On the free-boundary Incompressible Elastodynamics with and without surface tension

Longhui Xu

Abstract

We consider a free-boundary problem for the incompressible elastodynamics describing the motion of an elastic medium in a periodic domain with a moving boundary and a fixed bottom under the influence of surface tension. The local well-posedness in Lagrangian coordinates is proved by extending Gu-Luo-Zhang[8] on incompressible magnetohydrodynamics. We adapt the idea in Luo-Zhang[23] on compressible gravity-capillary water waves to obtain an energy estimate in graphic coordinates. The energy estimate is uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds and thus yields the zero-surface-tension limit.

1 Introduction

We consider the incompressible free-boundary elastodynamic equations

\begin{cases}D_{t}u+\nabla p=\text{div}(\mathbf{F}\mathbf{F}^{T})\qquad&\text{in }\mathcal{D},\\ \text{div }u=0\qquad&\text{in }\mathcal{D},\\ D_{t}\mathbf{F}=\nabla u\mathbf{F}\qquad&\text{in }\mathcal{D},\\ \text{div }\mathbf{F}^{T}=0\qquad&\text{in }\mathcal{D}.\end{cases}

(1.1)

Here $\mathcal{D}:=\bigcup_{0\leq t\leq T}\{t\}\times\mathcal{D}_{t}$ and $\mathcal{D}_{t}:=\{(x_{1},x_{2})\in\mathbb{T}^{2},-b<x_{3}<\psi(t,x_{1},x_{2})\}$ is the periodic domain occupied by the elastic medium at time $t$ . $u,p$ denotes the velocity and fluid pressure of the elastic medium. $\mathbf{F}:=(\mathbf{F}_{ij})_{3\times 3}$ is the deformation tensor and $\mathbf{F}^{T}:=(\mathbf{F}_{ji})$ is the transpose of $F$ . $\mathbf{F}\mathbf{F}^{T}$ is the Cauchy-Green tensor for neo-Hooken elsatic materials. $\nabla:=(\partial_{1},\partial_{2},\partial_{3})$ is the standard spatial derivative, div $X:=\nabla\cdot X=\partial_{i}X^{i}$ is the standard divergence for any vector filed $X$ . $D_{t}:=\partial_{t}+v\cdot\nabla$ is the material derivative. We consider the boundary conditions of (1.1)

\begin{cases}D_{t}|_{\partial\mathcal{D}}\in\mathcal{T}(\partial\mathcal{D}),\\ \mathbf{F}^{T}n=0\qquad&\text{on }\partial\mathcal{D}_{t},\\ p=\sigma\mathcal{H}\qquad&\text{on }\partial\mathcal{D}_{t}.\end{cases}

(1.2)

Here $\mathcal{T}(\partial\mathcal{D}_{t})$ is the tangent bundle of $\partial\mathcal{D}_{t}$ . The moving boundary $\partial\mathcal{D}_{t}$ consists of the moving top $\Sigma_{t}:=\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{3}=\psi(t,x_{1},x_{2})\}$ and the fixed flat bottom $\Sigma_{b}:=\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{3}=-b\}$ . The first condition is the kinematic boundary condition indicating the boundary moves with the velocity of the fluid. The second condition shows the deformation tensor is tangential on the boundary, where $n$ is the unit normal vector to the boundary. The third condition shows that the pressure is balanced by surface tension on the moving top $\Sigma_{t}$ , where $\sigma>0$ is the surface tension constant and $\mathcal{H}$ is the mean curvature of the moving surface.

Now we write the system in terms of columns of the deformation tensor. Let $A_{j}:=(A_{1j},A_{2j},A_{3j})^{T}$ be the $j$ -th column of $A$ for any $3\times 3$ matrix A. Then the last equation of (1.1) reads

0=(\text{div }\mathbf{F}^{T})_{i}=\partial_{j}(\mathbf{F}^{T})_{ij}=\partial_{j}\mathbf{F}_{ji}=\text{div }\mathbf{F}_{i}.

(1.3)

It follows that

(\text{div }(\mathbf{F}\mathbf{F}^{T}))_{i}=\partial_{j}(\mathbf{F}\mathbf{F}^{T})_{ij}=\partial_{j}(\mathbf{F}_{ik}\mathbf{F}_{jk})=\partial_{j}\mathbf{F}_{ik}\mathbf{F}_{kj}+\mathbf{F}_{ik}\partial_{j}\mathbf{F}_{jk}=\partial_{j}\mathbf{F}_{ik}\mathbf{F}_{kj}=(\mathbf{F}_{k}\cdot\nabla)\mathbf{F}_{k}.

(1.4)

Hence, the first equation reads

D_{t}u+\nabla p=(\mathbf{F}_{k}\cdot\nabla)\mathbf{F}_{k}.

(1.5)

Since $(\nabla u\mathbf{F}_{j})_{i}=\partial_{k}u_{i}\mathbf{F}_{kj}=(\mathbf{F}_{j}\cdot\nabla)u_{i}$ , the third equation reads

D_{t}\mathbf{F}_{j}=(\mathbf{F}_{j}\cdot\nabla)u.

(1.6)

Columnwisely , the boundary condition $\mathbf{F}^{T}n=0$ reads

0=(\mathbf{F}^{T}n)_{j}=\mathbf{F}_{kj}n_{k}=\mathbf{F}_{j}\cdot n.

(1.7)

The system (1.1) can be reformulated as

\begin{cases}D_{t}u+\nabla p=(\mathbf{F}_{k}\cdot\nabla)\mathbf{F}_{k}\qquad&\text{in }\mathcal{D},\\ \text{div }u=0\qquad&\text{in }\mathcal{D},\\ D_{t}\mathbf{F}_{j}=(\mathbf{F}_{j}\cdot\nabla)u\qquad&\text{in }\mathcal{D},\\ \text{div }\mathbf{F}_{j}=0\qquad&\text{in }\mathcal{D},\end{cases}

(1.8)

with the boundary condition

\begin{cases}D_{t}|_{\partial\mathcal{D}}\in\mathcal{T}(\partial\mathcal{D}),\\ \mathbf{F}_{j}\cdot n=0\qquad&\text{on }\partial\mathcal{D}_{t},\\ p=\sigma\mathcal{H}\qquad&\text{on }\partial\mathcal{D}_{t}.\\ \end{cases}

(1.9)

1.1 History and Background

Let us first briefly review the results of the free-boundary Euler equations, which have been intensively studied in recent decades. The first breakthrough was by Wu [25, 26] on the local well-posedness(LWP) of the irrotational water wave. For the general incompressible problem with non-zero vorticity, Christodoulou-Lindblad [1] established an apriori energy estimate and Lindblad [17, 20] proved the LWP for the case without surface tension by Nash-Moser iteration, which leads to a loss of regularity. Coutand-Shkoller [3, 4] proved the LWP for the case with surface tension by introducing tangential smoothing and avoided the loss of regularity.

The study of free-boundary compressible fluid is much less and most of the results only deal with the case without surface tension. Lindblad [18, 19] first proved the LWP by Nash-Moser and Trakhinin [24] extended the LWP to non-isentropic fluid in an unbounded domain, but both results have a loss of regularity. Luo [21] established an apriori estimate without loss of regularity for the isentropic case and Luo-Zhang [22] proved the LWP without using the Nash-Moser iteration. For the case with non-zero surface tension, Luo-Zhang [23] proved the LWP of the gravity-capillary water waves by tangential smoothing and artificial viscosity with an energy estimate uniform in Mach number and the surface tension coefficient, by which the incompressible limit and the zero surface tension limit can be simultaneously justified.

Now we review the developments of incompressible elastodynamics. The fixed-boundary problem of the elastodynamics is well-understood and the global existence is expected and we refer to [12, 11, 14, 16, 15, 2, 5]. However, the free-boundary problem is more difficult as we have limited boundary regularity and the boundary terms can enter the highest order in the energy estimate. Most of the existing literature neglected the effect of surface tension. Gu-Wang [9] and Li-Wang-Zhang [13] proved the LWP under a mixed stability condition. Hu-Huang [10] proved the LWP under the Rayleigh-Taylor sign condition. For the case with surface tension, Gu-Lei [7] proved the local well-posedness of the two-dimensional case by studying the viscoelastic system in Lagrangian coordinates and the vanishing viscosity limit.

Studying the case with nonzero surface tension with Lagrangian coordinates is painful due to its complicated description of the boundary. In this paper, we try to adapt the idea in Luo-Zhang [23] to establish an energy estimate for the incompressible elastodynamics with surface tension in graphical coordinates, by which the mean curvature of the boundary can be neatly formulated. Moreover, the energy estimate is uniform in surface tension coefficient and directly leads to zero surface tension limit. However, we have to point out that the LWP in graphical coordinates cannot be directly studied with tangential smoothing, since the divergence-free condition and the boundary condition of the deformation tensor can not be propagated from the constraints on the initial data in the tangentially smoothed system.

1.2 Outline of the paper

This paper is organised as follows. In Section 2, we first reformulate the system in the Lagrangian coordinates and establish the LWP by extending Gu-Luo-Zhang [8] on the LWP of Magnetohydrodynamics(MHD), then we introduce the formulation in the graphical coordinates and our main results. In Section 3, we recall some preliminary results that will be useful in the paper. Section 4 is the derivation of our energy estimate and Section 5 concerns the zero surface tension limit.

2 Reformulation of the system

2.1 Lagrangian Coordinates

The system (1.8)-(1.9) is similar to the free-boundary incompressible magnetohydrodynamics considered in Gu-Luo-Zhang[8] and thus the local wellposedness can be directly proven by extending their result.

2.1.1 Reformulation in Lagrangian Coordinates

We reformulate the equations in Lagrangian coordinates to transform the free-boundary problem to be a fixed boundary problem on

\displaystyle\Omega:=\mathbb{T}^{2}\times(-b,0).

Let $\Sigma:=\mathbb{T}^{2}\times\{0\}$ , $\Sigma_{b}:=\mathbb{T}^{2}\times\{-b\}$ . and $\eta:[0,T]\times\Omega\to\mathcal{D}$ be the flow map, i.e.,

\displaystyle\partial_{t}\eta(t,x)=u(t,\eta(t,x)),\qquad\eta_{0}=\eta(0,\cdot),

with

\displaystyle\eta_{0}\big{|}_{\Sigma}=\psi(0,\cdot)

We introduce the Lagrangian variables

\displaystyle v(t,x)=u(t,\eta(t,x)),\qquad q(t,x)=p(t,\eta(t,x)),\qquad F(t,x)=\mathbf{F}(t,\eta(t,x)).

We define the cofactor matrix $a=[\nabla\eta]^{-1}$ , it follows that the material derivative $D_{t}$ reduces to time derivative $\partial_{t}$ and the spacial derivative $\partial_{j}$ becomes $\nabla_{a}^{j}:=a^{ij}\partial_{i}$ . Let $N$ be the unit outer normal vector of $\partial\Omega$ , the system (1.8)-(1.9) becomes the following:

\begin{cases}\partial_{t}v+\nabla_{a}q=(F_{k}\cdot\nabla_{a})F_{k}\qquad&\text{in }\Omega,\\ \nabla_{a}\cdot v=0\qquad&\text{in }\Omega,\\ \partial_{t}F_{j}=(F_{j}\cdot\nabla_{a})v\qquad&\text{in }\Omega,\\ \nabla_{a}\cdot F_{j}=0\qquad&\text{in }\Omega\end{cases}

(2.1)

with the boundary condition

\begin{cases}v_{3}=F_{3j}=0\qquad&\text{on }\Sigma_{b}\\ a^{\mu\nu}F_{j\nu}N_{\mu}=0\qquad&\text{on }\Sigma,\\ a^{\mu\nu}N_{\mu}q=-\sigma(\sqrt{g}\Delta_{g}\eta^{\nu})\qquad&\text{on }\Sigma.\\ \end{cases}

(2.2)

where $g$ is the metric induced on $\Sigma_{t}=\eta(t,\Sigma)$ by the embedding $\eta$ and $\Delta_{g}$ is the Laplacian of $g$ .

2.1.2 Local Well-posedness in Lagrangian Coordinates

This system (2.1)- (2.2) is identical to the one considered in Gu-Luo-Zhang[8] by replacing $F_{j}$ by the magnetic field $b$ , hence we can apply their result to obtain the local well-posedness, provided the initial data satisfies certain compatibility conditions.

Theorem 2.1.

(Local existence) Let $v_{0},\eta_{0}\in H^{4.5}(\Omega)\cap H^{5}(\Sigma)$ and $F_{j}^{0}\in H^{4.5}$ be divergence-free vector fields with $(F_{j}^{0}\cdot N)|_{\Sigma}=0$ and define the initial data $q_{0}$ of $q$ to satisfies the following elliptic equation

\begin{cases}-\Delta q_{0}=(\partial v_{0})(\partial v_{0})-(\partial F_{j}^{0})(\partial F_{j}^{0})\qquad&\text{in }\Omega,\\ q_{0}=\sigma\mathcal{H}_{0}\qquad&\text{on }\Sigma,\\ \frac{\partial q_{0}}{\partial N}=0\qquad&\text{on }\Sigma_{b}.\end{cases}

(2.3)

Then there exists some $T>0$ depending on $\sigma,v_{0},F_{j}^{0}$ , such that the system (1.8)-(1.9) with initial data $(v_{0},F_{j}^{0},q_{0})$ has a unique strong solution $(\eta,v,q)$ with the energy estimates

\displaystyle\sup_{0\leq t\leq T}E(t)\leq C,

(2.4)

where $C$ depends on $||v_{0}||_{4.5},||F^{0}_{J}||_{4.5},|v_{0}|_{5}$ and

	$\displaystyle E(t):=$	$\displaystyle\|\|\eta(t)\|\|^{2}_{4.5}+\sum_{k=0}^{3}\bigg{(}\|\|\partial_{t}^{k}v(t)\|\|^{2}_{4.5-k},\|\|\partial_{t}^{k}(F_{j}^{0}\cdot\partial)\eta(t)\|\|^{2}_{4.5-k}\bigg{)}+\|\|\partial^{4}_{t}v(t)\|\|_{0}^{2}+\|\|\partial^{4}_{t}(F_{j}^{0}\cdot\partial)\eta(t)\|\|_{0}^{2}$		(2.5)
		$\displaystyle+\sum_{k=0}^{3}\big{(}\big{\|}\bar{\partial}(\Pi\bar{\partial}^{k}\partial_{t}^{3-k}v(t)\big{)}\big{\|}_{0}^{2}\big{)}+\big{\|}\bar{\partial}(\Pi\bar{\partial}^{3}(F_{j}^{0}\cdot\partial)\eta(t)\big{)}\big{\|}_{0}^{2},$		(2.6)

here $\Pi$ is the canonical normal projection defined on the moving interface.

2.2 Graphic Coordinates

The remaining parts of this paper are devoted to developing an $\sigma-$ uniform energy estimate, provided the Rayleigh-Taylor sign condition is satisfied, and thus we can obtain the zero-surface tension limit. This allow us to show the LWP for the case of $\sigma=0$

2.2.1 Reformulation in Graphic Coordinates

We introduce graphical coordinates to convert the free-boundary problem into a fixed domain problem on

\Omega:=\mathbb{T}^{2}\times(-b,0).

Let $\overline{x}=(x_{1},x_{2})$ . The moving top $\Sigma_{t}:=\{(\overline{x},x_{3})\in\Omega:x_{3}=\psi(t,\overline{x})\}$ is represented by the graph of $\psi$ . To fix the interior, we introduce the extension of $\psi$

\varphi(t,\overline{x},x_{3})=x_{3}+\chi(x_{3})\psi(t,\overline{x}),

(2.7)

where $\chi\in C_{c}^{\infty}(-b,0]$ satisfying

||\chi^{\prime}||_{L^{\infty}(-b,0]}\leq\frac{1}{1+||\psi_{0}||_{\infty}},\qquad\chi\equiv 1\text{ on }(-\delta_{0},0]

(2.8)

for some small constant $\delta_{0}>0$ .
Note that

\partial_{3}\varphi=1+\chi^{\prime}(x_{3})\psi(t,\overline{x}).

(2.9)

By (2.8), we have

|\chi^{\prime}(x_{3})\psi(t,\overline{x})|\leq\frac{||\psi(t)||_{\infty}}{1+||\psi_{0}||_{\infty}}<1,\qquad t\in[0,T]

(2.10)

for some small $T>0$ . It follows from (2.9) and (2.10)

\partial_{3}\varphi(t,\overline{x},x_{3})\geq c_{0}>0,\qquad t\in[0,T]

(2.11)

for some $c_{0}>0$ . Let

\Phi(t,\overline{x},x_{3})=(\overline{x},\varphi(t,\overline{x},x_{3})),

(2.12)

then (2.11) ensures us to define the diffeomorphism $\displaystyle\Phi(t,\cdot):\Omega\to\mathcal{D}_{t}$ .

Now we introduce the graphical variables

v(t,x)=u(t,\Phi(t,x)),\qquad q(t,x)=p(t,\Phi(t,x)),\qquad F(t,x)=\mathbf{F}(t,\Phi(t,x)).

