Local well-posedness of the capillary-gravity water waves with acute contact angles

Mei Ming^1 and Chao Wang^2

^1 School of Mathematics and Statistics,Yunnan University, Kunming 615000, P.R.China

E-mail<EMAIL_ADDRESS>
^2 School of  Mathematical Science, Peking University, Beijing 100871, P.R.China

E-mail<EMAIL_ADDRESS>
We consider the  two-dimensional capillary-gravity water waves problem where the free surface $\G_t$ intersects the bottom $\G_b$ at two contact points. In our previous works [53, 54],  the local well-posedness for this problem has been proved with the contact angles less than $\pi/16$. In this paper, we study the case where the contact angles belong to $(0, \pi/2)$. It involves much worse singularities generated from corresponding elliptic systems, which have a strong influence on the regularities for the free surface and the velocity field.
Combining the theory of  singularity decompositions for elliptic problems with the structure of the water waves system, we  obtain a priori energy estimates. Based on these estimates, we also prove the local well-posedness  of the solutions in a geometric formulation.

§ INTRODUCTION

We consider the irrotational incompressible capillary-gravity water waves problem in a two-dimensional  domain $\Om_t$, where $\Om_t$ is a bounded domain  with an  upper free surface $\Gamma_t$ and a fixed  bottom $\Gamma_b$.  This moving domain contains two moving contact points $p_{l}, p_{r}$ (left and right) with the contact angles $\om_l, \om_r\in(0, \pi/2)$, which are the intersection points of $\G_t, \G_b$:

\[
\G_t\cap \G_b=\{p_l, p_r\}.
\]

Moreover, the fixed bottom $\G_b$ is assumed to be smooth enough, and it becomes straight near the contact points $p_i$ ($i=l, r$) for the sake of simplicity.

The  water waves problem has been widely studied in centuries, see for example [64, 50]. This problem focuses on the motion of an ideal fluid and describes the evolution of the free surface $\G_t$ as well as the velocity field $v$. Mathematically, it is described by Euler's equation with boundary conditions and initial conditions, and in our case, we also need some boundary conditions at contact points.

We express the water waves problem on the corner domain $\Om_t$ as the  following  system $\mbox{(WW)}$ of  velocity $v$ and pressure $P$:

_t v+v·v=-P+g     in  _t,

v=0,  v=0    in  _t,

D_t=_t+v·  is tangent to {(t, X) | X ∈Ω_t},

v·n_b |_Γ_b=0,

β_c v_i=σ(cos_s-cos_i)    at  p_i (i=l,r).

Here (<ref>) from $\mbox{(WW)}$ is the Euler's equation where ${\bf g}=-g {\bf e_z}$ is the vertical gravity vector;  (<ref>) describes the incompressibility and irrotationality;
(<ref>) is the condition of the pressure on the free surface in the case with surface tension, where $\si$ is the coefficient of surface tension and $\kappa$ is the mean curvature of $\G_t$ (see Section 1.2); (<ref>) is the classical kinematic condition on the free surface $\G_t$ with $D_t$ the material derivative; Meanwhile, (<ref>) describes that the velocity along the fixed bottom $\G_b$ is always tangential, where $n_b$ is the unit outward normal vector of $\G_b$. These equations and conditions are  standard in water waves, see [46, 47].

In particular, (<ref>) gives the conditions at contact points, which come from  [59]. We denote by $v_i$ the upward tangential components of the velocity at the corner points along $\G_b$:

\[
v_l=-v\cdot\tau_b\quad\hbox{at}\  p_{l},\quad\hbox{and}\quad v_r=v\cdot\tau_b\quad\hbox{at}\  p_{r}.
\]

Here $\om_s$ is the stationary contact angle decided by the materials of the bottom and the fluid (see [77]), and $\beta_c$ denotes the  effective friction coefficient.  This condition shows that the slip velocity is dominated by the unbalanced Young stress,  and it is indeed  an effective variation of Young's law (1805) for  stationary contact angles [77]. In fact, this kind of conditions are commonly seen, see  [12, 13, 63, 28].

Before presenting our results, we  recall briefly earlier works on the well-posedness of classical water waves problem, where one has smooth surfaces $\G_t$ satisfying $\G_t\cap\G_b=\emptyset$.

We recall results on the local well-posedness. When the fluid is irrotational, some early works like Nalimov [56], Yosihara [75, 76] and Craig [23] established  the two-dimensional local well-posedness with small initial data in Sobolev spaces. In late 1990s,  Wu [69, 70]  proved for the first time the local well-posedness with general initial data in Sobolev spaces and showed  that the Taylor sign condition

-_n_t P≥c_0>0
held on $\G_t$ as long as $\Gamma_t$ was not self-intersecting.   Iguchi, Tanaka and Tani [39] and  Iguchi [37]   proved the local well-posedness in two-dimensional case respectively. Later on, Lannes [46] derived the local well-posedness of the gravity water waves under Zakharov formulation, which is convenient to link with approximate models.  Alazard, Burq and Zuily in [2, 3, 4, 5] used  paradifferential operators and Strichartz estimates to study the problem in a low-regularity space. On the other hand, when the fluid is rotational,  Christodoulou and Lindblad [19]  proved a priori estimates based on the geometry of the moving domain;  Lindblad [51] obtained the existence of solutions using Nash-Moser iteration. In 2007, Coutand and Shkoller [21] used Lagrangian coordinates to show the local well-posedness. Shatah and Zeng [61, 62] adopted a geometric point of view to reformulate the problem and prove the local well-posedness, while Beyer and Günther [14, 15] used a similar geometric approach to study the irrotational flow. Zhang and Zhang [78] proved the local well-posedness for rotational flow using a framework of Clifford analysis  introduced by Wu [70]. For more references, see  [57, 58, 9, 10, 60, 78, 68, 7, 47, 55] e.t.c..

For the global well-posedness of small data,   Wu [73] and  Germain, Masmoudi and Shatah [29] proved the global three-dimensional existence of gravity water waves respectively using different approaches.   One can check [72, 6, 40, 26, 34, 35, 36, 67, 74] e.t.c. and their references for more  results on   gravity or capillary-gravity water waves,.  Meanwhile, there are also some works concerning  geometric singularities on the free surfaces.  The authors in [17] proved the existence of a wave which is  given initially as the graph of a function and then can overturn at a later time. Later on, the authors in [18]  showed the existence for some  “splash" or “splat” singularities. This result was extended  to three-dimensional case and some other models in  [22].

Compared to the rich literature on the well-posedness of classical water waves, the research on the well-posedness of  water waves problem with non-smooth boundaries (we call it “non-smooth water waves”) just started several years ago and there are a lot of open questions . In general, there are two kinds of non-smooth water waves problems: The first kind of problems has contact points (or contact lines) between the free surface and the bottom, i.e. $\G_t\cap\G_b\neq\emptyset$; The other kind contains crests or cusps on the free surface, i.e. the surface is Lipschitz. We would like to mention that in the case with large crest angles,  the famous Stokes waves can  be dated back  to papers by G. Stokes [64, 65] which obtained  traveling-wave solutions with limit crest angle $2\pi/3$. Obviously, the main difference here compared to classical water waves lies in the corners on boundaries. As a result, the analysis involving the corners (i.e. domain singularities)  becomes  the key point in the non-smooth water waves.

Now, we are in a position to state an informal version of the main result.
To begin with,  we introduce the following compatibility conditions at $t=0$

β_c_t^k v_i(0)=σ_t^k(cos_s-cos_i0)  at  p_i  (i=l, r),   k=0,1,2,3.
These conditions are needed since  we will reduce system $\mbox(WW)$ into initial-boundary value problem (see (<ref>), (<ref>) and (<ref>)).

Let the initial data belong to a suitable space and the initial contact angles $\om_{i0}\in (0, \pi/2)$ for $i=l, r$. If the compatibility conditions (<ref>) are satisfied,  there exists a time interval depending on the initial data such that system $\mbox{(WW)}$ is locally well-posed in a suitable space.

The suitable space is defined by $\Sigma_h$ in Section 6.3 for some good unknowns. In fact, the proof of Theorem <ref> is divided into three parts. In Section 4 and Section 5, we obtain a priori estimates (see Theorem <ref> and Theorem <ref>). In the last section, we construct approximate solutions to prove the well-posedness based on the a priori estimates. The precise statement of local well-posedness for a geometric form of $\mbox(WW)$ is given in Theorem <ref> (Section 6.3).

When there is no surface tension, the authors of [41, 20] studied the case with contact angles. They  assume that the wall  $\G_b$ is vertical and the contact angle $\om\in(0, \pi/4)$. Then, by a symmetric extension,  they reduced the problem to the crest case with the crest angle less than $\pi/2$. As a result, [41, 20] do not contain the case of limit Stokes waves with the “ $2\pi/3$” crest. In that case,  one needs a contact angle of $\pi /3$ even with the help of a symmetric extension.  In this paper, we only require that  the contact angles  $\om_i\in(0, \pi /2)$, which brings us some useful experiences to deal with  domain singularities in a more general case for  water waves.

By continuity  argument of the contact angle in time similarly as in the last part of [53], one can see that as long as initial contact angles $\om_{i0}\in (0, \pi/2)$, we will have $\om_i(t)\in (0, \pi/2)$ in a short time interval.

We recall some works concerning the local well-posedness of non-smooth water waves.
In the case where there are crests or cusps on the free surface (the “crest” case),  Alazard, Burq and Zuily [5] study a special case (without surface tension) when the contact angle is equal to $\pi/2$ (the right angle), where they used symmetric and periodic extension  to turn this problem into a classical smooth periodic case.  A  breakthrough in this subject is made by Kinsey and Wu, see [41, 71].
They focus on  gravity water waves where the crest angle  is less than $\pi/2$.   The main difficulty is that the free surface is a non-$C^1$ interface with angled crest  and    the Taylor sign $-\na_{n_t}P$ degenerates at the crest point.  To be more precise,  they start  with reducing the water waves problem into the following equation in [41, 71]:

\[
\pa_t^2 u+\mathfrak a\na_{n} u= f(u, \pa_t u)
\]

where $\mathfrak a=-\na_{n_t}P$ is the Taylor sign. When $\mathfrak a$ degenerates at the crest point, the above system will loss its hyperbolicity, and classical analysis  does not apply any more. To solve the problem, Kinsey and Wu flatten the domain with a Riemann mapping, and some singular weights appear naturally in their equation. As a result, they introduce some weighted Sobolev spaces accordingly for  energy estimates to deal with these singular weights. Based on these works, Agrawal [1] show that these singularities are “rigid", which means that the angle of these crests can not change in time. Very recently,
Córdoba, Enciso and Grubic [20]  study a  similar case with cusps and crests without gravity, where the angles of these crests are less than $\pi/2$ and change in time.

For the other case where  there are  contact angles (that is $\G_t\cap\G_b\neq\emptyset$),  things become different from the “crest” case. First, the corners appear due to intersections of the free surface and the bottom (or wall); Second, there are different boundary conditions in corresponding elliptic systems compared to the crest case. In fact, various boundary conditions may have no big difference if we only focus on the elliptic theory, but there will be a series of consequences in water waves when boundary conditions change. For example, the evolution of the free surface is different from the crest case. Moreover, when there is surface tension, boundary conditions at the contact points as (<ref>) are needed  in order to close the system, and dissipations appear at the contact points too (See Section 4 and [53]).

In the case with contact angles,  de Poyferré [24] prove a priori estimates in bounded n-dimensional corner domains without surface tension. The contact angle is assumed to be small to ensure sufficient Sobolev regularity near the corner. Meanwhile, under a similar assumption of small contact angles, we  obtain the local well-posedness  in a two-dimensional corner domain (beach type) with surface tension, see [53, 54].  Meanwhile, we notice that [24] and [53, 54] use similar geometry formulations  introduced in [61].

To explain why  small contact angles are needed in [53, 54, 24],   we look at a typical mixed-boundary system in water waves:

\[
\begin{cases}{}
\D u=0\qquad in \quad \Om_t,\\
u|_{\G_t}=f,\quad \na_{n_b}u|_{\G_b}=0.
\end{cases}
\]

In fact, the elliptic theory on corner domains is well known already, see for example [44, 31, 45]. Generally, one still has variational solution $u\in H^1(\Om_t)$ if the right-side functions lie in proper Sobolev spaces. If one seeks for $H^2$ and above regularities,  singularity decompositions are  needed naturally for the solution, which decompose the solution into a singular  part $u_s$ (i.e. not good enough) near the corners and a regular part $u_r$:

\[
u=u_s+u_r\qquad\hbox{according to the required regularity.}
\]

For the mixed-boundary system above, the most singular part in $u_s$  is like $r^{\pi/2\om}$, where $r$ is the radius with respect to the corner point and $\om$ is the contact angle.   Consequently,  when  $\om$ is small enough, the singular part $u_s$ will be good enough so that we will have enough regularities from elliptic systems as in classical water waves to close the energy estimates. One can also find a singularity decomposition for $v$ in Proposition <ref> (Section 2).

In contrast, when the contact angle is larger or more general, the idea of taking small angles to improve  regularities in [24, 53, 54] does not work any more. Meanwhile,  there is no obvious weighted space to use  due to the structure of the water waves problem.

We want to show here the main  ingredients of this paper.
Similarly as in our previous works [53, 54], we still adopt the geometric formulation from Shatah and Zeng [61, 62]. In fact, we rewrite $(\hbox{WW})$ into an equation for $\mJ=\na \mK_\cH$ with boundary conditions, where $\mK$ is the modified mean curvature on $\G_t$ and $\cH$ means the harmonic extension in $\Om_t$  (see Section 3, and we only present a simpler form for this equation here):

D^2_t+=R  with  =(-__t^⊥),
where $R$ is some remainder part.

The  trouble here  is that  singularities from the domain $\Om_t$ affect directly the regularity of the solution to $\mbox{(WW)}$, which means  singularities always appear in  related elliptic systems   even if the boundary conditions are good enough in Sobolev spaces (see for example [31]). More precisely,  the natural  norm $\|\cA^{k}f \|_{L^2(\Om_t)}+\|f\|_{L^2(\Om_t)}$ with $k\geq 1$ arising from this equation above is not equivalent with  $\|f \|_{H^{3k}(\Om_t)}$ due to  larger contact angles, and apparently “singular parts" are contained in this norm.

Due to this kind of singularities from elliptic systems, we only have limited regularities for some quantities (for example, the velocity $v$) in Sobolev spaces.
Compared to [53, 54], we must make full use of the maximal regularity for each quantity (especially for $v$) while the contact angles $\om_i\in (0, \pi/2)$.  To do this, some  delicate estimates together with singular parts from singularity decompositions (see [52, 31]) are carefully used.    Meanwhile, it is also very important to gain more information from the  structure of $\mbox{(WW)}$.

The main part of the (lower-order) energy functional is  defined as

\[
E(t)=\|D_t \mJ\|^2_{L^2(\Om_t)}+\|  \mJ\cdot n_t\|^2_{H^1(\G_t)},
\]

which gives us the following estimate (see Section 4.1)

\[
\|v\|_{H^2(\Om_t)}+\|D_t v\|_{H^{3/2}(\Om_t)}+\|D_t^2 v\|_{L^2(\Om_t)}+\|\na v\|_{L^\infty(\Om_t)}+\|\kappa\|_{H^2(\G_t)} \leq P(E(t))
\]

with $P(E(t))$ the positive-coefficient polynomial of $E(t)$.
This means that the free surface $\G_t$ still has enough regularity and $v$ is Lipschitz.

In our previous work [53, 54], one main part from the remainder term $R$ in (<ref>) is  about the higher-order derivative terms of $P_{v,v}$, where $P_{v, v}$ is defined by an elliptic system with mixed-boundary conditions. According to the elliptic theory on corner domains (see for example [31]), the  system of $P_{v,v}$ only gives $P_{v,v}$ limited regularity around $H^2(\Om_t)$, when   the contact angles are less than $\pi/2$. To improve its regularity, we modify the definition of $P_{v, v}$ in this paper to have a Neumann-boundary system, see
(<ref>). In fact, thanks to the elliptic theory,  solutions of Neumann-boundary system (or Dirichlet system) may become more regular than solutions of mixed-boundary problem in Sobolev spaces, while the right-side data have the same regularity.  As a result, we obtain a bit more regularity from the elliptic system of $P_{v,v}$ (we have $P_{v,v}\in H^3(\Om_t)$ indeed), which is important in the energy estimates.

Moreover, another main part of $R$ lies in the higher-order derivative terms of the velocity  $v$. Here, we point out that compared to [53, 54], $v$ loses some regularity due to the existence of corners. Fortunately,  $D_t v$ and $D_t^2 v$ have the same regularities as before. As a result, when we deal with $v$, sometimes we need to  use material derivatives $D_t$ instead of spatial derivatives.  Meanwhile, we also need to apply singularity decompositions to $v$ and its potential $\phi$ in the estimates, see for example Lemma <ref>.

For the higher-order energy estimates, we use the material derivative $D_t$ instead of $\cA^{1/2}$ from [61]. The higher-order energy is defined as

\[
E_1(t)=\|\na_{\tau_t} D_t  (\mathfrak{J}\cdot n_t) \|^2_{L^2(\Gamma_t)} +\|D^2_t \mathfrak{J}\|^2_{L^2(\Om_t)}.
\]

In fact, one will see that $D_t$ is convenient to use when there are contact points. For example, taking $D_t$ on elliptic systems does not violate  boundary conditions, while it will change boundary conditions if one takes spatial derivatives. Moreover, one will find in the higher-order energy estimate that taking $D_t$ leads to better regularities for some quantities  than their own regularities (like $D_t v$ for the velocity $v$). However, due to the singularities of the boundary (contact points), it is not as convenient as before to turn these regularities into spatial regularities, which explains the reason why we  need to choose very carefully for “a suitable space” in Theorem <ref>.  As long as the energy estimates for $E(t), E_1(t)$ are finished, we  use the equation of $\mJ$ to gain more spatial regularities.

In the end,  we mention some other related works. Lannes and Métivier [49] studied the Green-Naghdi system in a beach-type domain, which is a shallow-water model of the water waves problem. Lannes [48] studied  the floating-body problem and proposed a new formulation  that can be easily  generalized in order to take into account the presence of a floating body.  Lannes and Iguchi [38] proved some sharp results for initial boundary value problem with a free boundary arising in wave-structure interaction, and it contains the floating problem in the shallow water regime. Besides, Guo and Tice [32] showed  a priori estimates for the contact line problem in the case of the stokes equations. Later on, Tice and Zheng proved the local well-posedness of the contact line problem in two-dimensional Stokes flow, see [66]. In 2020, Guo and Tice [33] proved a priori estimates for the contact line problem for two-dimensional Navire-Stokes flow. For  Darcy's flow, one can see [42, 43].

