}\overline{\partial}_{j}f)$
$\displaystyle=$
$\displaystyle\overline{\partial}_{i}D_{t}u_{3}^{\pm}-\overline{\partial}_{i}u_{j}^{\pm}(D_{t}\overline{\partial}_{j}f+\overline{\partial}_{j}u_{k}^{\pm}\overline{\partial}_{k}f)$
$\displaystyle-(\overline{\partial}_{i}D_{t}u_{j}^{\pm}-\overline{\partial}_{i}u_{k}^{\pm}\overline{\partial}_{k}u_{j}^{\pm})\overline{\partial}_{j}f-\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f$
$\displaystyle=$ $\displaystyle\overline{\partial}_{i}D_{t}u^{\pm}\cdot
N-2\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f.$

Furthermore, we can plug the momentum equation in (1.6) to (3.3) to get that

$\displaystyle\rho^{\pm}D_{t}^{2}\overline{\partial}_{i}f=$
$\displaystyle-\overline{\partial}_{i}\nabla p^{\pm}\cdot
N+\frac{\overline{\partial}_{i}\rho^{\pm}}{\rho^{\pm}}\nabla_{N}p^{\pm}-2\rho^{\pm}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f$
(3.4) $\displaystyle=$
$\displaystyle-\Big{\\{}\nabla\overline{\partial}_{i}p^{\pm}-\partial_{3}p^{\pm}\nabla\mathcal{H}^{\pm}(\overline{\partial}_{i}f)\Big{\\}}\cdot
N+\frac{\overline{\partial}_{i}\rho^{\pm}}{\rho^{\pm}}\nabla_{N}p^{\pm}-2\rho^{\pm}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f$
$\displaystyle=$
$\displaystyle\partial_{3}p^{\pm}\nabla_{N}\mathcal{H}^{\pm}(\overline{\partial}_{i}f)-\nabla_{N}\overline{\partial}_{i}p^{\pm}+\frac{\overline{\partial}_{i}\rho^{\pm}}{\rho^{\pm}}\nabla_{N}p^{\pm}-2\rho^{\pm}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f.$

Thus, by taking a sum of (3.4) with the positive index and the negative index
respectively, we have

$\displaystyle
D_{t}^{2}\overline{\partial}_{i}f+\frac{1}{\rho^{+}+\rho^{-}}\Big{\\{}\partial_{3}p^{+}\mathcal{G}^{+}-\partial_{3}p^{-}\mathcal{G}^{-}\Big{\\}}\overline{\partial}_{i}f=\frac{1}{\rho^{+}+\rho^{-}}\Big{\\{}\mathcal{M}^{+}+\mathcal{M}^{-}\Big{\\}},$
(3.5)

with the Dirichlet-Neumann operators $\mathcal{G}^{\pm}$ defined in (2.3) and

$\mathcal{M}^{\pm}=-\nabla_{N}\overline{\partial}_{i}p^{\pm}+\frac{\overline{\partial}_{i}\rho^{\pm}}{\rho^{\pm}}\nabla_{N}p^{\pm}-2\rho^{\pm}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f.$
(3.6)

Lastly, the decomposition of the DN operators
$\mathcal{G}^{\pm}=T_{\lambda}+R^{\pm}$ in (2.4) can be applied to get that

$D_{t}^{2}\overline{\partial}_{i}f+\mathfrak{a}T_{\lambda}\overline{\partial}_{i}f=\mathcal{N}^{+}+\mathcal{N}^{-},$
(3.7)

where the Taylor sign $\mathfrak{a}$ is

$\mathfrak{a}=\frac{\partial_{3}p^{+}-\partial_{3}p^{-}}{\rho^{+}+\rho^{-}},$
(3.8)

and the lower order terms are

$\displaystyle\mathcal{N}^{\pm}=$
$\displaystyle-\frac{1}{\rho^{+}+\rho^{-}}\nabla_{N}\overline{\partial}_{i}p^{\pm}+\frac{\overline{\partial}_{i}\rho^{\pm}}{(\rho^{+}+\rho^{-})\rho^{\pm}}\nabla_{N}p^{\pm}$
(3.9)
$\displaystyle-\frac{2\rho^{\pm}}{\rho^{+}+\rho^{-}}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f\mp\frac{\partial_{3}p^{\pm}}{\rho^{+}+\rho^{-}}R^{\pm}\overline{\partial}_{i}f.$

Since

$\partial_{3}=\frac{1}{1+|\nabla
f|^{2}}\Big{\\{}\nabla_{N}+\partial_{i}f\overline{\partial}_{i}\Big{\\}},$

the RH conditions (1.8) and the Taylor sign condition (1.12) imply that

$\displaystyle\mathfrak{a}=$
$\displaystyle\frac{1}{(\rho^{+}+\rho^{-})(1+|\nabla
f|^{2})}\Big{\\{}\llbracket\nabla_{N}p\rrbracket+\partial_{i}f\llbracket\overline{\partial}_{i}p\rrbracket\Big{\\}}$
(3.10) $\displaystyle=$
$\displaystyle\frac{\llbracket\nabla_{N}p\rrbracket}{(\rho^{+}+\rho^{-})(1+|\nabla
f|^{2})}>0.$

###  3.2. Evolution of the vorticity $\omega^{\pm}$ and the entropy $S^{\pm}$

By taking the curl of the both sides of the momentum equation in (1.6), we
have the evolution equation of the vorticity $\omega^{\pm}$ as

$D_{t}\omega^{\pm}=\omega^{\pm}\cdot\nabla u^{\pm}-\omega^{\pm}(\nabla\cdot
u^{\pm})-\nabla\frac{1}{\rho^{\pm}}\times\nabla p^{\pm}.$ (3.11)

The evolution of the entropy $S^{\pm}$ is given by the entropy equation in
(1.6) as

$D_{t}S^{\pm}=0.$ (3.12)

###  3.3. Evolution of the pressure $p^{\pm}$

To derive the evolution equation of the pressure $p^{\pm}$, we take
$D_{t}\eqref{eq: system: Euler: p u
S}_{1}-\nabla\cdot\Big{\\{}\frac{1}{\rho^{\pm}}\times\eqref{eq: system: Euler:
p u S}_{2}\Big{\\}}$ to get that

$D_{t}\Big{(}\frac{1}{\gamma
p^{\pm}}D_{t}p^{\pm}\Big{)}-\nabla\cdot\Big{(}\frac{1}{\rho^{\pm}}\nabla
p^{\pm}\Big{)}=\mathrm{tr}\,(\nabla u^{\pm})^{2}.$ (3.13)

This is a second order wave equation.

###  3.4. Compatibility conditions

Assume that $\kappa\geq 4$ is an integer. To estimate high order derivatives
of the piecewise smooth weak solutions, we need the following compatibility
conditions on the interface $\Gamma_{f}$:

$\llbracket D_{t}^{l}p\rrbracket=0,\quad\quad\llbracket
D_{t}^{l}u\rrbracket=0,\quad\quad 0\leq l\leq\kappa+1.$ (3.14)

Next we discuss an important consequence of the compatibility conditions
(3.14). Assume that the solution $(p^{\pm},\,u^{\pm},\,S^{\pm})\in
H^{3}(\Omega^{\pm})\subset C^{1}(\Omega^{\pm})$ and $f\in
H^{3}(\mathbb{T}^{2})\subset C^{1}(\mathbb{T}^{2})$. Recall the vectors
$\tau_{1}$ and $\tau_{2}$ in (1.4) which are tangential to the interface
$\Gamma_{f}$. The RH conditions (1.8) and the compatibility conditions (3.14)
can be applied to the momentum equation in (1.6) to get that

$\llbracket\frac{\nabla p}{\rho}\rrbracket=-\llbracket D_{t}u\rrbracket=0.$
(3.15)

Furthermore, since $\llbracket p\rrbracket=0$ and

$\llbracket\tau_{i}\cdot\nabla p\rrbracket=\tau_{i}\cdot\nabla\llbracket
p\rrbracket=0,\quad\quad i=1,\,2,$ (3.16)

by (3.15) there holds that

$0=\tau_{i}\cdot\llbracket\frac{\nabla
p}{\rho}\rrbracket=\llbracket\frac{1}{\rho}\rrbracket(\tau_{i}\cdot\nabla
p),\quad\quad i=1,\,2.$ (3.17)