(2.13)

Also, we introduce the induced differential operator $\displaystyle\partial^{\phi}$ so that for any function $g(t,x),x\in\mathcal{D}_{t}$ and $G:=g(t,\Phi(t,x))$ , we have

\partial_{\alpha}^{\varphi}G=\partial_{\alpha}g(t,\Phi(t,x)),\qquad\alpha=t,1,2,3.

(2.14)

It follows that

$\displaystyle\partial_{t}^{\varphi}$	$\displaystyle=\partial_{t}-\frac{\partial_{t}\varphi}{\partial_{3}\varphi}\partial_{3},$	(2.15)
$\displaystyle\nabla_{\tau}^{\varphi}=\partial_{\tau}^{\varphi}$	$\displaystyle=\partial_{\tau}-\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi}\partial_{3},\qquad\tau=1,2,$	(2.16)
$\displaystyle\nabla_{3}^{\varphi}=\partial_{3}^{\varphi}$	$\displaystyle=\frac{1}{\partial_{3}\varphi}\partial_{3}.$	(2.17)

We define the cofactor matrix

A:=\begin{pmatrix}1&0&0\\ 0&1&0\\ -\frac{\partial_{1}\varphi}{\partial_{3}\varphi}&-\frac{\partial_{2}\varphi}{\partial_{3}\varphi}&\frac{1}{\partial_{3}\varphi}\end{pmatrix}.

(2.18)

It follows that

\nabla_{i}^{\varphi}=A_{ji}\partial_{j}.

(2.19)

Next, we introduce the material derivative

$\displaystyle D_{t}^{\varphi}$	$\displaystyle:=\partial_{t}^{\varphi}+v\cdot\nabla^{\varphi}$	(2.20)
	$\displaystyle=(\partial_{t}-\frac{\partial_{t}\varphi}{\partial_{3}\varphi}\partial_{3})+v_{1}(\partial_{1}-\frac{\partial_{1}\varphi}{\partial_{3}\varphi}\partial_{3})+v_{2}(\partial_{1}-\frac{\partial_{2}\varphi}{\partial_{3}\varphi}\partial_{3})+\frac{v_{3}}{\partial_{3}\varphi}\partial_{3}$
	$\displaystyle=\partial_{t}+\overline{v}\cdot\overline{\partial}+(-\partial_{t}\varphi-v_{1}\partial_{1}\varphi-v_{2}\partial_{2}\varphi+v_{3})\frac{1}{\partial_{3}\varphi}\partial_{3}$
	$\displaystyle=\partial_{t}+\overline{v}\cdot\overline{\partial}+(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}^{\varphi},$	(2.21)

where $\mathbf{N}:=(-\partial_{1}\varphi,-\partial_{2}\varphi,1)$ and $\overline{\partial}=\overline{\nabla}=(\partial_{1},\partial_{2})$ .

Now, we reformulate the boundary conditions. Let $N:=(1,0,\partial_{1}\psi)\times(0,1,\partial_{2}\psi)=(-\partial_{1}\psi,-\partial_{2}\psi,1)$ be the standard normal vector to the moving top $\Sigma_{t}$ and $n=(0,0,1)$ the unit normal vector to the fixed bottom. By (2.21),

D_{t}^{\varphi}|_{\partial\Omega}=\partial_{t}+\overline{v}\cdot\overline{\partial}+\big{(}(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}^{\varphi}\big{)}|_{\Sigma\cup\Sigma_{b}}.

(2.22)

since

(v\cdot\mathbf{N}-\partial_{t}\varphi)|_{\Sigma}=v\cdot N-\partial_{t}\psi,

(2.23)

and

(v\cdot\mathbf{N}-\partial_{t}\varphi)|_{\Sigma_{b}}=v\cdot n-\partial_{t}(-b)=v_{3},

(2.24)

we have $D_{t}^{\varphi}$ is tangential if

	$\displaystyle v\cdot N-\partial_{t}\psi$	$\displaystyle=0\qquad\text{ on }\Sigma,$		(2.25)
	$\displaystyle v_{3}$	$\displaystyle=0\qquad\text{ on }\Sigma_{b}.$		(2.26)

Next, for the boundary condition of $q$ , by definition of mean curvature,

\displaystyle q=-\sigma\overline{\nabla}\cdot\big{(}\frac{\overline{\nabla\psi}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}})

\displaystyle\qquad\text{ on }\Sigma.

(2.27)

Consequently the systems (1.1) and (1.8) are converted into

\begin{cases}D_{t}^{\varphi}v+\nabla^{\varphi}q=(F_{k}\cdot\nabla^{\varphi})F_{k}\qquad&\text{in }\Omega,\\ \nabla^{\varphi}\cdot v=0\qquad&\text{in }\Omega,\\ D_{t}^{\varphi}F_{j}=(F_{j}\cdot\nabla^{\varphi})v\qquad&\text{in }\Omega,\\ \nabla^{\varphi}\cdot F_{j}=0\qquad&\text{in }\Omega\end{cases}

(2.28)

with the boundary conditions

\begin{cases}\partial_{t}\psi=v\cdot N&\qquad\text{ on }\Sigma,\\ q=-\sigma\overline{\nabla}\cdot\big{(}\frac{\overline{\nabla\psi}}{\sqrt{1+|\overline{\nabla}\psi|^{2}}})&\qquad\text{ on }\Sigma,\\ F_{j}\cdot N=0&\qquad\text{ on }\Sigma,\\ v_{3}=0&\qquad\text{ on }\Sigma_{b},\\ F_{3j}=0&\qquad\text{ on }\Sigma_{b},\\ q=0&\qquad\text{ on }\Sigma_{b}.\\ \end{cases}

(2.29)

We can show that div $F_{j}=0$ and the boundary condition $F_{j}\cdot N=0$ are only constraints on the initial data.

Proposition 2.2.

Let the initial data satisfies

\displaystyle\nabla^{\varphi}\cdot F_{j}=0,

(2.30)

and the boundary condition

\displaystyle F_{j}\cdot n=0\quad\text{on }\Sigma\cup\Sigma_{b}.

(2.31)

Then the solution satisfies (2.30) and (2.31) for all $t\in[0,T].$

Proof Taking $\nabla^{\varphi}\cdot$ on $D_{t}^{\varphi}F_{j}-(F_{j}\cdot\nabla^{\varphi})v=0$ , we have

	$\displaystyle 0$	$\displaystyle=\partial_{i}^{\varphi}(\partial_{t}^{\varphi}F_{ij}+\big{(}v\cdot\nabla^{\varphi})F_{ij}\big{)}-\partial_{i}^{\varphi}\big{(}(F_{k}\cdot\nabla^{\varphi})v_{i}\big{)}$
		$\displaystyle=D_{t}^{\varphi}\big{(}\nabla^{\varphi}\cdot F_{j}\big{)}+(\partial_{i}^{\varphi}v\cdot\nabla^{\varphi})F_{ij}-(\partial_{i}^{\varphi}F_{k}\cdot\nabla^{\varphi})v_{i}-(F_{k}\cdot\nabla^{\varphi})(\nabla^{\varphi}\cdot v),$

where the fourth term vanishes due to the divergence-free condition of $v$ and the third term cancels the second term. We have

\displaystyle D_{t}^{\varphi}\big{(}\nabla^{\varphi}\cdot F_{j}\big{)}=0,

(2.32)

which is a linear equation of $\nabla^{\varphi}\cdot F_{j}$ and the propagation of $\nabla^{\varphi}\cdot F_{j}$ follows from standard characteristic curve method. For the boundary condition, we consider $D_{t}^{\varphi}F_{j}-(F_{j}\cdot\nabla^{\varphi})v+F_{j}(\nabla^{\varphi}\cdot v)=0$ on the boundary, let $\tau=1,2$

	$\displaystyle 0$	$\displaystyle=(\partial_{t}+\overline{v}\cdot\overline{\partial})F_{j}-F_{\tau j}(\partial_{\tau}-\partial_{\tau}\varphi\partial_{3})v-F_{3j}\partial_{3}v+F_{j}(\partial_{\tau}-\partial_{\tau}\varphi\partial_{3})v_{\tau}+F_{j}\partial_{3}v_{3}$
		$\displaystyle=(\partial_{t}+\overline{v}\cdot\overline{\partial})F_{j}-F_{\tau j}\partial_{\tau}v-(F_{j}\cdot N)\partial_{3}v+F_{j}\partial_{\tau}v_{\tau}+F_{j}(N\cdot\partial_{3}v).$

We times $N$ on both sides to have

	$\displaystyle 0$	$\displaystyle=(\partial_{t}+\overline{v}\cdot\overline{\partial})F_{j}\cdot N-F_{\tau j}(\partial_{\tau}v\cdot N)-(F_{j}\cdot N)(\partial_{3}v\cdot N)+(F_{j}\cdot N)\partial_{\tau}v_{\tau}+(F_{j}\cdot N)(N\cdot\partial_{3}v)$
		$\displaystyle=(\partial_{t}+\overline{v}\cdot\overline{\partial})F_{j}\cdot N-F_{\tau j}(\partial_{\tau}v\cdot N)+(F_{j}\cdot N)\partial_{\tau}v_{\tau}$
		$\displaystyle=(\partial_{t}+\overline{v}\cdot\overline{\partial})(F_{j}\cdot N)-F_{j}\cdot\partial_{t}N-F_{j}\cdot(\overline{v}\cdot\overline{\partial})N-F_{\tau j}(\partial_{\tau}v\cdot N)+(F_{j}\cdot N)\partial_{\tau}v_{\tau},$

where the second term cancels exactly the third and fourth term,

\displaystyle-F_{j}\cdot\partial_{t}N=F_{\tau j}\partial_{t}\partial_{\tau}\psi=F_{\tau j}(\partial_{\tau}v\cdot N+v\cdot\partial_{\tau}N)=F_{\tau j}(\partial_{\tau}v\cdot N)+F_{\tau j}(\overline{v}\cdot\overline{\partial})N.

Hence, we have a linear equation of $F_{j}\cdot N$ on the boundary

\displaystyle 0=(\partial_{t}+\overline{v}\cdot\overline{\partial})(F_{j}\cdot N)+\partial_{\tau}v_{\tau}(F_{j}\cdot N).

(2.33)

2.3 The main result

Theorem 2.3.

Let $\sigma>0$ , $(v,q,F,\psi)$ satisfies the above system. Define

\displaystyle E(t)=\sum_{k=0}^{4}\bigg{(}||\partial_{t}^{k}F||_{4-k},||\partial_{t}^{k}v||_{4-k},||\sqrt{\sigma}\partial_{t}^{k}\overline{\nabla}\psi||_{4-k}\bigg{)}+\sum_{k=0}^{3}||\partial_{t}^{k}q||_{4-k},

(2.34)

Then there exists $T>0$ such that

\displaystyle E(t)\leq P(E(0))+\int_{0}^{T}P(E(t)).

(2.35)

Furthermore, if the Rayleigh-Taylor sign condition $-\partial_{3}q\geq c_{0}>0$ is assumed, then $\sum_{k=0}^{4}|\partial_{t}^{k}\psi|_{4-k}$ enters the energy and the estimate become $\sigma-$ uniform.

3 The auxiliary results

3.1 Preliminary Lemmas

Lemma 3.1.

For $\alpha,\beta=t,1,2,3$ and generic function $F$ ,

[\partial_{\alpha}^{\varphi},\partial_{\beta}^{\varphi}]F=0.

(3.1)

Proof It directly follows from the fact that $\partial_{\alpha}^{\varphi}F=\partial_{\alpha}f(t,\Phi(t,x))$ , where $F=f(t,\Phi(t,x))$ . More precisely

\displaystyle\partial_{\alpha}^{\varphi}(\partial_{\beta}^{\varphi}F)=\partial_{\alpha}^{\varphi}\big{(}\partial_{\beta}f(t,\Phi(t,x))\big{)}=\partial_{\alpha}\partial_{\beta}f(t,\Phi(t,x))=\partial_{\beta}\partial_{\alpha}f(t,\Phi(t,x))=\partial_{\beta}^{\varphi}(\partial_{\alpha}^{\varphi}F).

Lemma 3.2.

(Integration by parts) Let $g=g(t,x),f=f(t,x),x\in\Omega$ ,

\displaystyle\int_{\Omega}(\partial_{i}^{\varphi}f)g\partial_{3}\varphi=-\int_{\Omega}f(\partial_{i}^{\varphi}g)\partial_{3}\varphi+\int_{\Sigma}fgN_{i}+\int_{\Sigma_{b}}fgn_{i}.

(3.2)

where $N=(-\partial_{1}\psi,-\partial_{2}\psi,1),n=(0,0,1)$ are the normal vector function of $\Sigma,\Sigma_{b}$ respectively.

Proof The proof follows from change of variable and standard integration by parts. Let $G(t,x),F(t,x),x\in\mathcal{D}_{t}$ such that $g(t,x)=G(t,\Phi(t,x)),f=F(t,\Phi(t,x))$ , then

	$\displaystyle\int_{\Omega}(\partial_{i}^{\varphi}f)g\partial_{3}\varphi$	$\displaystyle=\int_{\Omega}\partial_{i}F(t,\Phi(t,x))G(t,\Phi(t,x))\partial_{3}\varphi$
		$\displaystyle=\int_{\mathcal{D}_{t}}\partial_{i}F(t,x)G(t,x)$
		$\displaystyle=-\int_{\mathcal{D}_{t}}F(t,x)\partial_{i}G(t,x)+\int_{\partial\mathcal{D}_{t}}F(t,x)G(t,x)\mathbf{n}_{i}$
		$\displaystyle=-\int_{\Omega}f(\partial_{i}^{\varphi}g)\partial_{3}\varphi+\int_{\Sigma}fgN_{i}+\int_{\Sigma_{b}}fgn_{i}.$

Lemma 3.3.

(Transport Theorem) Let $f=f(t,x),x\in\Omega$

\displaystyle\frac{d}{dt}\int_{\Omega}f\partial_{3}\varphi=\int_{\Omega}D_{t}^{\varphi}f\partial_{3}\varphi.

(3.3)

Proof The proof follows from definition of $D_{t}^{\varphi}$ , $\nabla^{\varphi}\cdot v=0$ and integration by parts,

	$\displaystyle\frac{d}{dt}\int_{\Omega}f\partial_{3}\varphi$	$\displaystyle=\int_{\Omega}\partial_{t}f\partial_{3}\varphi+\int_{\Omega}f\partial_{t}\partial_{3}\varphi$
		$\displaystyle=\int_{\Omega}\partial_{t}^{\varphi}f\partial_{3}\varphi+\int_{\Omega}\partial_{t}\varphi\partial_{3}f+\int_{\Omega}f\partial_{t}\partial_{3}\varphi$
		$\displaystyle=\int_{\Omega}D_{t}^{\varphi}f\partial_{3}\varphi-\int_{\Omega}(v\cdot\nabla^{\varphi})f\partial_{3}\varphi+\int_{\Sigma}f\partial_{t}\varphi$
		$\displaystyle=\int_{\Omega}D_{t}^{\varphi}\partial_{3}\varphi+\int_{\Omega}(\nabla^{\varphi}\cdot v)f\partial_{3}\varphi$
		$\displaystyle=\int_{\Omega}D_{t}^{\varphi}\partial_{3}\varphi.$

3.2 Elliptic Estimates

Lemma 3.4.

(The Hodge-type elliptic estimate) Let $X$ be a smooth vector field and $s\geq 1$ , then

||X||_{s}^{2}\lesssim C\big{(}|\psi|_{s},|\overline{\partial}\psi|_{W^{1,\infty}}\big{)}\big{(}||\nabla^{\varphi}\cdot X||_{s-1}^{2}+||\nabla^{\varphi}\times X||_{s-1}^{2}+||\overline{\partial}^{s}X||_{0}^{2}+||X||_{0}^{2}\big{)}.\\

(3.4)

We refer to lemma B.2 of [6] for the proof

Lemma 3.5.

(Low regularity elliptic estimate) Assume $W\in H^{1}$ with $W=0$ on $\Sigma_{b}$ satisfying

\begin{cases}-\Delta^{\varphi}W=\nabla^{\varphi}\cdot\pi\qquad&\text{in }\Omega,\\ \nabla^{\varphi}W\cdot N=h\qquad&\text{on }\partial\Omega,\end{cases}

(3.5)

where $\pi,\nabla^{\varphi}\cdot\pi\in L^{2}(\Omega)$ and $h\in H^{-0.5}(\partial\Omega)$ with the compatibility condition

\int_{\partial\Omega}(\pi\cdot N+h)dS=0.

(3.6)

Then, we have

||W||_{1}\lesssim_{vol(\Omega)}||\partial_{3}\varphi||_{\infty}||\pi||_{0}+|\pi\cdot N+h|_{-0.5}.

(3.7)

Proof Testing the first equation of (3.5) with $\omega\partial_{3}\phi,\omega\in H^{1}$ ,

\int_{\Omega}(-\partial_{i}^{\varphi}\partial_{i}^{\varphi}W)\omega\partial_{3}\phi=\int_{\Omega}(\partial_{i}^{\varphi}\pi)\omega\partial_{3}\phi.