§.§ Organization of the paper

In Section 2, we present various useful lemmas including  singularity decompositions and estimates for elliptic systems. In Section 3, we derive the equation for the good unknown $\mJ$ from $(\hbox{WW})$ with modified curvature $\mK$ and modified pressure $P_{v,v}$. The lower-order energy is constructed and the energy estimate is proved in Section 4, where estimates for various quantities like $\G_t$, $v$, $P_{v,v}$ are proved. Moreover, we consider the higher-order energy estimate using $D_t$  in Section 5. In Section 6, we present the precise main theorem in our paper and  show the local well-posedness.

§.§ Notations

- $X$ stands for a point in $\Om_t\subset \R^2$. $p_l, p_r$ are the left and right contact points. $n_j (j=t, b)$ are the unit outward normal vectors on $\G_j$, and $\tau_j$ are the corresponding unit tangential vectors obeying the right-hand rule with $n_j$.

- $\si$ is the surface tension coefficient. $\beta_c$ is the effective friction coefficient determined by interfacial widths, interactions between the fluid and the bottom, and the normal stress contributions.

- $\chi_{\om}$ is a characteristic function of  contact angles:

\[
\chi_{\om}(\theta)=
\begin{cases}
1,\quad \theta\in (\pi/3, \pi/2),\\
0,\quad \theta\in (0,\pi/3].
\end{cases}
\]

- $\chi_i$ ($i=l, r$) are cut-off functions near the corner points $p_i$:

\[
\chi_{i}(X)=
\begin{cases}
1,\quad \hbox{when}\ |X-p_i|\le r_0,\ X\in \Om_t;\\
0,\quad\ \hbox{otherwise}
\end{cases}
\]

with some small $r_0>0$.

- $f|_c=\chi_l (f|_{p_l})+\chi_r (f|_{p_r})$ stands for taking values of $f$ at the corner points.

- $\cS_{t,i}$ are straightened sector of $\Om_t$ with radius $r_0>0$ near the corner points $p_i$.

- $D_t=\p_t+\na_v$ is the material derivative.

- $M^*$ denotes the transport of a matrix $M$.

- $w^\perp$ on $\G_t$:   $w\cdot n_t$ for a vector $w\in T_X\G_t$.

- $w^\top$ on $\G_t$: $(w\cdot \tau_t)\,\tau_t$. Sometimes we also use $w^\top$ on $\G_b$ with a similar definition.

- $\Pi$ is the second fundamental form on $\G_t$, where $\Pi(w)=\na_w n_t\in T_X\G_t$ for $w\in T_X\G_t$.

- $\kappa=tr \Pi=\na_{\tau_t}n_t\cdot \tau_t$ is the mean curvature of the surface $\G_t$.

- We define on $\G_t$ that $\cD w=\cD_{\tau_t}w=(\na_{\tau_t}w)^\top=(\na_{\tau_t}w\cdot \tau_t\big)\tau_t$ for a vector $w\in T_X\G_t$.

- $\cD^2f(\tau_1, \tau_2)=D^2f(\tau_1,\tau_2)-\big(\Pi(\tau_1)\cdot \tau_2\big)\na_{n_t}f$ for any two vector $\tau_1, \tau_2\in T_X\G_t$.

- $\D_{\G_t}$ is the Beltrami-Laplace operator on $\G_t$:

\[
\D_{\G_t}f=\cD^2f(\tau_t, \tau_t)=\na_{\tau_t}\na_{\tau_t}f-\na_{(\na_{\tau_t}\tau_t)^\top}f.
\]

- $\cH(f)$ or $f_\cH$ is the harmonic extension for some function $f$ on $\G_t$, which is defined by the elliptic system

\[
\left\{\begin{array}{ll}
\D \cH(f)=0\qquad\hbox{in}\quad \Om_t,\\
\cH(f)|_{\G_t}=f,\quad \na_{n_b}\cH(f)|_{\G_b}=0.
\end{array}\right.
\]

- $\cN=\na_{n_t}\cH$ is the Dirichlet-Neumann operator on $\G_t$.

- $\D^{-1}(h,g)$ stands for  the solution $u$ to the system

\[
\left\{\begin{array}{ll}
\D u=h\qquad \hbox{in}\quad \Om_t\\
u|_{\G_t}=0,\qquad \na_{n_b}u|_{\G_b}=g.
\end{array}
\right.
\]

- The Sobolev norm $H^s$ for the boundary $\G_t$ or $\G_b$ is defined by local coordinates and local graphs.

- $\tilde H^{1/2}(\G_j)$ ($j=t,b$) (see [31]) is a subspace of $H^{1/2}(\G_j)$ related to corner domains

\[
\tilde H^{1/2}(\G_j)=\Big\{u\in \dot{H}^{1/2}(\G_j)\Big| \,\rho_i^{-1/2}u\in L^2(\G_j),\ i=l, r\Big\}\]
where $\dot H^{1/2}(\G_j)$ is the closure of $\mathscr D(\G_j)$ in $H^{1/2}(\G_j)$,  and $\rho_i=\rho_i(X)$ ($i=l, r$) is the distance (arc length) between the point $X\in \G_j$ and the end $p_i$.  We define the norm

\[
\|u\|^2_{\tilde H^{1/2}(\G_j)}=\|u\|^2_{H^{1/2}(\G_j)}+\int_{\G_j} \rho_l^{-1}|u|^2dX+\int_{\G_j} \rho_r^{-1}|u|^2dX.
\]

- $\tilde H^{-1/2}(\G_j)$ stands for the dual space of $\tilde H^{1/2}(\G_j)$. For more details, see [31].

- We define $P_{w,v}$ (with $w\neq v$) by the following Neumann system:

P_w,v=-tr (w v),    in  _t

_n_t P_w,v|__t=C_w,v(t)-(w·τ_t) _τ_t v·n_t,
_n_b  P_w,v|__b=w·_v n_b.
Here $C_{w,v}$ is a function of $t$ satisfying the compatibility condition

\[
|\G_t|C_{w, v}(t)= -\int_{\Om_t}tr (\na w \na v)dX+\int_{\G_t}(w\cdot\tau_t) \na_{\tau_t} v\cdot n_t  ds-\int_{\G_b}w\cdot \na_v n_b ds.
\]

- $P_{v,v}$ is defined in (<ref>) from Section 3.

- $C(a)$ stands for a positive constant $C$ depending on a quantity $a$. $P(E(t))$ stands for a polynomial of the energy $E(t)$ with positive constant coefficients.

§ PRELIMINARIES

To get started, we recall some useful estimates for Sobolev space.
(Sobolev Embeddings) One has the following inclusion:

\[
W^{s, p}(\Om)\subseteq L^q( \Om)
\]
for $1/q=1/p-s/n$ and $\Om$ any bounded open subset of $\R^n$ with a Lipschitz boundary. Moreover, one also has

\[
W^{s, p}(\Om)\subseteq W^{t,q}(\bar\Om)
\]

for $t\le s, q\ge p$ such that  $s-n/p=t-n/q$.

(Product estimates)
(1) For functions $f\in H^{1/2}(\Om_t)$ and $g\in H^1(\Om_t)\cup L^\infty(\Om_t)$, one has the following product estimate:

\[
\|f\,g\|_{H^{1/2}(\Om_t)}\le C\|f\|_{H^{1/2}(\Om_t)}\big(\|g\|_{H^1(\Om_t)}+\|g\|_{L^\infty(\Om_t)}\big)
\]

with a constant $C$ independent of $f, g$;

(2) For functions $f, g\in L^\infty(\G_t)\cup H^{1/2}(\G_t)$, one has

\[
\|f\,g\|_{H^{1/2}(\G_t)}\le C \big(\|f\|_{L^\infty(\G_t)}\|g\|_{H^{1/2}(\G_t)}+\|f\|_{H^{1/2}(\G_t)}\|g\|_{L^\infty(\G_t)}\big)
\]

with a constant $C$ independent of $f, g$.

(1) In fact, one firstly extends $f, g$  to be defined on the full plane $\R^2$ with a control of their corresponding norms. Secondly, one can apply standard para-product analysis to prove the estimate on $\R^2$. The details are omitted here. (2) The proof is similar to the proof of (1).

We quote some Hardy inequalities here.
(Hardy inequalities) (1)(Corollary 2.3[16]) Let $f\in H^1(0, d)\cup C^0(0, d)$ with $d>0$ and  $f(0)=0$. Then there exists  a positive  constant $C=C(\eps, d)$ such that

\[
\int^d_0 r^{-2\epsilon}|f(r)|^2dr\le C\int^d_0r^{-2\epsilon+2}|f'(r)|^2dr
\]

for $\eps\in (1/2, 1)$;

(2) (Fractional-order version, see [27, 30]) For a number $\epsilon\in (0, 1)$ and any function $f\in H^\epsilon(\R^+)$ with $f(0)=0$, there exists a positive constant
$C=C(\epsilon)$ such that

\[
\int_{\R^+}r^{-2\epsilon}|f(r)|^2dr\le C\int_{\R^+}\int_{\R^+}\frac{|f(r)-f(\rho)|^2}{|r-\rho|^{1+2\epsilon}}drd\rho.
\]

(Traces on $\G_t$ or $\G_b$, Theorem 5.3 [53])
The maps

\[
u\mapsto \{u, \na_{n_j} u\}|_{\G_j},\quad \hbox{for}\quad j=t, b
\]

have  unique continuous extensions as  operators from
$H^{s}(\Om_t)$ onto $\Pi^1_{i=1} H^{s-i-1/2}(\Gamma_j)$ for $s>3/2$.

Moreover, one has the estimate :

\[
\|u\|_{H^{s-1/2}(\G_j)}+\| \na_{n_j} u\|_{H^{s-3/2 }(\Gamma_j )}\le    C\big(\|\Gamma_t\|_{H^{s-1/2} }\big)\|u\|_{H^{s}(\Om_t)}.
\]

Next,  we present some some special trace theorems on corner domains involving $\tilde H^{1/2}(\G_j)$ and $\tilde H^{-1/2}(\G_j)$ ($j=t, b$).

Assume that $u|_{\G_t}=f$ with $f|_{p_i}=0$ ($i=l,r$) for a function $u\in H^1(\Om_t)$. Then one has $f\in \tilde H^{1/2}(\G_t)$ and

\[
\|f\|_{\tilde H^{1/2}(\G_t)}\le C(\|\Gamma_t\|_{H^{5/2} }) \|u\|_{H^1(\Om_t)}.
\]

The case on $\G_b$ holds similarly.

Proof. First, one has $f\in H^{1/2}(\G_t)$ immediately by Lemma <ref>. Second, noticing $f|_{p_i}=0$ and applying Lemma <ref>(2) with some straightening localizations near $p_i$, one can see  that $f\in \tilde H^{1/2}(\G_t)$ with the desired estimate. Moreover, a similar lemma can be found as Lemma 5.5 [53].

(Lemma 5.6[53])
Let $u\in H^{1/2}(\G_j)$ ($j=t, b$), then $\na_{\tau_j} u$ belongs to  $\tilde H^{-1/2}(\G_j)$ and satisfies the estimate

\[
\|\na_{\tau_b}u\|_{\tilde H^{-1/2}(\G_j)}\le C(\|\Gamma_j\|_{H^{5/2} } ) \|u\|_{H^{1/2}(\G_j)}.
\]

We now recall  some elliptic systems and estimates including singular decompositions in corner domains.
First, for  the  mixed-boundary system

\[
\mbox{(MBVP)}\quad\left\{\begin{array}{ll}
\Delta u=h,\qquad \hbox{in}\quad \Omega_t\\
u\,|_{\G_t}=f,\qquad \na_{n_b}u \,|_{\G_b}=g,
\end{array}\right.
\]

we quote directly the following variational result.
(Lemma 5.9 [53])
For a given function $f\in H^{1/2}(\G_t)$, the system

(f)=0    in  _t,

(f)|__t=f,  _n_b(f)|__b=0
admits a unique  solution  $f_\cH=\cH(f)\in H^1(\Om_t)$,  and  there holds

\[
\|\cH(f)\|_{H^1(\Om_t)}\le C(\|\Gamma_t\|_{H^{5/2} }) \|f\|_{H^{1/2}(\G_t)}.
\]

The following proposition shows the existence and estimate for solutions in $H^2(\Om_t)$. Notice that when the contact angles are  below $\pi/2$, no singularity appears.

Let $h\in L^2(\Om_t)$, $f\in H^{3/2}(\G_t)$, $g\in H^{1/2}(\Gamma_b)$ and $\G_t \in H^{5/2}$ be given in $\mbox{(MBVP)}$. The contact angles $\om_i\in(0, \pi/2)$.
Then there exists a unique solution $u\in H^2(\Om_t)$ to $\mbox{(MBVP)}$.
Moreover, one  has

\[
\|u\|_{H^2(\Om_t)}\le C\big(\|h\|_{L^2(\Om)}+\|f\|_{H^{3/2}(\G_t)}+\|g\|_{H^{1/2}(\G_b)}\big)
\]

with the constant $C=C(\|\G_t\|_{H^{5/2}})$.

This proposition is a direct conclusion from Proposition 5.1, Lemma 5.2 and Theorem 5.3 in [52].

Second, we consider  the Neumann-boundary system:

Δu=h  in  _t,

_n_tu|_Γ_t=f,    _n_b u |_Γ_b=g

satisfying the compatibility condition
∫__t hdX=∫_Γ_t fds+∫_Γ_bgds.

The following results  for $H^3$-and-below case are needed in our paper.

(1)  Let $h\in L^2(\Om_t)$, $f\in L^2(\G_t)$, $g\in L^2(\Gamma_b)$  in $\mbox{(NBVP)}$.
Then there exists a unique  (up to an additive constant) variational solution $u\in H^1(\Om_t)$ to  $\mbox{(NBVP)}$ satisfying

\[
\|u\|_{H^1(\Om_t)}\le C(\|\G_t\|_{H^{5/2}})\big(\|h\|_{L^2(\Om_t)}+\|f\|_{L^{2}(\G_t)}+\|g\|_{L^{2}(\G_b)}+\|u\|_{L^2(\Om_t)}\big);
\]

(2) Let $h\in H^s(\Om_t)$, $f\in H^{1/2+s}(\G_t)$, $g\in H^{1/2+s}(\Gamma_b)$  in $\mbox{(NBVP)}$. The contact angles $\om_i\in(0, \pi/2)$.
Then there exists a unique  (up to an additive constant) solution $u\in H^3(\Om_t)$  to  $\mbox{(NBVP)}$ satisfying

\[
\|u\|_{H^{2+s}(\Om_t)}\le C(\|\G_t\|_{H^{3}})\big(\|h\|_{H^s(\Om_t)}+\|f\|_{H^{1/2+s}(\G_t)}+\|g\|_{H^{1/2+s}(\G_b)}+\|u\|_{L^2(\Om_t)}\big)
\]

for a constant $s\in [0,1]$.

Proof. We only write a sketch for the proof here, since one can find similar details in our previous papers [52, 53, 54].  First, the existence of the variational solution $u\in H^1(\Om_t)$ can be proved directly by a standard variation procedure based on  Lemma 4.4.3.1[31]. Note that here one doesn't require that the contact angles are below $\pi/2$.
Second, one needs to notice that, there is no singular part  since the most singular part for $\mbox{(NBVP)}$ behaves like $r^{\pi/\om}$ near the corners (see for example [31]). Therefore, one can have  directly  $H^2$ estimate from Theorem 3.2.5[11] and Proposition 5.1[52], Theorem 4.3.1.4[31]. Similarly to the proof for Proposition 5.13[52], one has $H^3$ estimate without singular part. As a result, the desired estimate in (2) can be proved by an interpolation.

For the first estimate in Proposition <ref>, when one has additionally

\[
\int_{\Om_t}udX=0,
\]

the norm $\|u\|_{L^2(\Om_t)}$ can be deleted from the right side during the proof.

Moreover, we present the $H^4$ singularity decomposition and estimate.

Let $h\in H^2(\Om_t)$, $f\in H^{5/2}(\G_t)$, $g\in H^{5/2}(\Gamma_b)$ and $\G_t \in H^{4}$ in $\mbox{(NBVP)}$. The contact angles $\om_i\in(0, \pi/2)$.
Then there exists a unique  (up to an additive constant) solution $u\in H^3(\Om_t)$ to  $\mbox{(NBVP)}$ such that

\[
u=u_r+ u_s\quad\hbox{with the singular part}\ u_s=\chi_{\om}(\om_l)\chi_{l}\,c_l  S_l\circ T_l+\chi_{\om}(\om_r)\chi_{r} \, c_r  S_r\circ T_r,
\]

and the regular part $u_r\in H^4(\Om_t)$. Here the cut-off functions $\chi_\om, \chi_i$ are defined in the notation part.
$T_i\in H^{4}(\Om_t)$ are  boundary-straightening diffeomorphisms from $\Om_t$ onto the sectors $\cS_{t,i}$ near the corner points $p_i$ (from [52]), and $S_i=  r^{\pi/\om_i}s(\theta)$ with $r$ the radius with respect to $p_i$ in $\cS_{t, i}$ and $s_i(\theta)$ some fixed sine or cosine functions.

Moreover, one  has  estimates for the singular coefficients $c_i$ ($i=l,r$) and the regular part $u_r$:

\[
|c_l|+\|u_r\|_{H^4(\Om_t)}\le C\big(\|h\|_{H^2(\Om_t)}+\|f\|_{H^{5/2}(\G_t)}+\|g\|_{H^{5/2}(\G_b)}+\|u\|_{L^2(\Om_t)}\big)
\]

with the constant $C=C(\|\G_t\|_{H^{4}})$.

Proof.  Similarly to the proofs of Proposition 5.17, Proposition 5.18[52] for the mixed-boundary problem, the proof can be checked and follows line by line thanks to [31], so we omit the details here.

A direct conclusion from this proposition is that when the contact angles $\om_i\in (0,\pi/3]$, the solution $u$ to $\mbox{(NBVP)}$ with the same right side will be in $H^4(\Om_t)$ with the corresponding estimate.

Based on the  proposition above and estimate (9.15) in [25], when the contact angles $\om_i\in (\pi/3, \pi/2)$,  one
can have a more delicate and also natural estimate for $\mbox{(NBVP)}$  with $h\in H^{1+\eps}(\Om_t)$, $f\in H^{3/2+\eps}(\G_t)$, $g\in H^{3/2+\eps}(\Gamma_b)$ and $\G_t \in H^{4}$:

\[
\|u\|_{H^{3+\eps}(\Om_t)}\le C(\|\G_t\|_{H^{4}})\big(\|h\|_{H^{1+\eps}(\Om_t)}+\|f\|_{H^{3/2+\eps}(\G_t)}+\|g\|_{H^{3/2+\eps}(\G_b)}+\|u\|_{L^2(\Om_t)}\big),
\]

where $\eps\in (0, \pi/\om-2)\subset (0,1)$.