That is, when $\llbracket\rho\rrbracket\neq 0$, by (3.17) we must have

$\overline{\partial}_{i}p\big{|}_{\Gamma_{f}}=\tau_{i}\cdot\nabla
p\big{|}_{\Gamma_{f}}=0\quad\quad(i=1,\,2),$ (3.18)

which implies that the pressure on the interface
$p|_{\Gamma_{f}}=p(t,\overline{x},f(t,\,\overline{x}))$ is independent on the
space variables $\overline{x}$. Thus, we may assume that

$p\big{|}_{\Gamma_{f}}=q(t),\quad\quad q(0)=0.$ (3.19)

######  Remark 3.1.

The consequence (3.18) is necessary for the piecewise smooth weak solution to
be an entropy wave. If (3.18) fails, the tangential component of the pressure
force is not trivial. Since the densities have a jump
$\llbracket\rho\rrbracket\neq 0$, the accelerations on both sides must have a
jump:

$\llbracket D_{t}u\rrbracket\cdot\tau_{j}=-\llbracket\frac{\tau_{j}\cdot\nabla
p}{\rho}\rrbracket=-\llbracket\frac{1}{\rho}\rrbracket(\tau_{j}\cdot\nabla
p)\neq 0.$

The entropy wave could evolve to a vortex sheet immediately.

###  3.5. Main result

Assume that $\kappa\geq 4$ is an integer. The full energy norm is defined as

$\displaystyle\mathcal{E}(t)=$
$\displaystyle\big{\|}f\big{\|}_{H^{\kappa}(\mathbb{T}^{2})}+\big{\|}D_{t}f\big{\|}_{H^{\kappa-\frac{1}{2}}(\mathbb{T}^{2})}$
(3.20)
$\displaystyle+\big{\|}u\big{\|}_{H^{\kappa}(\Omega)}+\big{\|}D_{t}u\big{\|}_{H^{\kappa-1}(\Omega)}+\sum_{l=2}^{\kappa+1}\big{\|}D_{t}^{l}u\big{\|}_{H^{\kappa+1-l}(\Omega)}$
$\displaystyle+\big{\|}(p,\,S)\big{\|}_{H^{\kappa}(\Omega)}+\sum_{l=1}^{\kappa+1}\big{\|}(D_{t}^{l}p,\,D_{t}^{l}S)\big{\|}_{H^{\kappa+1-l}(\Omega)}.$

For some $T>0$, we shall denote by

$M=\sup_{0<t<T}\mathcal{E}(t),\quad\quad M_{0}=\mathcal{E}(0).$

The lower order energy norm is

$\displaystyle\mathcal{F}(t)=\big{\|}(p,\,u,\,S)\big{\|}_{H^{\kappa-1}(\Omega)}+\sum_{l=1}^{\kappa}\big{\|}(D_{t}^{l}p,\,D_{t}^{l}u,\,D_{t}^{l}S)\big{\|}_{H^{\kappa-l}(\Omega)}.$
(3.21)

######  Theorem 3.2.

Let $\kappa\geq 4$ be an integer. Suppose that the initial data
$(f_{\mathrm{in}},\,p_{\mathrm{in}}^{\pm},\,u_{\mathrm{in}}^{\pm},\,S_{\mathrm{in}}^{\pm})$
satisfy the bound $\mathcal{E}(0)=M_{0}<\infty$. Furthermore, assume that
there are two constants $0<c_{0}<C_{0}$ such that

  1. (1)

$c_{0}\leq\rho_{\mathrm{in}}^{\pm}\leq C_{0}$, $c_{0}\leq
p_{\mathrm{in}}^{\pm}\leq C_{0}$;

  2. (2)

$-1+c_{0}\leq f_{\mathrm{in}}\leq 1-c_{0}$;

  3. (3)

$\frac{N_{\mathrm{in}}}{|N_{\mathrm{in}}|}\cdot\llbracket\nabla
p_{\mathrm{in}}^{\pm}\rrbracket\geq c_{0}$.

Then for $T>0$, the solution $(f,\,p^{\pm},\,u^{\pm},\,S^{\pm})$ to the
problem (1.6)–(1.11) under the compatibility conditions (3.14) satisfies that

  1. (1)

$M\leq C(M_{0},\,c_{0},\,C_{0})+TC(M,\,c_{0},\,C_{0})$;

  2. (2)

$-1+\frac{c_{0}}{2}\leq f\leq 1-\frac{c_{0}}{2}$;

  3. (3)

$\frac{N}{|N|}\cdot\llbracket\nabla p^{\pm}\rrbracket\geq\frac{c_{0}}{2}$.

The constants $C(M_{0},\,c_{0},\,C_{0})$ are continuous functions of
$M_{0},\,c_{0},\,C_{0}$. In the energy estimates in the following sections, we
shall take $c_{0}$ and $C_{0}$ to be fixed and just use $C(M_{0})$ to denote
constants from line to line.

##  4\. Basic energy estimates

In this section, we prove some basic energy estimates.

###  4.1. Lower order estimates

For the lower order energy norm $\mathcal{F}$ defined in (3.21), we have the
following estimates.

######  Proposition 4.1.

For $t\in[0,\,T]$, there holds that

$\mathcal{F}(t)\leq C(M_{0})+TC(M).$ (4.1)

Furthermore,

$\big{\|}(p,\,u,\,S)\big{\|}_{W^{\kappa-3,\infty}(\Omega)}+\sum_{l=1}^{\kappa-2}\big{\|}(D_{t}^{l}p,\,D_{t}^{l}u,\,D_{t}^{l}S)\big{\|}_{W^{\kappa-2-l,\infty}(\Omega)}\leq
C(M_{0})+TC(M).$ (4.2)

###### Proof.

For $1\leq l\leq\kappa$, since

$\begin{cases}D_{t}\Lambda^{\kappa-l}D_{t}^{l}p=\Lambda^{\kappa-l}D_{t}^{l+1}p-[\Lambda^{\kappa-l},\,D_{t}]D_{t}^{l}p,\\\
D_{t}\Lambda^{\kappa-l}D_{t}^{l}u=\Lambda^{\kappa-l}D_{t}^{l+1}u-[\Lambda^{\kappa-l},\,D_{t}]D_{t}^{l}u,\\\
D_{t}\Lambda^{\kappa-l}D_{t}^{l}S=0,\end{cases}$

we have

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\big{\|}(D_{t}^{l}p,\,D_{t}^{l}u,\,D_{t}^{l}S)\big{\|}_{H^{\kappa-l}(\Omega)}^{2}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\Big{\\{}D_{t}\Lambda^{\kappa-l}D_{t}^{l}p\cdot\Lambda^{\kappa-l}D_{t}^{l}p+D_{t}\Lambda^{\kappa-l}D_{t}^{l}u\cdot\Lambda^{\kappa-l}D_{t}^{l}u+D_{t}\Lambda^{\kappa-l}D_{t}^{l}S\cdot\Lambda^{\kappa-l}D_{t}^{l}S\Big{\\}}\mathrm{d}x$
$\displaystyle+\int_{\Omega}\frac{\nabla\cdot
u}{2}\Big{\\{}|\Lambda^{\kappa-l}D_{t}^{l}p|^{2}+|\Lambda^{\kappa-l}D_{t}^{l}u|^{2}+|\Lambda^{\kappa-l}D_{t}^{l}S|^{2}\Big{\\}}\mathrm{d}x$
$\displaystyle\leq$ $\displaystyle C(M).$

The case when $l=0$ follows in a similar way.

Application of the Sobolev inequalities to (4.1) proves (4.2).