(3.8)

By integration by parts,

	$\displaystyle\int_{\Omega}(-\partial_{i}^{\varphi}\partial_{i}^{\varphi}W)\omega\partial_{3}\phi$	$\displaystyle=\int_{\Omega}(\partial_{i}^{\varphi}W)\partial_{i}^{\varphi}\omega\partial_{3}\phi-\int_{\partial\Omega}(\partial_{i}^{\varphi}W)\omega N_{i}$
		$\displaystyle=\int_{\Omega}A^{\mu i}A^{\nu i}\partial_{\mu}W\partial_{\nu}\omega\partial_{3}\phi-\int_{\partial\Omega}h\omega,$
	$\displaystyle\int_{\Omega}(\partial_{i}^{\varphi}\pi)\omega\partial_{3}\phi$	$\displaystyle=-\int_{\Omega}\pi\cdot\nabla^{\varphi}\omega\partial_{3}\phi+\int_{\partial\Omega}(\pi\cdot N)\omega.$

It follows from (3.8) and $\partial_{3}\phi>0$ that

	$\displaystyle\int_{\Omega}A^{\mu i}A^{\nu i}\partial_{\mu}W\partial_{\nu}\omega\leq\int_{\Omega}A^{\mu i}A^{\nu i}\partial_{\mu}W\partial_{\nu}\omega\partial_{3}\phi$	$\displaystyle=-\int_{\Omega}\pi\cdot\nabla^{\varphi}\omega\partial_{3}\phi+\int_{\partial\Omega}(\pi\cdot N+h)\omega$
		$\displaystyle\lesssim\|\|\pi\|\|_{0}\|\|\partial_{3}\phi\|\|_{\infty}\|\|\omega\|\|_{1}+\|\pi\cdot N+h\|_{-0.5}\|\omega\|_{0.5}.$		(3.9)

Since $\displaystyle AA^{T}$ is positive definite and $||AA^{T}||_{\infty}\leq K$ , by elliptic estimate

||\partial W||_{0}\lesssim||\partial_{3}\phi||_{\infty}||\pi||_{0}+|\pi\cdot N+h|_{-0.5}.

(3.10)

By Poincaré’s inequality

||W||_{0}\lesssim_{vol(\Omega)}||\partial W||_{0}+\int_{\Omega}W,

(3.11)

where

\int_{\Omega}Wdx=\int_{\Omega}\partial_{3}x_{3}Wdx=-\int_{\Omega}x_{3}\partial_{3}Wdx\leq||x_{1}||_{0}||\partial W||_{0}\lesssim_{Vol(\Omega)}||\partial W||_{0},

(3.12)

the boundary terms vanishes since $x_{3}=0$ on $\Sigma$ and $W=0$ on $\Sigma_{b}$ .

4 Energy estimate

4.1 Pressure Estimate

By the first equation of (2.28),

-\nabla^{\varphi}q=D_{t}^{\varphi}v-(F_{k}\cdot\nabla^{\varphi})F_{k}.

(4.1)

Taking $\displaystyle\nabla^{\varphi}\cdot$ , we have

-\Delta^{\varphi}q=\nabla^{\varphi}\cdot(D_{t}^{\varphi}v-(F_{k}\cdot\nabla^{\varphi})F_{k})\qquad\text{ on }\Omega

(4.2)

with the boundary condition

\nabla^{\varphi}q\cdot N=-(D_{t}^{\varphi}v-(F_{k}\cdot\nabla^{\varphi})F_{k})\cdot N\qquad\text{ on }\Sigma\cup\Sigma_{b},

(4.3)

which is exactly the form of the elliptic system in lemma (3.5) and satisfies the compatibility condition. It follows that

||q||_{1}\lesssim_{vol(\Omega)}||\partial_{3}\varphi||_{\infty}(||D_{t}^{\varphi}v||_{0}+||F_{k}||_{0}||\nabla^{\varphi}F_{k}||_{0}).

(4.4)

For the higher order term, by definition or $\partial_{3}^{\varphi}$ and taking the normal component of (4.1),

\displaystyle\partial_{3}q=\partial_{3}\varphi\partial_{3}^{\varphi}q=-\partial_{3}\varphi D_{t}^{\varphi}v+\partial_{3}\varphi(F_{k}\cdot\nabla^{\varphi})F_{3k}.

(4.5)

Let $\partial_{\gamma}=\partial_{t},\partial_{1},\partial_{2},\partial_{3}$ and $k=1,2,3$ . We have

\displaystyle||\partial_{\gamma}^{k}\partial_{3}q||_{0}=||\partial_{\gamma}^{k}\big{(}-\partial_{3}\varphi D_{t}^{\varphi}v+\partial_{3}\varphi(F_{k}\cdot\nabla^{\varphi})F_{3k}\big{)}||_{0}\leq P\bigg{(}\sum_{l=0}^{3}|\partial_{t}^{l}\overline{\nabla}\psi|_{3-l},\sum_{l=0}^{4}||\partial_{t}^{k}v||_{4-k},\sum_{l=0}^{4}||\partial_{t}^{k}F||_{4-k}\bigg{)}.

(4.6)

Next, by definition of $\partial_{\tau}^{\varphi},\tau=1,2$ ,

\displaystyle\partial_{\tau}q=\partial_{\tau}^{\varphi}q+\frac{1}{\partial_{3}\varphi}\partial_{3}q=D_{t}^{\varphi}v_{\tau}-(F_{k}\cdot\nabla^{\varphi})F_{\tau k}+\frac{1}{\partial_{3}\varphi}\partial_{3}q.

(4.7)

Let $D=\overline{\partial}$ or $\partial_{t},k=1,2,3$ , we have

\displaystyle||D^{k}\partial_{\tau}q||_{0}=||D^{k}\big{(}D_{t}^{\varphi}v_{\tau}-(F_{k}\cdot\nabla^{\varphi})F_{\tau k}+\frac{1}{\partial_{3}\varphi}\partial_{3}q\big{)}||_{0}\leq P\bigg{(}\sum_{l=0}^{3}|\partial_{t}^{l}\overline{\nabla}\psi|_{3-l},\sum_{l=0}^{4}||\partial_{t}^{k}v||_{4-k},\sum_{l=0}^{4}||\partial_{t}^{k}F||_{4-k}\bigg{)}.

(4.8)

4.2 Div-Curl Estimate

We adopt Hodge-type elliptic estimates to study

||\partial_{t}^{k}v||_{4-k}^{2},\qquad||\partial_{t}^{k}F_{j}||_{4-k}^{2}\qquad\text{for }k=0,1,2,3,4.

By lemma (3.4), replacing $\displaystyle X$ by $\partial_{t}^{k}v$ and $\displaystyle\partial_{t}^{k}F_{j}$ and $s$ by $3-k$ , we have

	$\displaystyle\|\|\partial_{t}^{k}v\|\|_{4-k}^{2}$	$\displaystyle\lesssim C\big{(}\|\psi\|_{4-k},\|\overline{\partial}\psi\|_{W^{1,\infty}}\big{)}\big{(}\|\|\nabla^{\varphi}\cdot\partial_{t}^{k}v\|\|_{3-k}^{2}+\|\|\nabla^{\varphi}\times\partial_{t}^{k}v\|\|_{3-k}^{2}+\|\|\overline{\partial}^{4-k}\partial_{t}^{k}v\|\|_{0}^{2}+\|\|v\|\|_{0}^{2}\big{)},$		(4.9)
	$\displaystyle\|\|\partial_{t}^{k}F_{j}\|\|_{4-k}^{2}$	$\displaystyle\lesssim C\big{(}\|\psi\|_{4-k},\|\overline{\partial}\psi\|_{W^{1,\infty}}\big{)}\big{(}\|\|\nabla^{\varphi}\cdot\partial_{t}^{k}F_{j}\|\|_{3-k}^{2}+\|\|\nabla^{\varphi}\times\partial_{t}^{k}F_{j}\|\|_{3-k}^{2}+\|\|\overline{\partial}^{4-k}\partial_{t}^{k}F_{j}\|\|_{0}^{2}+\|\|F_{j}\|\|_{0}^{2}\big{)}.$		(4.10)

4.2.1 $L^{2}$ -estimates

Testing $v\partial_{3}\varphi$ with the first equation of (2.28), we have

\displaystyle\int_{\Omega}(D_{t}^{\varphi}v+\nabla^{\varphi}q)\cdot v\partial_{3}\varphi=\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})F_{k}\cdot v\partial_{3}\varphi.

(4.11)

On the left-hand side, by the transport theorem (3.3) and integration $\nabla^{\varphi}$ by parts (3.2),

\displaystyle\int_{\Omega}(D_{t}^{\varphi}v+\nabla^{\varphi}q)\cdot v\partial_{3}\varphi=\frac{1}{2}\frac{d}{dt}\int_{\Omega}|v|^{2}\partial_{3}\varphi-\int_{\Omega}q(\nabla^{\varphi}\cdot v)\partial_{3}\varphi+\int_{\Sigma}q(v\cdot N),

(4.12)

where the boundary term on $\Sigma_{b}$ vanishes due to the slip boundary condition $\displaystyle v\cdot n=v_{3}=0$ on $\Sigma_{b}$ and the second term $\displaystyle\int_{\Omega}q(\nabla^{\varphi}\cdot v)\partial_{3}\varphi$ vanishes due to the incompressible condition $\nabla^{\varphi}\cdot v=0$ .
Plugging in the kinematic boundary condition and the boundary condition for $q$ , then integrate $\overline{\nabla}$ by parts

$\displaystyle\int_{\Sigma}qv\cdot N$	$\displaystyle=-\sigma\int_{\Sigma}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\sqrt{1+\|\overline{\nabla}\psi\|^{2}}}\partial_{t}\psi$	(4.13)
	$\displaystyle=\sigma\int_{\Sigma}\frac{\overline{\nabla}\psi}{\sqrt{1+\|\overline{\nabla}\psi\|^{2}}}\cdot\overline{\nabla}\partial_{t}\psi$	(4.14)
	$\displaystyle=\sigma\frac{d}{dt}\int_{\Sigma}\sqrt{1+\|\overline{\nabla}\psi\|^{2}}.$	(4.15)

On the right-hand side of (4.11), since $\nabla^{\varphi}\cdot F_{j}=0$ we can integrate $(F_{k}\cdot\nabla^{\varphi})$ by parts to get

\displaystyle\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})F_{k}\cdot v\partial_{3}\varphi

\displaystyle=-\int_{\Omega}F_{k}\cdot(F_{k}\cdot\nabla^{\varphi})v\partial_{3}\varphi,

(4.16)

where the boundary term vanishes since $F_{j}\cdot N=0$ on $\Sigma\cup\Sigma_{b}.$ Since $D_{t}^{\varphi}F_{j}=(F_{j}\cdot\nabla^{\varphi})v$ , it follows from (4.16) that

\displaystyle\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})F_{k}\cdot v\partial_{3}\varphi=-\int_{\Omega}F_{k}\cdot D_{t}^{\varphi}F_{k}\partial_{3}\varphi=-\frac{1}{2}\frac{d}{dt}\sum_{k=1}^{3}\int_{\Omega}|F_{k}|^{2}\partial_{3}\varphi.

(4.17)

Therefore,

\frac{d}{dt}\bigg{(}\frac{1}{2}\sum_{k=1}^{3}\int_{\Omega}|F_{k}|^{2}\partial_{3}\varphi+\frac{1}{2}\int_{\Omega}|v|^{2}\partial_{3}\varphi+\sigma\int_{\Sigma}\sqrt{1+|\overline{\nabla}\psi|^{2}}\bigg{)}=0.

(4.18)

4.2.2 Curl estimates

In this section, we control $\displaystyle||\nabla^{\varphi}\times\partial_{t}^{k}v||_{3-k}^{2}$ and $||\nabla^{\varphi}\times\partial_{t}^{k}F_{j}||_{3-k}^{2}$ . Recall in the system (2.28) that we have

	$\displaystyle D_{t}^{\varphi}v+\nabla^{\varphi}q$	$\displaystyle=(F_{k}\cdot\nabla^{\varphi})F_{k},$		(4.19)
	$\displaystyle D_{t}^{\varphi}F_{j}$	$\displaystyle=(F_{j}\cdot\nabla^{\varphi})v.$		(4.20)

Taking the curl operator, we have

	$\displaystyle D_{t}^{\varphi}(\operatorname{curl}^{\varphi}v)$	$\displaystyle=(F_{k}\cdot\nabla^{\varphi})(\operatorname{curl}^{\varphi}F_{k})+J_{1},$		(4.21)
	$\displaystyle D_{t}^{\varphi}(\operatorname{curl}^{\varphi}F_{j})$	$\displaystyle=(F_{j}\cdot\nabla^{\varphi})(\operatorname{curl}^{\varphi}v)+K_{1},$		(4.22)

where $J_{1}:=[\operatorname{curl}^{\varphi},D_{t}^{\varphi}]v+[\operatorname{curl}^{\varphi},(F_{k}\cdot\nabla^{\varphi})]F_{k}$ and $K_{1}:=[\operatorname{curl}^{\varphi},D_{t}^{\varphi}]F_{j}+[\operatorname{curl}^{\varphi},(F_{j}\cdot\nabla^{\varphi})]v$ . Next taking $\partial^{\alpha}$ , $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ with $\alpha\leq 3$ ,

	$\displaystyle D_{t}^{\varphi}(\partial^{\alpha}\operatorname{curl}^{\varphi}v)$	$\displaystyle=(F_{k}\cdot\nabla^{\varphi})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})+\partial^{\alpha}J_{1}+J_{2},$		(4.23)
	$\displaystyle D_{t}^{\varphi}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{j})$	$\displaystyle=(F_{j}\cdot\nabla^{\varphi})(\partial^{\alpha}\operatorname{curl}^{\varphi}v)+\partial^{\alpha}K_{1}+K_{2},$		(4.24)

where $J_{2}:=[\partial^{\alpha},D^{\varphi}_{t}]\operatorname{curl}^{\varphi}v+[\partial^{\alpha},(F_{k}\cdot\nabla^{\varphi})]\operatorname{curl}^{\varphi}F_{k}$ and $K_{2}=[\partial^{\alpha},D_{t}^{\varphi}]\operatorname{curl}^{\varphi}F_{j}+[\partial^{\alpha},(F_{k}\cdot\nabla^{\varphi})]\operatorname{curl}^{\varphi}v$ .

Now we test (4.23) with $\partial^{\alpha}\operatorname{curl}^{\varphi}v\partial_{3}\varphi$

\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}|\partial^{\alpha}\operatorname{curl}^{\varphi}v|^{2}\partial_{3}\varphi=\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi+\int_{\Omega}(\partial^{\alpha}J_{1}+J_{2})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi.

(4.25)

Then we integrate $(F_{k}\cdot\nabla^{\varphi})$ by parts to get

	$\displaystyle\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi$		(4.26)
	$\displaystyle=-\int_{\Omega}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(F_{k}\cdot\nabla^{\varphi})(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi-\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi$
	$\displaystyle=-\int_{\Omega}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot D_{t}^{\varphi}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{j})\partial_{3}\varphi+\int_{\Omega}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})(\partial^{\alpha}K_{1}+K_{2})\partial_{3}\varphi$
	$\displaystyle\quad-\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi$
	$\displaystyle=-\frac{1}{2}\frac{d}{dt}\int_{\Omega}\|\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k}\|^{2}\partial_{3}\varphi+\int_{\Omega}(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})(\partial^{\alpha}K_{1}+K_{2})\partial_{3}\varphi$
	$\displaystyle\quad-\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi,$		(4.27)

where the boundary term vanishes as $F_{j}\cdot N=0.$ It follows from (4.25) and (4.27) that

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}(\|\partial^{\alpha}\operatorname{curl}^{\varphi}v\|^{2}+\|\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k}\|^{2})\partial_{3}\varphi=$	$\displaystyle\int_{\Omega}(\partial^{\alpha}J_{1}+J_{2})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi$
	$\displaystyle+\int_{\Omega}(\partial^{\alpha}K_{1}+K_{2})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\partial_{3}\varphi$
	$\displaystyle-\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi.$	(4.28)

It remains to control the error terms on the right-hand side of (4.28). We first control $\partial^{\alpha}J_{1}$ . Since

	$\displaystyle\operatorname{curl}^{\varphi}(D_{t}^{\varphi}v)_{i}$	$\displaystyle=\epsilon^{i\alpha\beta}\partial_{\alpha}^{\varphi}(D_{t}^{\varphi})v_{\beta}$
		$\displaystyle=\epsilon^{i\alpha\beta}\partial_{\alpha}^{\varphi}(\partial_{t}^{\varphi}+v_{k}\partial_{k}^{\varphi})v_{\beta}$
		$\displaystyle=\epsilon^{ij\beta}(\partial_{t}^{\varphi}+v_{k}\partial_{k}^{\varphi})\partial_{\alpha}^{\varphi}v_{\beta}+\epsilon^{i\alpha\beta}\partial_{\alpha}^{\varphi}v_{k}\partial_{k}^{\varphi}v_{\beta},$

we have

[\operatorname{curl}^{\varphi},D_{t}^{\varphi}]v=\epsilon^{i\alpha\beta}\partial_{\alpha}^{\varphi}v_{k}\partial_{k}^{\varphi}v_{\beta}.