In the end, we recall  some useful expressions and commutators from  [61, 53, 54].

- $D_tn_t$ and $D_t\tau_t$. One has

D_t n_t=-((v)^*n_t)^⊤,  D_tτ_t=(_τ_tv·n_t)n_t.

- $[D_t,\cH]$. One has for a smooth function $f$ on $\G_t$ that

[D_t,]f=^-1(2v·^2f_+v·f_,  (_N_bv-_v N_b)·f_)   in  _t,
where $\D^{-1}(h,g)$ and $\cH$ are defined in the notation part.

- $[D_t, \cN]$.
One has

[D_t, ]f=   _n_t^-1(2v·^2f_+v·f_, (_n_bv-_vn_b)·f_)

-_n_tv·f_-_(f_)^⊤ v·n_t        on  _t.

- $[D_t,\D_{\G_t}]$.  For a smooth function $f$ on $\G_t$, there holds

[D_t, __t]f=2^2f(τ_t,(_τ_tv)^⊤)-(f)^⊤·__tv+κ_(f)^⊤ v·n_t      on  _t.

- $[D_t,\,\D^{-1}]$. We have

D_t^-1(h,g)=^-1(D_th, D_tg)+^-1(h_1, g_1)

\[
h_1=2\na v\cdot\na^2\D^{-1}(h,g)+\D v\cdot \na \D^{-1}(h,g),\quad
g_1=(\na_{N_b}v-\na_{v}N_b)\cdot \na\D^{-1}(h,g).
\]

-  $[D_t, \na_{\tau_t}]$. Direct computations lead to

[D_t, _τ_t]=(D_tτ_t-_τ_tv)·=(n_t _τ_tv·n_t-_τ_tv)·=-(_τ_tv·τ_t)_τ_t.

§ REFORMULATION OF THE PROBLEM

In this section, we derive the equation for a good unknown $\mathfrak J$, which is slightly different from the quantity $J=\na\kappa_{\cH}$ introduced in [61] and used in our previous papers [53, 54].

To begin with, we define $P_{v, v}$ by the following Neumann-boundary system:

P_v, v=-tr (v v)    in  _t

_n_tP_v, v|__t=C_v,v(t),    _n_bP_v, v|__b=v·_v n_b.

where $C_{v,v}(t)$ satisfies

\[
|\G_t| C_{v,v}(t) = -\int_{\Om_t}tr (\na v \na v)dX-\int_{\G_b}v\cdot \na_v n_b ds.
\]

Moreover, due to the non-uniqueness of the variational solution to this Neumann problem,  we assume that $P_{v, v}$ satisfies

∫__t P_v, vdX=0.
In this way, we will have  a unique solution  to (<ref>) in Sobolev space.

Let the pressure $P$ be decomposed into

P=_+  P_v,v,
where  $\mK$ is the modified curvature on $\G_t$ defined by

=σκ-P_v,v     _t.

Now,  we  can define the new quantity

\[
\mathfrak J=\na \mathfrak{K}_{\cH}
\]

and we are ready to derive its equation from $\hbox{(WW)}$. In fact, since the derivation for  the equation of $\mJ$ is much similar to
the derivation for $J$ in  [53],  many details are omitted here.

First, recall the equation for the curvature $\kappa$ from [53]:

or equivalently

D_tκ=-__tv·n_t-2Π(τ_t)·_τ_tv     on  _t.

Applying $D_t$ to (<ref>) and using (<ref>), we have

D^2_t=-__tD_tv ·n_t +2Π(τ_t)·_τ_t J+R_1- D_t^2 P_v,v    on  _t

\[
\begin{split}
R_1= - [D_t, \D_{\G_t}]v\cdot n_t-\D_{\G_t}v\cdot D_tn_t+2 \Pi(\tau_t)\cdot\na_{\tau_t}\na P_{v,v}
-2 \Pi(\tau_t)\cdot[D_t, \na_{\tau_t}]v-2  D_t\big(\Pi(\tau_t)\big)\cdot\na_{\tau_t}v.
\end{split}
\]

Moreover, direct computations almost the same as in [53] lead to

where the remainder terms

A_1=[D_t,]D_t=   ^-1(2v·+v·,(_n_bv-_vn_b)·)

(D_t-^-1(2v·+v·, (_n_bv-_v n_b)·)+(v)^*),

A_2   =D_t^-1(2v·+v·,(_n_bv-_vn_b)·)

=^-1(2v·^2w_A2+v·w_A2,(_n_bv-_vn_b)·w_A2) +^-1(h_A2,  g_A2)

\[
\begin{split}
w_{A2}=&\D^{-1}\big(2\na v\cdot \na \mJ+\D v\cdot \mJ,\,(\na_{n_b}v-\na_v n_b)\cdot \mJ\big),\\
h_{A2}=&2\na v\cdot(\na D_t\mJ-(\na v)^*\mJ)+2(\na D_tv-(\na v)^*\na v)\cdot \na \mJ+D_t\mJ\cdot\D v\\
&+\mJ\cdot (\D D_tv-\D v\cdot\na v-2\na v\cdot \na^2 v),\\
g_{A2}=&(\na_vn_b-\na_{n_b}v)\cdot \na v\cdot \mJ+\na_{n_b}D_t v\cdot \mJ-(D_t v-\na_v v)\cdot \na n_b\cdot \mJ\\
& +\na_v\big((\na v)^*n_b\big)^\top\cdot \mJ+(\na_{n_b}v-\na_vn_b)\cdot D_t\mJ,
\end{split}
\]

A_3=-2(v)^*D_t-(D_t v)^* -((v)^2)^*+((v)^*v).
Here we note that the leading-order terms in $A_1, A_2$  are like $\mJ,D_t\mJ,\na v,\na D_tv$.

Substituting (<ref>) into (<ref>) and applying Euler's equation, we can have after some more computations that

D^2_t=(__t^⊥)+R_0+D_t(  D_t P_v,v ) .

\[
R_0=-\si\na \cH(\mJ\cdot \D_{\G_t}n_t)+\si \na \cH(n_t\cdot \D_{\G_t}\na P_{v,v})+\si \na\cH(R_1 ) +A_1+A_2+A_3+[D_t, \na \cH](  D_t P_{v,v}).
\]

Here, one observation is that since $\na_{n_t} P_{v,v}\big|_{\G_t} =C_{v,v}(t)$ from (<ref>), we find immediately in $R_0$ that

\[
n_t\cdot \D_{\G_t}\na P_{v,v} =[n_t,\D_{\G_t} ] \cdot\na P_{v,v} .
\]

Moreover, we use the following Hodge decomposition from [61, 53]:

_t =D_t +_,v
such that $\cD_t \mJ$ satisfies

\[
\na\cdot \cD_t\mJ=0\quad\mbox{on} \ \Om_t,\quad\hbox{and}\quad  n_b\cdot\cD_t\mJ\big|_{\G_b}=0,
\]

$\cP_{\mJ,v}$ satisfies the Neumann-boundary system

_,v=-tr (v)    in  _t

_n_t _,v|__t=C_,v(t)-_τ_t  _τ_t v·n_t,  _n_b  _,v|__b= ·_v n_b

\[
|\G_t| C_{\mJ,v}(t) = -\int_{\Om_t}tr (\na \mJ \na v)dX+\int_{\G_t}\na_{\tau_t}  \mK \na_{\tau_t} v\cdot n_t  ds-\int_{\G_b}\mJ\cdot \na_v n_b ds.
\]

Here the assumption

\[
\int_{\Om_t} \cP_{\mJ, v}dX=0
\]

holds similarly to guarantee the uniqueness as for $P_{v,v}$ system.

This implies immediately

\[
\begin{split}
D_t\cD_t\mJ= D_t(D_t\mJ+\na \cP_{\mJ,v})=D^2_t\mJ+D_t\na \cP_{\mJ,v}.
\end{split}
\]

As a result, we finally conclude the equation for $\mJ$ in the following form based on (<ref>):

D_t[_t -(  D_t P_v,v -v·(P_v,v|_c))]+=where the third-order elliptic operator $\cA$ is defined in  the same way  as in [61, 53]:

\[
\cA(w)=\na\cH\big(-\D_{\G_t}(w|_{\G_t})^\perp\big)
\]

for any smooth-enough function $w$ defined on $\Om_t$,
and the remainder term $\cR$ is

\[
\cR=R_0  +D_t\na \cP_{\mJ,v}+D_t\na\cH\big( v\cdot(\na P_{v,v}|_c)\big).
\]

Moreover,  one can find the definition of $\na P_{v,v}|_c$ in the notation part.

Instead of (<ref>), we write the equation of $\mJ$ in a more complicated form (<ref>). This happens because of technical reasons. In fact, $\cD_t \mJ$ is more handy to use in the energy estimates, while as a price we have the remainder $\cP_{\mJ,v}$ part to deal with. Meanwhile, the part $v\cdot(\na P_{v,v}|_c)$ is added to derive better estimate for $D_tP_{v,v}$, see Lemma <ref>.

§ LOWER-ORDER ENERGY ESTIMATES

The lower-order energy $E(t)$ is defined as

\[
E(t)=\|\na_{\tau_t}\mJ^\bot\|^2_{L^2(\Gamma_t)} +\|\cD_t \mJ\|^2_{L^2(\Om_t)}+\|\Gamma_t\|^2_{H^{5/2}}+\|v\|^2_{H^{3/2}(\Om_t)},
\]

where  we write

\[
E_l(t)= \|\Gamma_t\|^2_{H^{5/2}}+\|v\|^2_{H^{3/2}(\Om_t)}.
\]

Moreover, recalling from [53, 54] that our $\mbox{(WW)}$ system has some dissipation at corner points,  we have the following dissipation term at corner points

\[
F(t)=\sum_{i=l,r}\big|(\sin \om_i)\na_{{\bf \tau}_t}\mJ^\perp |_{p_i}\big|^2.
\]

Let the contact angles $\om_i\in (0,\pi/2)$. Assume that  $E(t), \int_0^T F(t)dt$ are both bounded above in $[0, T]$ for some  $T>0$. Then  the following a priori estimate holds for system $\mbox{(WW)}$:

\[
\sup_{0\leq t\leq T} E(t)+\int_0^T F(t)dt\le P(E(0))+\int_0^T P(E(t))dt,
\]

where $P(\cdot)$ is some polynomial with positive constant coefficients depending on $\si, \beta_c$.

One can see immediately that, our energy is defined mainly in forms of $\mJ$. As a result, we need to related $(v, P)$, $\G_t$ and other quantities to $\mJ$ firstly before we prove this energy estimate.

§.§ Dependence on $E(t)$

We show in this part that all the quantities related to our problem can be controlled by $E(t)$.
To start with, the following proposition focuses on the estimates for $P_{v,v}, n_t, \mK_\cH, \mJ$.

Assuming that $E(t)\in L^\infty[0,T]$ for some $T>0$, one has the following estimates

\[
\|P_{v, v}\|_{H^{2}(\Om_t)} \leq P(E_l(t))
\]

\[
\|n_t\|_{H^2(\Gamma_t)}+\|\mJ\|_{H^{3/2}(\Om_t)}+\|\mK_\cH\|_{H^{5/2}(\Om_t)}\leq  P(E(t)).
\]

Proof. (1) The first inequality. Applying Proposition <ref> with an interpolation to $P_{v,v}$ system (<ref>), we get

\[
\|P_{v, v}\|_{H^{2}(\Om_t)} \leq C(\|\Gamma_t\|_{H^{5/2}})\big(\|tr(\na v \na v)\|_{L^2(\Om_t)}+|\G_t|^{1/2}|C_{v,v}(t)|+\|v\cdot \na_v n_b\|_{H^{1/2}(\G_b)}+\|P_{v,v}\|_{L^2(\Om_t)}\big).
\]

For the right-hand side, we have firstly  by Lemma <ref> that

\[
\|tr(\na v \na v)\|_{L^2(\Om_t)}\le C(\|\Gamma_t\|_{H^{5/2}})\|v\|^2_{H^{3/2}(\Om_t)}.
\]

Second, it's straightforward to find

\[
|C_{v,v}(t)|\le C\|v\|^2_{H^{3/2}(\Om_t)}
\]

\[
\begin{split}
\|P_{v,v}\|_{L^2(\Om_t)}&=\big\|P_{v,v}-\int_{\Om_t}P_{v,v}dX\big\|_{L^2(\Om_t)}\le C\|\na P_{v,v}\|_{L^2(\Om_t)}\\
&\le C(\|\Gamma_t\|_{H^{5/2}})\big( \|tr(\na v \na v)\|_{L^2(\Om_t)}+|\G_t|^{1/2}|C_{v,v}(t)|+\|v\cdot \na_v n_b\|_{L^2(\G_b)}\big),
\end{split}
\]

where (<ref>) is used and the variational estimate above is proved similarly as Proposition <ref>(1).

Consequently, summing up all these inequalities above, we conclude that

\[
\|P_{v, v}\|_{H^{2}(\Om_t)} \le C(\|\Gamma_t\|_{H^{5/2}})\|v\|^2_{H^{3/2}(\Om_t)}\leq P(E_l(t)).
\]

(2) The second inequality. In fact, the key point of the proof  lies in the estimates for mean curvature $\kappa$ and $\mK=\sigma\kappa-P_{v,v}$.

To begin with, one can see immediately that $\mK_\cH$ satisfies the system

_=0    in  _t

_n_t_|__t=^⊥|__t∈H^1(Γ_t),    _n_b _|__b=0.
Applying Proposition <ref>(2), we find

__H^5/2(_t)   ≤C(Γ_t_H^5/2)(^⊥_H^1(Γ_t)+ __L^2(_t))

≤C(Γ_t_H^5/2)( _τ_t ^_L^2(Γ_t) + __L^2(_t)),

where interpolation for $\mK_\cH$ is applied to $\|\mJ^\perp\|_{L^2}$.

Moreover, from the definition of $\mK$ and Lemma <ref> we have

\[
\| \mK_\cH\|_{L^2(\Om_t)}\le C(\|\Gamma_t\|_{H^{5/2}})\big(\|\kappa\|_{H^{1/2}(\G_t)}+\|P_{v,v}\|_{H^{1/2}(\G_t)}\big),
\]

which together with the previous inequality leads to

__H^5/2(_t)≤C(Γ_t_H^5/2) (1+ _τ_t ^_L^2(Γ_t)+  P_v, v_H^1/2(_t) ).
This implies immediately the desired $H^{3/2}$ estimate for $\mJ$.

For the estimate for $n_t$, we find by Lemma <ref>, (<ref>) and part (1) that

κ_H^1(Γ_t)    ≤^-1( _H^1(Γ_t)+ P_v,v_H^1(Γ_t))

≤C(Γ_t_H^52) (1+ _τ_t ^_L^2(Γ_t)+  P_v, v_H^3/2(_t))

As a result, the proof is finished thanks to  the fact

\[
\kappa=\na_{\tau_t}n_t\cdot \tau_t\quad\hbox{where}\ \na_{\tau_t}n_t\| \tau_t\quad\hbox{on}\ \G_t.
\]

Assuming that $E(t)\in L^\infty[0,T]$ for some $T>0$, one has the following estimates for $v$:

\[
\|v\|_{H^2(\Om_t)}+\|v^\perp\|_{H^{5/2}(\G_t)}\leq P(E(t)).
\]

Meanwhile, one also has

\[
\|D_tP_{v, v}\|_{H^{1}(\Om_t)} +\|\cP_{\mJ, v}\|_{H^1(\Om_t)}\leq P(E(t))
\]

Proof. - $H^1$  estimate for $\cP_{\mJ, v}$.
From  system (<ref>) of $\cP_{\mJ, v}$, we can have the following variational estimate by Proposition <ref>(1) and Remark <ref>:

\[
\begin{split}
\|\cP_{\mJ, v}\|_{H^{1}(\Om_t)} \leq &C(\|\Gamma_t\|_{H^{5/2}})\big(\|tr(\na \mJ \na v) \|_{L^2(\Om_t)}+|\G_t|^{1/2}|C_{\mJ,v}(t)|+\|\na_{\tau_t}  \mK\cdot \na_{\tau_t} v\cdot n_t\|_{L^2(\Gamma_t)}\\
+\|\mJ\cdot \na_v n_b\|_{L^2(\G_b)}\big),
\end{split}
\]

which together with Lemma <ref> leads to the desired estimate for $\cP_{\mJ, v}$.

- $v^\perp$ estimate. In fact, it is straightforward  to get by Lemma <ref> that

\[
\|D_t \kappa\|_{H^{1/2}(\G_t)}  \le C(\|\G_t\|_{H^{5/2}})\|D_t\kappa_\cH\|_{H^1(\Om_t)}
\leq C(\|\G_t\|_{H^{5/2}})(\|\na D_t \kappa_\cH\|_{L^2(\Om_t)} +\|D_t \kappa\|_{L^2(\G_t)} ).
\]

Rewriting $\na\kappa_\cH$ in terms of $\mJ, \na P_{v,v}$ leads to

\[
\begin{split}
\|D_t \kappa\|_{H^{1/2}(\G_t)}
&\leq C(\|\G_t\|_{H^{5/2}})\big(\|D_t \mJ\|_{L^2(\Om_t)}+\|D_t\na P_{v,v}\|_{L^2(\Om_t)}+\|\na v \cdot\mJ\|_{L^2(\Om_t)}+\|\na v\cdot \na P_{v,v}\|_{L^2(\Om_t)}\\
&\qquad  +\|D_t \kappa\|_{L^2(\G_t)} \big) \\
&\leq P(E(t)) \big( 1+\|\na \cP_{\mJ, v}\|_{L^2(\Om_t)}+\|D_tP_{v,v}\|_{H^1(\Om_t)}+\|P_{v,v}\|_{H^{3/2}(\Om_t)}+\|\mJ\|_{H^{3/2}(\Om_t)} \\
&\qquad  +\|D_t \kappa\|_{L^2(\G_t)}  \big).
\end{split}
\]

Applying Lemma <ref>, we have

\[
\|D_t \kappa\|_{H^{1/2}(\G_t)}\le P(E(t)) \big( 1+\|\na \cP_{\mJ, v}\|_{L^2(\Om_t)}+\|D_tP_{v,v}\|_{H^1(\Om_t)}
+\|D_t \kappa\|_{L^2(\G_t)}  \big).
\]

On the other hand, we rewrite (<ref>) into
which together with the inequality above for $\|D_t \kappa\|_{H^{1/2}(\G_t)}$ and the estimate for $\cP_{\mJ, v}$  imply

__tv^⊥_H^1/2(_t)    ≤C(_t_H^5/2,n_t_H^5/2(_t))(D_tκ_H^1/2(_t)  + v_H^2(_t))

≤P(E(t))(1+_, v_L^2(_t)+D_tP_v,v_H^1(_t)
+D_t κ_L^2(_t) + v_H^2(_t))

≤P(E(t))(1 + D_tP_v, v_H^1(_t)  + v_H^2(_t)).