∎

###  4.2. Tangential energy of $(p,\,u)$

Recall the equations for the pressure and velocity $(p,\,u)$ in (1.6):

$\begin{cases}\frac{1}{\gamma p}D_{t}p+\nabla\cdot u=0,\\\ \rho D_{t}u+\nabla
p=0.\end{cases}$ (4.3)

For $0\leq l\leq\kappa+1$, by taking $D_{t}^{l}$ to both sides of (4.3), we
have the system for $(D_{t}^{l}p,\,D_{t}^{l}u)$ as

$\begin{cases}\frac{1}{\gamma p}D_{t}^{l+1}p+\nabla\cdot
D_{t}^{l}u=\mathcal{N}_{p}^{l},\\\ \rho D_{t}^{l+1}u+\nabla
D_{t}^{l}p=\mathcal{N}_{u}^{l},\end{cases}$ (4.4)

where

$\mathcal{N}_{p}^{l}=[\frac{1}{\gamma
p},\,D_{t}^{l}]D_{t}p+[\nabla\cdot,\,D_{t}^{l}]u,\quad\quad\mathcal{N}_{u}^{l}=[\rho,\,D_{t}^{l}]D_{t}u+[\nabla,\,D_{t}^{l}]p.$

######  Proposition 4.2.

For $t\in[0,\,T]$, there holds that

$\sum_{l=0}^{\kappa+1}\big{\|}(D_{t}^{l}p,\,D_{t}^{l}u)\big{\|}_{L^{2}(\Omega)}\leq
C(M_{0})+TC(M).$ (4.5)

###### Proof.

Since $\kappa\geq 4$ and

$\displaystyle\big{\|}(\mathcal{N}_{p}^{l},\,\mathcal{N}_{u}^{l})\big{\|}_{L^{2}(\Omega)}\leq$
$\displaystyle C(M),$

energy estimates of (4.4) yield that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\Big{\\{}\frac{1}{\gamma
p}\frac{|D_{t}^{l}p|^{2}}{2}+\rho\frac{|D_{t}^{l}u|^{2}}{2}\Big{\\}}\mathrm{d}x$
(4.6) $\displaystyle=$
$\displaystyle\int_{\Omega}D_{t}\Big{\\{}\frac{1}{\gamma
p}\frac{|D_{t}^{l}p|^{2}}{2}+\rho\frac{|D_{t}^{l}u|^{2}}{2}\Big{\\}}\mathrm{d}x+\int_{\Omega}(\nabla\cdot
u)\Big{\\{}\frac{1}{\gamma
p}\frac{|D_{t}^{l}p|^{2}}{2}+\rho\frac{|D_{t}^{l}u|^{2}}{2}\Big{\\}}\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}\Big{\\{}D_{t}\big{(}\frac{1}{\gamma
p}\big{)}\frac{|D_{t}^{l}p|^{2}}{2}+D_{t}\rho\frac{|D_{t}^{l}u|^{2}}{2}\Big{\\}}\mathrm{d}x+\int_{\Omega}(\nabla\cdot
u)\Big{\\{}\frac{1}{\gamma
p}\frac{|D_{t}^{l}p|^{2}}{2}+\rho\frac{|D_{t}^{l}u|^{2}}{2}\Big{\\}}\mathrm{d}x$
$\displaystyle+\int_{\Omega}\Big{\\{}D_{t}^{l}p\mathcal{N}_{p}^{l}+D_{t}^{l}u\cdot\mathcal{N}_{u}^{l}\Big{\\}}\mathrm{d}x+\int_{\Omega}\nabla\cdot(D_{t}^{l}uD_{t}^{l}p)\mathrm{d}x$
$\displaystyle\leq$ $\displaystyle C(M),$

where we have used the boundary conditions (1.11) and the compatibility
conditions (3.14) .

∎

###  4.3. Estimates of $\omega$ and $S$

From (3.11)–(3.12), the vorticity $\omega$ and the entropy $S$ satisfy the
following transport equations respectively:

$D_{t}\omega=\omega\cdot\nabla u-\omega\nabla\cdot
u-\nabla\frac{1}{\rho}\times\nabla p,$ (4.7)

and

$D_{t}S=0.$ (4.8)

Direct energy estimates yield the following result.

######  Proposition 4.3.

For $t\in[0,\,T]$, there hold that

$\big{\|}\omega\big{\|}_{H^{\kappa-1}(\Omega)}+\sum_{l=1}^{\kappa}\big{\|}D_{t}^{l}\omega\big{\|}_{H^{\kappa-l}(\Omega)}\leq
M_{0}+TC(M),$ (4.9)

and

$\big{\|}S\big{\|}_{H^{\kappa}(\Omega)}+\sum_{l=1}^{\kappa+1}\big{\|}D_{t}^{l}S\big{\|}_{H^{\kappa+1-l}(\Omega)}\leq
M_{0}+TC(M).$ (4.10)

###### Proof.

Taking $\Lambda^{\kappa-l}D_{t}^{l}$ $(1\leq l\leq\kappa)$ to both sides of
(4.7), we have

$D_{t}\Lambda^{\kappa-l}D_{t}^{l}\omega=[D_{t},\,\Lambda^{\kappa-l}]D_{t}^{l}\omega+\Lambda^{\kappa-l}D_{t}^{l}(\omega\cdot\nabla
u-\omega\nabla\cdot u-\nabla\frac{1}{\rho}\times\nabla p).$ (4.11)

Then, the Sobolev inequalities and (4.11) can be applied to get that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}|\Lambda^{\kappa-l}D_{t}^{l}\omega|^{2}\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{\Omega}D_{t}\Lambda^{\kappa-l}D_{t}^{l}\omega\cdot\Lambda^{\kappa-l}D_{t}^{l}\omega\mathrm{d}x+\int_{\Omega}\frac{\nabla\cdot
u}{2}|\Lambda^{\kappa-l}D_{t}^{l}\omega|^{2}\mathrm{d}x$ $\displaystyle\leq$
$\displaystyle
C(M)\big{\|}D_{t}\Lambda^{\kappa-l}D_{t}^{l}\omega\big{\|}_{L^{2}(\Omega)}+C(M)\big{\|}\nabla\cdot
u\big{\|}_{L^{\infty}({\Omega})}$ $\displaystyle\leq$ $\displaystyle C(M).$

This proves (4.9) with $1\leq l\leq\kappa+1$. The case when $l=0$ follows
similarly.

For the entropy $S$ in (4.8), since

$D_{t}\Lambda^{\kappa+1-l}D_{t}^{l}S=[D_{t},\,\Lambda^{\kappa+1-l}]D_{t}^{l}S,$

the estimates in (4.10) can be proved just as those in (4.9). ∎

##  5\. Estimates of the interface $f$

In this section, we shall derive the estimates of the interface $f$.

######  Proposition 5.1.

For $t\in[0,\,T]$, there holds that

$\big{\|}f\big{\|}_{H^{\kappa}(\mathbb{T}^{2})}+\big{\|}D_{t}f\big{\|}_{H^{\kappa-\frac{1}{2}}(\mathbb{T}^{2})}\leq
M_{0}+TC(M).$ (5.1)

Recall the equation of $f$ in (3.7):

$D_{t}^{2}\overline{\partial}_{i}f+\mathfrak{a}T_{\lambda}\overline{\partial}_{i}f=\mathcal{N}^{+}+\mathcal{N}^{-},$
(5.2)

where $\lambda$, $\mathfrak{a}$, and $\mathcal{N}^{\pm}$ are given by (2.5),
(3.8), and (3.9) respectively. The rest of the section is devoted to the proof
of Proposition 5.1.

Set

$F=\Upsilon^{\kappa-\frac{3}{2}}\overline{\partial}_{i}f=\langle\overline{\partial}\rangle^{\kappa-\frac{3}{2}}\overline{\partial}_{i}f.$

Taking $\Upsilon^{\kappa-\frac{3}{2}}$ to both sides of (5.2), we have the
equation for $F$ as

$\begin{split}D_{t}^{2}F+\mathfrak{a}T_{\lambda}F=&-[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}^{2}]\overline{\partial}_{i}f-[\Upsilon^{\kappa-\frac{3}{2}},\,\mathfrak{a}T_{\lambda}]\overline{\partial}_{i}f+\Upsilon^{\kappa-\frac{3}{2}}(\mathcal{N}^{+}+\mathcal{N}^{-}).\end{split}$
(5.3)