(4.29)

Also

	$\displaystyle\operatorname{curl}^{\varphi}\big{(}(F_{k}\cdot\nabla^{\varphi})F_{k}\big{)}_{i}$	$\displaystyle=\epsilon^{i\alpha\beta}\partial^{\varphi}_{\alpha}\big{(}(F_{k}\cdot\nabla^{\varphi})F_{\beta k}\big{)}$
		$\displaystyle=\epsilon^{i\alpha\beta}(F_{k}\cdot\nabla^{\varphi})\partial_{\alpha}^{\varphi}F_{\beta k}+\epsilon^{i\alpha\beta}\big{(}(\partial^{\varphi}_{\alpha}F_{k})\cdot\nabla^{\varphi}\big{)}F_{\beta k},$

it follows that

[\operatorname{curl}^{\varphi},(F_{k}\cdot\nabla^{\varphi})]F_{k}=\epsilon^{i\alpha\beta}\big{(}(\partial^{\varphi}_{\alpha}F_{k})\cdot\nabla^{\varphi}\big{)}F_{\beta k}.

(4.30)

Since the terms of (4.29) and (4.30) are up to the first order, they can be easily controlled by the energy even after taking $\partial^{\alpha}$ . For $J_{2}$ , since $D_{t}^{\varphi}$ only contains the first order of $\varphi$ and $\operatorname{curl}^{\varphi}$ has only 1 derivative on $v$ , $[\partial^{\alpha},D_{t}^{\varphi}]\operatorname{curl}^{\varphi}v$ can have at most $4$ derivative on $\varphi$ and $v$ . More precisely, up to lower order terms,

	$\displaystyle\partial^{\alpha}(D_{t}^{\varphi}\operatorname{curl}^{\varphi}v)$	$\displaystyle=\partial^{\alpha}\big{(}(\partial_{t}+\overline{v}\cdot\overline{\partial}+(v\cdot\mathbf{N}-\partial_{t}\varphi)\frac{1}{\partial_{3}\varphi}\partial_{3}\big{)}\operatorname{curl}^{\varphi}v$
		$\displaystyle\stackrel{{\scriptstyle\text{L}}}{{=}}D_{t}^{\varphi}(\partial^{\alpha}\operatorname{curl}^{\varphi}v)+(v\cdot\partial^{\alpha}\mathbf{N}-\partial^{\alpha}\partial_{t}\varphi)\frac{1}{\partial_{3}\varphi}\partial_{3}\operatorname{curl}^{\varphi}v-(v\cdot\mathbf{N}-\partial_{t}\varphi)\frac{\partial^{\alpha}\partial_{3}\varphi}{(\partial_{3}\varphi)^{2}}\partial_{3}\operatorname{curl}^{\varphi}v,$

where $\partial^{\alpha}\mathbf{N}=(-\partial^{\alpha}\partial_{1}\varphi,-\partial^{\alpha}\partial_{2}\varphi,0)$ . The reasoning for $[\partial^{\alpha},(F_{k}\cdot\nabla^{\varphi})]$ is the same. Hence,

\int_{\Omega}(\partial^{\alpha}J_{1}+J_{2})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi\lesssim P(||v||_{4},||F_{j}||_{4},|\psi|_{4},|\partial_{t}\psi|_{3}).

(4.31)

The argument for the control of $\int_{\Omega}(\partial^{\alpha}K_{1}+K_{2})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\partial_{3}\varphi$ is similar by replacing $v$ by $F_{k}$ .

Now, for $\displaystyle\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi$ , again, we can see that the terms of $(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})$ and $(\partial^{\alpha}\operatorname{curl}^{\varphi}v)$ are up to forth order of $F_{k},v$ and $\varphi$ . We can then control the highest order term by $L^{2}$ ,

\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})(\partial^{\alpha}\operatorname{curl}^{\varphi}F_{k})\cdot(\partial^{\alpha}\operatorname{curl}^{\varphi}v)\partial_{3}\varphi\lesssim P(||v||_{4},||F||_{4},|\psi|_{4}).

(4.32)

Let $k=1,2,|\alpha|=3-k$ . The estimate for the general cases for $\displaystyle||\nabla^{\varphi}\times\partial_{t}^{k}v||_{3-k}^{2}$ and $||\nabla^{\varphi}\times\partial_{t}^{k}F_{j}||_{3-k}^{2}$ follows from a parallel argument by taking $\partial^{\alpha}\partial_{t}^{k}$ on $(\ref{curl:0.1})$ and $(\ref{curl:0.1})$ with the number of time derivatives of $\varphi$ up to 4. Consequently,

\frac{1}{2}\frac{d}{dt}\int_{\Omega}(|\partial^{\alpha}\partial_{t}^{k}\operatorname{curl}^{\varphi}v|^{2}+|\partial^{\alpha}\partial_{t}^{k}\operatorname{curl}^{\varphi}F_{k}|^{2})\partial_{3}\varphi\lesssim P(||\partial_{t}^{k}v||_{4-k},||\partial_{t}^{k}F_{j}||_{4-k},|\partial_{t}^{k}\psi|_{4-k}).

(4.33)

4.3 Tangential Estimate

To get the tangential estimate $||\overline{\partial}^{4-k}\partial_{t}^{k}v||_{0}^{2}$ , we have to take $D^{4}=\overline{\partial}^{4-k}\partial_{t}^{k}$ on the first equation of (2.28), the process produces a trouble term $D^{4}\partial_{\tau}^{\varphi}q,\tau=1,2$ . Since $\partial_{\tau}^{\varphi}$ contains $\partial_{\tau}\varphi$ , the commutator $[D^{4},\partial_{\tau}^{\varphi}]q$ contains $D^{4}\partial_{\tau}\varphi$ , a term with 4 derivatives on $\varphi$ , which can be controlled by $|D^{4}\partial_{\tau}\psi|_{0}$ . However, the highest energy term of $\varphi$ from the tangential esimtae is $|\sqrt{\sigma}D^{4}\overline{\nabla}\psi|_{0}$ . It follows that the commutator $[D^{4},\partial_{\tau}^{\varphi}]q$ cannot be controlled $\sigma-$ uniformly.

4.3.1 Alinhac’s Good Unknown

To avoid the appearance of the top order of $\varphi$ , we introduce the Alinhac’s good unknown. Let $D^{\alpha}=\partial_{t}^{\alpha_{0}}\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$ , $|\alpha|\leq 4$ ,

$\displaystyle D^{\alpha}\partial_{\tau}^{\varphi}f$	$\displaystyle=D^{\alpha}(\partial_{\tau}f-\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi}\partial_{3}f)$
	$\displaystyle=\partial^{\varphi}_{\tau}D^{\alpha}f-\frac{\partial_{\tau}D^{\alpha}\varphi}{\partial_{3}\varphi}\partial_{3}f+\frac{\partial_{\tau}\varphi\partial_{3}D^{\alpha}\varphi}{(\partial_{3}\varphi)^{2}}\partial_{3}f+\mathcal{C}_{\tau}^{\prime}(f)$
	$\displaystyle=\partial^{\varphi}_{\tau}D^{\alpha}f-(\partial_{\tau}-\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi}\partial_{3})D^{\alpha}\varphi\partial_{3}^{\varphi}f+\mathcal{C}_{\tau}^{\prime}(f)$
	$\displaystyle=\partial^{\varphi}_{\tau}(D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f)+\mathcal{C}_{\tau}(f),$	(4.34)

where for $\tau=1,2$ and $|\beta|=1$

	$\displaystyle\mathcal{C}_{\tau}^{\prime}(f)$	$\displaystyle=-[D^{\alpha},\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi},\partial_{3}f]-\partial_{3}f[D^{\alpha},\partial_{\tau}\varphi,\frac{1}{\partial_{3}\varphi}]+\partial_{3}f\partial_{\tau}\varphi[D^{\alpha-\beta},\frac{1}{(\partial_{3}\varphi)^{2}}]D^{\beta}\partial_{3}\varphi,$		(4.35)
	$\displaystyle\mathcal{C}_{\tau}(f)$	$\displaystyle=D^{\alpha}\varphi\partial_{\tau}^{\varphi}\partial_{3}^{\varphi}f+\mathcal{C}_{\tau}^{\prime}(f).$		(4.36)

Then by similar calculation, we get

$\displaystyle D^{\alpha}\partial_{3}^{\varphi}f$	$\displaystyle=D^{\alpha}(\frac{1}{\partial_{3}\varphi}\partial_{3}f)$
	$\displaystyle=\partial_{3}^{\varphi}D^{\alpha}f-\frac{\partial_{3}D^{\alpha}\varphi}{(\partial_{3}\varphi)^{2}}\partial_{3}f+\mathcal{C}_{3}^{\prime}(f)$
	$\displaystyle=\partial_{3}^{\varphi}D^{\alpha}f-\partial_{3}^{\varphi}D^{\alpha}\varphi\partial_{3}^{\varphi}f+\mathcal{C}_{3}^{\prime}(f)$
	$\displaystyle=\partial_{3}^{\varphi}(D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f)+D^{\alpha}\varphi(\partial_{3}^{\varphi})^{2}f+\mathcal{C}_{3}^{\prime}(f)$
	$\displaystyle=\partial_{3}^{\varphi}(D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f)+\mathcal{C}_{3}(f),$	(4.37)

where

	$\displaystyle\mathcal{C}_{3}^{\prime}(f)$	$\displaystyle=[D^{\alpha},\frac{1}{\partial_{3}\varphi},\partial_{3}f]-\partial_{3}f[D^{\alpha-\beta},\frac{1}{(\partial_{3}\varphi)^{2}}]D^{\beta}\partial_{3}\varphi,$		(4.38)
	$\displaystyle\mathcal{C}_{3}(f)$	$\displaystyle=D^{\alpha}\varphi(\partial_{3}^{\varphi})^{2}f+\mathcal{C}_{3}^{\prime}(f).$		(4.39)

Next, we study

$\displaystyle D^{\alpha}D_{t}^{\varphi}f$	$\displaystyle=D^{\alpha}(\partial_{t}+\overline{v}\cdot\overline{\nabla}+\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3})f$
	$\displaystyle=D_{t}^{\varphi}D^{\alpha}f-\frac{\partial_{3}D^{\alpha}\varphi}{(\partial_{3}\varphi)^{2}}(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}f+\frac{1}{\partial_{3}\varphi}(v\cdot D^{\alpha}\mathbf{N}-\partial_{t}D^{\alpha}\varphi)\partial_{3}f+\mathcal{D}^{\prime}(f)$
	$\displaystyle=D_{t}^{\varphi}D^{\alpha}f-\big{[}\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}D^{\alpha}\varphi+(\overline{v}\cdot\overline{\nabla}D^{\alpha}\varphi+\partial_{t}D^{\alpha}\varphi)\big{]}\partial_{3}^{\varphi}f+\mathcal{D}^{\prime}(f)$
	$\displaystyle=D_{t}^{\varphi}D^{\alpha}f-D_{t}^{\varphi}D^{\alpha}\varphi\partial_{3}^{\varphi}f+\mathcal{D}^{\prime}(f)$
	$\displaystyle=D_{t}^{\varphi}(D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f)+D^{\alpha}\varphi D_{t}^{\varphi}\partial_{3}^{\varphi}f+\mathcal{D}^{\prime}(f)$
	$\displaystyle=D_{t}^{\varphi}(D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f)+\mathcal{D}(f),$	(4.40)

where

$\displaystyle\mathcal{D}^{\prime}(f)$	$\displaystyle=[D^{\alpha},\overline{v}]\cdot\overline{\nabla}f+[D^{\alpha},\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi),\partial_{3}f]+[D^{\alpha},\frac{1}{\partial_{3}\varphi},v\cdot\mathbf{N}-\partial_{t}\varphi]\partial_{3}f$
	$\displaystyle\quad-(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}f[D^{\alpha-\beta},\frac{1}{(\partial_{3}\varphi)^{2}}]D^{\beta}\partial_{3}\varphi+\frac{1}{\partial_{3}\varphi}\partial_{3}f[D^{\alpha},v]\mathbf{N},$	(4.41)
$\displaystyle\mathcal{D}(f)$	$\displaystyle=D^{\alpha}\varphi D_{t}^{\varphi}\partial_{3}^{\varphi}f+\mathcal{D}^{\prime}(f).$	(4.42)

The quantity $\displaystyle\mathbf{F}:=D^{\alpha}f-D^{\alpha}\varphi\partial_{3}^{\varphi}f$ is called the Alinhac’s Good Unknown of $f$ . It follows that the control of $||F||_{0}$ yields the control for $||D^{\alpha}f||_{0}$ , more precisely

\displaystyle||D^{\alpha}f||_{0}\leq||F||_{0}+||\partial_{3}^{\varphi}f||_{0}||D^{\alpha}\varphi||_{0}.

(4.43)

4.3.2 Reformulation in Alinhac’s Good Unknown

Let

\displaystyle\mathbf{V}=D^{\alpha}v-D^{\alpha}\varphi\partial_{3}^{\varphi}v,\quad\mathbf{F}=D^{\alpha}F-D^{\alpha}\varphi\partial_{3}^{\varphi}F,\quad\mathbf{Q}=D^{\alpha}q-D^{\alpha}\varphi\partial_{3}^{\varphi}q.

(4.44)

Taking $D^{\alpha}$ on the first equation of (2.28), then by (4.36), (4.37) and (4.40),

$\displaystyle D_{t}^{\varphi}\mathbf{V}_{i}+\partial_{i}^{\varphi}\mathbf{Q}$	$\displaystyle=D^{\alpha}\big{(}(F_{k}\cdot\nabla^{\varphi})F_{ik}\big{)}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q)$
	$\displaystyle=F_{lk}D^{\alpha}\partial_{l}^{\varphi}F_{ik}+[D^{\alpha},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q)$
	$\displaystyle=(F_{k}\cdot\nabla^{\varphi})\mathbf{F}_{ik}+F_{lk}C_{l}(F_{ik})+[D^{\alpha},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q)$
	$\displaystyle=(F_{k}\cdot\nabla^{\varphi})\mathbf{F}_{ik}+\mathcal{R}_{i}^{1},$	(4.45)

where

\displaystyle\mathcal{R}_{i}^{1}=F_{lk}\mathcal{C}_{l}(F_{ik})+[D^{\alpha},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q).

(4.46)

Similarly, we have

	$\displaystyle D_{t}^{\varphi}\mathbf{F}_{ij}$	$\displaystyle=(F_{j}\cdot\nabla^{\varphi})\mathbf{V}_{i}+\mathcal{R}_{ij}^{2},$		(4.47)
	$\displaystyle\partial_{k}^{\varphi}\mathbf{V}_{k}$	$\displaystyle=-\mathcal{C}_{k}(v_{k}),$		(4.48)

where

\displaystyle\mathcal{R}_{ij}^{2}=F_{kj}\mathcal{C}_{k}(v_{i})+[D^{\alpha},F_{kj}]\partial_{k}v_{i}-\mathcal{D}(F_{ij}).

(4.49)

Next, we reformulate the boundary conditions. By definition of $\mathbf{Q}$ ,

	$\displaystyle\mathbf{Q}\|_{\Sigma}$	$\displaystyle=(D^{\alpha}q-D^{\alpha}\varphi\partial_{3}^{\varphi}q)\|_{\Sigma}$
		$\displaystyle=-\sigma D^{\alpha}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}-D^{\alpha}\psi\partial_{3}q\qquad\text{on }\Sigma.$		(4.50)

For the kinematic boundary condition, on $\Sigma$ ,

$\displaystyle\partial_{t}D^{\alpha}\psi$	$\displaystyle=D^{\alpha}(v\cdot N)$
	$\displaystyle=D^{\alpha}(-\overline{v}\cdot\overline{\partial}\psi+v_{3})$
	$\displaystyle=-D^{\alpha}\overline{v}\cdot\overline{\partial}\psi-\overline{v}\cdot D^{\alpha}\overline{\partial}\psi+D^{\alpha}v_{3}-\sum_{\begin{subarray}{c}\|\beta_{1}\|+\|\beta_{2}\|=3\\ \|\beta_{1}\|,\|\beta_{2}\|>0\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi$
	$\displaystyle=D^{\alpha}v\cdot N-\overline{v}\cdot D^{\alpha}\overline{\partial}\psi-\sum_{\begin{subarray}{c}\|\beta_{1}\|+\|\beta_{2}\|=3\\ \|\beta_{1}\|,\|\beta_{2}\|>0\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi$
	$\displaystyle=\mathbf{V}\cdot N-\overline{v}\cdot D^{\alpha}\overline{\partial}\psi+\mathcal{S}_{1},$	(4.51)

where

\displaystyle\mathcal{S}_{1}:=D^{\alpha}\psi\partial_{3}v\cdot N-\sum_{\begin{subarray}{c}|\beta_{1}|+|\beta_{2}|=4\\ |\beta_{1}|,|\beta_{2}|>0\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi.

(4.52)

For the slip boundary condition, since $D^{\alpha}\varphi|_{\Sigma_{b}}=0$ and $v_{3}|_{\Sigma_{b}}=0$ on $\Sigma_{b}$ ,

\displaystyle\mathbf{V}_{3}=\mathbf{V}\cdot n=D^{\alpha}v\cdot n=0.