Here we use (<ref>) to obtain

\[
\|D_t \kappa\|_{L^2(\G_t)}\le C(\|\G_t\|_{H^{5/2}}) \|v\|_{H^2(\Om_t)}.
\]

As a result, the estimate for $v^\perp$ depends on the estimates for $\| D_tP_{v, v}\|_{H^1(\Om_t)}, \|v\|_{H^2(\Om_t)}$.

- $H^2$ estimate for $v$. Recalling  system (<ref>) for velocity potential $\phi$ with $v=\na \phi$, we have by  Proposition <ref>  that

\[
\begin{split}
\|v\|_{H^2(\Om_t)} &\le \|\phi\|_{H^3(\Om_t)}\leq  C(\|\Gamma_t\|_{H^{5/2}})\big(\|v^\perp\|_{H^{3/2}(\G_t)}+\|\phi\|_{L^2(\Om_t)}\big)\\
&\le C(\|\Gamma_t\|_{H^{5/2}})\big(\|v^\perp\|_{H^{3/2}(\G_t)}+\|v\|_{L^2(\Om_t)}\big)\\
&\leq P(E(t))\big(1 +\|v^\perp\|_{H^{3/2}(\G_t)}\big).
\end{split}
\]

Bringing the above estimate into (<ref>) and applying interpolations to $\|v^\perp\|_{H^{3/2}(\G_t)}$,  we arrive at

__tv^⊥_H^1/2(_t)+v_H^2(_t)≤P(E(t))(1+ D_tP_v, v_H^1(_t) ).
Therefore, it remains to deal with $\| D_tP_{v, v}\|_{H^1(\Om_t)}$.

- $D_tP_{v,v}$ estimates.  From the definition of $P_{v, v}$, we derive the system for $D_tP_{v,v}$:

D_t P_v, v=-D_t tr(v v)+ 2tr(v  ^2  P_v, v)    in  _t

_n_t D_t P_v, v|__t=C'_v,v(t)+_n_t v·P_v, v|__t,  _n_b D_tP_v, v|__b=D_t(v·_v n_b)+_n_b v·P_v, v|__b.

\[
\int_{\Om_t}D_tP_{v,v}dX=0.
\]

It is easy to see from Euler's equation that

D_t tr(v v)=-2tr (v·v  v)-2tr ((P_v,v+_)v),
and similar expression can be derived for $D_t(v\cdot\na_v n_b)$ on $\G_b$.

Moreover, we have

\[
C'_{v,v} =|\Gamma_t|^{-1}\int_{\Om_t} 2tr(D_t \na v\na v)dX
-|\G_t|^{-2}\pa_t|\G_t|\int_{\Om_t}tr(\na v\na v)dX,
\]

where $\f d{dt}|\G_t|=\int_{\G_t}\big(v^\perp \kappa+\na_{\tau_t}(v\cdot\tau_t)\big)ds$. So we obtain by Lemma <ref>, (<ref>) and (<ref>) that

|C'_v,v|≤C(Γ_t_H^5/2,  v_H^3/2(_t), P_v,v_H^2(_t),
__H^2(_t), κ_L^2(_t))≤P(E(t)).

To prove $H^1$ estimate for $D_tP_{v,v}$, we use the following form of variation equation:

\[
\begin{split}
\int_{\Om_t}\big|\na D_tP_{v,v}\big|^2dX=&-\int_{\Om_t}(\Delta D_tP_{v,v})D_tP_{v,v}dX+\int_{\G_t}(\na_{n_t}D_tP_{v,v})D_tP_{v,v}ds\\
\end{split}
\]

Substituting the expressions in (<ref>) into the equality above, we can deal with the integrals one by one, where Lemma <ref> is applied. For example,  for the highest-order terms of $v$ on the boundary, we use Green's Formula and Lemma <ref> to find

\[
\begin{split}
&\int_{\G_t}\na_{n_t}v\cdot P_{v,v} \,D_tP_{v,v}ds+\int_{\G_b}\na_{n_b}v\cdot P_{v,v} \,D_tP_{v,v}ds\\
&=\int_{\Om_t}tr(\na v \na^2 P_{v,v}) D_tP_{v,v}dX+\int_{\Om_t}\na v\cdot\na P_{v,v}\cdot \na D_tP_{v,v}dX\\
&\le \|\na v\|_{L^4(\Om_t)}\|\na^2P_{v,v}\|_{L^2(\Om_t)}\|D_tP_{v,v}\|_{L^4(\Om_t)}+|\na v\|_{L^4(\Om_t)}\|\na P_{v,v}\|_{L^4(\Om_t)}\|\na D_tP_{v,v}\|_{L^2(\Om_t)}\\
&\le C(\|\Gamma_t\|_{H^{5/2}})\|v\|_{H^{3/2}(\Om_t)}\|P_{v,v}\|_{H^2(\Om_t)}\|D_tP_{v,v}\|_{H^1(\Om_t)}.
\end{split}
\]

The other integrals above can be handled similarly and the details are omitted.

As a result, we  obtain by Lemma <ref>  the following desired estimate:

\[
\|D_t P_{v, v}\|_{H^1(\Om_t)} \leq P(E(t))\big(\|P_{v,v}\|_{H^2(\Om_t)}+\|\mK_\cH\|_{H^{5/2}(\Om_t)}+|C'_{v,v} |\big)\le P(E(t)).
\]

Consequently,  applying  (<ref>) leads to the estimates for $v^\perp$ and $v$.

We know immediately from the proof of Lemma <ref> the following estimate

\[
\|D_t\kappa\|_{H^{1/2}(\G_t)}\le P(E(t)).
\]

Moreover, we also have

\[
\|D_t\mK_\cH\|_{H^1(\Om_t)}\le \sigma\|D_t\kappa_\cH\|_{H^1(\Om_t)}+\|D_tP_{v,v}\|_{H^1(\Om_t)}\le P(E(t)).
\]

Based on these lemmas above, we are ready to show some higher-order estimates.

Let $E(t)\in L^\infty[0,T]$ for some $T>0$, then one has

\[
\|P_{v, v}\|_{H^{5/2}(\Om_t)}+\|n_t\|_{H^3(\G_t)}\le P(E(t)).
\]

Proof. For $P_{v,v}$, similar arguments as in the proof of Lemma <ref> lead to

\[
\begin{split}
\|P_{v, v}\|_{H^{5/2}(\Om_t)} &\leq C(\|\Gamma_t\|_{H^{5/2}})\big(\|tr(\na v \na v)\|_{H^{1/2}(\Om_t)}+|\G_t|^{1/2}|C_{v,v}(t)|+\|v\cdot \na_v n_b\|_{H^{1}(\G_b)}\big)\\
&\leq P(E(t))(1+\|   v\|_{H^2(\Om_t)} ).
\end{split}
\]

Applying Lemma <ref>, we have the desired estimate.

Again, similar arguments as (<ref>) in the proof of Lemma <ref> lead directly to the estimate
κ_H^2(Γ_t) ≤^-1( _H^2(Γ_t)+ P_v,v_H^2(Γ_t))≤C(Γ_t_H^5/2) (1+ _τ_t ^_L^2(Γ_t)+  P_v, v_H^5/2(_t)),
which implies

\[
\|n_t\|_{H^3(\Gamma_t)}\leq  P(E(t))( 1+\|  P_{v, v}\|_{H^{5/2}(\Om_t)}).
\]

Therefore, combining this with the estimate for $P_{v,v}$, the proof is finished.

We also present more estimates for $v$.

Assuming that $E(t)\in L^\infty[0,T]$ for some $T>0$,  one has

\[
\|D_tv\|_{H^{3/2}(\Om_t)} +\|D^2_tv\|_{L^2(\Om_t)}  \leq  P(E(t))
\]

Proof. First, one recalls Euler's equation to get that

\[
\|D_tv\|_{H^{3/2}(\Om_t)} \leq C\big(1+\|\mJ\|_{H^{3/2}(\Om_t)}+ \|\na  P_{v, v}\|_{H^{3/2}(\Om_t)}\big),
\]

and applying Lemma <ref> leads to the estimate for $D_tv$.

Second, taking $D_t$ on Euler's equation leads to the following estimate:

\[
\|D^2_tv\|_{L^2(\Om_t)} \leq C(\|\cD_t\mJ\|_{L^2(\Om_t)} +\|\na  \cP_{\mJ, v}\|_{L^2(\Om_t)} + \|D_t\na  P_{v, v}\|_{L^2(\Om_t)} ).
\]

Using Lemma <ref> again, we can finish the estimate for $D^2_t v$.

We now consider the following Neumann system for velocity potential $\phi$:

ϕ=0,    on  _t

_n_tϕ|__t=v^⊥,  _n_bϕ|__b=0,
where in order to obtain the uniqueness we chose $\phi$ to satisfy

\[
\int_{\Om_t}\phi dX=0
\]

without loss of generality.

We show by singularity decompositions that $\na v$ lies in $L^\infty(\Om_t)$, which is a key ingredient in the following estimates.

Assume that $E(t)\in L^\infty[0,T]$ for some $T>0$, then there exists a unique $\phi\in H^3(\Om_t)$ to system (<ref>) and one finds the following singular decomposition:

\[
\phi=\phi_r+\phi_s
\]

where the regular part $\phi_r\in H^4(\Om_t)$, and the singular part is expressed in the same way as $u_s$ in Proposition <ref>.
Moreover, one has

\[
\|\na v\|_{L^{\infty}(\Om_t)}\le P(E(t)).
\]

In fact, applying Proposition <ref>, we know immediately about the existence of $\phi\in H^3(\Om_t)$ and the singular decomposition. Therefore, we have

v=ϕ_r+ϕ_s=v_r+v_s,    with  v_r∈H^3(_t).

It only remains to show the estimate for $\na v$. Since we have

\[
\phi_s=\chi_{\om}(\om_l)\chi_{l}\,c_l  r^{\pi/\om_l}\circ T_l+\chi_{\om}(\om_r)\chi_{r} \, c_r r^{\pi/\om_r}\circ T_r
\]

where the singular part exists when $\om_i\in (\pi/3, \pi/2)$. In this case, we find by Proposition <ref> that

\[
\begin{split}
\|\na^2\phi_s\|_{L^\infty(\Om_t)}&\le C(\|\G_t\|_{H^4})\big(\|v^\perp\|_{H^{5/2}(\G_t)}+\|\phi\|_{L^2(\Om_t)}\big) \big(\|\chi_{l}r^{\pi/\om_l-2}\circ T_l\|_{L^\infty(\Om_t)}\\
&\quad+\|\chi_{r}r^{\pi/\om_r-2}\circ T_r\|_{L^\infty(\Om_t)}\big)
\end{split}
\]

with $\pi/\om_i-2\in (0,1)$. Consequently, we obtain

\[
\|\na^2\phi_s\|_{L^\infty(\Om_t)}\le C(\|\G_t\|_{H^4})\big(\|v^\perp\|_{H^{5/2}(\G_t)}+\|\phi\|_{L^2(\Om_t)}\big)
\le C(\|\G_t\|_{H^4})\|v^\perp\|_{H^{5/2}(\G_t)}
\]

thanks to a direct variational estimate

\[
\|\phi\|_{L^2(\Om_t)}\le C(\|\G_t\|_{H^{5/2}}) \|\na\phi\|_{L^2(\Om_t)}\le C(\|\G_t\|_{H^{5/2}})\|v^\perp\|_{L^2(\G_t)}.
\]

Moreover, we also have by Proposition <ref>  the following estimate:

\[
\|\phi_r\|_{H^4(\Om_t)}\le C(\|\G_t\|_{H^4})\big(\|v^\perp\|_{H^{5/2}(\G_t)}+\|\phi\|_{L^2(\Om_t)}\big)
\le C(\|\G_t\|_{H^4})\|v^\perp\|_{H^{5/2}(\G_t)}.
\]

As a result, we find that

\[
\|\na v\|_{L^\infty(\Om_t)}\le\|\na^2 \phi_r\|_{L^\infty(\Om_t)} +\|\na^2 \phi_s\|_{L^\infty(\Om_t)}
\le C(\|\G_t\|_{H^4})\|v^\perp\|_{H^{5/2}(\G_t)},
\]

which together with Lemma <ref> lead to the desired estimate.

As long as we have   $\|\na v\|_{L^\infty(\Om_t)}$  estimate, we are able to deal with more higher-order estimates.

Let $E(t)\in L^\infty[0,T]$ for some $T>0$, one has the following estimate:

\[
\|P_{v, v}\|_{H^{3}(\Om_t)}+ \|P_{\mJ, v}\|_{H^{5/2}(\Om_t)}+\|D_tP_{v,v}\|_{H^{3/2}(\Om_t)}\le P(E(t))
\]

Proof. - $P_{v,v}$ estimate. In fact, similar arguments as in the proof of Lemma <ref> show that

\[
\|P_{v, v}\|_{H^{3}(\Om_t)} \leq P(E(t))(1+\|  v\|_{H^2(\Om_t)})(1+\|\na v\|_{L^\infty(\Om_t)}).
\]

Consequently, applying Lemma <ref> and Lemma <ref> leads to the desired estimate.

- Estimate for $P_{\mJ, v}$. Apply Proposition <ref>, we have

\[
\begin{split}
\|P_{\mJ, v}\|_{H^{5/2}(\Om_t)} \leq &C(\|\Gamma_t\|_{H^{5/2}})\big(\|tr(\na \mJ \na v) \|_{H^{1/2}(\Om_t)}+|C_{\mJ,v}(t)|+\|\na_{\tau_t}  \mK  \na_{\tau_t} v\cdot n_t\|_{H^{1}(\Gamma_t)}\\
+\|\mJ\cdot \na_v n_b\|_{H^{1}(\G_b)}+\|P_{\mJ, v}\|_{L^{2}(\Om_t)}\big).
\end{split}
\]

Using Lemma <ref>, Lemma <ref>,  Lemma <ref>, Lemma <ref> and Lemma <ref>, we can finish the esitmate.

- $H^{3/2}$ estimate for $D_tP_{v,v}$. Following the proof of Lemma <ref> and using Lemma <ref> again to improve  estimates for $v$, one can easily see that we have the desired estimate.

We now give a high-order estimate for $D_t P_{v, v}$.

Assuming $E(t)\in L^\infty[0,T]$ for some $T>0$,  one has
D_t P_v, v-v·(P_v,v|_c)_H^5/2(_t) ≤P(E(t)).

We denote by

\[
w=D_t P_{v, v}-v\cdot (\na P_{v,v}|_c)\quad\hbox{with}\quad \na P_{v,v}|_c=\chi_l(\na P_{v,v}|_{p_l})+\chi_r(\na P_{v,v}|_{p_r}).
\]

A direct computation using (<ref>)  leads to

w=-trD_t(v v) +[,v]·(P_v, v- P_v,v|_c)-v·(P_v,v|_c)   in  _t

_n_tw|__t=C_v,v'(t)+_n_t v·(P_v, v-P_v,v|_c)-v·_n_t(P_v,v|_c)|__t,

_n_bw|__b=D_t(v·_v n_b)+_n_b v·(P_v, v-P_v,v|_c)-v·_n_b(P_v,v|_c)|__b.

Applying Proposition <ref> to system (<ref>), we have

\[
\|w\|_{H^{5/2}(\Om_t)}\le C(\|\G_t\|_{H^{5/2}})\big(\|\Delta w\|_{H^{1/2}(\Om_t)}+\|\na_{n_t}w\|_{H^1(\G_t)}
\]

We need to deal with the terms on the right side above. First, thanks to (<ref>), Lemma <ref>, Lemma <ref> and Lemma <ref>,  it is straightforward to check that
D_t tr(v v)_H^1/2(_t)+[,v]·(P_v, v- P_v,v|_c) _H^1/2(_t)
+v·(P_v,v|_c)_H^1/2(_t) ≤P(E(t)).
Notice that here we need to use Lemma <ref> for the product estimate of $\|\na^2\mK_\cH\na v\|_{H^{1/2}(\Om_t)}$ from $\|D_t tr(\na v \na v)\|_{H^{1/2}(\Om_t)}$:

\[
\|\na^2\mK_\cH\na v\|_{H^{1/2}(\Om_t)}\le C\|\na^2\mK_\cH\|_{H^{1/2}(\Om_t)}\big(\|\na v\|_{H^1(\Om_t)}+
\|\na v\|_{L^\infty(\Om_t)}\big)\le P(E(t)).
\]

The estimate for  $\|\na^2 P_{v,v}\na v\|_{H^{1/2}(\Om_t)}$ also follows in a similar way.

Second, for the boundary terms, we only need to take care of the following estimate from $\|\na_{n_t} v\cdot (\na P_{v, v}-\na P_{v,v}|_{c})\|_{H^1(\G_t)}$:

_τ_t_n_t v·(P_v, v-P_v,v|_c)_L^2(_t)

≤_τ_t_n_t v·χ_i(P_v, v-P_v,v|_p_i)_L^2(_t)
+_τ_t_n_t v·(1-χ_l-χ_r)P_v, v_L^2(_t)

≤_τ_t_n_t v·χ_i(P_v, v-P_v,v|_p_i)_L^2(_t) +P(E(t)).

Here the estimate for the second term in the equality above holds thanks to (<ref>) and Proposition <ref>.

To handle the first term in (<ref>), using a straightening differeomorphism as $T_i$ from Proposition <ref> and (<ref>), we know immediately

_τ_t_n_t v·χ_i(P_v, v-P_v,v|_p_i)_L^2(_t)

≤C(_t_H^5/2)(_τ_t_n_t v)∘T^-1_i·(P_v,v-P_v,v|_p_i)∘T^-1_i_L^2(0, r_0)

≤C(_t_H^3)(c_ir^π/-3 (P_v,v-P_v,v|_p_i)∘T^-1_i_L^2(0, r_0)

+(^2ϕ_r)∘T^-1_i·(P_v,v-P_v,v|_p_i)∘T^-1_i_L^2(0, r_0))

≤P(E(t))(r^π/-3+3/4_L^2(0,r_0)r^-3/4(P_v,v-P_v,v|_p_i)∘T^-1_i_L^∞(0, r_0)+1).