Direct computation shows that

$\begin{split}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{T}^{2}}\Big{\\{}|D_{t}F|^{2}+\mathfrak{a}|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}\\\
=&\frac{1}{2}\int_{\mathbb{T}^{2}}D_{t}\Big{\\{}|D_{t}F|^{2}+\mathfrak{a}|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}+\frac{1}{2}\int_{\mathbb{T}^{2}}(\overline{\partial}_{j}u_{j})\Big{\\{}|D_{t}F|^{2}+\mathfrak{a}|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}\\\
=&\int_{\mathbb{T}^{2}}\Big{\\{}D_{t}^{2}F\cdot D_{t}F+\mathfrak{a}\cdot
D_{t}T_{\sqrt{\lambda}}F\cdot
T_{\sqrt{\lambda}}F\Big{\\}}\mathrm{d}\overline{x}\\\
&+\frac{1}{2}\int_{\mathbb{T}^{2}}\Big{\\{}(\overline{\partial}_{j}u_{j})|D_{t}F|^{2}+(\overline{\partial}_{j}u_{j}\mathfrak{a}+D_{t}\mathfrak{a})|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}\\\
=&\int_{\mathbb{T}^{2}}\Big{\\{}D_{t}^{2}F+\mathfrak{a}T_{\lambda}F\Big{\\}}D_{t}F\mathrm{d}\overline{x}+\int_{\mathbb{T}^{2}}\mathfrak{a}\Big{\\{}T_{\sqrt{\lambda}}^{*}T_{\sqrt{\lambda}}F-T_{\lambda}F\Big{\\}}D_{t}F\mathrm{d}\overline{x}\\\
&+\int_{\mathbb{T}^{2}}[\mathfrak{a}D_{t},\,T_{\sqrt{\lambda}}]F\cdot
T_{\sqrt{\lambda}}F\mathrm{d}\overline{x}+\frac{1}{2}\int_{\mathbb{T}^{2}}\Big{\\{}(\overline{\partial}_{j}u_{j})|D_{t}F|^{2}+(\overline{\partial}_{j}u_{j}\mathfrak{a}+D_{t}\mathfrak{a})|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}.\end{split}$

Therefore,

$\begin{split}&\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\Big{\\{}|D_{t}F|^{2}+\mathfrak{a}|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}\\\
=&\frac{1}{2}\int_{\mathbb{T}^{2}}\Big{\\{}(\overline{\partial}_{j}u_{j})|D_{t}F|^{2}+(\overline{\partial}_{j}u_{j}\mathfrak{a}+D_{t}\mathfrak{a})|T_{\sqrt{\lambda}}F|^{2}\Big{\\}}\mathrm{d}\overline{x}\\\
&+\int_{\mathbb{T}^{2}}\mathfrak{a}\Big{\\{}T_{\sqrt{\lambda}}^{*}T_{\sqrt{\lambda}}F-T_{\lambda}F\Big{\\}}D_{t}F\mathrm{d}\overline{x}+\int_{\mathbb{T}^{2}}[\mathfrak{a}D_{t},\,T_{\sqrt{\lambda}}]F\cdot
T_{\sqrt{\lambda}}F\mathrm{d}\overline{x}\\\
&-\int_{\mathbb{T}^{2}}[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}^{2}]\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}-\int_{\mathbb{T}^{2}}[\Upsilon^{\kappa-\frac{3}{2}},\,\mathfrak{a}T_{\lambda}]\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}\\\
&+\int_{\mathbb{T}^{2}}\Upsilon^{\kappa-\frac{3}{2}}(\mathcal{N}^{+}+\mathcal{N}^{-})\cdot
D_{t}F\mathrm{d}\overline{x}\\\
:=&I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}.\end{split}$ (5.4)

For $I_{1}$ in (5.4), it is direct to verify that

$I_{1}\leq
C(M)\Big{\\{}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}^{2}+\big{\|}T_{\sqrt{\lambda}}F\big{\|}_{L^{2}(\mathbb{T}^{2})}^{2}\Big{\\}}\leq
C(M).$ (5.5)

For $I_{2}$ in (5.4), it follows from Lemma C.4 that

$I_{2}\leq
C(M)\big{\|}(T_{\sqrt{\lambda}}^{*}T_{\sqrt{\lambda}}-T_{\lambda})F\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}\leq
C(M).$ (5.6)

For $I_{3}$ in (5.4), an application of Lemma C.5 gives

$\displaystyle I_{3}=$
$\displaystyle\int_{\mathbb{T}^{2}}[\mathfrak{a},\,T_{\sqrt{\lambda}}]D_{t}F\cdot
T_{\sqrt{\lambda}}F\mathrm{d}\overline{x}+\int_{\mathbb{T}^{2}}\mathfrak{a}[D_{t},\,T_{\sqrt{\lambda}}]F\cdot
T_{\sqrt{\lambda}}F\mathrm{d}\overline{x}$ (5.7) $\displaystyle\leq$
$\displaystyle\big{\|}[\mathfrak{a},\,T_{\sqrt{\lambda}}]D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}T_{\sqrt{\lambda}}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle+\big{\|}\mathfrak{a}\big{\|}_{L^{\infty}(\mathbb{T}^{2})}\big{\|}[D_{t},\,T_{\sqrt{\lambda}}]F\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}T_{\sqrt{\lambda}}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle\leq$ $\displaystyle C(M).$

Similarly,

$\displaystyle I_{4}=$
$\displaystyle-\int_{\mathbb{T}^{2}}[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}]D_{t}\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}-\int_{\mathbb{T}^{2}}D_{t}[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}]\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}$ (5.8) $\displaystyle\leq$
$\displaystyle\big{\|}[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}]D_{t}\overline{\partial}_{i}f\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle+\big{\|}D_{t}[\Upsilon^{\kappa-\frac{3}{2}},\,D_{t}]\overline{\partial}_{i}f\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle\leq$ $\displaystyle C(M).$

For $I_{5}$ in (5.4), there holds that

$\displaystyle I_{5}=$
$\displaystyle-\int_{\mathbb{T}^{2}}[\Upsilon^{\kappa-\frac{3}{2}},\,\mathfrak{a}]T_{\lambda}\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}-\int_{\mathbb{T}^{2}}\mathfrak{a}[\Upsilon^{\kappa-\frac{3}{2}},\,T_{\lambda}]\overline{\partial}_{i}f\cdot
D_{t}F\mathrm{d}\overline{x}$ (5.9) $\displaystyle\leq$
$\displaystyle\big{\|}[\Upsilon^{\kappa-\frac{3}{2}},\,\mathfrak{a}]T_{\lambda}\overline{\partial}_{i}f\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle+\big{\|}\mathfrak{a}[\Upsilon^{\kappa-\frac{3}{2}},\,T_{\lambda}]\overline{\partial}_{i}f\big{\|}_{L^{2}(\mathbb{T}^{2})}\big{\|}D_{t}F\big{\|}_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle\leq$ $\displaystyle C(M).$

Next we estimate $I_{6}$ with $\mathcal{N}^{\pm}$ given by (3.9). Since
$\overline{\partial}_{i}p=\partial_{i}p+\mathcal{H}(\overline{\partial}_{i}f)\partial_{3}p$,
it can be derived from (2.2) and (3.18) that

$\begin{cases}\Delta\overline{\partial}_{i}p^{\pm}=\partial_{i}\Delta
p^{\pm}+\mathcal{H}^{\pm}(\overline{\partial}_{i}f)\partial_{3}\Delta
p^{\pm}+2\nabla\mathcal{H}^{\pm}(\overline{\partial}_{i}f)\cdot\nabla\partial_{i}p^{\pm},&\text{in
$\Omega^{\pm}$},\\\ \overline{\partial}_{i}p^{\pm}=0,&\text{on
$\Gamma_{f}$},\\\ \partial_{3}\overline{\partial}_{i}p^{\pm}=0,&\text{on
$\Gamma^{\pm}$}.\end{cases}$ (5.10)

Then we can use the elliptic system (6.1), the estimates (5.1) and (6.3) to
get that

$\big{\|}\Delta\overline{\partial}_{i}p^{\pm}\big{\|}_{H^{\kappa-2}(\Omega^{\pm})}\leq
C(M).$ (5.11)

Thus, (5.11) can be applied to (5.10) to yield that

$\big{\|}\nabla_{N}\overline{\partial}_{i}p^{\pm}\big{\|}_{H^{\kappa-\frac{3}{2}}(\Gamma_{f})}\leq
C(M)\big{\|}\Delta\overline{\partial}_{i}p^{\pm}\big{\|}_{H^{\kappa-2}(\Omega^{\pm})}\leq
C(M).$ (5.12)