(4.53)

4.3.3 Energy estimate with full spatial derivatives

In this section, we study the equations with full spacial derivative, more precisely, $D^{\alpha}=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$ .

Theorem 4.1.

There exists $T>0$ so that

\displaystyle||D^{\alpha}v_{i}||_{0}^{2}+||D^{\alpha}F_{ik}||_{0}^{2}+|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.54)

Furthermore, if the Rayleigh-Taylor sign condition $\-\partial_{3}q\geq c_{0}>0$ is assumed on $\Sigma$ , the term $|D^{\alpha}\psi|_{0}^{2}$ enters the energy and the estimate become $\sigma$ -uniform

\displaystyle||D^{\alpha}v_{i}(T)||_{0}^{2}+||D^{\alpha}F_{ik}||_{0}^{2}+|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2}+|D^{\alpha}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.55)

Testing (4.45) with $\mathbf{V}_{i}\partial_{3}\varphi$ , we have

	$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}\|\mathbf{V}_{i}\|^{2}\partial_{3}\varphi$	$\displaystyle=\int_{\Omega}\mathbf{V}_{i}D_{t}^{\varphi}\mathbf{V}_{i}\partial_{3}\varphi$
		$\displaystyle=-\int_{\Omega}\partial_{i}^{\varphi}\mathbf{Q}\mathbf{V}_{i}\partial_{3}\varphi+\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})\mathbf{F}_{ik}\mathbf{V}_{i}\partial_{3}\varphi+\int_{\Omega}\mathcal{R}_{i}^{1}\mathbf{V}_{i}\partial_{3}\varphi.$		(4.56)

Integrating $\partial_{i}^{\varphi}$ by parts, then by $\mathbf{V}\cdot n=0$ on $\Sigma_{b}$ and (4.48),

	$\displaystyle-\int_{\Omega}\partial_{i}^{\varphi}\mathbf{Q}\mathbf{V}_{i}\partial_{3}\varphi$	$\displaystyle=\int_{\Omega}\mathbf{Q}\partial_{i}^{\varphi}\mathbf{V}_{i}\partial_{3}\varphi-\int_{\Sigma}\mathbf{Q}\mathbf{V}\cdot N$
		$\displaystyle=-\int_{\Omega}\mathcal{C}_{i}(v_{i})\mathbf{Q}\partial_{3}\varphi-\int_{\Sigma}\mathbf{Q}\mathbf{V}\cdot N.$		(4.57)

Next, integrating $\nabla^{\varphi}$ by parts, then by (4.47) and $\nabla^{\varphi}\cdot F_{k}=0$ ,

$\displaystyle\int_{\Omega}(F_{k}\cdot\nabla^{\varphi})\mathbf{F}_{ik}\mathbf{V}_{i}\partial_{3}\varphi$	$\displaystyle=-\int_{\Omega}\mathbf{F}_{ik}(F_{k}\cdot\nabla^{\varphi})\mathbf{V}_{i}\partial_{3}\varphi-\int_{\Omega}(\nabla^{\varphi}\cdot F_{k})\mathbf{F}_{ik}\mathbf{V}_{i}\partial_{3}\varphi$
	$\displaystyle=-\int_{\Omega}\mathbf{F}_{ik}D_{t}^{\varphi}\mathbf{F}_{ik}\partial_{3}\varphi+\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}^{2}_{ik}\partial_{3}\varphi$
	$\displaystyle=-\frac{1}{2}\frac{d}{dt}\int_{\Omega}\|\mathbf{F}_{ik}\|^{2}\partial_{3}\varphi+\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}^{2}_{ik}\partial_{3}\varphi,$	(4.58)

where the boundary term vanishes due to $F_{k}\cdot N=0$ on $\Sigma\cup\Sigma_{b}$ . Summing up (4.56), (4.57) and (4.58), we have

\displaystyle\frac{1}{2}\frac{d}{dt}(\int_{\Omega}|\mathbf{V}_{i}|^{2}\partial_{3}\varphi+\int_{\Omega}|\mathbf{F}_{ik}|^{2}\partial_{3}\varphi)=-\int_{\Sigma}\mathbf{Q}\mathbf{V}\cdot N-\int_{\Omega}\mathcal{C}_{i}(v_{i})\mathbf{Q}\partial_{3}\varphi+\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}^{2}_{ik}\partial_{3}\varphi+\int_{\Omega}\mathcal{R}_{i}^{1}\mathbf{V}_{i}\partial_{3}\varphi.

(4.59)

Control of the error terms
Let $\partial_{3}\varphi\geq c_{0}>0$ and $D^{\alpha}=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}$ . It directly follows from the definition of $\mathcal{C}_{i}(f)$ and $\mathcal{D}(f)$ that

	$\displaystyle\|\|\mathcal{C}_{i}(f)\|\|_{0}$	$\displaystyle\leq P(c_{0}^{-1},\|\psi\|_{4})\|\|f\|\|_{4},$		(4.60)
	$\displaystyle\|\|\mathcal{D}(f)\|\|_{0}$	$\displaystyle\leq P(c_{0}^{-1},\|\|v\|\|_{4},\|\psi\|_{4},\|\partial_{t}\psi\|_{3})(\|\|f\|\|_{4}+\|\|\partial_{t}f\|\|_{2}).$		(4.61)

Then

\displaystyle-\int_{\Omega}\mathcal{C}_{i}(v_{i})\mathbf{Q}\partial_{3}\varphi\leq||\mathcal{C}_{i}(v_{i})||_{0}||\mathbf{Q}||_{0}||\partial_{3}\varphi||_{\infty},

(4.62)

where

\displaystyle||\mathbf{Q}||_{0}\leq||D^{\alpha}q||_{0}+||D^{\alpha}\varphi||_{0}||\partial_{3}^{\varphi}q||_{\infty}.

(4.63)

Next,

\displaystyle\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}_{ik}^{2}\partial_{3}\varphi\leq||\partial_{3}\varphi||_{\infty}||\mathbf{F}||_{0}||\mathcal{R}^{2}_{ik}||_{0},

(4.64)

where

	$\displaystyle\|\|\mathcal{R}_{ik}^{2}\|\|_{0}$	$\displaystyle=\|\|F_{kj}\mathcal{C}_{k}(v_{i})+[D^{\alpha},F_{kj}]\partial_{k}v_{i}-\mathcal{D}(F_{ij})\|\|_{0}$		(4.65)
		$\displaystyle\lesssim\|\|F_{kj}\|\|_{\infty}\|\|\mathcal{C}_{k}(v_{i})\|\|_{0}+\|\|F_{kj}\|\|_{4}\|\|v\|\|_{4}+\|\|\mathcal{D}(F_{ij})\|\|_{0}.$		(4.66)

Similarly,

\displaystyle\int_{\Omega}\mathcal{R}_{i}^{1}\mathbf{V}_{i}\partial_{3}\varphi\leq||\mathcal{R}_{i}^{1}||_{0}||\mathbf{V}_{i}||_{0}||\partial_{3}\varphi||_{\infty},

(4.67)

where

	$\displaystyle\|\|\mathcal{R}_{i}^{1}\|\|_{0}$	$\displaystyle=\|\|F_{lk}\mathcal{C}_{l}(F_{ik})+[D^{\alpha},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q)\|\|_{0}$		(4.68)
		$\displaystyle\lesssim\|\|F_{ik}\|\|_{\infty}\|\|\mathcal{C}_{l}(F_{ik})\|\|_{0}+\|\|F_{lk}\|\|_{4}^{2}+\|\|\mathcal{D}(v_{i})\|\|_{0}+\|\|\mathcal{C}_{i}(q)\|\|_{0}.$		(4.69)

Control of $-\int_{\Sigma}\mathbf{Q}(\mathbf{V}\cdot N)$
Plugging in the higher order kinematic boundary condition (4.51), we have

\displaystyle-\int_{\Sigma}\mathbf{Q}(\mathbf{V}\cdot N)=-\int_{\Sigma}\mathbf{Q}(\partial_{t}^{\varphi}D^{\alpha}\psi+\overline{v}\cdot D^{\alpha}\overline{\partial}\psi-\mathcal{S}_{1}).

(4.70)

For the first term, invoking the boundary condition for pressure (4.50),

$\displaystyle I:$	$\displaystyle=-\int_{\Sigma}\mathbf{Q}\partial_{t}^{\varphi}D^{\alpha}\psi$
	$\displaystyle=\int_{\Sigma}\sigma D^{\alpha}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\partial_{t}^{\varphi}D^{\alpha}\psi+\partial_{3}qD^{\alpha}\psi\partial_{t}^{\varphi}D^{\alpha}\psi$
	$\displaystyle=:ST+RT.$	(4.71)

We first deal with the term contributed by the surface tension. Let $|\beta|=1$ , integrating $\overline{\nabla}$ by parts,

$\displaystyle ST$	$\displaystyle=-\int_{\Sigma}\sigma D^{\alpha}\big{(}\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi$
	$\displaystyle=-\int_{\Sigma}\sigma\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi$
	$\displaystyle\quad-\int_{\Sigma}\sigma\big{(}[D^{\alpha},\overline{\nabla}\psi,\frac{1}{\|N\|}]-\overline{\nabla}\psi[D^{\alpha-\beta},\frac{\overline{\nabla}\psi}{\|N\|^{3}}]\cdot D^{\beta}\overline{\nabla}\psi\big{)}\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi$
	$\displaystyle=:ST_{1}+ST^{R}_{2}.$	(4.72)

Observing the symmetry in the first term, we expect it to contribute to energy terms,

$\displaystyle ST_{1}$	$\displaystyle=-\frac{\sigma}{2}\frac{d}{dt}\int_{\Sigma}\big{(}\frac{\|D^{\alpha}\overline{\nabla}\psi\|^{2}}{\|N\|}-\frac{\|\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi\|^{2}}{\|N\|^{3}}\big{)}$
	$\displaystyle\quad+\frac{\sigma}{2}\int_{\Sigma}\partial_{t}(\frac{1}{\|N\|})\|D^{\alpha}\overline{\nabla}\psi\|^{2}-\partial_{t}(\frac{1}{\|N\|^{3}})\|\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi\|^{2}$
	$\displaystyle=:ST_{11}+ST_{12}^{R}.$	(4.73)

We do the following calculation to check that $ST_{11}$ indeed give rise to the energy of surface tension, we do the following calculation.

\displaystyle\frac{|D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|}-\frac{|\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|^{3}}\geq\frac{|D^{\alpha}\overline{\nabla}\psi|^{2}(1+|\overline{\nabla}\psi|^{2})-|\overline{\nabla}\psi|^{2}|D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|^{3}}=\frac{|D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|^{3}}.

(4.74)

Since $|N|$ is bounded in short time, it follows that

\displaystyle\frac{\sigma}{2}\int_{\Sigma}|D^{\alpha}\overline{\nabla}\psi|^{2}\leq\frac{\sigma}{2}\int_{\Sigma}\big{(}\frac{|D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|}-\frac{|\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|^{3}}\big{)}.

(4.75)

Now, it remains to control $ST_{2}^{R}$ and $ST_{12}^{R}$ . For $ST_{12}^{R}$ , observe that $\partial_{t}(\frac{1}{|N|}),\partial_{t}(\frac{1}{|N|^{3}})$ contributes to $|\partial_{t}\overline{\nabla}\psi|_{\infty}$ . Although $|D^{\alpha}\overline{\nabla}\psi|_{0}^{2}$ contributes to the top order of $\psi$ , it is attached with $\sigma$ . Hence,

\displaystyle ST_{12}^{R}\leq P(|\overline{\nabla}\psi|_{\infty})|\overline{\nabla}\partial_{t}\psi|_{\infty}|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2}.

(4.76)

For $ST_{2}^{R}$ , we first deal with

\displaystyle ST_{21}^{R}:=-\int_{\Sigma}\sigma[D^{\alpha},\overline{\nabla}\psi,\frac{1}{|N|}]\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi.

(4.77)

Let $|\gamma|=|\zeta|=1$ . Notice that in the commutator $[D^{\alpha},\overline{\nabla}\psi,\frac{1}{|N|}]$ , the top order terms appears when $D^{\alpha-\gamma}$ lands on $\overline{\nabla}\psi$ or $\frac{1}{|N|}$ , more precisely, $ST_{21}^{R}$ contributes to

$\displaystyle ST_{211}^{R}$	$\displaystyle:=-\sigma\int_{\Sigma}D^{\alpha-\gamma}\overline{\nabla}\psi D^{\gamma}(\frac{1}{\|N\|})\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi,$	(4.78)
$\displaystyle ST_{212}^{R}$	$\displaystyle:=-\sigma\int_{\Sigma}D^{\alpha-\gamma-\zeta}\overline{\nabla}\psi D^{\gamma+\zeta}(\frac{1}{\|N\|})\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi,$	(4.79)
$\displaystyle ST_{213}^{R}$	$\displaystyle:=-\sigma\int_{\Sigma}D^{\gamma}\overline{\nabla}\psi D^{\alpha-\gamma}(\frac{1}{\|N\|})\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi.$	(4.80)

The first term and the last term have similar structures, the second term is even easier as it only contains the lower order term of the commutator, so we only deal with $R_{211}^{R}$ . Integrating $\overline{\nabla}$ by parts,

$\displaystyle ST_{211}^{R}$	$\displaystyle=\sigma\int_{\Sigma}\overline{\nabla}\cdot(D^{\alpha-\gamma}\overline{\nabla}\psi D^{\gamma}(\frac{1}{\|N\|})\big{)}\partial_{t}^{\varphi}D^{\alpha}\psi$
	$\displaystyle=\sigma\int_{\Sigma}(D^{\alpha-\gamma}\overline{\nabla}\cdot\overline{\nabla}\psi)D^{\gamma}(\frac{1}{\|N\|})\partial_{t}^{\varphi}D^{\alpha}\psi+\sigma\int_{\Sigma}\big{(}D^{\alpha-\gamma}\overline{\nabla}\psi\cdot\overline{\nabla}D^{\gamma}(\frac{1}{\|N\|})\big{)}\partial_{t}^{\varphi}D^{\alpha}\psi$
	$\displaystyle\lesssim\|\psi\|_{4}\|\sqrt{\sigma}\overline{\nabla}D^{\alpha}\psi\|_{0}\|\sqrt{\sigma}\partial_{t}^{\varphi}D^{\alpha}\psi\|_{0}.$	(4.81)

The control of $ST_{22}^{R}:=\int_{\Sigma}\sigma\big{(}[D^{\alpha-\beta},\frac{\overline{\nabla}\psi}{|N|^{3}}]\cdot D^{\beta}\overline{\nabla}\psi\big{)}\big{(}\overline{\nabla}\psi\cdot\partial_{t}^{\varphi}\overline{\nabla}D^{\alpha}\psi\big{)}$ is similar by noticing that the highest order term of the commutator occurs when $D^{\alpha-\beta}$ lands on $\frac{\overline{\nabla}\psi}{|N|^{3}}$ or when $D^{\alpha-\beta-\gamma}$ lands on $D^{\beta}\overline{\nabla}\psi$ ,

\displaystyle ST_{22}^{R}\lesssim P(|\psi|_{4})|\sqrt{\sigma}\overline{\nabla}D^{\alpha}\psi|_{0}|\sqrt{\sigma}\partial_{t}^{\varphi}D^{\alpha}\psi|_{0}.

(4.82)

By (4.72), (4.73), (4.76), (4.81), (4.82), we have

\displaystyle ST+\frac{\sigma}{2}\frac{d}{dt}\int_{\Sigma}\big{(}\frac{|D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|}-\frac{|\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi|^{2}}{|N|^{3}}\big{)}\lesssim P(E(t)).

(4.83)

It follows from (4.75) that

\displaystyle\int_{0}^{T}ST+\frac{\sigma}{2}\int_{\Sigma}|D^{\alpha}\overline{\nabla}\psi|^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.84)

Next, we deal with the term $RT=\int_{\Sigma}\partial_{3}qD^{\alpha}\psi\partial_{t}^{\varphi}D^{\alpha}\psi$ . If we assume the Rayleigh-Taylor sign condition $-\partial_{3}q\geq c_{0}>0$ on $\Sigma$ , then the term $\int_{\Sigma}|D^{\alpha}\psi|^{2}$ enters the energy. More precisely, by observing the symmetry, we have

	$\displaystyle RT$	$\displaystyle=-\int_{\Sigma}(-\partial_{3}q)D^{\alpha}\psi\partial_{t}^{\varphi}D^{\alpha}\psi$
		$\displaystyle=-\frac{d}{dt}\int_{\Sigma}(-\partial_{3}q)\|D^{\alpha}\psi\|^{2}+\int_{\Sigma}\partial_{3}\partial_{t}q\|D^{\alpha}\psi\|^{2},$		(4.85)

where

\displaystyle\int_{\Sigma}\partial_{3}\partial_{t}q|D^{\alpha}\psi|^{2}\leq|\partial_{3}\partial_{t}q|_{\infty}|D^{\alpha}\psi|^{2}_{0}.