Moreover, applying Lemma <ref> leads to

\[
\begin{split}
&\|r^{-3/4}(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\|_{L^2(0, r_0)}\le C\|(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\|_{H^{3/4}(0, r_0)}
\quad\hbox{and}\\
&\|r^{-3/4}(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\|_{H^1(0, r_0)}\le C\|(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\|_{H^{1+3/4}(0, r_0)},
\end{split}
\]

so we obtain by an interpolation the following inequality

\[
\begin{split}
\big\|r^{-3/4}(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\big\|_{L^\infty(0, r_0)}&\le C(\|\G_t\|_{H^{5/2}})\big\|r^{-3/4}(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\big\|_{H^{3/4}(0, r_0)}\\
&\le C(\|\G_t\|_{H^{5/2}})\|(\na P_{v,v}-\na P_{v,v}|_{p_i})\circ T_i\|_{H^{3/2}(0, r_0)}.
\end{split}
\]

As a result, combining this with (<ref>), we conclude that

\[
\big\|\na_{\tau_t}\na_{n_t} v\cdot \chi_i(\na P_{v, v}-\na P_{v,v}|_{p_i})\big\|_{L^2(\G_t)}\le P(E(t))(\|P_{v,v}\|_{H^3(\Om_t)}+1)\le P(E(t)).
\]

Substituting this estimate into (<ref>), we finally obtain

\[
\big\|\na_{n_t} v\cdot (\na P_{v, v}-\na P_{v,v}|_{c})\big\|_{H^1(\G_t)}\le P(E(t)).
\]

In the end, the  estimate for $\|\na_{n_b} v\cdot (\na P_{v, v}-\na P_{v,v}|_{c})\|_{H^1(\G_b)}$ follows in a similar way, and the proof can be finished.

In the end, we give the estimate of $\int_{\Om_t} \na D_t\cP_{\mathfrak{J}, v}\cdot D_t \mathfrak{J}dX$.

Let $E(t)\in L^\infty[0,T]$ for some $T>0$. Then, we have the following estimate:

\[
\Big|\int_{\Om_t} \na D_t\cP_{\mathfrak{J}, v} \cdot D_t \mathfrak{J}dX-\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX \Big|\leq P(E(t)).
\]

Proof. Recalling system (<ref>), we get

D_t_ 𝔍,v=- D_ttr(𝔍v)+2tr(v ^2 _ 𝔍,v)    in  _t

_n_t D_t_,v|__t= C'_ 𝔍, v-D_t(_τ_t𝔎_·_τ_t v ·n_t)+_n_tv·_ 𝔍,v,

_n_b  D_tP_ 𝔍,v|__b=D_t( ·_v n_b)+_n_bv·P_ 𝔍,v.
Therefore, we have directly

\[
\begin{split}
&\int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v} \cdot D_t \mathfrak{J}dX= \int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v}\cdot  D_t \na \mathfrak{K}_{\cH}dX\\
&= \int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v}\cdot \na D_t \mathfrak{K}_{\cH}dX+\int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX\\
&= \int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v}\cdot \na D_t \mathfrak{K}_{\cH}dX+\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX-\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot D_t(\na v\cdot\na \mathfrak{K}_{\cH})dX\\
+\int_{\Om_t}\na v\cdot\na \cP_{\mJ,v}\cdot\na v\cdot\na\mK_\cH dX,
\end{split}
\]

which leads to the following equality:

\[
\begin{split}
&\int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v} \cdot D_t \mathfrak{J}dX-\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX\\
&=\int_{\Om_t} \na D_t \cP_{\mathfrak{J}, v}\cdot \na D_t \mathfrak{K}_{\cH}dX-\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot D_t(\na v\cdot\na \mathfrak{K}_{\cH})dX+\int_{\Om_t}\na v\cdot\na \cP_{\mJ,v}\cdot\na v\cdot\na\mK_\cH dX.
\end{split}
\]

To finish the proof, the key lies in the analysis for the first integral. The remainder part is controlled by $P(E(t))$  thanks to Lemma <ref>, Lemma <ref>, Remark <ref> and  Lemma <ref>.

For the first integral on the right side of the equality above, we have by Green's Formula that

∫__t D_t _𝔍, v·D_t 𝔎_dX

=∫__t(_n_t D_t _𝔍, v)  D_t 𝔎_ds+∫__b(_n_b D_t_𝔍, v)  D_t 𝔎_ds-∫__tΔD_t_,v D_t 𝔎_dX

=∫__t[C'_, v-D_t(_τ_t𝔎_·_τ_t v ·n_t)+_n_tv·_ 𝔍,v] D_t 𝔎_ds
+∫__bD_t( ·_v n_b)D_t 𝔎_ds

+∫__b_n_bv·P_ 𝔍,v D_t 𝔎_ds
+∫__t D_ttr(𝔍v) D_t 𝔎_dX-∫__t2tr(v ^2 _ 𝔍,v) D_t 𝔎_dX

=C'_, v∫__tD_t 𝔎ds-∫__t(_τ_tD_t𝔎·_τ_t v ·n_t) D_t 𝔎ds+∫__b(D_t·_v n_b) D_t 𝔎_ds

+∫__ttr(D_tv)D_t_dX +l.o.t.,

where the remainder lower-order terms can be controlled by $P(E(t))$ in a similar way as before.

Now we deal with the terms in (<ref>) one by one. To begin with, we write from the definition of $C_{\mJ,v}$ in system (<ref>) that

\[
\begin{split}
\f d{dt}\big(|\G_t|C_{\mJ,v}\big)&=-\int_{\Om_t}D_t tr(\na\mJ\na v)dX+\int_{\G_t}D_t(\na_{\tau_t}\mK\na_{\tau_t}v\cdot n_t ds)
-\int_{\G_b}D_t(\mJ\cdot \na_v n_b ds)\\
&=-\int_{\Om_t}tr(\na D_t\mJ\na v)dX+\int_{\G_t}\na_{\tau_t}D_t\mK\na_{\tau_t}v\cdot n_tds-\int_{\G_b}D_tJ\cdot\na_v n_bds+\hbox{l.o.t.},
\end{split}
\]

where the lower-order terms can be controlled again and hence the details are omitted. Applying Green's Formula again and using the decompositions

\[
\begin{split}
&\na D_t\mK_\cH=(\na_{\tau_t}D_t\mK)\tau_t+(\na_{n_t}D_t\mK_\cH)n_t\qquad\hbox{on}\quad \G_t,\\
&\na D_t\mK_\cH=(\na_{\tau_b}D_t\mK_\cH)\tau_b+(\na_{n_b}D_t\mK_\cH)n_b\qquad\hbox{on}\quad \G_b,
\end{split}
\]

we find

=-∫__tD_t·v·n_tds-∫__bD_t·v·n_bds+∫__t_τ_tD_t_τ_tv·n_tds-∫__bD_tJ·_v n_bds+l.o.t.

-∫__bD_t_·_v n_bds+l.o.t.

=-∫__t_n_tD_t__n_tv·n_tds-∫__b_n_bD_t__n_bv·n_bds-∫__b_τ_bD_t·_τ_b v·n_bds

-∫__bD_t_·_v n_bds+l.o.t..

Here, for the last three terms on $\G_b$, thanks to the assumption that $n_b$ is constant near the contact points $p_i$ and $v\cdot n_b|_{\G_b}=0$, we know that $\na_{\tau_b} v\cdot n_b=-v\cdot\na_{\tau_b}n_b$ and $\na_v n_b=(\na_v n_b\cdot \tau_b)\tau_b$ vanish near $p_i$. Moreover, we also know from the definition of $\mK_\cH$ that

\[
\na_{n_b}D_t\mK_\cH|_{\G_b}=[\na_{n_b}, D_t]\mK_\cH=\na_{n_b}v\cdot \na\mK_\cH.
\]

Consequently, applying Lemma <ref> and Lemma <ref>, we have

\[
\begin{split}
&\int_{\G_b}\na_{\tau_b}D_t\mK_\cH\cdot\na_{\tau_b} v\cdot n_bds+\int_{\G_b}\na D_t\mK_\cH\cdot\na_v n_bds\\
&\le \|\na_{\tau_b}D_t\mK_\cH\|_{\tilde H^{-1/2}(\G_b)}\big(\|v\cdot\na_{\tau_b}n_b\|_{\tilde H^{1/2}(\G_b)}+\|\na_v n_b\cdot \tau_b\|_{\tilde H^{1/2}(\G_b)}\big)\\
&\le C(\|\G_t\|_{H^{5/2}})\|D_t\mK_\cH\|_{H^1(\Om_t)}\big(\|v\cdot\na_{\cH(\tau_b)}\cH(n_b)\|_{H^1(\Om_t)}+\|\na_v \cH(n_b)\cdot \cH(\tau_b)\|_{H^1(\Om_t)}\big)\le P(E(t))
\end{split}
\]

\[
\int_{\G_b}\na_{n_b}D_t\mK_\cH\na_{n_b}v\cdot n_bds\le P(E(t)).
\]

For the first integral in (<ref>), applying Green's Formula again leads to

\[
\int_{\G_t}\na_{n_t}D_t\mK_\cH\na_{n_t}v\cdot n_tds=\int_{\Om_t}[\Delta, D_t]\mK_\cH\na_{\cH(n_t)}v\cdot \cH(n_t)ds-\int_{\G_b}[\na_{n_b},D_t]\mK_\cH\na_{\cH(n_t)}v\cdot \cH(n_t)ds,
\]

so this term is  controlled by $P(E(t))$ again.

Therefore, summing up all these estimates above and going back to (<ref>), we arrive at

\[
\f d{dt}\big(|\G_t|C_{\mJ,v}\big)= \hbox{l.o.t.}
\]

with the lower-order terms controlled by $P(E(t))$. This leads to the following estimate

\[
C'_{\mJ,v}\le P(E(t)).
\]

For the moment, we can go back and deal with the other terms in (<ref>). In fact, using similar arguments as above, we also conclude that

\[
\begin{split}
\int_{\G_t}\big(\na_{\tau_t}D_t\mathfrak{K}_{\cH}\cdot  \na_{\tau_t} v \cdot n_t\big) D_t \mathfrak{K}_{\cH}ds+\int_{\G_b}(D_t\mJ\cdot\na_v n_b) D_t \mathfrak{K}_{\cH}ds-\int_{\Om_t}tr(\na D_t\mJ\na v)D_t\mK_\cH dX\le P(E(t))
\end{split}
\]

Therefore,  the proof is finished.

§.§ Boundary terms at the contact points

We have the following equation on $\G_b$:

D_t= -σ^2 β_c(n_t·τ_b) (_τ_t )^τ_b+R_c1,
and there holds at the contact points $p_i$ ($i=l, r$) that

\[
(D_t\mJ)^\perp\,( \na_{{\bf \tau}_t}\mJ)^\perp \big|_{p_l}=-\f{\sigma^2}{\beta_c}F_l(t) +R_{c2,l},\quad (D_t\mJ)^\perp\,( \na_{{\bf \tau}_t}\mJ)^\perp \big|_{p_r}=\f{\sigma^2}{\beta_c}F_r(t) + R_{c2,r}
\]

\[
\begin{split}
&R_{c1}=r_c\,\tau_b-  ( \mJ \cdot  D_t n_b)\,n_b,\quad
R_{c2,i}= \big(-r_c +\cot \om(\mJ \cdot  D_t n_b)\big)\, (\sin\om) \na_{{\bf \tau}_t}\mJ^\perp\big|_{p_i},\qquad\hbox{and}\\
&r_c=-\sigma  \sin\om (\na_{\tau_t}v\cdot D_t n_t)+\sigma  \tau_b\cdot D_t n_t\,(\na_{\tau_t}v\cdot n_t)-\sigma  \sin\om  \big(\na P_{v,v}\cdot\na_{\tau_t}n_t +[D_t, \na_{\tau_t} ] v\cdot n_t\big)\\
- \beta_c D_t \na P_{v,v}\cdot \tau_b
\end{split}
\]

\[
|R_{c1}|\le P(E(t)),\quad |R_{c2,i}|\le P(E(t))F(t)^{1/2}.
\]

Proof. The proof follows the proof of Lemma 7.1 in [53]. For the estimates, we apply Lemma <ref>, Lemma <ref>, Lemma <ref> and  Lemma <ref>. Here the differences compared to the proof of Lemma 7.1 in [53] lie in that  we have different $\mJ$  and

\[
\na_{\tau_t}\na P_{v,v}\cdot n_t=-\na P_{v,v}\cdot\na_{\tau_t}n_t
\]

in $r_c$ due to the definition of  $P_{v,v}$.

§.§ Proof of Theorem <ref>

We are ready to show the a priori energy estimate in Theorem <ref>. In fact,  taking $L^2(\Om_t)$ inner product with $\cD_t J-\na \cH(  D_t P_{v,v} - v\cdot \big(\na P_{v,v}|_{c})\big)$ on both sides of (<ref>), we have

∫__t D_t [ _t -(  D_t P_v,v - v·(P_v,v|_c))]·[ _t -(  D_t P_v,v - v·(P_v,v|_c))]dX

+σ∫__t ·[ _t -(  D_t P_v,v - v·(P_v,v|_c))]dX =∫__t·[ _t -(  D_t P_v,v - v·(P_v,v|_c))]dX.

We deal with these integrals above one by one. To get started, for the first term on the left side, we rewrite it as

\[
\begin{split}
&\int_{\Om_t} D_t \big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]\cdot\big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]dX\\
&=\f12 \f d{dt}\int_{\Om_t} \big| \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX .
\end{split}
\]

For the second term on the left side of  (<ref>),  we have by Green's Formula  that

∫__t ·[ _t -(  D_t P_v,v - v·(P_v,v|_c))]dX

=-∫_Γ_t Δ_Γ_t ^   _t ·n_tds +∫_Γ_t Δ_Γ_t ^  (  D_t P_v,v - v·(P_v,v|_c))·n_tds.

For the first term on the right side of (<ref>), one deduces from Hodge decomposition (<ref>) and integration by parts as in [54] that

\[
\begin{split}
&-\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot \,  \cD_t \mJ\cdot n_tds\\
&=\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, \na_{\tau_t}(D_t \mJ\cdot n_t )ds-(D_t \mJ)^\bot \na_{\tau_t} \mJ^\bot \big|^{p_l}_{p_r} -\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot  (\na  \cP_{\mJ,v}\cdot n_t)ds\\
&=\f12\,\f d{dt}\int_{\Gamma_t}| \na_{\tau_t} \mJ^\bot|^2ds-\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, [  D_t, \na_{\tau_t}] \mJ^\bot ds- \int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, \na_{\tau_t}(\mJ\cdot  D_t  n_t )ds
+\f{\sigma^2}{\beta_c}F(t) \\
&\quad -R_{c2,l}+R_{c2,r}-\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot  (\na  \cP_{\mJ,v}\cdot n_t)ds,
\end{split}
\]

where we apply Lemma <ref> at corner points.

For the second term in (<ref>), one has in a similar way as above that

\[
\begin{split}
&\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot \, \na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\cdot n_tds\\
&=-\int_{\Gamma_t}\na_{\tau_t} J^\bot  \cdot   \na_{\tau_t}\na_{n_t}\cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)ds +\na_{\tau_t} J^\bot  \cdot    \na_{n_t}\cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^{p_r}_{p_l}\\
&\leq P(E(t))\big(1+ F(t)^{1/2}\big),
\end{split}
\]

where Lemma <ref>  and Lemma <ref> are used.

As a result, going back to (<ref>), we summerise that

\[
\begin{split}
&\f12 \f d{dt}\int_{\Om_t} \big| \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX+\f\si2\,\f d{dt}\int_{\Gamma_t}| \na_{\tau_t} \mJ^\bot|^2ds
+ \f{\sigma^3}{2\beta_c}  F(t)\\
&\le P(E(t))+\si\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, [  D_t, \na_{\tau_t}] \mJ^\bot ds+\si\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, \na_{\tau_t}(\mJ\cdot  D_t  n_t )ds
+\si\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot  (\na  \cP_{\mJ,v}\cdot n_t)ds\\
&\quad +\int_{\Om_t}\cR\cdot \big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]dX.
\end{split}
\]

Moreover, direct estimates similarly as in [54] lead to the following estimate

\[
\begin{split}
&\si\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, [  D_t, \na_{\tau_t}] \mJ^\bot ds+\si\int_{\Gamma_t} \na_{\tau_t} \mJ^\bot \, \na_{\tau_t}(\mJ\cdot  D_t  n_t )ds
+\si\int_{\Gamma_t} \Delta_{\Gamma_t} \mJ^\bot  (\na  \cP_{\mJ,v}\cdot n_t)ds\\
&\le P(E(t))+\f{\sigma^3}{4\beta_c}  F(t),
\end{split}
\]

so we obtain

12 ddt(∫__t | _t -(  D_t P_v,v - v·(P_v,v|_c))|^2dX
+∫_Γ_t| _τ_t ^|^2ds)
+ σ^34β_c  F(t)

+∫__t·[ _t -(  D_t P_v,v - v·(P_v,v|_c))]dX.

In order to finish the energy estimate, we still need to deal with the integral on the right side involving the remainder term $\cR$ in (<ref>).

- Estimates for the part $R_0-\si\na\cH(R_1)$ in $\cR$.

For the remainder term $R_0$ defined in (<ref>), we have the estimate

\[
\|R_0-\si\na\cH(R_1) \|_{L^2(\Om_t)}\leq  P(E(t)) .
\]

Recalling from (<ref>), we know that

\[
R_0-\si\na\cH(R_1) =-\s\na \cH(J\cdot \D_{\G_t}n_t)+\si\na \cH(n_t\cdot \D_{\G_t}\na P_{v,v}) +[D_t, \na \cH](  D_t P_{v,v}  ) +A_1+A_2+A_3
\]

where $R_1$ and $A_1,A_2,A_3$ are defined in (<ref>) and (<ref>), (<ref>), (<ref>) respectively.

For the term $\na \cH(n_t\cdot \D_{\G_t}\na P_{v,v})$, we have

\[
\|\na \cH(n_t\cdot \D_{\G_t}\na P_{v,v})\|_{L^2(\Om_t)}=\|\na \cH([n_t, \D_{\G_t}]\na P_{v,v})\|_{L^2(\Om_t)}\le  P(E(t)),
\]

where the boundary condition $\na_{n_t}P_{v,v}|_{\G_t}=C_{v,v}(t)$ and Lemma <ref> are applied.

For the term $[D_t, \na \cH](  D_t P_{v,v}  )$, direct computations using (<ref>) and similar arguments as in the proof of Lemma <ref> lead to

\[
\|[D_t, \na \cH](  D_t P_{v,v}  )\|_{L^2(\Om_t)}\leq\|\na v\cdot \na \cH(  D_t P_{v,v}  )\|_{L^2(\Om_t)}+\| [D_t, \cH](  D_t P_{v,v}  )\|_{H^1(\Om_t)}\leq P(E(t)).
\]

The estimates for the other terms follow from lemmas in the previous section and can be done similarly as [54], so we omit the details here.