Therefore, we have

$\big{\|}\mathcal{N}^{\pm}\big{\|}_{H^{\kappa-\frac{3}{2}}(\mathbb{T}^{2})}\leq
C(M)\big{\|}(\nabla_{N}\overline{\partial}_{i}p^{\pm},\,\overline{\partial}\rho^{\pm},\,\overline{\partial}u^{\pm},\,D_{t}\overline{\partial}_{i}f)\big{\|}_{H^{\kappa-\frac{3}{2}}(\mathbb{T}^{2})}\leq
C(M).$

Thus,

$I_{6}\leq C(M).$ (5.13)

Combining all the estimates (5.5)–(5.13), we have

$\frac{\mathrm{d}}{\mathrm{d}t}\big{\|}(D_{t}\overline{\partial}_{i}f,\,\mathfrak{a}^{1/2}T_{\sqrt{\lambda}}\overline{\partial}_{i}f)\big{\|}_{H^{\kappa-\frac{3}{2}}(\mathbb{T}^{2})}\leq
C(M).$ (5.14)

As for $\big{\|}f\big{\|}_{L^{2}(\mathbb{T}^{2})}$, we use (3.1) to get that

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\big{\|}f\big{\|}_{L^{2}(\mathbb{T}^{2})}^{2}=\int_{\mathbb{T}^{2}}D_{t}f\cdot
f\mathrm{d}\overline{x}+\int_{\mathbb{T}^{2}}\frac{\overline{\partial}_{j}u_{j}}{2}|f|^{2}\mathrm{d}\overline{x}\leq
C(M).$ (5.15)

Assuming the Taylor sign condition, (5.14)–(5.15) and (3.10) prove (5.1).

##  6\. Full estimates of $(p,\,u)$

In this section, we shall recover the full estimates of $(p,\,u)$ from the
tangential energy estimates in Section 4 by some elliptic estimates.

###  6.1. Full estimates of $p$

To recover the full estimates of the pressure $p$ from the tangential
estimates of $D_{t}^{l}p$ in Section 4, we shall rewrite the wave equation of
$p$ in (3.13) as

$\begin{cases}\Delta p^{\pm}=\frac{\rho^{\pm}}{\gamma
p^{\pm}}D_{t}^{2}p^{\pm}-\rho^{\pm}\mathrm{tr}\,(\nabla
u^{\pm})^{2}+\mathcal{M}_{p}^{\pm},&\text{in $\Omega^{\pm}$},\\\
p^{\pm}=q(t),&\text{on $\Gamma_{f}$},\\\ \partial_{3}p^{\pm}=0,&\text{on
$\Gamma^{\pm}$},\end{cases}$ (6.1)

where $q(t)$ is given by (3.19) and

$\mathcal{M}_{p}^{\pm}=-\frac{\rho^{\pm}}{\gamma(p^{\pm})^{2}}(D_{t}p^{\pm})^{2}+\frac{1}{\rho^{\pm}}\nabla\rho^{\pm}\cdot\nabla
p^{\pm}.$ (6.2)

######  Proposition 6.1.

For $t\in[0,\,T]$, there holds that

$\big{\|}p\big{\|}_{H^{\kappa}(\Omega)}+\sum_{l=1}^{\kappa+1}\big{\|}D_{t}^{l}p\big{\|}_{H^{\kappa+1-l}(\Omega)}\leq
C(M_{0})+TC(M).$ (6.3)

Furthermore,

$\big{\|}D_{t}u\big{\|}_{H^{\kappa-1}(\Omega)}+\sum_{l=2}^{\kappa+1}\big{\|}D_{t}^{l}u\big{\|}_{H^{\kappa+1-l}(\Omega)}\leq
C(M_{0})+TC(M).$ (6.4)

###### Proof.

To prove (6.3), we shall use an induction over the index $l$. When
$l=\kappa+1$, it follows from (4.5) that

$\big{\|}(D_{t}^{\kappa+1}p,\,D_{t}^{\kappa+1}u)\big{\|}_{L^{2}(\Omega)}\leq
C(M_{0})+TC(M).$ (6.5)

For $D_{t}^{\kappa}p$, since

$\nabla D_{t}^{\kappa}p=\rho D_{t}^{\kappa}(\frac{\nabla
p}{\rho})-\rho[D_{t}^{\kappa},\,\frac{1}{\rho}]\nabla
p-[D_{t}^{\kappa},\,\nabla]p,$

we have from the momentum equation $D_{t}u=-\frac{\nabla p}{\rho}$ that

$\displaystyle\big{\|}D_{t}^{\kappa}p\big{\|}_{H^{1}(\Omega)}\leq$
$\displaystyle\big{\|}\nabla
D_{t}^{\kappa}p\big{\|}_{L^{2}(\Omega)}+\big{\|}D_{t}^{\kappa}p\big{\|}_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle\big{\|}\rho
D_{t}^{\kappa+1}u\big{\|}_{L^{2}(\Omega)}+\big{\|}\rho[D_{t}^{\kappa},\,\frac{1}{\rho}]\nabla
p\big{\|}_{L^{2}(\Omega)}$
$\displaystyle+\big{\|}[D_{t}^{\kappa},\,\nabla]p\big{\|}_{L^{2}(\Omega)}+\big{\|}D_{t}^{\kappa}p\big{\|}_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\big{\|}D_{t}^{\kappa+1}u\big{\|}_{L^{2}(\Omega)}+C(\mathcal{F}).$

Thus, by (4.1) and (6.5), we have

$\big{\|}D_{t}^{\kappa}p\big{\|}_{H^{1}(\Omega)}\leq C(M_{0})+TC(M).$ (6.6)

Assume that $1\leq l\leq\kappa-1$ and

$\sum_{k=l+1}^{\kappa+1}\big{\|}D_{t}^{k}p\big{\|}_{H^{\kappa+1-k}(\Omega)}\leq
C(M_{0})+TC(M).$ (6.7)

Then we shall prove that

$\big{\|}D_{t}^{l}p\big{\|}_{H^{\kappa+1-l}(\Omega)}\leq C(M_{0})+TC(M).$
(6.8)

When $1\leq l\leq\kappa-1$, the equation for $D_{t}^{l}p$ is

$\begin{cases}\Delta D_{t}^{l}p=\frac{\rho}{\gamma
p}D_{t}^{l+2}p+[\Delta,\,D_{t}^{l}]p-[\frac{\rho}{\gamma
p},\,D_{t}^{l}]D_{t}^{2}p\\\ \quad\quad\quad\quad-
D_{t}^{l}\Big{\\{}\rho\mathrm{tr}\,(\nabla
u)^{2}\Big{\\}}+D_{t}^{l}\mathcal{M}_{p},&\text{in $\Omega^{\pm}$},\\\
D_{t}^{l}p=\partial_{t}^{l}q(t),&\text{on $\Gamma_{f}$},\\\
\partial_{3}D_{t}^{l}p=[\partial_{3},D_{t}^{l}]p,&\text{on
$\Gamma^{\pm}$},\end{cases}$ (6.9)

Since

$\displaystyle p=$
$\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}[\Delta,\,D_{t}]D_{t}^{l-1-m}p$
$\displaystyle=$ $\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}\Big{\\{}\Delta
u_{J}\partial_{J}D_{t}^{l-1-m}p+2\nabla
u_{J}\partial_{J}D_{t}^{l-1-m}p\Big{\\}}$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{l-1}\sum_{n=0}^{m}\Big{\\{}D_{t}^{n}\Delta
u_{J}\cdot D_{t}^{m-n}\partial_{J}D_{t}^{l-1-m}p+2D_{t}^{n}\nabla u_{J}\cdot
D_{t}^{m-n}\partial_{J}D_{t}^{l-1-m}p\Big{\\}},$
$\displaystyle-[\frac{\rho}{\gamma p},\,D_{t}^{l}]D_{t}^{2}p=$
$\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}[D_{t},\,\frac{\rho}{\gamma
p}]D_{t}^{l+1-m}p$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}\Big{\\{}D_{t}(\frac{\rho}{\gamma
p})D_{t}^{l+1-m}p\Big{\\}}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{l-1}D_{t}^{n+1}(\frac{\rho}{\gamma p})\cdot
D_{t}^{l+1-n}p,$ $\displaystyle D_{t}^{l}(\rho\mathrm{tr}\,(\nabla u)^{2})=$
$\displaystyle D_{t}^{l}(\rho\partial_{J}u_{K}\partial_{K}u_{J})$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{l}\sum_{n=0}^{l-m}D_{t}^{l-m-n}\rho\cdot
D_{t}^{m}\partial_{J}u_{K}\cdot D_{t}^{n}\partial_{K}u_{J},$

we have

$\big{\|}\Delta D_{t}^{l}p\big{\|}_{H^{\kappa-1-l}(\Omega)}\leq
C(\mathcal{F})\big{\|}D_{t}^{l+2}p\big{\|}_{H^{\kappa-1-l}(\Omega)}+C(\mathcal{F}).$
(6.10)