(4.86)

However, if we drop the Rayleigh-Taylor sign condition, we can only control the term depending on $\sigma$ ,

\displaystyle RT

\displaystyle=\int_{\Sigma}(\partial_{3}q)D^{\alpha}\psi\partial_{t}^{\varphi}D^{\alpha}\psi\leq|\partial_{3}q|_{\infty}|D^{\alpha}\psi|_{0}|\partial_{t}D^{\alpha}\psi|_{0},

(4.87)

where

\displaystyle|\partial_{t}D^{\alpha}\psi|_{0}\leq\sigma^{-\frac{1}{2}}\sqrt{E(t)}.

(4.88)

For the second term,

$\displaystyle II:$	$\displaystyle=-\int_{\Sigma}\mathbf{Q}(\overline{v}\cdot D^{\alpha}\overline{\partial}\psi)$
	$\displaystyle=\int_{\Sigma}\sigma D^{\alpha}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}(\overline{v}\cdot D^{\alpha}\overline{\partial}\psi)+\int_{\Sigma}\partial_{3}qD^{\alpha}\psi(\overline{v}\cdot D^{\alpha}\overline{\partial}\psi)$
	$\displaystyle=:II_{1}+II_{2}.$	(4.89)

For $II_{2}$ , it is not hard to see the symmetry. Integrating $\overline{\partial}$ by parts,

	$\displaystyle II_{2}$	$\displaystyle=-\int_{\Sigma}\partial_{3}q(\overline{v}\cdot D^{\alpha}\overline{\partial}\psi)D^{\alpha}\overline{\partial}\psi-\int_{\Sigma}\overline{\partial}\cdot(\partial_{3}q\overline{v})\|D^{\alpha}\psi\|^{2}$
		$\displaystyle=-II_{2}-\int_{\Sigma}\overline{\partial}\cdot(\partial_{3}q\overline{v})\|D^{\alpha}\psi\|^{2}.$		(4.90)

It follows that

\displaystyle II_{2}=-\frac{1}{2}\int_{\Sigma}\overline{\partial}\cdot(\partial_{3}q\overline{v})|D^{\alpha}\psi|^{2}\leq P(||p||_{4},||v||_{3})|D^{\alpha}\psi|_{0}^{2}.

(4.91)

For $II_{1}$ , there is a symmetric structure of $D^{\alpha}\overline{\partial}\psi$ . Let $|\beta|=1$ , we expand $D^{\alpha}(\frac{\overline{\nabla}\psi}{|N|})$

$\displaystyle II_{1}$	$\displaystyle=\sigma\int_{\Sigma}\overline{\nabla}\cdot\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}(\overline{v}\cdot\overline{\nabla})D^{\alpha}\psi$
	$\displaystyle\quad+\sigma\int_{\Sigma}\overline{\nabla}\cdot\big{(}[D^{\alpha},\overline{\nabla}\psi,\frac{1}{\|N\|}]-\overline{\nabla}\psi[D^{\alpha-\beta},\frac{\overline{\nabla}\psi}{\|N\|^{3}}]\cdot D^{\beta}\overline{\nabla}\psi\big{)}(\overline{v}\cdot\overline{\nabla})D^{\alpha}\psi$
	$\displaystyle=:II_{11}+II_{12}^{R}.$	(4.92)

The commutator term $II_{12}^{R}$ can be directly controlled similar to $ST_{2}^{R}$ , we have

\displaystyle II_{12}^{R}\lesssim P(|\psi|_{4})|v|_{0}|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|^{2}_{0}.

(4.93)

For $II_{11}$ , we first integrate $\overline{\nabla}$ by parts,

$\displaystyle II_{11}$	$\displaystyle=-\sigma\int_{\Sigma}\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}\cdot(\overline{v}\cdot\overline{\nabla})D^{\alpha}\overline{\nabla}\psi$
	$\displaystyle\quad-\sigma\int_{\Sigma}\bigg{(}\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}\cdot\overline{\nabla}\bigg{)}\overline{v}\cdot\overline{\nabla}D^{\alpha}\psi$
	$\displaystyle=:II_{111}+II_{112}^{R},$	(4.94)

where $II_{112}^{R}$ can be directly controlled,

\displaystyle II_{112}^{R}=-\sigma\int_{\Sigma}\bigg{(}\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{|N|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{|N|^{3}}\overline{\nabla}\psi\big{)}\cdot\overline{\nabla}\bigg{)}\overline{v}\cdot\overline{\nabla}D^{\alpha}\psi\lesssim P(|\psi|_{3})|\overline{\nabla}\overline{v}|_{\infty}|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2},

(4.95)

then we integrate $(\overline{v}\cdot\overline{\nabla})$ in $II_{111}$ by parts to get symmetric terms,

$\displaystyle II_{111}$	$\displaystyle=\sigma\int_{\Sigma}\big{(}\frac{(\overline{v}\cdot\overline{\nabla})D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot(\overline{v}\cdot\overline{\nabla})D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}\cdot D^{\alpha}\overline{\nabla}\psi$
	$\displaystyle\quad+\sigma\int_{\Sigma}\big{(}\frac{D^{\alpha}\overline{\nabla}\psi}{\|N\|}-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi\big{)}\cdot(\overline{\nabla}\cdot\overline{v})D^{\alpha}\overline{\nabla}\psi+\sigma\int_{\Sigma}\|D^{\alpha}\overline{\nabla}\psi\|^{2}(\overline{v}\cdot\overline{\nabla})(\frac{1}{\|N\|})$
	$\displaystyle\quad-\int_{\Sigma}\bigg{(}\frac{(\overline{v}\cdot\overline{\nabla})\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}\overline{\nabla}\psi-\frac{\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi}{\|N\|^{3}}(\overline{v}\cdot\overline{\nabla})\overline{\nabla}\psi-(\overline{\nabla}\psi\cdot D^{\alpha}\overline{\nabla}\psi)(\overline{\nabla}\psi\big{)}(\overline{v}\cdot\overline{\nabla})(\frac{1}{\|N\|^{3}})\bigg{)}\cdot D^{\alpha}\overline{\nabla}\psi$
	$\displaystyle=-II_{111}+II_{1112}^{R}+II_{1113}^{R}+II_{1114}^{R}.$	(4.96)

Hence,

\displaystyle II_{111}=\frac{1}{2}(II_{1112}^{R}+II_{1113}^{R}+II_{1114}^{R})\lesssim P(|\psi|_{4},||v||_{3})|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2},

(4.97)

which closes the control of $II$ . By (4.89), (4.92), (4.91), (4.92), (4.93), (4.94), (4.95), (4.97), we have

\displaystyle\int_{0}^{T}II\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.98)

It remains to control the last term

$\displaystyle III:$	$\displaystyle=\int_{\Sigma}\mathbf{Q}\mathcal{S}_{1}$
	$\displaystyle=\int_{\Sigma}D^{\alpha}q\mathcal{S}_{1}-\int_{\Sigma}D^{\alpha}\psi\partial_{3}q\mathcal{S}_{1}$
	$\displaystyle=:III_{1}+III_{2}.$	(4.99)

Since $\mathcal{S}_{1}$ consists of only lower order terms, we have

\displaystyle III_{2}\leq|\partial_{3}q|_{\infty}|D^{\alpha}\psi|_{0}|\mathcal{S}_{1}|_{0}\lesssim P(|\psi|_{4},||v||_{4})|\partial_{3}q|_{\infty}|D^{\alpha}\psi|_{0}.

(4.100)

For $III_{1}$ , since $D^{\alpha}q$ has 4 derivatives on the boundary, we have to remove at least half of the derivative of $q$ to close the estimate. Let $|\beta_{1}^{\prime}|=3,|\beta_{2}^{\prime}|=1$ ,

	$\displaystyle III_{1}$	$\displaystyle=\int_{\Sigma}D^{\alpha}qD^{\beta_{1}^{\prime}}\overline{v}\cdot D^{\beta_{2}^{\prime}}\overline{\partial}\psi+\int_{\Sigma}D^{\alpha}q\bigg{(}D^{\alpha}\psi\partial_{3}v\cdot N-\sum_{\begin{subarray}{c}\|\beta_{1}\|+\|\beta_{2}\|=4\\ \beta_{1}>0,\beta_{2}>0\\ \beta_{1}\neq 3,\beta_{2}\neq 1\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi\bigg{)}$
		$\displaystyle=III_{11}+III_{12}.$		(4.101)

For $III_{12}$ , we plug in the boundary condition of $q$ , then integrate $\overline{\nabla}$ by parts to get

$\displaystyle III_{12}$	$\displaystyle=-\sigma\int_{\Sigma}D^{\alpha}(\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|})\bigg{(}D^{\alpha}\psi\partial_{3}v\cdot N-\sum_{\begin{subarray}{c}\|\beta_{1}\|+\|\beta_{2}\|=4\\ \beta_{1}>0,\beta_{2}>0\\ \beta_{1}\neq 3,\beta_{2}\neq 1\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi\bigg{)}$
	$\displaystyle=\sigma\int_{\Sigma}D^{\alpha}(\frac{\overline{\nabla}\psi}{\|N\|})\cdot\overline{\nabla}\bigg{(}D^{\alpha}\psi\partial_{3}v\cdot N-\sum_{\begin{subarray}{c}\|\beta_{1}\|+\|\beta_{2}\|=4\\ \beta_{1}>0,\beta_{2}>0\\ \beta_{1}\neq 3,\beta_{2}\neq 1\end{subarray}}D^{\beta_{1}}\overline{v}\cdot D^{\beta_{2}}\overline{\partial}\psi\bigg{)}$
	$\displaystyle\lesssim P(\|\psi\|_{4},\|\|v\|\|_{4},\|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi\|_{0}).$	(4.102)

$III_{11}$ can be directly controlled by

\displaystyle III_{11}\lesssim|\psi|_{4}|D^{\alpha}q|_{-\frac{1}{2}}|D^{\beta_{1}^{\prime}}v|_{\frac{1}{2}}\lesssim|\psi|_{4}||q||_{4}||v||_{4}.

(4.103)

Combining (4.59), (4.84), (4.85), (4.86), (4.87), (4.88), (4.98), (4.99), (4.100), (4.101),(4.103), (4.102), we have

\displaystyle||\mathbf{V}_{i}||_{0}^{2}+||\mathbf{F}_{ik}||_{0}^{2}+|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.104)

Furthermore, if the Rayleigh-Taylor sign condition is assumed, the term $|D^{\alpha}\psi|_{0}^{2}$ enters the energy and we have the $\sigma-$ uniform estimate

\displaystyle||\mathbf{V}_{i}||_{0}^{2}+||\mathbf{F}_{ik}||_{0}^{2}+|\sqrt{\sigma}D^{\alpha}\overline{\nabla}\psi|_{0}^{2}+|D^{\alpha}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.105)

By (4.43), we can replace $\mathbf{V}$ and $\mathbf{F}$ by $D^{\alpha}v$ and $D^{\alpha}F$ respectively. Hence, we conclude the proof of Theorem 4.1.

The case when there is at least one spatial derivative in $D^{\alpha}$ , i.e. $0<\alpha_{0}\leq 3$ can be studied with the same analysis as above.

4.3.4 Energy estimate with time derivatives

In this section, we focus on the fully times differentiated equations, i.e $D^{\alpha}=\partial_{t}^{4}$ .

Theorem 4.2.

There exists $T>0$ such that

\displaystyle||\partial_{t}^{4}v_{i}||_{0}^{2}+||\partial_{t}^{4}F_{ik}||_{0}^{2}+|\sqrt{\sigma}\partial_{t}^{4}\overline{\nabla}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.106)

Furthermore, if the Rayleigh-Taylor sign condition $\-\partial_{3}q\geq c_{0}>0$ is assumed on $\Sigma$ , the term $|\partial_{t}^{4}\psi|_{0}^{2}$ enters the energy and the estimate become $\sigma$ -uniform,

\displaystyle||\partial_{t}^{4}v_{i}||_{0}^{2}+||\partial_{t}^{4}F_{ik}||_{0}^{2}+|\sqrt{\sigma}\partial_{t}^{4}\overline{\nabla}\psi|_{0}^{2}+|\partial_{t}^{4}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.107)

Parallel to the case with spatial derivatives (4.59), we have

\displaystyle\frac{1}{2}\frac{d}{dt}(\int_{\Omega}|\mathbf{V}_{i}|^{2}\partial_{3}\varphi+\int_{\Omega}|\mathbf{F}_{ik}|^{2}\partial_{3}\varphi)=-\int_{\Sigma}\mathbf{Q}\mathbf{V}\cdot N-\int_{\Omega}\mathcal{C}_{i}(v_{i})\mathbf{Q}\partial_{3}\varphi+\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}^{2}_{ik}\partial_{3}\varphi+\int_{\Omega}\mathcal{R}_{i}^{1}\mathbf{V}_{i}\partial_{3}\varphi.

(4.108)

The Alinhac good unknowns reads

\displaystyle\mathbf{V}=\partial_{t}^{4}v-\partial_{t}^{4}\varphi\partial_{3}^{\varphi}v,\quad\mathbf{F}=\partial_{t}^{4}F-\partial_{t}^{4}\varphi\partial_{3}^{\varphi}F,\quad\mathbf{Q}=\partial_{t}^{4}q-\partial_{t}^{4}\varphi\partial_{3}^{\varphi}q

(4.109)

with the following properties, for any function $g$ and $i=1,2,3,$

	$\displaystyle\partial_{t}^{4}\partial_{i}^{\varphi}g$	$\displaystyle=\partial_{i}^{\varphi}\mathbf{G}+\mathcal{C}_{i}(g),$		(4.110)
	$\displaystyle\partial_{t}^{4}D_{t}^{\varphi}g$	$\displaystyle=D_{t}^{\varphi}\mathbf{G}+\mathcal{D}(g),$		(4.111)

where, for $\tau=1,2,$

$\displaystyle\mathcal{C}_{\tau}(g)$	$\displaystyle=\partial_{t}^{4}\varphi\partial_{\tau}^{\varphi}\partial_{3}^{\varphi}g-[\partial_{t}^{4},\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi},\partial_{3}g]-\partial_{3}g[\partial_{t}^{4},\partial_{\tau}\varphi,\frac{1}{\partial_{3}\varphi}]+\partial_{3}g\partial_{\tau}\varphi[\partial_{t}^{3},\frac{1}{(\partial_{3}\varphi)^{2}}]\partial_{t}\partial_{3}\varphi,$	(4.112)
$\displaystyle\mathcal{C}_{3}(g)$	$\displaystyle=\partial_{t}^{4}\varphi(\partial_{3}^{\varphi})^{2}g+[\partial_{t}^{4},\frac{1}{\partial_{3}\varphi},\partial_{3}g]-\partial_{3}g[\partial_{t}^{3},\frac{1}{(\partial_{3}\varphi)^{2}}]\partial_{t}\partial_{3}\varphi,$	(4.113)
$\displaystyle\mathcal{D}(g)$	$\displaystyle=\partial_{t}^{4}\varphi D_{t}^{\varphi}\partial_{3}^{\varphi}g+[\partial_{t}^{4},\overline{v}]\cdot\overline{\nabla}g+[\partial_{t}^{4},\frac{1}{\partial_{3}\varphi}(v\cdot\mathbf{N}-\partial_{t}\varphi),\partial_{3}g]+[\partial_{t}^{4},\frac{1}{\partial_{3}\varphi},v\cdot\mathbf{N}-\partial_{t}\varphi]\partial_{3}g$
	$\displaystyle\quad-(v\cdot\mathbf{N}-\partial_{t}\varphi)\partial_{3}g[\partial_{t}^{3},\frac{1}{(\partial_{3}\varphi)^{2}}]\partial_{t}\partial_{3}\varphi+\frac{1}{\partial_{3}\varphi}\partial_{3}g[\partial_{t}^{4},v]\mathbf{N}.$	(4.114)

Also, the remaining terms read

	$\displaystyle\mathcal{R}_{i}^{1}$	$\displaystyle=F_{lk}\mathcal{C}_{l}(F_{ik})+[\partial_{t}^{4},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q),$		(4.115)
	$\displaystyle\mathcal{R}_{ij}^{2}$	$\displaystyle=F_{kj}\mathcal{C}_{k}(v_{i})+[\partial_{t}^{4},F_{kj}]\partial_{k}v_{i}-\mathcal{D}(F_{ij}).$		(4.116)

Control of the error terms
Let $\partial_{3}\varphi\geq c_{0}>0$ , then

	$\displaystyle\|\|\mathcal{C}_{i}(g)\|\|_{0}$	$\displaystyle\leq P\bigg{(}c_{0}^{-1},\|\overline{\nabla}\psi\|_{\infty},\sum_{k=1}^{3}\|\overline{\nabla}\partial_{t}^{k}\psi\|_{3-k},\|\partial_{t}^{4}\psi\|_{0}\bigg{)}\cdot\bigg{(}\|\|\partial g\|\|_{\infty}+\|\|\partial^{2}g\|\|_{\infty}+\sum_{k=1}^{3}\|\|\partial_{t}^{k}g\|\|_{4-k}\bigg{)},$		(4.117)
	$\displaystyle\|\|\mathcal{D}(g)\|\|_{0}$	$\displaystyle\leq P\bigg{(}c_{0}^{-1},\sum_{k=0}^{3}\|\|\partial_{t}^{k}v\|\|_{0},\|\overline{\nabla}\psi\|_{\infty},\sum_{k=1}^{3}\|\overline{\nabla}\partial_{t}^{k}\psi\|_{3-k},\|\partial_{t}^{4}\psi\|_{0}\bigg{)}\cdot\bigg{(}\|\|\partial g\|\|_{\infty}+\|\|\partial^{2}g\|\|_{\infty}+\sum_{k=1}^{3}\|\|\partial_{t}^{k}g\|\|_{4-k}\bigg{)},$		(4.118)

where $|\partial_{t}^{4}\psi|_{0}$ can be controlled by taking $\partial_{t}^{3}$ on the kinematic boundary condition. More precisely,

$\displaystyle\partial_{t}^{4}\psi$	$\displaystyle=\partial_{t}^{3}(v\cdot N)$
	$\displaystyle=-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\psi-\partial_{t}^{3}\overline{v}\cdot\overline{\partial}\psi+\partial_{t}^{3}v_{3}-[\partial_{t}^{3},\overline{v},\overline{\partial}\psi]$
	$\displaystyle=-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\psi-\partial_{t}^{3}v\cdot N-[\partial_{t}^{3},\overline{v},\overline{\partial}\psi].$	(4.119)

It follows that

\displaystyle|\partial_{t}^{4}\psi|_{0}\leq P\big{(}\sum_{k=0}^{3}|\overline{\nabla}\partial_{t}^{k}\psi|_{3-k},\sum_{k=0}^{3}||\partial_{t}^{k}v||_{3-k}\big{)}.