- The part $\si\na\cH(R_1 )$ in $\cR$. In fact,  the following integral from the right side of (<ref>) is rewritten as

\[
\begin{split}
&\int_{\Om_t}\na\cH(R_1 )\cdot \big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]dX\\
&= \int_{\G_t}R_1\, \cD_t \mJ\cdot n_t ds-\int_{\G_t}R_1 \na_{n_t} \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big) ds\\
&=\int_{\G_t}R_1\, D_t \mJ\cdot n_t ds+\int_{\G_t}R_1 \,\na \cP_{\mJ,v}\cdot n_t ds-\int_{\G_t}R_1 \na_{n_t} \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big) ds
\end{split}
\]

On the other hand, noticing  from  (<ref>) that $R_1$ contains terms like $\na^2 v ,\na n_t, \kappa$ and $\na^2 P_{v, v}$ and using (<ref>) and lemmas in the previous subsections,  we have

\[
\int_{\G_t}R_1 \na_{n_t} \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big) ds
\le \int_{\G_t}|R_1 |ds\, \|\na_{n_t} \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\|_{L^\infty(\G_t)}\le P(E(t)).
\]

The details in the estimate above are omitted and we only note that for $\pa^2 v$ terms in $R_1$, we can have the following estimate thanks to (<ref>):

\[
\int_{\G_t}|\na^2 v|ds\le \int_{\G_t}|\na^3\phi_r|ds+\int_{\G_t}|\na^3\phi_s|ds\le P(E(t)).
\]

Similarly but more easily, we also  obtain

\[
\int_{\G_t}R_1 \,\na \cP_{\mJ,v}\cdot n_t ds\le P(E(t)).
\]

Besides, for the part $\int_{\G_t}R_1 D_t \mJ\cdot n_t ds$, we put $D_t$ out of the integral:

\[
\begin{split}
&\int_{\G_t}R_1 D_t \mJ\cdot n_t ds=\int_{\G_t}R_1 D_t(\mJ-\mJ|_c)ds+\int_{\G_t}R_1 D_t(\mJ|_c)ds\\
&=\f d{dt}\int_{\G_t} R_1\,(\mJ-\mJ|_c)ds-\int_{\G_t}D_t R_1\,(\mJ-\mJ|_c)ds-\int_{\G_t}R_1\,(\mJ-\mJ|_c)D_tds+\int_{\G_t}R_1 D_t(\mJ|_c)ds.
\end{split}
\]

Here we need to take care of the terms in $\int_{\G_t}D_t R_1\,(\mJ-\mJ|_c)ds$ and $\int_{\G_t}R_1\,(\mJ-\mJ|_c)D_tds$. In fact, similarly to the analysis in (<ref>), the key terms like $\|\pa^2 v (\mJ-\mJ|_c)\|_{L^2(\G_t)}$ can be handled as follows:

^2 v (-|_c)_L^2(_t)≤C(_t_H^5/2)r^δ (^2 v)∘T^-1_i _L^∞(_t) r^-δ(-|_p_i)∘T^-1_i_L^2(_t)≤P(E(t)),
where the constant $\delta\in (0,1)$ is chosen to satisfy

\[
\delta+\pi/\om_i-3>0,\quad\hbox{ when}\quad \om_i\in (\pi/3, \pi/2).
\]

Consequently, we derive

\[
\int_{\G_t}R_1 D_t \mJ\cdot n_t ds=\f d{dt}\int_{\G_t} R_1 (\mJ-
\mJ|_c)\cdot n_t ds+l.o.t.,
\]

where all the lower-order terms are controlled by $P(E(t))$.

As a result, we conclude that

\[
\int_{\Om_t}\na\cH(R_1 )\cdot \big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]dX
=\f d{dt}\int_{\G_t} R_1 (\mJ-
\mJ|_c)\cdot n_t ds+l.o.t.
\]

with the lower-order terms controlled by $P(E(t))$.

- The terms $D_t\na \cP_{\mJ,v}+D_t\na\cH\big(v\cdot(\na P_{v,v}|_c)\big)$ in $\cR$.
First,  one has directly from Lemma <ref> that

\[
\int_{\Om_t} \na D_t\cP_{\mathfrak{J}, v} \cdot D_t \mathfrak{J}dX=\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX +l.o.t.
\]

where the lower-order terms are all controlled by $P(E(t))$.

Second, for the integral

\[
\int_{\Om_t} \na D_t\cP_{\mathfrak{J}, v} \cdot \na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)dX,
\]

applying Green's formula and similar calculations as in (<ref>) show directly that it can be controlled by $P(E(t))$, and we omit the details.

Therefore, we conclude that

\[
\int_{\Om_t} \na D_t\cP_{\mathfrak{J}, v}
\cdot \big[ \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big]dX
=\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX+l.o.t.,
\]

where all the lower-order terms are controlled by $P(E(t))$.

Summing up all these estimates related to $\cR$ above and going back to (<ref>), we obtain the following estimate

\[
\begin{split}
&\f12 \f d{dt}\Big(\int_{\Om_t} \big| \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX
+\si\int_{\Gamma_t}| \na_{\tau_t} \mJ^\bot|^2ds\Big)
+ \f{\sigma^3}{4\beta_c}  F(t)\\
&\le P(E(t))+\f d{dt}\int_{\G_t} R_1 (\mJ-
\mJ|_c)\cdot n_t ds
+\f d{dt}\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX.
\end{split}
\]

Integrating on both sides with respect to time  on $[0, t]$, we have

\[
\begin{split}
&\f12\int_{\Om_t} \big| \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX
+\f\si 2\int_{\Gamma_t}| \na_{\tau_t} \mJ^\bot|^2ds
+ \f{\sigma^3}{4\beta_c} \int^t_0 F(t')dt'\\
&\le \f12\int_{\Om_0} \big| \cD_t \mJ-\na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|_{t'=0}\big|^2dX
+\f\si 2\int_{\Gamma_{t0}}| \na_{\tau_t} \mJ^\bot|^2\big|_{t'=0}ds\\
&\quad+\int^t_0 P(E(t'))dt'+\int_{\G_{t'}} R_1 (\mJ-
\mJ|_c)\cdot n_tds\Big|^t_0
+\int_{\Om_{t'}} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX\Big|^t_0\\
&\le P(E(0))+\int^t_0 P(E(t'))dt'+\int_{\G_{t'}} R_1 (\mJ-
\mJ|_c)\cdot n_tds\Big|^t_0
+\int_{\Om_{t'}} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX\Big|^t_0.
\end{split}
\]

Replacing the first integral with $\|\cD_t \mJ\|^2_{L^2(\Om_t)}$, we arrive at the following inequality:

\[
\begin{split}
E(t)+ \int^t_0 F(t')dt'
&\le P(E(0))+\int^t_0 P(E(t'))dt'+C\int_{\G_{t'}} R_1 (\mJ-
\mJ|_c)\cdot n_tds\Big|^t_0\\
+C\int_{\Om_{t'}} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX\Big|^t_0
+C\int_{\Om_{t}} \big| \na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX,
\end{split}
\]

where the constant $C$ depends on $\si, \beta_c$.

We now deal with the last three integrals on the right side above one by one. First, for the terms in $R_1$, similarly as in (<ref>), we can have by careful estimates with interpolations that

\[
\|r^{\delta}( \pa^2 v)\circ T^{-1}_i \|_{L^\infty(0, r_0)}\le P(E_l(t))E(t)^{1/2}
\]

\[
\|r^{-\delta}(\mJ-\mJ|_{p_i})\circ T^{-1}_i\|_{L^2(0, r_0)}\le C(\|\G_t\|_{H^{5/2}})\big\|\mJ-\mJ|_c\big\|_{H^\delta(\G_t)}\le C(\|\G_t\|_{H^{5/2}})\|\mJ\|_{H^{\delta+1/2}(\Om_t)},
\]

where $1>\delta>3-\pi/\om$ when $\om\in (\pi/3, \pi/2)$.

Therefore, we can prove by interpolations that there exist $\delta_0\in (1/2, 1)$ and $\epsilon$ small enough  such that

\[
\int_{\G_t} R_1 (\mJ-\mJ|_c)\cdot n_tds\le P(E_l(t))E(t)^{\delta_0}\le \epsilon E(t)+C_{\epsilon, \delta_0} P(E_l(t)).
\]

Moreover, the following estimates can also be proved in a similar but easier way:

\[
\int_{\Om_t} \na  \cP_{\mathfrak{J}, v}\cdot \na v\cdot\na \mathfrak{K}_{\cH}dX
+\int_{\Om_t} \big| \na \cH\big(  D_t P_{v,v} - v\cdot (\na P_{v,v}|_{c})\big)\big|^2dX
\le \epsilon E(t)+C_{\epsilon, \delta_0} P(E_l(t)).
\]

As a result, summing up the estimates above, we are able to conclude that

\[
E(t)+ \int^t_0 F(t')dt'\le P(E(0))+\int^t_0 P(E(t'))dt' +P(E_l(t)).
\]

In the end, to close the energy estimates, we  deal with $E_l(t)$.  First, we show by direct calculations using Euler's equation and lemmas from the previous section that

\[
\|v\|^2_{H^k(\Om_t)}\le C(T) \|v(0)\|^2_{H^k(\Om_0)}+C(T)\|\mJ\|^2_{L^2([0,T],H^k(\Om_t))}+\int^t_0P(E(t'))dt'
\]

for $k=1,2$.
Then we take the square root on both sides of the inequalities above and apply an interpolation between $H^1(\Om_t), H^2(\Om_t)$ as well as Lemma <ref> to obtain

\[
\|v\|_{H^{3/2}(\Om_t)}\le C(T) \|v(0)\|_{H^{3/2}(\Om_0)}+\int^t_0P(E(t'))dt'.
\]

Second,  the estimate for $\|\G_t\|_{H^{5/2}}$ can be derived in a similar way as above.
Consequently, we have

\[
E_l(t)\le P(E(0))+\int^t_0P(E(t'))dt'.
\]

Combining this estimate with the estimate above for $E(t)$, we  finish the lower-order energy estimates.

§ HIGHER-ORDER TIME-DERIVATIVE ENERGY ESTIMATES

In this section, we prove  higher-order energy estimates with respect to $D_t$, which is needed in the local well-posedness part.  To start with, we define the  energy functional

\[
E_1(t)=\|\na_{\tau_t} D_t  \mathfrak{J}^\bot \|^2_{L^2(\Gamma_t)} +\|D^2_t \mathfrak{J}\|^2_{L^2(\Om_t)},
\]

and the dissipation

\[
F_1(t)=\sum_{i=l,r}\big|(\sin \om_i)\na_{{\bf \tau}_t}D_t \mathfrak{J}^\perp |_{p_i}\big|^2.
\]

The main result of this section is as follows:

Let the contact angles $\om_i\in (0,\pi/2)$ and  $E(t), \int_0^T F(t)dt, E_1(t), \int_0^T F_1(t)dt$ be bounded above in $[0, T]$ for some  $T>0$. Then the following higher-order a priori estimate holds

\[
\sup_{0\leq t\leq T} E_1(t)+\int_0^T F_1(t)dt\le P(E_1(0))+\int_0^T P(E_1(t))dt.
\]

To prove this theorem, we begin with  a higher-order equation for $\mJ$ and more delicate estimates involving $D_t$ based on Section <ref>. With these preparations, we are able to finish the energy estimate in the last subsection of this part.

§.§ The higher-order equation for $\mJ$

Firstly, we recall system (<ref>) and rewrite it as follows:

D^2_t 𝔍=(__t  𝔍^⊥-h_v)+R̃_0,

\[
\tilde{R}_0 =-\si\na \cH(\mathfrak{J} \cdot \D_{\G_t}n_t)+\si\na \cH([n_t, \D_{\G_t}]\cdot\na P_{v,v})+\s\na\cH(R_1+h_v ) +A_1+A_2+A_3+ \na \cH(  D^2_t P_{v,v}  ),
\]

and $h_v$ contains all the second-order terms of $v$ in $R_1$ which come from the commutator $[D_t, n_t\D_{\G_t}]\cdot v$.  More precisely, we have

h_v= __t v·D_t n_t  +2D^2v(τ_t, (_τ_tv)^⊤)·n_t.
As a result,  $R_1+h_v$ only contains lower-order derivatives like $\pa v, \pa n_t, \kappa$.

Acting $D_t$ on both sides of (<ref>), we obtain

D_t (^2_t 𝔍) +(_Γ_t  D_t 𝔍^-D_th_v)

= _2,
with the right side

\[
\begin{split}
\tilde\cR_2&=D_t \tilde{R}_0  +[\na, D_t]\cH \big(\D_{\Gamma_t} \mathfrak{J}^\bot-h_v)\big)  +\na [\cH, D_t]\big(\D_{\Gamma_t} \mathfrak{J}^\bot-h_v\big)+\na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot\\
&\quad+D_t\big(D_t\na P_{\mathfrak{J}, v}+\na \cP_{\na P_{\mathfrak{J}, v}, v}+\na P_{\cD_t \mathfrak{J}, v}\big).
\end{split}
\]

Here we use

\[
-[\cA, D_t]=[\na, D_t]\cH \D_{\Gamma_t}+\na [\cH, D_t]\D_{\Gamma_t}+\na \cH [\D_{\Gamma_t}, D_t]
\]

with all these commutators from the end of Section 2,

^2_t 𝔍= D_t^2 𝔍+D_tP_𝔍, v+P_P_𝔍, v, v+P__t 𝔍, v,
where $P_{w, v}$ is defined in (<ref>).

§.§ More preliminary estimates.

Based on Section <ref>, we are going to prove higher-order estimates for different quantities using our higher-order energy.

- Higher-order estimates for $\mathfrak{K}_{\cH}$ and $D_t\mJ$.
To get started, we recall system (<ref>) to find

\[
\left\{\begin{array}{ll}
\D D_t\mathfrak{K}_\cH=2tr(\na v \na^2 \mathfrak{K}_\cH)\qquad\hbox{in}\quad\Om_t\\
\na_{n_t} D_t \mathfrak{K}_\cH|_{\G_t} = D_t \mJ^\perp +\na_{n_t} v \cdot \na \mathfrak{K}_\cH|_{\G_t},\qquad  \na_{n_b}D_t\mathfrak{K}_\cH|_{\G_b}=\na_{n_b}v\cdot \na\mK_\cH|_{\G_b}.
\end{array}\right.
\]

Applying Proposition <ref> and lemmas in Section <ref>, we have

D_t 𝔎__H^2(_t)   ≤C(_t_H^5/2)(2tr(v ^2 𝔎_)_L^2(_t)+ D_t ^⊥+_n_t v ·𝔎__H^1/2(_t)

≤P(E(t))(1+ D_t  𝔍^_H^1/2(_t)).

Here we notice that the regularity of $D_t\mathfrak{K}_\cH$ is constrained due to the regularity of $v$.  To get a higher-order estimate, we need to get rid of the worst part.

Similarly as in Lemma <ref>, we consider a good quantity $D_t\mathfrak{K}_\cH-v\cdot (\na \mathfrak{K}_\cH|_{c})$ which satisfies

\[
\left\{\begin{array}{ll}
\D \big(D_t\mathfrak{K}_\cH-v\cdot (\na\mathfrak{K}_\cH|_{c})\big)=2tr(\na v \na^2 \mathfrak{K}_\cH)-\D\big(v\cdot (\na\mathfrak{K}_\cH|_{c})\big) \qquad\hbox{in}\quad\Om_t\\
\na_{n_t}(D_t\mathfrak{K}_\cH-v\cdot (\na\mathfrak{K}_\cH|_{c}))\big|_{\G_t}=D_t \mJ^\perp +\na_{n_t} v \cdot \big(\na \mathfrak{K}_\cH-\na\mathfrak{K}_\cH|_{c}\big)-v\cdot \na_{n_t}(\na\mathfrak{K}_\cH|_{c})\big|_{\G_t},\\
\na_{n_b}(D_t\mathfrak{K}_\cH-v\cdot (\na\mathfrak{K}_\cH|_{c}))\big|_{\G_b}=\na_{n_b}v\cdot \big(\na\mK_\cH-\na\mathfrak{K}_\cH|_{c}\big)
-v\cdot \na_{n_b}(\na\mathfrak{K}_\cH|_{c})\big|_{\G_b}.
\end{array}\right.
\]

As a result, similar arguments as in (<ref>) applied for the part $\na v\cdot\big(\na\mK_\cH-\na\mathfrak{K}_\cH|_{c}\big)$ , we can  have

D_t𝔎_-v·(𝔎_|_c)_H^5/2(_t)  ≤P(E(t))(1+E_1(t)^1/2).
where $\|D_t\mJ^\perp\|_{L^2(\G_t)}$ is handled by (<ref>).

On the other hand, we deal with $D_t\mJ$.  First, we know directly from (<ref>) that

D_t_H^1(_t)≤P(E(t))(1+ D_t  𝔍^_H^1/2(_t)).
Second, we notice that

\[
D_t\mJ=D_t\na \mathfrak{K}_\cH=\na \big(D_t\mathfrak{K}_\cH-v\cdot (\mJ|_{c})\big)-\na v\cdot \big(\mJ-\mJ|_{c}\big).
\]

Similarly as in (<ref>),  we apply Lemma <ref>(1) and (<ref>) to obtain the estimate

\[
\begin{split}
&\|\na v\cdot (\mJ-\mJ|_{c})\|_{H^{3/2}(\Om_t)}\\
&\le P(E(t))\big(1+\|(\pa\na v)\cdot (\mJ-\mJ|_{c})\|_{H^{1/2}(\Om_t)}\big)\\
&\le P(E(t)\Big(1+\sum_{i=l,r}\big(\|r^{\pi/\om_i-2}\|_{L^\infty(0,r_0)}+\|r^{\pi/\om_i-2}\|_{H^1(0,r_0)}\big)
\|r^{-1}(\mJ-\mJ|_{p_i})\circ T^{-1}_i\|_{H^{1/2}(\cS_{t,i})}\Big)\\
&\le P(E(t)),
\end{split}
\]

while recall that $\cS_{t,i}$ are straightened sector of $\Om_t$ with radius $r_0$ near the corners.

Therefore, combining this estimate with (<ref>), we have

D_t_H^3/2(_t) ≤P(E(t))(1+E_1(t)^1/2)
and also

\[
\|D_t\mK_\cH\|_{H^2(\Om_t)}\le P(E(t))(1+E_1(t)^{1/2}).
\]

- Estimates for $D_tP_{v,v}$. Checking system (<ref>) carefully and applying lemmas from Section <ref>, one has immediately

where we use arguments similar to (<ref>) on boundary terms.

- Estimates for $v$.
To begin with,  applying Remark <ref>, we obtain for a small number $\eps\in (0, \pi/\om_i-2)$ (when $\om_i\in (\pi/3, \pi/2)$) that

Moreover, we improve the regularity of $v^\perp$. In fact, similarly as in (<ref>), we have thanks to (<ref>),(<ref>) and (<ref>) the estimate

__tv^⊥_H^1/2+(_t)    ≤C(_t_H^4)(D_tκ_H^1/2+(_t)  + v_H^1/2+(_t))

- Estimates for $D^2_tv, D^3_tv$.  First, acting $D_t$ on both sides of Euler's equation leads to

\[
D_t^2 v =-D_t \mathfrak{J}-D_t\na P_{v,v}.
\]

Applying Lemma <ref>, Lemma <ref>, (<ref>), and (<ref>), we find

≤D_t 𝔍_H^3/2(_t)+D_tP_v,v_H^3/2(_t) ≤P(E(t))(1+E_1(t)^1/2).