Similarly,

$\displaystyle p=$
$\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}[\partial_{3},\,D_{t}]D_{t}^{l-1-m}p$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{l-1}D_{t}^{m}\Big{\\{}\partial_{3}u_{J}\partial_{J}D_{t}^{l-1-m}p\Big{\\}}$
$\displaystyle=$
$\displaystyle\sum_{m=0}^{l-1}\sum_{n=0}^{m}D_{t}^{n}\partial_{3}u_{J}\cdot
D_{t}^{m-n}\partial_{J}D_{t}^{l-1-m}p,$

yields that

$\big{\|}\partial_{3}D_{t}^{l}p\big{\|}_{H^{\kappa-\frac{1}{2}-l}(\Gamma^{\pm})}\leq
C(\mathcal{F}).$ (6.11)

On the interface $\Gamma_{f}$, the fact that $p|_{\Gamma_{f}}=q(t)$ which is
independent on $\overline{x}$ infers that

$\displaystyle\big{\|}D_{t}^{l}p\big{\|}_{H^{\kappa+\frac{1}{2}-l}(\Gamma_{f})}=$
$\displaystyle\big{\|}D_{t}^{l}p\big{\|}_{L^{2}(\Gamma_{f})}$ (6.12)
$\displaystyle\leq$ $\displaystyle
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\big{\|}D_{t}^{l}p\big{\|}_{H^{1}(\Omega)}\leq
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})C(\mathcal{F}).$

Therefore, the standard elliptic theory can be applied to (6.9) to get that

$\displaystyle\big{\|}D_{t}^{l}p\big{\|}_{H^{\kappa+1-l}(\Omega)}\leq$
$\displaystyle
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\Big{\\{}\big{\|}\Delta
D_{t}^{l}p\big{\|}_{H^{\kappa-1-l}(\Omega)}$ (6.13)
$\displaystyle+\big{\|}\partial_{3}D_{t}^{l}p\big{\|}_{H^{\kappa-\frac{1}{2}-l}(\Gamma^{\pm})}+\big{\|}D_{t}^{l}p\big{\|}_{H^{\kappa+\frac{1}{2}-l}(\Gamma_{f})}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C(M_{0})+TC(M).$

where we have used (6.10)–(6.12), the induction assumption (6.7), (4.1) and
(5.1).

The case of $l=0$ follows in a similar way. Notice that we can only get
$\big{\|}p\big{\|}_{H^{\kappa}(\Omega)}$ instead of
$\big{\|}p\big{\|}_{H^{\kappa+1}(\Omega)}$ due to limited regularity of the
interface $f$. Thus, (6.3) is proved.

To prove (6.4), since $D_{t}u=-\frac{\nabla p}{\rho}$ and

$D_{t}^{l+1}u=-D_{t}^{l}(\frac{1}{\rho}\nabla p)=-\frac{1}{\rho}\nabla
D_{t}^{l}p-[D_{t}^{l},\,\frac{1}{\rho}\nabla]p,$

the estimates of $u$ in (6.4) follow from (6.3) and (4.1).

∎

###  6.2. Full estimates of $u$

The full estimates of $u$ can be recovered by Lemma B.2 and the fact that

$\overline{\partial}_{i}u\cdot
N=\overline{\partial}_{i}D_{t}f-\overline{\partial}_{j}f\overline{\partial}_{i}u_{j}=D_{t}\overline{\partial}_{i}f,\quad\quad
i=1,\,2.$ (6.14)

######  Proposition 6.2.

For $t\in[0,\,T]$, there holds that

$\big{\|}u\big{\|}_{H^{\kappa}(\Omega)}\leq C(M_{0})+TC(M).$ (6.15)

###### Proof.

We apply (B.3) and (6.14) to get that

$\displaystyle\big{\|}u\big{\|}_{H^{\kappa}(\Omega)}\leq$ $\displaystyle
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\Big{\\{}\big{\|}\nabla\times
u\big{\|}_{H^{\kappa-1}(\Omega)}+\big{\|}\nabla\cdot
u\big{\|}_{H^{\kappa-1}(\Omega)}$
$\displaystyle\quad\quad\quad\quad+\big{\|}\overline{\partial}_{i}u\cdot
N\big{\|}_{H^{\kappa-\frac{3}{2}}(\mathbb{T}^{2})}+\big{\|}u\big{\|}_{L^{2}(\Omega)}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\Big{\\{}\big{\|}\omega\big{\|}_{H^{\kappa-1}(\Omega)}+\big{\|}\frac{1}{\gamma
p}D_{t}p\big{\|}_{H^{\kappa-1}(\Omega)}$
$\displaystyle\quad\quad\quad\quad+\big{\|}D_{t}\overline{\partial}_{i}f\big{\|}_{H^{\kappa-\frac{3}{2}}(\mathbb{T}^{2})}+\big{\|}u\big{\|}_{L^{2}(\Omega)}\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}C(M_{0})+TC(M)\Big{)}\cdot\Big{\\{}C(M_{0})+TC(M)\Big{\\}}$
$\displaystyle\leq$ $\displaystyle C(M_{0})+TC(M),$

where we have used (4.1), (4.5), (4.9) and (5.1). ∎

##  Appendix A Remark on the incompressible case

In this section, we consider the interface problem for the incompressible
Euler equations in $\Omega=\mathbb{T}^{2}\times(-1,\,1)$,

$\begin{cases}\rho^{\pm}(\partial_{t}u^{\pm}+u^{\pm}\cdot\nabla
u^{\pm})+\nabla p^{\pm}=0,\\\ \nabla\cdot u^{\pm}=0,\end{cases}\text{in
$\Omega^{\pm}$},$ (A.1)

where $\rho^{\pm}$ are constants and

$\Omega^{\pm}=\\{(\overline{x},\,x_{3})\in\Omega:\,\overline{x}\in\mathbb{T}^{2},\,x_{3}\gtrless
f(t,\,\overline{x})\\}.$

On the interface
$\Gamma_{f}=\\{(\overline{x},\,f(t,\,\overline{x}):\,\overline{x}\in\mathbb{T}^{2})\\}$,
there holds that

$\llbracket\rho\rrbracket\neq 0,\quad\quad\llbracket
u\rrbracket=0,\quad\quad\llbracket p\rrbracket=0,$ (A.2)

and

$u^{\pm}\cdot N=\partial_{t}f.$ (A.3)

On the fixed upper and lower boundaries
$\Gamma^{\pm}=\mathbb{T}^{2}\times\\{\pm 1\\}$, there holds that

$u^{\pm}\cdot n^{\pm}\big{|}_{\Gamma^{\pm}}=0.$ (A.4)

The compatibility condition on the interface $\Gamma_{f}$ is

$\llbracket D_{t}u\rrbracket=0.$ (A.5)

We can derive in a similar way as in Section 3.4 that

$\overline{\partial}_{i}p\big{|}_{\Gamma_{f}}=0.$ (A.6)

Thus $p|_{\Gamma_{f}}=q(t)$ is independent of the spatial variables
$\overline{x}=(x_{1},\,x_{2})$. By taking $p-q(t)$ as the pressure, we may
just assume that

$p\big{|}_{\Gamma_{f}}=0.$ (A.7)

###  A.1. Reformulation of the incompressible problem

The evolution of the interface can be derived just as in Section 3.1:

$D_{t}^{2}\overline{\partial}_{i}f+\mathfrak{a}T_{\lambda}\overline{\partial}_{i}f=\mathcal{N}^{+}+\mathcal{N}^{-},$
(A.8)

where the Taylor sign $\mathfrak{a}$ is

$\mathfrak{a}=\frac{\partial_{3}p^{+}-\partial_{3}p^{-}}{\rho^{+}+\rho^{-}}=\frac{\llbracket\nabla_{N}p\rrbracket}{(\rho^{+}+\rho^{-})(1+|\nabla
f|^{2})}>0,$ (A.9)

and the lower order terms are

$\displaystyle\mathcal{N}^{\pm}=$
$\displaystyle-\frac{1}{\rho^{+}+\rho^{-}}\nabla_{N}\overline{\partial}_{i}p^{\pm}-\frac{2\rho^{\pm}}{\rho^{+}+\rho^{-}}\overline{\partial}_{i}u_{j}^{\pm}D_{t}\overline{\partial}_{j}f\mp\frac{\partial_{3}p^{\pm}}{\rho^{+}+\rho^{-}}R^{\pm}\overline{\partial}_{i}f.$
(A.10)

The evolution of the vorticity $\omega^{\pm}=\nabla\times u^{\pm}$ is given by

$D_{t}\omega^{\pm}=\omega^{\pm}\cdot\nabla u^{\pm}.$ (A.11)

The velocity $u^{\pm}$ can be recovered by the div-curl lemmas in Appendix B
from

$\begin{cases}\nabla\times u^{\pm}=\omega^{\pm},\quad\quad\nabla\cdot
u^{\pm}=0,&\text{in $\Omega^{\pm}$},\\\ u^{\pm}\cdot
N=0,\quad\quad(\overline{\partial}_{i}u^{\pm}\cdot
N=D_{t}\overline{\partial}_{i}f,)&\text{on $\Gamma_{f}$},\\\ u^{\pm}\cdot
n^{\pm}=0,&\text{on $\Gamma^{\pm}$}.\end{cases}$ (A.12)

The pressure $p^{\pm}$ can be solved by

$\begin{cases}-\Delta p^{\pm}=\rho^{\pm}\mathrm{tr}\,(\nabla
u^{\pm})^{2},&\text{in $\Omega^{\pm}$},\\\ p^{\pm}=0,&\text{on
$\Gamma_{f}$},\\\ \partial_{3}p=0,&\text{on $\Gamma^{\pm}$}.\end{cases}$
(A.13)

###  A.2. Main result for the incompressible problem

Assume that $\kappa\geq 4$ is an integer. We still denote the full energy norm
by

$\displaystyle\mathcal{E}(t)=$
$\displaystyle\big{\|}f\big{\|}_{H^{\kappa}(\mathbb{T}^{2})}+\big{\|}D_{t}f\big{\|}_{H^{\kappa-\frac{1}{2}}(\mathbb{T}^{2})}$
$\displaystyle+\big{\|}u\big{\|}_{H^{\kappa}(\Omega)}+\big{\|}p\big{\|}_{H^{\kappa}(\Omega)}$
(A.14)

For some $T>0$, we shall denote by

$M=\sup_{0<t<T}\mathcal{E}(t),\quad\quad M_{0}=\mathcal{E}(0).$

######  Theorem A.1.

Let $\kappa\geq 4$ be an integer. Suppose that the initial data
$(f_{\mathrm{in}},\,p_{\mathrm{in}}^{\pm},\,u_{\mathrm{in}}^{\pm})$ satisfy
the bound $\mathcal{E}(0)=M_{0}<\infty$. Furthermore, assume that there is a
constant $c_{0}>0$ such that

  1. (1)

$-1+c_{0}\leq f_{\mathrm{in}}\leq 1-c_{0}$;

  2. (2)

$\frac{N_{\mathrm{in}}}{|N_{\mathrm{in}}|}\cdot\llbracket\nabla
p_{\mathrm{in}}^{\pm}\rrbracket\geq c_{0}$.

Then for $T>0$, the solution $(f,\,p^{\pm},\,u^{\pm})$ to the problem
(A.1)–(A.4) under the compatibility conditions (A.5) satisfies that

  1. (1)

$M\leq C(M_{0},\,c_{0})+TC(M,\,c_{0})$;

  2. (2)

$-1+\frac{c_{0}}{2}\leq f\leq 1-\frac{c_{0}}{2}$;

  3. (3)

$\frac{N}{|N|}\cdot\llbracket\nabla p^{\pm}\rrbracket\geq\frac{c_{0}}{2}$.

Compared with the proof of Theorem 3.2, the proof of Theorem A.1 can be
simplified since the pressure $p$ can be estimated by the elliptic equations
(A.13) directly. We shall omit the detailed proof. Furthermore, the solution
to the incompressible system can be constructed by a vanishing surface tension
argument as in [22]. This proves the well-posedness of the interface problem
(A.1)–(A.4) of the incompressible Euler equations.

##  Appendix B Elliptic estimates

For the one-phase elliptic system

$\begin{cases}\nabla\times u=\omega,\quad\nabla\cdot u=\sigma,&\text{in
$\Omega_{f}^{-}$},\\\ u\cdot N=\theta,&\text{on $\Gamma_{f}$},\\\ u\cdot
n_{-}=0,\quad\int_{\mathbb{T}^{2}}u_{j}\mathrm{d}\overline{x}=\alpha_{j}\,(j=1,\,2),&\text{on
$\Gamma^{-}$},\\\ \end{cases}$ (B.1)

we have the following existence result given by Proposition 5.1 in [33] (see
also [8, 32]):

######  Lemma B.1.

Assume that $f\in H^{\kappa-\frac{1}{2}}(\mathbb{T}^{2})$ with
$\kappa>\frac{5}{2}$. For $s\in[2,\,\kappa]$, let $(\omega,\,\sigma)\in
H^{s-2}(\Omega_{f}^{-})$ and $\theta\in H^{s-\frac{3}{2}}(\mathbb{T}^{2})$ be
such that

$\int_{\Omega_{f}^{-}}\sigma\mathrm{d}x=\int_{\mathbb{T}^{2}}\theta\mathrm{d}\overline{x},$
$\nabla\cdot\omega=0,\,\text{in
$\Omega_{f}^{-}$},\quad\int_{\Gamma^{-}}\omega_{3}\mathrm{d}\overline{x}=0.$

Then there exists a unique $u\in H^{s-1}(\Omega_{f}^{-})$ to the system (B.1)
such that

$\big{\|}u\big{\|}_{H^{s-1}(\Omega_{f}^{-})}\leq
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\Big{\\{}\big{\|}(\omega,\,\sigma)\big{\|}_{H^{s-2}(\Omega_{f}^{-})}+\big{\|}\theta\big{\|}_{H^{s-\frac{3}{2}}(\Gamma_{f})}+|\alpha_{1}|+|\alpha_{2}|\Big{\\}}.$
(B.2)

The regularity of the solution of the one-phase elliptic system (B.1) was
improved in [8] (see also [38]) by using tangential derivatives for the
boundary condition on the surface $\Gamma_{f}$:

######  Lemma B.2.

Assume that $f\in H^{\kappa-\frac{1}{2}}(\mathbb{T}^{2})$ with
$\kappa>\frac{5}{2}$. For $s\in[2,\,\kappa]$, there holds that

$\displaystyle\big{\|}u\big{\|}_{H^{s}(\Omega_{f}^{-})}\leq$ $\displaystyle
C(\big{\|}f\big{\|}_{H^{\kappa-\frac{1}{2}}})\Big{\\{}\big{\|}\nabla\times
u\big{\|}_{H^{s-1}(\Omega_{f}^{-})}+\big{\|}\nabla\cdot
u\big{\|}_{H^{s-1}(\Omega_{f}^{-})}$ (B.3)
$\displaystyle\quad\quad+\sum_{i=1,2}\big{\|}\overline{\partial}_{i}u\cdot
N\big{\|}_{H^{s-\frac{3}{2}}(\Gamma_{f})}+\big{\|}u\big{\|}_{L^{2}(\Omega_{f}^{-})}\Big{\\}}.$

Clearly, these two results also hold for the one-phase elliptic systems in
$\Omega^{+}_{f}$ in a similar fashion.

##  Appendix C Paradifferential operators and commutator estimates

In this appendix, we shall recall some basic facts on paradifferential
operators from [25].

We first introduce the symbols with limited spatial smoothness. Let
$W^{k,\infty}(\mathbb{R}^{d})$ be the usual Sobolev spaces for
$k\in\mathbb{N}$.