(4.120)

Next,

	$\displaystyle\|\|\mathcal{R}_{i}^{1}\|\|_{0}$	$\displaystyle\leq P\bigg{(}\sum_{k=0}^{4}\|\|\partial_{t}^{k}F\|\|_{4-k}\bigg{)}\cdot\bigg{(}\|\|\mathcal{C}_{l}(F_{ik})\|\|_{0}+\|\|\mathcal{D}(v_{i})\|\|_{0}+\mathcal{C}_{i}(q)\|\|_{0}\bigg{)},$		(4.121)
	$\displaystyle\|\|\mathcal{R}_{ij}^{2}\|\|_{0}$	$\displaystyle\leq P\bigg{(}\sum_{k=0}^{4}\|\|\partial_{t}^{k}F\|\|_{4-k},\sum_{k=0}^{3}\|\|\partial_{t}^{k}v\|\|_{4-k}\bigg{)}\cdot\bigg{(}\|\|\mathcal{C}_{l}(v_{i})\|\|_{0}+\|\|\mathcal{D}(F_{ij})\|\|_{0}\bigg{)}.$		(4.122)

Then, we can directly control the error terms

	$\displaystyle\int_{\Omega}\mathbf{F}_{ik}\mathcal{R}^{2}_{ik}\partial_{3}\varphi$	$\displaystyle\leq\|\|\partial_{3}\varphi\|\|_{\infty}\|\|\mathbf{F}_{ik}\|\|_{0}\|\|\mathcal{R}_{ik}^{2}\|\|_{0},$		(4.123)
	$\displaystyle\int_{\Omega}\mathcal{R}_{i}^{1}\mathbf{V}_{i}\partial_{3}\varphi$	$\displaystyle\leq\|\|\partial_{3}\varphi\|\|_{\infty}\|\|\mathbf{V}_{i}\|\|_{0}\|\|\mathcal{R}_{i}^{1}\|\|_{0}.$		(4.124)

As for

\displaystyle-\int_{\Omega}\mathcal{C}_{i}(v_{i})\mathbf{Q}\partial_{3}\varphi=-\int_{\Omega}\partial_{t}^{4}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi+\int_{\Omega}\partial_{t}^{4}\psi\partial_{3}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi,

(4.125)

we can only control the second term

\displaystyle\int_{\Omega}\partial_{t}^{4}\psi\partial_{3}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi\leq|\partial_{t}^{4}\psi|_{0}|\partial_{3}q|_{\infty}||\mathcal{C}_{i}(v_{i})||_{0}|\partial_{3}\varphi|_{\infty},

(4.126)

and $-\int_{\Omega}\partial_{t}^{4}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi$ cannot be directly controlled as above, since we do not have the control of $||\partial_{t}^{4}q||_{0}$ . However, it can be cancelled later in the control of $\int_{\Sigma}\mathbf{Q}\mathcal{S}_{1}$ .

Control of $-\int_{\Sigma}\mathbf{Q}\mathbf{V}\cdot N$
Plugging in $D^{\alpha}=\partial_{t}^{4}$ to (4.51), (4.52), we have the time differentiated kinematic boundary condition

\displaystyle\partial_{t}^{5}\psi=\mathbf{V}\cdot N-\overline{v}\cdot\partial_{t}^{4}\overline{\nabla}\psi+\mathcal{S}^{*}_{1},

(4.127)

where

\displaystyle\mathcal{S}^{*}_{1}=\partial_{t}^{4}\psi\partial_{3}v\cdot N-[\partial_{t}^{4},\overline{v},\overline{\partial}\psi].

(4.128)

Also, we have the boundary condition for $\mathbf{Q}$ on $\Sigma$ ,

\displaystyle\mathbf{Q}|_{\Sigma}=-\sigma\partial_{t}^{4}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{|N|}\big{)}-\partial_{t}^{4}\psi\partial_{3}q.

(4.129)

Now, we plug in the kinematic boundary condition to get

	$\displaystyle-\int_{\Sigma}\mathbf{Q}\big{(}\mathbf{V}\cdot N\big{)}$	$\displaystyle=-\int_{\Sigma}\mathbf{Q}\partial_{t}^{5}\psi-\int_{\Sigma}\mathbf{Q}\big{(}\overline{v}\cdot\partial_{t}^{4}\overline{\nabla}\psi\big{)}+\int_{\Sigma}\mathbf{Q}\mathcal{S}^{*}_{1}$
		$\displaystyle=:I^{}+II^{}+III^{*}.$		(4.130)

For $I^{*}$ , plugging in the boundary condition,

	$\displaystyle I^{*}$	$\displaystyle=\int_{\Sigma}\sigma\partial_{t}^{4}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\partial_{t}^{5}\psi+\int_{\Sigma}\partial_{3}q\partial_{t}^{4}\psi\partial_{t}^{5}\psi$
		$\displaystyle=:ST^{}+RT^{}.$		(4.131)

The control of $ST^{*}$ is parallel to the estimate of $ST$ replacing $D^{\alpha}$ by $\partial_{t}^{4}$ , we have

\displaystyle\int_{0}^{T}ST^{*}+\frac{\sigma}{2}\int_{\Sigma}|\partial_{t}^{4}\overline{\nabla}\psi|_{0}^{2}\lesssim P(E(0))+\int_{0}^{T}P(E(t)).

(4.132)

For $RT^{*}$ , if we assume the Rayleigh-Taylor sign condition $-\partial_{3}q\geq c_{0}>0$ , then we can handle $RT^{*}$ similarly to $RT$ by generating an energy term $|\partial_{t}^{4}\psi|_{0}^{2}$ . However, if we drop the sign condition, we will need some new estimate to bound $RT^{*}$ , since it has a top order term $\partial_{t}^{5}\psi$ which is not in the energy.

To study $RT^{*}$ , we use the kinematic boundary condition,

\displaystyle\partial_{t}^{5}\psi=\partial_{t}^{4}(v\cdot N)=-(\overline{v}\cdot\overline{\partial})\partial_{t}^{4}\psi+\partial_{t}^{4}v\cdot N-[\partial_{t}^{4},\overline{v},\overline{\partial}\psi].

(4.133)

It follows that

	$\displaystyle RT^{*}$	$\displaystyle=-\int_{\Sigma}\partial_{3}q\partial_{t}^{4}\psi(\overline{v}\cdot\overline{\partial})\partial_{t}^{4}\psi+\int_{\Sigma}\partial_{3}q\partial_{t}^{4}\psi\partial_{t}^{4}v\cdot N-\int_{\Sigma}\partial_{3}q\partial_{t}^{4}\psi[\partial_{t}^{4},\overline{v},\overline{\partial}\psi]$		(4.134)
		$\displaystyle=:RT^{}_{1}+RT_{2}^{}+RT^{*}_{3}.$		(4.135)

Note that we do not need to estimate $RT_{1}^{*}$ and $RT^{*}_{3}$ , since they will be cancelled by parts of $II^{*}$ and $III^{*}$ . As for $RT^{*}_{2}$ , the term $\partial_{t}^{4}v$ has the top order of $v$ on the boundary, which we certainly cannot directly control. We first plug in (4.119) for $\partial_{t}^{4}\psi$ and use Green’s formula to drag the term from boundary to the domain $\Omega$ ,

$\displaystyle RT_{2}^{*}$	$\displaystyle=\int_{\Sigma}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\psi-\partial_{t}^{3}v\cdot N-[\partial_{t}^{3},\overline{v},\overline{\partial}\psi]\big{)}\partial_{t}^{4}v\cdot N$
	$\displaystyle=\int_{\Omega}\partial_{3}\bigg{(}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{t}^{4}v\cdot\mathbf{N}\bigg{)}$
	$\displaystyle=\int_{\Omega}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{3}\partial_{t}^{4}v\cdot\mathbf{N}$
	$\displaystyle\quad+\int_{\Omega}\partial_{3}q\partial_{3}\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{t}^{4}v\cdot\mathbf{N}$
	$\displaystyle\quad+\int_{\Omega}\partial_{3}^{2}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{t}^{4}v\cdot\mathbf{N}$
	$\displaystyle\quad+\int_{\Omega}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{t}^{4}v\cdot\partial_{3}\mathbf{N}$
	$\displaystyle=RT^{}_{21}+RT^{}_{22}+RT^{}_{23}+RT^{}_{24},$	(4.136)

where $RT^{*}_{22},RT^{*}_{23},RT^{*}_{24}$ can be directly controlled

$\displaystyle RT^{*}_{22}$	$\displaystyle\leq P\bigg{(}\sum_{k=0}^{3}\|\|\partial_{t}^{k}v\|\|_{4-k},\sum_{k=0}^{3}\|\partial_{t}^{k}\overline{\nabla}\psi\|_{4-k}\bigg{)}\|\|\partial_{3}q\|\|_{\infty}\|\overline{\nabla}\psi\|_{\infty},$	(4.137)
$\displaystyle RT^{*}_{23}$	$\displaystyle\leq P\bigg{(}\sum_{k=0}^{3}\|\|\partial_{t}^{k}v\|\|_{3-k},\sum_{k=0}^{3}\|\partial_{t}^{k}\overline{\nabla}\psi\|_{3-k}\bigg{)}\|\|\partial^{2}_{3}q\|\|_{\infty}\|\overline{\nabla}\psi\|_{\infty},$	(4.138)
$\displaystyle RT^{*}_{24}$	$\displaystyle\leq P\bigg{(}\sum_{k=0}^{3}\|\|\partial_{t}^{k}v\|\|_{3-k},\sum_{k=0}^{3}\|\partial_{t}^{k}\overline{\nabla}\psi\|_{3-k}\bigg{)}\|\|\partial_{3}q\|\|_{\infty}\|\partial_{3}\overline{\nabla}\psi\|_{\infty}.$	(4.139)

For $RT^{*}_{21}$ , we invoke the time integral and integrate $\partial_{t}$ by parts,

$\displaystyle\int_{0}^{T}RT^{*}_{21}$	$\displaystyle=-\int_{0}^{T}\int_{\Omega}\partial_{t}\bigg{(}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\mathbf{N}\bigg{)}\cdot\partial_{3}\partial_{t}^{3}v$
	$\displaystyle\quad+\bigg{(}\int_{\Omega}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\partial_{3}\partial_{t}^{3}v\cdot\mathbf{N}\bigg{)}\bigg{\|}_{0}^{T}$
	$\displaystyle=J_{1}+J_{2},$	(4.140)

where

	$\displaystyle J_{1}$	$\displaystyle\leq\int_{0}^{T}\bigg{\|}\bigg{\|}\partial_{t}\bigg{(}\partial_{3}q\big{(}-\overline{v}\cdot\overline{\partial}\partial_{t}^{3}\varphi-\partial_{t}^{3}v\cdot\mathbf{N}-[\partial_{t}^{3},\overline{v},\overline{\partial}\varphi]\big{)}\mathbf{N}\bigg{)}\bigg{\|}\bigg{\|}_{0}\|\|\partial_{3}\partial_{t}^{3}v\|\|_{0}$
		$\displaystyle\leq\int_{0}^{T}P\bigg{(}\sum_{k=0}^{4}\|\|\partial_{t}^{k}v\|\|_{4-k},\sum_{k=0}^{4}\|\partial_{t}^{k}\overline{\nabla}\psi\|_{4-k},\|\|\partial_{3}q\|\|_{\infty},\|\|\partial_{t}\partial_{3}q\|\|_{\infty}\bigg{)}.$		(4.141)

As for $J_{2}$ , the only trouble term is $||\partial_{3}\partial_{t}^{3}v||_{0}$ , which we can make absorbed to the left-hand side by Cauchy inequality,

\displaystyle J_{2}\leq P(E(0))+\epsilon||\partial_{3}\partial_{t}^{3}v||_{0}^{2}+C_{\epsilon}P\bigg{(}\sum_{k=0}^{3}||\partial_{t}^{k}v||_{3-k},\sum_{k=0}^{3}|\partial_{t}^{k}\overline{\nabla}\psi|_{3-k}\bigg{)}||\partial_{3}q||_{\infty}|\overline{\nabla}\psi|_{\infty}.

(4.142)

Combining (4.132), (4.137), (4.138), (4.139), (4.141), (4.142), we get the estimate of $I^{*}$ ,

\displaystyle\int_{0}^{T}I^{*}+\frac{\sigma}{2}|\overline{\nabla}\partial_{t}^{4}\psi|_{0}^{2}\lesssim\epsilon||\partial_{3}\partial_{t}^{3}v||_{0}^{2}+P(E(0))+\int_{0}^{T}P(E(t)).

(4.143)

Next, we expand $II^{*}+III^{*}$ ,

	$\displaystyle II^{}+III^{}=$	$\displaystyle-\int_{\Sigma}\partial_{t}^{4}q(\overline{v}\cdot\partial_{t}^{4}\overline{\nabla}\psi)+\int_{\Sigma}\partial_{t}^{4}\psi\partial_{3}q(\overline{v}\cdot\partial_{t}^{4}\overline{\nabla}\psi)$
		$\displaystyle+\int_{\Sigma}\partial_{t}^{4}q\mathcal{S}_{1}^{*}-\int_{\Sigma}\|\partial_{t}^{4}\psi\|^{2}\partial_{3}q(\partial_{3}v\cdot N)+\int_{\Sigma}\partial_{t}^{4}\psi\partial_{3}q[\partial_{t}^{4},\overline{v},\overline{\partial}\psi],$		(4.144)

where the second term cancels $RT^{*}_{1}$ and the fifth term cancels $RT^{*}_{3}$ . The first term can be controlled exactly as the analysis of $II_{11}$ replacing $D^{\alpha}$ by $\partial_{t}^{4}$ and the fourth term can be directly controlled as follows,

\displaystyle-\int_{\Sigma}|\partial_{t}^{4}\psi|^{2}\partial_{3}q(\partial_{3}v\cdot N)\leq|\partial_{t}^{4}\psi|_{0}^{2}|\partial_{3}q|_{\infty}|\partial_{3}v|_{\infty}|N|_{\infty}.

(4.145)

It remains to control the third term,

\displaystyle\int_{\Sigma}\partial_{t}^{4}q\mathcal{S}_{1}^{*}=\int_{\Sigma}\partial_{t}^{4}q\partial_{t}^{4}\psi\partial_{3}v\cdot N-4\int_{\Sigma}\partial_{t}^{4}q(\partial_{t}^{3}\overline{v}\cdot\overline{\partial}\partial_{t}\psi)-\int_{\Sigma}\partial_{t}^{4}q\sum_{k=1}^{2}\begin{pmatrix}4\\ k\end{pmatrix}\partial_{t}^{k}v\cdot\partial_{t}^{4-k}\overline{\partial}\psi.