Next, we give the estimates of $D_t^3 v$.  Before that, we firstly prove the following lemma.

Let $P_{\mathfrak{J}, v}, P_{\na P_{\mathfrak{J}, v}, v}$ and $P_{\cD_t \mathfrak{J}, v}$ be defined by (<ref>). Then there hold

\[
\|D_t\na P_{\mathfrak{J}, v}+\na P_{\na P_{\mathfrak{J}, v}, v}+\na P_{\cD_t \mathfrak{J}, v}\|_{H^1(\Om_t)} \leq   P(E(t))\big(1+ E_1(t)^{1/2}\big)
\]

D^2_tP_v,v_L^2(_t) ≤P(E(t))(1+ E_1(t)^1/2).

First, we deal with $D_t P_{\mathfrak{J}, v}$. Recalling system (<ref>), we have by Proposition <ref>,  (<ref>), (<ref>), (<ref>) and lemmas from Section <ref> that

\[
\begin{split}
&\|D_t P_{\mathfrak{J}, v}\|_{H^2(\Om_t)}\\
&\leq C(\|\G_t\|_{H^{5/2}})\Big(\|- D_ttr(\na\mathfrak{J}\na v)+2tr(\na v \na^2 \cP_{ \mathfrak{J},v})\|_{L^2(\Om_t)} + \|D_t( \mJ\cdot\na_v n_b)+\na_{n_b}v\cdot \na P_{ \mathfrak{J},v}\|_{H^{1/2}(\G_b)}\\
&\quad+\|C'_{ \mathfrak{J}, v}-D_t(\na_{\tau_t}\mathfrak{K}_{\cH}\cdot  \na_{\tau_t} v \cdot n_t)+\na_{n_t}v\cdot \na P_{ \mathfrak{J},v}\|_{H^{1/2}(\G_t)} +\|D_t P_{\mathfrak{J}, v}\|_{L^2(\Om_t)}\Big)\\
&\leq P(E(t))(1+E_1(t)^{1/2}).
\end{split}
\]

Second, checking the definition (<ref>) and applying Lemma <ref> (1) and Lemma <ref>, we can have  for $ P_{\na P_{\mathfrak{J}, v}, v}$ the following estimate immediately:

P_P_𝔍, v, v_H^5/2(_t) ≤P(E(t)).
Moreover,  a similar argument also leads to

\[
\|P_{\cD_t \mathfrak{J}, v}\|_{H^2(\Om_t)} \leq  P(E(t))\big(1+ E_1(t)^{1/2}\big).
\]

In the end, for  the part $D_t^2 P_{v,v}$, we deduce from (<ref>) the following system:

(D^2_t P_v, v )=-tr D^2_t (v v) +2trD_t  (v  ^2  P_v, v)    in  _t

_n_t( D^2_t P_v, v )|__t=C”_v,v(t)+D_t(_n_t v·P_v, v )+_n_t v·D_t P_v, v|__t,

_n_b ( D^2_t P_v, v) |__b=D^2_t(v·_v n_b)+D_t(_n_b v·P_v, v )+_n_b v·D_t P_v, v |__b

\[
\int_{\Om_t}D^2_tP_{v,v}dX=0.
\]

Applying (<ref>), (<ref>) and checking term by term, we obtain the variational estimate

D^2_t P_v, v _H^1(_t) ≤P(E(t))(1+E_1(t)^1/2),
and the proof is finished.

Now, we are in a position to give the estimate for $D_t^3 v$. In fact, acting $D_t^2$ on both sides of Euler's equation, we derive by the previous lemma and lemmas from Section <ref>  that

D_t^3v_L^2(_t) ≤D^2_t 𝔍_L^2(_t) +D^2_tP_v,v_L^2(_t)  ≤P(E(t))(1+E_1(t)^1/2).

- Estimate for $v^\perp$.
We obtain by the definition of $\mathfrak{J}$ that

\[
\na (D_t^2 \mathfrak{K}_\cH) =D_t^2 \mathfrak{J}+ \na v\cdot \na D_t \mathfrak{K}_\cH+ D_t(\na v\cdot \na  \mathfrak{K}_\cH),
\]

and applying (<ref>) and (<ref>) implies

D_t^2 𝔎__H^1(_t) ≤P(E(t))(1+E_1(t)^1/2).
Recalling (<ref>) again, we know

\[
\si D_t\kappa=D_t (\mathfrak{K}+P_{v,v})=-\D_{\G_t}v^\perp-v^\perp|\na_{\tau_t}n_t|^2+\na_{\tau_t}\na_{v^\top}n_t\cdot \tau_t \qquad\hbox{on}\quad \G_t,
\]

so we obtain by (<ref>) and (<ref>) the estimate

D_t _Γ_tv^⊥_L^2(Γ_t)   ≤P(E(t))(1+E_1(t)^1/2).

- Some more higher-order estimates for $\mK_\cH$, $v$ and $\mJ$.

Assuming that $E(t), E_1(t)\in L^\infty[0,T]$ for some $T>0$, we have the following estimate:

\[
\|\mathfrak{K}_\cH\|_{H^3(\Om_t)}+\|D_t v\|_{H^2(\Om_t)} +\|\na^2 \mathfrak{K}_\cH\|_{L^\infty(\Om_t)}  +\|D_t \na v\|_{L^\infty(\Om_t)}\leq P(E(t))(1+E_1(t)^{1/2}).
\]

Proof. (1) Higher regularity for $\mK_\cH$.  To begin with, we use Euler's equation to rewrite (<ref>) as follows:

\[
D^2_t\mK= \big(\D_{\G_t}J^\perp-h_v\big)+[n_t,\D_{\G_t}](J+\na P_{v,v}+{\bf g})+2\s \Pi(\tau_t)\cdot \na_{\tau_t} J+(R_1+h_v)- D_t^2 P_{v,v}\quad\hbox{on}\  \G_t
\]

where we notice that $\D_{\G_t}(\na_{n_t} P_{v,v})|_{\G_t}=0$ thanks to the definition of $P_{v,v}$ and $(R_1+h_v)$ contains only $\pa v$ terms instead of $\pa^2 v$ terms.

Meanwhile, we know from (<ref>) and (<ref>) that

\[
\|D^2_t\mK\|_{H^{1/2}(\G_t)}+\| D_t^2 P_{v,v}\|_{H^{1/2}(\G_t)}\le P(E(t))(1+E_1(t)^{1/2}),
\]

so checking term by term in the equation above, we have immediately

On the other hand, we know from (<ref>) that (when $\om_i\in (\pi/3, \pi/2)$)

\[
h_v=h_{v,r}+h_{v,s}\quad\hbox{with}\quad h_{v,r}\in H^{1/2}(\G_t),\ h_s=\sum_i\big[\chi_{\om} a_{i,1}r^{2\alpha_i-3}+\chi_{\om}a_{i,2}(\pa v_r\circ T^{-1}_i)r^{\alpha_i-2}\big]\circ T_i,
\]

where we note $\al_i=\pi/\om_i-1\in (1,2)$,
$a_{i,k}$ ($i=l, r$) contain $n_t, \tau_t, \pa n_t, \pa\tau_t$ and the singular coefficient from  (<ref>). Moreover, $a_{i,3}(\pa v_r\circ T^{-1}_i)$ is linear with respect to $\pa v_r\circ T^{-1}_i$, where we recall that $v_r=\na\phi_r\in H^3(\Om_t)$ and $\|v_r\|_{H^3(\Om_t)}$ is controlled by $P(E(t))$.

As a result, we find

\[
2\alpha_i-3\in (-1,1),\quad \alpha_i-2\in (-1,0),
\]

which implies immediately that

\[
\|h_v\|_{L^p(\G_t)}\le P(E(t)),\qquad\hbox{for}\quad 1<p<\min\{(2-\alpha_i)^{-1}, |3-2\alpha_i|^{-1}\}.
\]

Summing up  the estimates above for $\|\D_{\G_t}J^\perp-h_v\|_{H^{1/2}(\G_t)}$ and $\|h_v\|_{L^p(\G_t)}$, we obtain

\[
\begin{split}
\|\D_{\G_t}J^\perp\|_{L^p(\G_t)}&\le \|\D_{\G_t}J^\perp-h_v\|_{L^p(\G_t)}+\|h_v\|_{L^p(\G_t)}\\
&\le C(\|\G_t\|_{H^{5/2}})\|\D_{\G_t}J^\perp-h_v\|_{H^{1/2}(\G_t)}+\|h_v\|_{L^p(\G_t)}\\
&\le P(E(t))(1+E_1(t)^{1/2}),
\end{split}
\]

and this leads to

\[
\|\na_{\tau_t}J^\perp\|_{W^{1,p}(\G_t)}\le P(E(t))(1+E_1(t)^{1/2}).
\]

Applying Lemma <ref>, we finally show that

with $\epsilon=1-1/p$.

Consequently, applying Proposition <ref>, we have the desired estimate for $\|\mK_\cH\|_{H^{3}(\Om_t)}$. Notice that we have in fact the estimate by Remark <ref>:

Moreover, we apply Lemma <ref> to have  $\mK=\si\kappa-P_{v,v}\in H^{5/2+\epsilon}(\G_t)$ with the estimate

\[
\|\mK\|_{H^{5/2+\epsilon}(\G_t)}\le P(E(t))(1+E_1(t)^{1/2}).
\]

(2) $H^2$ estimate for $D_tv$. Recalling the Euler's equations and $P_{v,v}$ estimate from Lemma <ref>,
we get
D_t v_H^2(_t)≤_H^2(_t)+P_v,v_H^2(_t)+g_H^2(_t) ≤P(E(t))(1+E_1(t)^1/2).

(3) $L^\infty$ estimates. Applying Remark <ref> and (<ref>) with the same $\epsilon=1-1/p$ as above lead immediately to

\[
\begin{split}
\|\na^2  \mathfrak{K}_\cH\|_{L^\infty} &\le C(\|\G_t\|_{H^{5/2}})\|\na^2\mK_\cH\|_{H^{1+\epsilon}(\Om_t)}
\le P(E(t))(1+E_1(t)^{1/2}).
\end{split}
\]

Applying Remark <ref> to $P_{v,v}$, we obtain

\[
\begin{split}
&\|\na^2 P_{v,v}\|_{L^\infty(\Om_t)}\le C(\|\G_t\|_{H^{5/2}})\|\na^2P_{v,v}\|_{H^{1+\eps}(\Om_t)}\\
&\le C(\|\G_t\|_{H^4})\big(\|tr(\na v\na v)\|_{H^{1+\epsilon}(\Om_t)}+\|C_{v,v}(t)\|_{H^{3/2+\eps}(\G_t)}+\|v\cdot\na_vn_b\|_{H^{1/2+\epsilon}(\G_b)} +\|P_{v,v}\|_{L^2(\Om_t)}\big)\\
&\leq P(E(t))\big(1+\|\na v\|^2_{H^{1+\epsilon}(\Om_t)}\big).
\end{split}
\]

Using (<ref>) and a similar argument as in (1), we have

\[
\|\na v\|_{H^{1+\epsilon}(\Om_t)}\le \|\na^2\phi_s\|_{H^{1+\epsilon}(\Om_t)}+\|\na^2\phi_r\|_{H^2(\Om_t)}
\le P(E(t)),
\]

which implies

\[
\|\na^2 P_{v,v}\|_{L^\infty(\Om_t)}\le P(E(t)).
\]

In the end,  apply Euler's equation again leads to

\[
\|\na D_t v\|_{L^\infty(\Om_t)}\le \|\na^2\mK_\cH\|_{L^\infty(\Om_t)}+\|\na^2 P_{v,v}\|_{L^\infty(\Om_t)}\leq P(E(t))(1+E_1(t)^{1/2}),
\]

which finishes the proof.

- Estimate for $D^2_tP_{v,v}$.
Based the above estimates, we firstly improve the estimate for $D_tP_{v,v}$. Using system (<ref>) and Lemma <ref>,
we improve the estimate in Lemma <ref> and get

D_t P_v, v-v·(P_v,v|_c) _H^3(_t)

≤C(_t_H^3)(-tr D_t (v v) +[,v]·(P_v, v- P_v,v|_c)+v·(P_v,v|_c) _H^1(_t)+| C'_v,v(t)|

+_n_t v·(P_v, v-P_v,v|_c)-v·_n_t(P_v,v|_c)_H^3/2(_t)
+D_t(v·_v n_b)_H^3/2(_b)

+_n_b v·(P_v, v-P_v,v|_c)-v·_n_b(P_v,v|_c)_H^3/2(_b))

Here we use Lemma <ref> (2) for the boundary terms like $\|\na_{n_t} v\cdot (\na P_{v, v}-\na P_{v,v}|_{c})\|_{H^{3/2}(\G_t)}$, which are handled similarly as in the proof for (<ref>).

Next, we derive the equation of $D_t(D_t P_{v, v}-v\cdot (\na P_{v,v}|_{c}))$. To simplify the notation, we define

\[
\mathcal P_{t,1}=D_t P_{v, v}-v\cdot (\na P_{v,v}|_{c}),
\]

and we rewrite (<ref>) as

𝒫_t,1 _H^3(_t)≤P(E(t))(1+E_1(t)^1/2).
Direct computations lead to the following system for $D_t\mathcal P_{t,1}$:

(D_t𝒫_t,1)=-tr D^2_t (v v) +2trD_t(v(P_v, v- P_v,v|_c) )-D_t(v·(P_v,v|_c))

+[D_t, ]𝒫_t,1    in  _t

_n_t( D_t 𝒫_t,1)|__t=C”_v,v(t)+D_t(_n_t v·(P_v, v-P_v,v|_c))-D_t(v·_n_t(P_v,v|_c))+_n_t v·𝒫_t,1|__t,

_n_b( D_t 𝒫_t,1)|__b= D^2_t(v·_v n_b)+D_t(_n_b v·(P_v, v-P_v,v|_c))-D_t(v·_n_b(P_v,v|_c))

+_n_b v·𝒫_t,1|__b.

Moreover, we define

𝒫_t,2=D_t 𝒫_t,1-v·(𝒫_t,1|_c)= D^2_tP_v,v-D_t(v·(P_v,v|_c))-v·(𝒫_t,1|_c)
and we modify this system above as in Lemma <ref> into a new system for $\mathcal P_{t,2}$ below:

𝒫_t,2=-tr D^2_t (v v) +2trD_t(v(P_v, v- P_v,v|_c) )-D_t(v·(P_v,v|_c))

+[D_t, ]𝒫_t,1-v·(𝒫_t,1|_c)    in  _t

_n_t𝒫_t,2|__t=C”_v,v(t)+D_t(_n_t v·(P_v, v-P_v,v|_c))-D_t(v·_n_t(P_v,v|_c))

+_n_t v·(𝒫_t,1-𝒫_t,1|_c)-v·_n_t(𝒫_t,1|_c)|__t,

_n_b𝒫_t,2|__b= D^2_t(v·_v n_b)+D_t(_n_b v·(P_v, v-P_v,v|_c))-D_t(v·_n_b(P_v,v|_c))

+_n_b v·(𝒫_t,1-𝒫_t,1|_c)-v·_n_b(𝒫_t,1|_c)|__b.

Thanks to Lemma <ref>, (<ref>) and (<ref>),  it is straightforward to show that

- The boundary condition for $D_t^2 \mathfrak{J}$ at corner points.

We have at the contact points $p_i (i=l,r)$ the following equations

D^2_t  𝔍 =σ^2 β_c(sin_i) D_t(_τ_t  𝔍^)
τ_b+R_c, h, i,
where the remainder terms

\[
R_{c, h, i}=D_tR_{c1} + \f{\sigma^2 }{\beta_c}D_t(\tau_b\sin\om_i )  (\na_{\tau_t}  \mathfrak{J})^\bot
- \f{\sigma^2 }{\beta_c}(\sin\om_i )  D_t(\na_{\tau_t} n_t\cdot \mathfrak{J})\tau_b\big|_{p_i}
\]

with $R_{c1}$ defined in Lemma <ref>.
Moreover, there hold for $i=l, r$ that

\[
| R_{c, h,i}|\leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

Proof. First, Acting $D_t$ directly on both sides of the equations in Lemma  <ref>, one has the desired equations immediately. Next, apply lemmas from Section <ref>,  Lemma <ref> (<ref>) and (<ref>), the estimates follow.
Moreover,  we notice that the higher-order term $\na^2_{\tau_t}v^\perp(\tau_t\cdot\mJ)$ from $ D_t(\na_{\tau_t} n_t\cdot \mathfrak{J})\tau_b$ in $R_{c,h,i}$ can be  controlled thanks to (<ref>), where we use

D_t n_t=-((v)^*n_t)^ ⊤=-(n_t·_τ_tv)τ_t=-(_τ_tv^⊥)τ_t+(_τ_tn_t·v)τ_t.

§.§ Proof of Theorem <ref>

At this moment, we are finally ready to prove the higher-order energy estimate in Theorem <ref>.
To begin with, we rewrite equation (<ref>) of $\cD_t^2 \mathfrak{J}$ into:

D_t (^2_t 𝔍 -(_t,2) )+(_Γ_t  D_t 𝔍^-D_t h_v)= _2 +D_t _tP_𝔍, v+D_t P__t 𝔍, v,

\[
\begin{split}
\cR_2&= D_t \big(\tilde{R}_0-\na\cH(D_t^2P_{v,v}) \big )+[\na, D_t]\cH (\D_{\Gamma_t} \mathfrak{J}^\bot-h_v)+\na [\cH, D_t](\D_{\Gamma_t} \mathfrak{J}^\bot-h_v)+\na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot    \\
&\quad+D_t( \na P_{\na P_{\mathfrak{J}, v}, v} )  -D_t \na \cH \big[D_t\big(v\cdot (\na P_{v,v}|_{c})\big)+ v\cdot (\na \mathcal P_{t,1}|_{c})\big].
\end{split}
\]

In fact,  this complicated form is used due to similar technical reasons as (<ref>), see Remark <ref>, where  the terms $\na \cH(\cP_{t,2})$,  $D_t h_v$ are added on the left side to improve the estimates.