######  Definition C.1.

Given $\mu\in[0,\,1]$ and $m\in\mathbb{R}$, we denote by
$\Gamma^{m}_{\mu}(\mathbb{R}^{d})$ the space of locally bounded functions
$a(x,\,\xi)$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\\{0\\}$, which
are $C^{\infty}$ with respect to $\xi$ for $\xi\neq 0$ such that, for all
$\alpha\in\mathbb{N}^{d}$ and $\xi\neq 0$, the function
$x\to\partial_{\xi}^{\alpha}a(x,\,\xi)$ belongs to $W^{\mu,\infty}$ and there
exists a constant $C_{\alpha}$ such that

$\big{\|}\partial_{\xi}^{\alpha}a(\cdot,\,\xi)\big{\|}_{W^{\mu,\infty}}\leq
C_{\alpha}(1+|\xi|)^{m-|\alpha|},\quad\quad\forall\,|\xi|\geq\frac{1}{2}.$

The seminorm of the symbol is defined as

$M_{\mu}^{m}(a):=\sup_{|\alpha|\leq\frac{3d}{2}+1+\mu}\sup_{|\xi|\geq\frac{1}{2}}\big{\|}(1+|\xi|)^{-m+|\alpha|}\partial_{\xi}^{\alpha}a(\cdot,\,\xi)\big{\|}_{W^{\mu,\infty}}$

If $a$ is a function independent of $\xi$, then

$M^{0}_{\mu}(a)=\big{\|}a\big{\|}_{W^{\mu,\infty}}.$

######  Definition C.2.

Given a symbol $a$, the paradifferential operator $T_{a}$ is defined by

$\widehat{T_{a}u}(\xi):=(2\pi)^{-d}\int_{\mathbb{R}^{d}}\chi(\xi-\eta,\,\eta)\widehat{a}(\xi-\eta,\,\eta)\psi(\eta)\widehat{u}(\eta)\mathrm{d}\eta,$
(C.1)

where $\widehat{a}$ is the Fourier transform of $a$ with respect to the first
variable. $\chi(\xi,\,\eta)\in C^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$
is an admissible cutoff function, that is, there exist
$0<\varepsilon_{1}<\varepsilon_{2}$ such that

$\chi(\xi,\,\eta)=1\quad\text{for
$|\xi|\leq\varepsilon_{1}|\eta|$},\quad\quad\chi(\xi,\,\eta)=0\quad\text{for
$|\xi|\geq\varepsilon_{2}|\eta|$},$

and

$|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\chi(\xi,\,\eta)|\leq
C_{\alpha,\beta}(1+|\eta|)^{-|\alpha|-|\beta|}\quad\text{for
$(\xi,\,\eta)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$}.$

The cutoff function $\psi(\eta)\in C^{\infty}(\mathbb{R}^{d})$ satisfies

$\psi(\eta)=0\quad\text{for $|\eta|\leq
1$},\quad\quad\psi(\eta)=1\quad\text{for $|\eta|\geq 2$}.$

The admissible cutoff function $\chi(\xi,\,\eta)$ can be chosen as

$\chi(\xi,\,\eta)=\sum_{k=0}^{\infty}\zeta_{k-3}(\xi)\varphi(\eta),$

where $\zeta(\xi)=1$ for $|\xi|\leq 1.1$, $\zeta(\xi)=0$ for $|\xi|\geq 1.9$,
and

$\begin{cases}\zeta_{k}(\xi)=\zeta(2^{-k}\xi)&\text{for $k\in\mathbb{Z}$},\\\
\varphi_{0}=\zeta,\quad\varphi_{k}=\zeta_{k}-\zeta_{k-1}&\text{for $k\geq
1$}.\end{cases}$

We also introduce the Littlewood-Paley operators $\Delta_{k},\,S_{k}$ defined
as

$\Delta_{k}u=\mathcal{F}^{-1}(\varphi_{k}\widehat{u})\quad\text{for $k\geq
0$},\quad\quad\Delta_{k}u=0\quad\text{for $k<0$},$ $S_{k}u=\sum_{l\leq
k}\Delta_{l}u\quad\text{for $k\in\mathbb{Z}$}.$

When the symbol $a$ depends only on the first variable $x$ in $T_{a}u$, we
take $\psi=1$ in (C.1). Then $T_{a}u$ is just usual Bony’s paraproduct defined
as

$T_{a}u=\sum_{k=0}S_{k-3}a\Delta_{k}u.$ (C.2)

We have the following Bony’s paraproduct decomposition:

$au=T_{a}u+T_{u}a+R(a,\,u),$ (C.3)

where the remainder term $R(a,\,u)$ is

$R(a,\,u)=\sum_{|k-l|\leq 2}\Delta_{k}a\Delta_{l}u.$

######  Lemma C.3.

There holds that

  1. (1)

If $s\in\mathbb{R}$ and $\sigma<\frac{d}{2}$, then

$\big{\|}T_{a}u\big{\|}_{H^{s}}\lesssim\min\\{\big{\|}a\big{\|}_{L^{\infty}}\big{\|}u\big{\|}_{H^{s}},\,\big{\|}a\big{\|}_{H^{\sigma}}\big{\|}u\big{\|}_{H^{s+\frac{d}{2}-\sigma}},\,\big{\|}a\big{\|}_{H^{\frac{d}{2}}}\big{\|}u\big{\|}_{H^{s+}}\\}.$

  2. (2)

If $s>0$ and $s_{1},\,s_{2}\in\mathbb{R}$ with $s_{1}+s_{2}=s+\frac{d}{2}$,
then

$\big{\|}R(a,\,u)\big{\|}_{H^{s}}\lesssim\big{\|}a\big{\|}_{H^{s_{1}}}\big{\|}u\big{\|}_{H^{s_{2}}}.$

  3. (3)

If $s>0$, $s_{1}\geq s$, $s_{2}\geq s$ and $s_{1}+s_{2}=s+\frac{d}{2}$, then

$\big{\|}au\big{\|}_{H^{s}}\lesssim\big{\|}a\big{\|}_{H^{s_{1}}}\big{\|}u\big{\|}_{H^{s_{2}}}.$
(C.4)

There is also the symbolic calculus of paradifferential operator in Sobolev
spaces.

######  Lemma C.4.

Let $m,\,m^{\prime}\in\mathbb{R}$.

  1. (1)

If $a\in\Gamma^{m}_{0}(\mathbb{R}^{d})$, then for any $s\in\mathbb{R}$,

$\big{\|}T_{a}\big{\|}_{H^{s}\to H^{s-m}}\lesssim M^{m}_{0}(a).$

  2. (2)

if $a\in\Gamma^{m}_{\rho}(\mathbb{R}^{d})$ and
$b\in\Gamma^{m^{\prime}}_{\rho}(\mathbb{R}^{d})$ for $\rho>0$, then for any
$s\in\mathbb{R}$,

$\big{\|}T_{a}T_{b}-T_{a\sharp b}\big{\|}_{H^{s}\to
H^{s-m-m^{\prime}+\rho}}\lesssim
M^{m}_{\rho}(a)M^{m^{\prime}}_{0}(b)+M^{m}_{0}(a)M^{m^{\prime}}_{\rho}(b),$

where

$a\sharp
b=\sum_{|\alpha|<\rho}\partial_{\xi}^{\alpha}a(x,\,\xi)D_{x}^{\alpha}b(x,\,\xi),\quad\quad
D_{x}=-\mathrm{i}\partial_{x}.$

  3. (3)

If $a\in\Gamma^{m}_{\rho}(\mathbb{R}^{d})$ for $\rho\in(0,\,1]$, then for any
$s\in\mathbb{R}$,

$\big{\|}(T_{a})^{*}-T_{a^{*}}\big{\|}_{H^{s}\to H^{s-m+\rho}}\lesssim
M^{m}_{\rho}(a),$

where $(T_{a})^{*}$ is the adjoint operator of $T_{a}$ and $a^{*}$ is the
complex conjugate of the symbol $a$.

To estimate commutators, we recall a lemma from [1] (Lemma 2.15).

######  Lemma C.5.

Consider a symbol $p=p(t,\,x,\,\xi)$ which is homogeneous of order $m$. There
holds that

$\big{\|}[T_{p},\,\partial_{t}+T_{u}\cdot\nabla]u\big{\|}_{H^{m}}\lesssim\Big{\\{}M_{0}^{m}(p)\big{\|}u\big{\|}_{C^{1+}_{\star}}+M_{0}^{m}(D_{t}p)\Big{\\}}\big{\|}u\big{\|}_{H^{m}}.$
(C.5)

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