(4.146)

For the first term, invoking the boundary condition and integration by parts,

$\displaystyle\int_{\Sigma}\partial_{t}^{4}q\partial_{t}^{4}\psi\partial_{3}v\cdot N$	$\displaystyle=-\int_{\Sigma}\sigma\partial_{t}^{4}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\partial_{t}^{4}\psi\partial_{3}v\cdot N$
	$\displaystyle=\int_{\Sigma}\sigma\partial_{t}^{4}\big{(}\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\cdot\overline{\nabla}\big{(}\partial_{t}^{4}\psi\partial_{3}v\cdot N\big{)}$
	$\displaystyle\leq P\big{(}\|\sqrt{\sigma}\partial_{t}^{4}\overline{\nabla}\psi\|_{0},\|\partial_{t}^{4}\psi\|_{0},\|\partial_{3}\overline{\nabla}v\|_{\infty},\|\partial_{3}v\|_{\infty},\|\overline{\nabla}^{2}\psi\|_{\infty},\|\overline{\nabla}\psi\|_{\infty}\big{)}.$	(4.147)

Similarly, for the last term,

	$\displaystyle-\int_{\Sigma}\partial_{t}^{4}q\sum_{k=1}^{2}\begin{pmatrix}4\\ k\end{pmatrix}\partial_{t}^{k}v\cdot\partial_{t}^{4-k}\overline{\partial}\psi$
	$\displaystyle=\int_{\Sigma}\sigma\partial_{t}^{4}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\sum_{k=1}^{2}\begin{pmatrix}4\\ k\end{pmatrix}\partial_{t}^{k}v\cdot\partial_{t}^{4-k}\overline{\partial}\psi$
	$\displaystyle=-\int_{\Sigma}\sigma\partial_{t}^{4}\big{(}\frac{\overline{\nabla}\psi}{\|N\|}\big{)}\cdot\overline{\nabla}\sum_{k=1}^{2}\begin{pmatrix}4\\ k\end{pmatrix}\partial_{t}^{k}v\cdot\partial_{t}^{4-k}\overline{\partial}\psi$
	$\displaystyle\leq P\big{(}\|\partial_{t}^{2}\overline{\nabla}v\|_{0},\|\partial_{t}^{2}v\|_{\infty},\|\partial_{t}^{2}\overline{\partial}^{2}\psi\|_{0},\|\partial_{t}^{2}\overline{\partial}\psi\|_{\infty},\|\partial_{t}\overline{\nabla}v\|_{\infty},\|\partial_{t}v\|_{\infty},\|\sqrt{\sigma}\overline{\nabla}^{2}\partial_{t}^{3}\psi\|_{0},\|\overline{\nabla}\partial_{t}^{3}\psi\|_{0}\big{)}\|\sqrt{\sigma}\partial_{t}^{4}\overline{\nabla}\psi\|_{0}.$		(4.148)

As for the second term

\displaystyle J^{*}_{1}:=-4\int_{\Sigma}\partial_{t}^{4}q(\partial_{t}^{3}\overline{v}\cdot\overline{\partial}\partial_{t}\psi),

(4.149)

we seek for cancellation from the interior term $-\int_{\Omega}\partial_{t}^{4}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi$ . Note that the possible trouble terms in $-\int_{\Omega}\partial_{t}^{4}q\mathcal{C}_{i}(v_{i})\partial_{3}\varphi$ are

$\displaystyle J_{2}^{*}$	$\displaystyle=-\int_{\Omega}\partial_{t}^{4}q\partial_{t}^{4}\varphi\partial_{i}^{\varphi}\partial_{3}^{\varphi}v_{i}\partial_{3}\varphi,\qquad i=1,2,3,$	(4.150)
$\displaystyle J_{3}^{*}$	$\displaystyle=4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\big{(}\frac{\partial_{\tau}\varphi}{\partial_{3}\varphi})\partial_{t}^{3}\partial_{3}v_{\tau}\partial_{3}\varphi,\qquad\tau=1,2,$	(4.151)
$\displaystyle J_{4}^{*}$	$\displaystyle=-4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\big{(}\frac{1}{\partial_{3}\varphi}\big{)}\partial_{t}^{3}\partial_{3}v_{3}\partial_{3}\varphi,$	(4.152)

while the remaining terms are contributed by the lower order terms of $\mathcal{C}_{i}(v_{i})$ and can be controlled by integrating $\partial_{t}$ by parts under time integral. Due to the divergence free of $v$ , $J_{2}^{*}$ vanishes,

\displaystyle J_{2}^{*}=-\int_{\Omega}\partial_{t}^{4}q\partial_{t}^{4}\varphi\partial_{3}^{\varphi}(\nabla^{\varphi}\cdot v)\partial_{3}\varphi=0.

(4.153)

Next, we expand $J_{3}^{*}$ ,

	$\displaystyle J_{3}^{*}$	$\displaystyle=4\int_{\Omega}\partial_{t}^{4}q(\partial_{t}\overline{\partial}\varphi\cdot\partial_{t}^{3}\partial_{3}v)-4\int_{\Omega}\partial_{t}^{4}q\frac{\partial_{\tau}\varphi\partial_{t}\partial_{3}\varphi}{\partial_{3}\varphi}\partial_{t}^{3}\partial_{3}v_{\tau}$
		$\displaystyle=:J_{31}^{}+J_{32}^{}.$		(4.154)

We can easily see the similarity between $J_{1}^{*}$ and $J_{31}^{*}$ . After integrating $\partial_{3}$ by parts in $J_{31}^{*}$ , the boundary term exactly cancels $J_{1}^{*}$ .

\displaystyle J_{1}^{*}+J_{31}^{*}=4\int_{\Omega}\partial_{t}^{4}\partial_{3}q(\partial_{t}\overline{\partial}\varphi\cdot\partial_{t}^{3}v)+\partial_{t}^{4}q(\partial_{t}\partial_{3}\overline{\partial}\varphi\cdot\partial_{t}^{3}v),

(4.155)

which can be handled by integrating $\partial_{t}$ by parts

	$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}^{4}\partial_{3}q(\partial_{t}\overline{\partial}\varphi\cdot\partial_{t}^{3}v)+\partial_{t}^{4}q(\partial_{t}\partial_{3}\overline{\partial}\varphi\cdot\partial_{t}^{3}v)$
	$\displaystyle=-\int_{0}^{T}\int_{\Omega}\bigg{(}\partial_{t}^{3}\partial_{3}q\partial_{t}(\partial_{t}\overline{\partial}\varphi\cdot\partial_{t}^{3}v)+\partial_{t}^{3}q\partial_{t}(\partial_{t}\partial_{3}\overline{\partial}\varphi\cdot\partial_{t}^{3}v)\bigg{)}+\bigg{(}\int_{\Omega}\partial_{t}^{3}\partial_{3}q(\partial_{t}\overline{\partial}\varphi\cdot\partial_{t}^{3}v+\partial_{t}^{3}q(\partial_{t}\partial_{3}\overline{\partial}\varphi\cdot\partial_{t}^{3}v))\bigg{)}\bigg{\|}_{0}^{T}$
	$\displaystyle\leq\epsilon\|\|\partial_{t}^{3}\partial_{3}q\|\|_{0}^{2}+P(E(0))+\int_{0}^{T}P(E(t)).$		(4.156)

Finally, it remains to control $J_{32}^{*}$ and $J_{4}^{*}$ . Recall that $\partial_{\tau}^{\varphi}=\partial_{\tau}-\partial_{\tau}\varphi\frac{1}{\partial_{3}\varphi}\partial_{3}$ and $\partial_{3}^{\varphi}=\frac{1}{\partial_{3}\varphi}\partial_{3}$ , so

$\displaystyle J^{}_{4}+J_{32}^{}$	$\displaystyle=4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\partial_{3}^{\varphi}\partial_{t}^{3}v_{3}+4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\big{(}\partial_{\tau}^{\varphi}\partial_{t}^{3}v_{\tau}-\partial_{\tau}\partial_{t}^{3}v_{\tau}\big{)}$
	$\displaystyle=4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\big{(}\nabla^{\varphi}\cdot\partial_{t}^{3}v\big{)}-4\int_{\Omega}\partial_{t}^{4}q\varphi\partial_{t}\partial_{3}\varphi\partial_{\tau}\partial_{t}^{3}v_{\tau}$
	$\displaystyle=4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\partial_{t}^{3}\big{(}\nabla^{\varphi}\cdot v\big{)}-4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\big{(}\big{[}\partial_{t}^{3},\nabla^{\varphi}\big{]}\cdot v\big{)}-4\int_{\Omega}\partial_{t}^{4}q\partial_{t}\partial_{3}\varphi\partial_{\tau}\partial_{t}^{3}v_{\tau},$	(4.157)

where the first term vanishes due to $\nabla^{\varphi}\cdot v=0$ . The second term can be controlled after integration $\partial_{t}$ by parts since $[\partial_{t}^{3},\nabla^{\varphi}\big{]}\cdot v$ only contains up to 2 time derivatives of $v$ and 3 time derivatives of $\varphi$ . For the last term, after integration $\partial_{t}$ by parts and integration $\partial_{\tau}$ by parts, it contributes to the top order term,

\displaystyle-4\int_{0}^{T}\int_{\Omega}\partial_{t}^{3}\partial_{\tau}q\partial_{t}\partial_{3}\varphi\partial_{t}^{4}v_{\tau}-\bigg{(}4\int_{\Omega}\partial_{t}^{3}q\partial_{t}\partial_{3}\varphi\partial_{\tau}\partial_{t}^{3}v_{\tau}\bigg{)}\bigg{|}_{0}^{T}\leq\epsilon||\partial_{\tau}\partial_{t}^{3}v||_{0}^{2}+P(E(0))+\int_{0}^{T}P(E(t)).

(4.158)

Combining (4.143), (4.144), (4.145), (4.147),(4.148), (4.156), (4.158), after choosing $\epsilon$ carefully, we conclude the estimate

\displaystyle||\mathbf{V}_{i}||_{0}^{2}+||\mathbf{F}_{ik}||_{0}^{2}+|\sqrt{\sigma}\partial_{t}^{4}\overline{\nabla}\psi|_{0}^{2}\leq P(E(0))+\int_{0}^{T}P(E(t)).

(4.159)

5 Zero surface tension limit

In this section, we study the behaviour of the solution of (2.28)-(2.29) as the surface tension coefficient $\sigma$ tends to 0 and thus show that the solution of (2.28)-(2.29) can be passed to the zero surface tension limit. Consider the following incompressible elastodynamics without surface tension reformulated in graphic coordinates:

\begin{cases}D_{t}^{\varphi}v+\nabla^{\varphi}q=(F_{k}\cdot\nabla^{\varphi})F_{k}\qquad&\text{in }\Omega,\\ \nabla^{\varphi}\cdot v=0\qquad&\text{in }\Omega,\\ D_{t}^{\varphi}F_{j}=(F_{j}\cdot\nabla^{\varphi})v\qquad&\text{in }\Omega,\\ \nabla^{\varphi}\cdot F_{j}=0\qquad&\text{in }\Omega\\ \partial_{t}\psi=v\cdot N\qquad&\text{on }\Sigma,\\ q=0\qquad&\text{on }\Sigma,\\ F_{j}\cdot N=0\qquad&\text{on }\Sigma,\\ v_{3}=0\qquad&\text{on }\Sigma_{b},\\ F_{3j}=0\qquad&\text{on }\Sigma_{b},\\ q=0\qquad&\text{on }\Sigma_{b},\\ \end{cases}

(5.1)

with the boundary conditions

\begin{cases}\partial_{t}\psi=v\cdot N\qquad&\text{on }\Sigma,\\ q=0\qquad&\text{on }\Sigma,\\ F_{j}\cdot N=0\qquad&\text{on }\Sigma,\\ v_{3}=0\qquad&\text{on }\Sigma_{b},\\ F_{3j}=0\qquad&\text{on }\Sigma_{b},\\ q=0\qquad&\text{on }\Sigma_{b}.\\ \end{cases}

(5.2)

To differentiate between the the solution of (5.1)-(5.2) and (2.28)-(2.29), we denote the solution of (2.28)-(2.29) by $(\psi^{\sigma},v^{\sigma},F_{j}^{\sigma})$ to emphasise the dependence of the solution on $\sigma$ . We have shown that $|\psi^{\sigma}(t)|_{4}^{2}$ $||v^{\sigma}(t)||_{4}^{2},||F_{j}^{\sigma}(t)||_{4}^{2}$ are bounded uniformly in $\sigma$ , provided the Rayleigh-Taylor sign condition holds. Hence the Morrey-type embeddings imply that $v^{\sigma}(t),F_{j}^{\sigma}(t)$ are equicontinuous and uniformly bounded in $C^{2}(\Omega)$ and $\psi^{\sigma}(t)$ is equicontinuous and uniformly bounded in $C^{2}(\Sigma)$ , which implies $(\psi^{\sigma},v^{\sigma},F_{j}^{\sigma})\to(\psi,v,F_{j})$ in $C^{0}([0,t],C^{2}(\Sigma)\times C^{2}(\Omega)\times C^{2}(\Omega))$ as $\sigma$ tends to zero.

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	$\displaystyle\|\|\partial_{t}^{k}v\|\|_{4-k}^{2}$	$\displaystyle\lesssim C\big{(}\|\psi\|_{4-k},\|\overline{\partial}\psi\|_{W^{1,\infty}}\big{)}\big{(}\|\|\nabla^{\varphi}\cdot\partial_{t}^{k}v\|\|_{3-k}^{2}+\|\|\nabla^{\varphi}\times\partial_{t}^{k}v\|\|_{3-k}^{2}+\|\|\overline{\partial}^{4-k}\partial_{t}^{k}v\|\|_{0}^{2}+\|\|v\|\|_{0}^{2}\big{)},$		(4.9)
	$\displaystyle\|\|\partial_{t}^{k}F_{j}\|\|_{4-k}^{2}$	$\displaystyle\lesssim C\big{(}\|\psi\|_{4-k},\|\overline{\partial}\psi\|_{W^{1,\infty}}\big{)}\big{(}\|\|\nabla^{\varphi}\cdot\partial_{t}^{k}F_{j}\|\|_{3-k}^{2}+\|\|\nabla^{\varphi}\times\partial_{t}^{k}F_{j}\|\|_{3-k}^{2}+\|\|\overline{\partial}^{4-k}\partial_{t}^{k}F_{j}\|\|_{0}^{2}+\|\|F_{j}\|\|_{0}^{2}\big{)}.$		(4.10)

	$\displaystyle\mathbf{Q}\|_{\Sigma}$	$\displaystyle=(D^{\alpha}q-D^{\alpha}\varphi\partial_{3}^{\varphi}q)\|_{\Sigma}$
		$\displaystyle=-\sigma D^{\alpha}\big{(}\overline{\nabla}\cdot\frac{\overline{\nabla}\psi}{\|N\|}\big{)}-D^{\alpha}\psi\partial_{3}q\qquad\text{on }\Sigma.$		(4.50)

	$\displaystyle\|\|\mathcal{C}_{i}(f)\|\|_{0}$	$\displaystyle\leq P(c_{0}^{-1},\|\psi\|_{4})\|\|f\|\|_{4},$		(4.60)
	$\displaystyle\|\|\mathcal{D}(f)\|\|_{0}$	$\displaystyle\leq P(c_{0}^{-1},\|\|v\|\|_{4},\|\psi\|_{4},\|\partial_{t}\psi\|_{3})(\|\|f\|\|_{4}+\|\|\partial_{t}f\|\|_{2}).$		(4.61)

	$\displaystyle\|\|\mathcal{R}_{ik}^{2}\|\|_{0}$	$\displaystyle=\|\|F_{kj}\mathcal{C}_{k}(v_{i})+[D^{\alpha},F_{kj}]\partial_{k}v_{i}-\mathcal{D}(F_{ij})\|\|_{0}$		(4.65)
		$\displaystyle\lesssim\|\|F_{kj}\|\|_{\infty}\|\|\mathcal{C}_{k}(v_{i})\|\|_{0}+\|\|F_{kj}\|\|_{4}\|\|v\|\|_{4}+\|\|\mathcal{D}(F_{ij})\|\|_{0}.$		(4.66)

	$\displaystyle\|\|\mathcal{R}_{i}^{1}\|\|_{0}$	$\displaystyle=\|\|F_{lk}\mathcal{C}_{l}(F_{ik})+[D^{\alpha},F_{lk}]\partial_{l}^{\varphi}F_{ik}-\mathcal{D}(v_{i})-\mathcal{C}_{i}(q)\|\|_{0}$		(4.68)
		$\displaystyle\lesssim\|\|F_{ik}\|\|_{\infty}\|\|\mathcal{C}_{l}(F_{ik})\|\|_{0}+\|\|F_{lk}\|\|_{4}^{2}+\|\|\mathcal{D}(v_{i})\|\|_{0}+\|\|\mathcal{C}_{i}(q)\|\|_{0}.$		(4.69)

On the free-boundary Incompressible Elastodynamics with and without surface tension

Abstract

1 Introduction

1.1 History and Background

1.2 Outline of the paper

2 Reformulation of the system

2.1 Lagrangian Coordinates

2.1.1 Reformulation in Lagrangian Coordinates

2.1.2 Local Well-posedness in Lagrangian Coordinates

Theorem 2.1.

2.2 Graphic Coordinates

2.2.1 Reformulation in Graphic Coordinates

Proposition 2.2.

2.3 The main result

Theorem 2.3.

3 The auxiliary results

3.1 Preliminary Lemmas

Lemma 3.1.

Lemma 3.2.

Lemma 3.3.

3.2 Elliptic Estimates

Lemma 3.4.

Lemma 3.5.

4 Energy estimate

4.1 Pressure Estimate

4.2 Div-Curl Estimate

4.2.1 L2L^{2}-estimates

4.2.2 Curl estimates

4.3 Tangential Estimate

4.3.1 Alinhac’s Good Unknown

4.3.2 Reformulation in Alinhac’s Good Unknown

4.3.3 Energy estimate with full spatial derivatives

Theorem 4.1.

4.3.4 Energy estimate with time derivatives

Theorem 4.2.

5 Zero surface tension limit

References

4.2.1 $L^{2}$ -estimates