Next, we apply $L^2(\Om_t)$ inner product with $\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})$ on both sides of (<ref>) to get

\[
\begin{split}
&\f12\f d{dt} \|\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\|^2_{L^2(\Om_t)} -\int_{\Om_t} \na\cH\big(\D_{\Gamma_t} D_t \mathfrak{J}^\bot-D_t h_v\big)\cdot \big(\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big) dX \\
&=\int_{\Om_t}   \cR_2    \cdot \big(\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big)dX +\int_{\Om_t}\big(  D_t \cD_t\na P_{\mJ, v}+D_t \na P_{\cD_t \mJ, v}   \big)\cdot \big(\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big)dX.
\end{split}
\]

Consequently, for any $t'\in [0,T]$ there holds

12^2_t 𝔍(t') -(_t,2)(t')^2_L^2(_t') -∫_0^t'∫__t (_Γ_t D_t 𝔍^-D_th_v)·(^2_t 𝔍 -(_t,2)) dXdt

=12 ^2_t 𝔍(0) -(_t,2)(0)^2_L^2(_0) +∫_0^t'∫__t   _2    ·(^2_t 𝔍 -(_t,2))dXdt

+∫_0^t'∫__t(  D_t _tP_J, v+D_t P__t J, v   )·(^2_t 𝔍 -(_t,2))dXdt.

§.§.§ Left side of (<ref>).

In this subsection, we  prove  estimates for the second term on the left side of (<ref>).
One has for some $\delta_0\in (0,1)$ the following estimate:

\[
\begin{split}
&-\int_0^T\int_{\Om_t} \na\cH\big(\D_{\Gamma_t} D_t \mathfrak{J}^\bot-D_t h_v\big)\cdot \big(\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big)dXdt \\
&\ge \f12  \sup_{t\in [0, T]}\int_{\Gamma_t}|\na_{\tau_t}D_t \mathfrak{J}^\bot |^2ds + \f14\int_0^T F_1(t)dt -\int_0^TP(E(t))\big(1+E_1(t)^{3/2}\big)dt\\
&\quad -\sup_{t\in [0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big)-P(E(0))\big(1+E_1(0)^{\delta_0}\big).
\end{split}
\]

Applying Green's Formula, we have

- ∫__t (_Γ_t D_t 𝔍^- D_t h_v)·(^2_t 𝔍 -(_t,2))dX

=-∫_Γ_t  (_Γ_t_t 𝔍^-D_t h_v)  (^2_t 𝔍 -(_t,2))·n_tds

= -∫_Γ_t  (_Γ_t D_t 𝔍^- D_t h_v)  (D_t^2 𝔍+D_tP_𝔍, v+P_P_𝔍, v, v+P__t 𝔍, v)·n_tds

+∫_Γ_t  (_Γ_t D_t 𝔍^- D_t h_v) _n_t (_t,2) ds

=-∫_Γ_t  _Γ_t D_t 𝔍^  D_t^2 𝔍·n_tds+∫_Γ_t D_t h_v    D_t^2 𝔍  ·n_tds +I_R,

\[
\begin{split}
I_R&=-\int_{\Gamma_t}  \big(\D_{\Gamma_t} D_t \mathfrak{J}^\bot -D_t h_v\big) \big( D_t\na P_{\mathfrak{J}, v}+\na P_{\na P_{\mathfrak{J}, v}, v}+\na P_{\cD_t \mathfrak{J}, v}\big)\cdot n_tds\\
+\int_{\Gamma_t}  \big(\D_{\Gamma_t} D_t \mathfrak{J}^\bot- D_t h_v\big)\cdot   \na_{n_t} \cH(\cP_{t,2}) ds.
\end{split}
\]

We deal with these integrals above one by one.  First, integrating by parts and taking one $D_t$ out of the first integral in (<ref>), we  get

-∫_Γ_t  _Γ_t D_t 𝔍^  D_t^2 𝔍·n_t ds

=  12ddt∫_Γ_t|_τ_tD_t 𝔍^|^2ds-12 ∫_Γ_t|_τ_tD_t 𝔍^|^2 D_tds+ _τ_t D_t 𝔍^ (D_t^2 𝔍·n_t)|^p_r_p_l

+∫_Γ_t  _τ_t D_t 𝔍^  [_τ_t,D_t] D_t𝔍^ds  +∫_Γ_t  _τ_t D_t 𝔍^ _τ_t ([n_t, D_t^2]  ·𝔍)ds.

Here,   Lemma <ref> is applied at the corner terms:

\[
\begin{split}
&\na_{\tau_t} D_t \mathfrak{J}^\bot \,(D_t^2 \mathfrak{J}\cdot n_t)\big|^{p_r}_{p_l}\\
\na_{\tau_t} D_t \mathfrak{J}^\bot\ \f{\sigma^2 }{\beta_c}(\sin\om_l) D_t\na_{\tau_t}  \mathfrak{J}^\bot
(\tau_b\cdot n_t)\big|_{p_r}-\na_{\tau_t} D_t \mathfrak{J}^\bot\ \f{\sigma^2 }{\beta_c}(\sin\om_r) D_t\na_{\tau_t}  \mathfrak{J}^\bot(\tau_b\cdot n_t)\big|_{p_l}\\
&\quad+\na_{\tau_t} D_t \mathfrak{J}^\bot (R_{c,h,l}\cdot n_t)\big|_{p_r}-\na_{\tau_t} D_t \mathfrak{J}^\bot (R_{c,h,r}\cdot n_t)\big|_{p_l}\\
&=F_1(t)+\na_{\tau_t} D_t \mathfrak{J}^\bot\ \f{\sigma^2 }{\beta_c}(\sin\om_l) [D_t,\na_{\tau_t}]  \mathfrak{J}^\bot
(\tau_b\cdot n_t)\big|_{p_r}-\na_{\tau_t} D_t \mathfrak{J}^\bot\ \f{\sigma^2 }{\beta_c}(\sin\om_r) [D_t,\na_{\tau_t} ] \mathfrak{J}^\bot(\tau_b\cdot n_t)\big|_{p_l}\\
&\quad+\na_{\tau_t} D_t \mathfrak{J}^\bot (R_{c,h,l}\cdot n_t)\big|_{p_r}-\na_{\tau_t} D_t \mathfrak{J}^\bot (R_{c,h,r}\cdot n_t)\big|_{p_l},
\end{split}
\]

so one has by (<ref>) and lemmas from Section <ref> that

\[
\na_{\tau_t} D_t \mathfrak{J}^\bot \,(D_t^2 \mathfrak{J}\cdot n_t)\big|^{p_r}_{p_l}\ge
\f12 F_1(t)-P(E(t))\big(1+E_1(t)\big).
\]

On the other hand, since $D_tds=(v^\perp\kappa+\na_{\tau_t}(v\cdot\tau_t)\big)ds$, we obtain directly the estimate:

\[
\f12\int_{\Gamma_t}|\na_{\tau_t}D_t \mathfrak{J}^\bot |^2\,D_tds\le  P(E(t))E_1(t).
\]

Moreover, a direct computation using (<ref>) shows that

\[
\begin{split}
[n_t, D_t^2]  \cdot\mathfrak{J}&=- D_t(D_tn_t)\cdot\mJ-2(D_tn_t)\cdot D_t\mJ\\
&=-D_t\big(-(\na_{\tau_t}v^\perp)\tau_t+(\na_{\tau_t}n_t\cdot v)\tau_t\big)\cdot\mJ-2\big(-(\na_{\tau_t}v^\perp)\tau_t+(\na_{\tau_t}n_t\cdot v)\tau_t\big)\cdot D_t\mJ.
\end{split}
\]

Therefore, thanks to (<ref>), Lemma <ref> and checking term by term, we  have  the following estimate for the last two integrals in (<ref>):

\[
\int_{\Gamma_t}  \na_{\tau_t} D_t \mathfrak{J}^\bot\, [\na_{\tau_t},D_t] D_t\mathfrak{J}^\bot ds  +\int_{\Gamma_t}  \na_{\tau_t} D_t \mathfrak{J}^\bot \,\na_{\tau_t} \big([n_t, D_t^2]  \cdot\mathfrak{J}\big)ds \leq P(E(t))\big(1+E_1(t)\big) .
\]

As a result,  the proof is finished as long as we have Lemma <ref> and Lemma <ref>.

We deal with the remainder integrals in (<ref>) in the following two lemmas.
One has for the second term of the right hand of (<ref>) that

\[
\begin{split}
\Big|\int_0^T\int_{\Gamma_t} D_t h_v \   D_t^2 \mathfrak{J}  \cdot n_tds \Big|  &\leq P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) \\
&\quad+\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt+ \f{1}{8} \int_0^T F_1(t).
\end{split}
\]

To begin with, one recalls the definition of $h_v$ in (<ref>) to obtain

\[
\int_{\Gamma_t} D_t h_v \   D_t^2 \mathfrak{J}  \cdot n_tds=\int_{\Gamma_t} D_t\big(\D_{\G_t} v\cdot D_t n_t  \big) D_t^2 \mathfrak{J}  \cdot n_tds +\int_{\Gamma_t} D_t\big[2D^2v\big(\tau_t, (\na_{\tau_t}v)^\top\big)\cdot n_t\big] D_t^2 \mathfrak{J}  \cdot n_tds.
\]

We only deal with the first integral in the integral above, since the second one can be handled in a similar way.

In fact, it is straightforward to see by (<ref>) that

\[
\begin{split}
&\int_{\Gamma_t} D_t\big(\D_{\G_t} v\cdot D_t n_t  \big) D_t^2 \mathfrak{J}  \cdot n_tds\\
&=\int_{\Gamma_t} D_t\big[\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v )\big] D_t^2 \mathfrak{J}  \cdot n_tds\\
&= \int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 \mathfrak{J}  \cdot n_tds
+ \int_{\Gamma_t}\D_{\G_t} v\cdot D_t\big[\tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v)\big] D_t^2 \mathfrak{J}  \cdot n_tds \\
&\triangleq I_1+I_2.
\end{split}
\]

The estimates for $I_1, I_2$ are proved  in the following lines.

- Estimates of $I_1$.  First, direct computations lead to

\[
\begin{split}
I_1 =&\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}-\mJ|_c)  \cdot n_tds
\\
&  +\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}|_c)  \cdot n_tds  \\
= & \f d{dt} \int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&-\int_{\Gamma_t} D^2_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t(\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&-\int_{\Gamma_t} D_t\D_{\G_t} v\cdot D_t\big[\tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v)\big] D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&-\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) \,D_t (\mathfrak{J}-\mJ|_c)  \cdot D_t(n_tds)\\
&  +\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}|_c)  \cdot n_tds  \\
\triangleq & \f d{dt} \int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds+\sum_{i=1}^4I_{1i}.
\end{split}
\]

We deal with $I_{1i}$ one by one.  In fact, applying integration by parts, we have

\[
\begin{split}
&=D_t\na_{\tau_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) \,D_t^2 (\mathfrak{J}|_c)  \cdot n_t\big|^{p_l}_{p_r}\\
&\quad-\int_{\Gamma_t} D_t\na_{\tau_t} v\cdot \na_{\tau_t}\big[ \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}|_c)  \cdot n_t\big]ds\\
&\quad + \int_{\Gamma_t} [D_t,\na_{\tau_t}]\na_{\tau_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}|_c)  \cdot n_tds\\
&\quad - \int_{\Gamma_t} D_t\big((\na_{\tau_t}\tau_t)^\top\cdot\na v\big)\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t^2 (\mathfrak{J}|_c)  \cdot n_tds.
\end{split}
\]

Using (<ref>) and similar arguments as in (<ref>) for $\pa^2v$ and $v^\perp$ terms and applying Lemma <ref> to $D^2\mJ|_{p_i}$, we derive

\[
|I_{14}|\le \f1{16}F_1(t)+P(E(t))\big(1+E(t)\big).
\]

For $I_{11}$, since

\[
\begin{split}
&D^2_t\D_{\G_t} v=D^2_t\big(\na_{\tau_t}\na_{\tau_t}v-(\na_{\tau_t}\tau_t\cdot\tau_t)\na_{\tau_t} v\big)\\
&=\na_{\tau_t}D^2_t\na_{\tau_t}v+[D^2_t, \na_{\tau_t}]\na_{\tau_t}v-D^2_t \big((\na_{\tau_t}\tau_t\cdot\tau_t)\na_{\tau_t} v\big)\\
+(D_t\tau_t-\na_{\tau_t}v)\cdot\na D_t\na_{\tau_t}v-D^2_t \big((\na_{\tau_t}\tau_t\cdot\tau_t)\na_{\tau_t} v\big)\\
&\triangleq h_{v,2}+l.o.t.
\end{split}
\]

\[
h_{v,2}=\na_{\tau_t}D^2_t\na_{\tau_t}v+ 2\na_{\tau_t}\big((D_t\tau_t-\na_{\tau_t}v)\cdot\na D_t v\big)
-\na^2_{\tau_t}D_tv\cdot n_t(n_t\cdot\tau_t)\na_{\tau_t}v
-(\na_{\tau_t}\tau_t\cdot\tau_t)\na_{\tau_t}D^2_t v
\]
contains higher-order terms of $v$, and
the remainder part contains products like $\pa D_tv\,\pa^2v$ and other lower-order terms and can be controlled by (<ref>), (<ref>), Lemma <ref> and similar arguments as in (<ref>).

Therefore, we arrive at

\[
I_{11}=\int_{\Gamma_t} h_{v,2}\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t(\mathfrak{J}-\mJ|_c)  \cdot n_tds+l.o.t..
\]

To finish the estimate for $I_{11}$, we firstly take care of the first term $\na_{\tau_t}D^2_t\na_{\tau_t}v$ in $h_{v,2}$.  In fact,  we use integration by parts on $\G_t$ to obtain

\[
\begin{split}
&\int_{\Gamma_t} \na_{\tau_t}D^2_t\na_{\tau_t}v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t(\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&=-\int_{\Gamma_t}D^2_t\na_{\tau_t}v\cdot  \na_{\tau_t}\big(\tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v)\big) D_t(\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&\quad-\int_{\Gamma_t} D^2_t\na_{\tau_t}v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) \na_{\tau_t}\big(D_t(\mathfrak{J}-\mJ|_c)  \cdot n_t\big)ds.
\end{split}
\]

Similar arguments as in(<ref>) and  applying (<ref>), (<ref>), we derive

\[
\int_{\Gamma_t} \na_{\tau_t}D^2_t\na_{\tau_t}v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t(\mathfrak{J}-\mJ|_c)  \cdot n_tds\le P(E(t))\big(1+E_1(t)\big).
\]

Meanwhile, the other terms in $I_{11}$  can also be handled similarly, so we conclude

\[
|I_{11}|\le P(E(t))\big(1+E_1(t)\big).
\]

For $I_{12}$, we have

\[
\begin{split}
I_{12}=&-\int_{\Gamma_t} \big([D_t,\na_{\tau_t}]\na_{\tau_t}v-D_t(\na_{\tau_t}\tau_t\cdot\tau_t\na_{\tau_t} v)\big)\cdot D_t\big[\tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v)\big] D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds \\
&-\int_{\Gamma_t} \na_{\tau_t} D_t\na_{\tau_t}v\cdot D_t\big[\tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v)\big] D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds,
\end{split}
\]

where the first integral can be handled as above. For the second integral, we know from Lemma <ref> that

\[
D_t (\mathfrak{J}-\mJ|_c)\in \tilde H^{1/2}(\G_t),
\]

so we apply Lemma <ref>, Lemma <ref>, (<ref>) and Lemma <ref> to find

-∫_Γ_t _τ_t D_t_τ_tv·D_t[τ_t(-_τ_tv^⊥+_τ_tn_t·v)] D_t (𝔍-|_c)  ·n_tds

≤D_t[τ_t(-_τ_tv^⊥+_τ_tn_t·v)]_L^∞(_t)_τ_t D_t_τ_tv_H̃^-1/2(_t)D_t (𝔍-|_c)  ·n_t_H̃^1/2(_t)

As a result, we summarize that

\[
|I_{12}|\le P(E(t))\big(1+E_1(t)^{3/2}\big).
\]

Moreover, similar arguments as for $I_{12}$, we also have

\[
| I_{13}| \leq P(E(t))\big(1+E_1(t)\big).
\]

Together with all these estimates above for $I_{11}$ to $I_{14}$, we can go back to $I_1$ expression and integrate on both sides with respect to time $t$ on $[0,T]$ to find

\[
\begin{split}
\int^T_0 I_1 dt&\le \int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds\big|^T_0\\
&\quad+\int^T_0P(E(t))\big(1+E_1(t)^{3/2}\big)dt+\f{1}{16} \int_0^T F_1(t).
\end{split}\]

Moreover, we have for the first integral on the right side above the following estimate

\[
\begin{split}
&\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&= \int_{\Gamma_t} \big([D_t,\na_{\tau_t}]\na_{\tau_t}v-D_t(\na_{\tau_t}\tau_t\cdot\tau_t\na_{\tau_t} v)\big)\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds\\
&\quad +\int_{\Gamma_t} \na_{\tau_t}D_t\na_{\tau_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds.
\end{split}
\]

Similar arguments as in (<ref>) and checking carefully on the highest-order terms, we  derive

\[
\int_{\Gamma_t} D_t\D_{\G_t} v\cdot \tau_t(-\na_{\tau_t}v^\perp+\na_{\tau_t}n_t\cdot v) D_t (\mathfrak{J}-\mJ|_c)  \cdot n_tds
\le P(E(t))\big(1+E_1(t)^{\delta_0}\big)
\]

with the number $\delta_0\in (0,1)$ as above.

Consequently, we have the following estimate for $I_1$:

|∫_0^T I_1 dt  |
≤P(E(0))(1+E_1(0)^δ_0)+sup_t∈[0, T]P(E(t))(1+E_1(t)^δ_0)

+∫_0^T P(E(t))(1+E_1(t)^3/2)dt+ 116 ∫_0^T F_1(t).

- Estimates of $I_2$. The integral $I_2$ can be  handled in the same way as $I_1$. We simply conclude that

|∫_0^T I_2 dt  |
≤P(E(0))(1+E_1(0)^δ_0)+sup_t∈[0, T]P(E(t)(1+E_1(t)^δ_0)

+∫_0^T P(E(t))(1+E_1(t)^3/2)dt+ 116 ∫_0^T F_1(t).

In the end, combing (<ref>) with (<ref>),  the proof is finished.

Next, we deal with $I_R$.
One has

|I_R|    ≤    18 F_1(t)+P(E_1(t)).

Proof. First, to simplify the notations in the first integral of $I_R$, we denote by

\[
C_P =(D_t\na \cP_{\mathfrak{J}, v}+\na \cP_{\na \cP_{\mathfrak{J}, v}, v}+\na \cP_{\cD_t \mathfrak{J}, v})\cdot n_t.
\]

Thanks to the definition of $P_{w, v}$ (see (<ref>)), we have

\[
C_P=C'_{\mathfrak{J}, v}+\cC_{\na P_{\mathfrak{J}, v}, v}+C_{\cD_t \mathfrak{J}, v}   -D_t  n_t\cdot\na P_{\mathfrak{J}, v}- D_t \big((\mJ\cdot{\tau_t})\na_{\tau_t} v\cdot n_t\big)-(\na_{\tau_t}P_{\mathfrak{J}, v}+\cD_t\mathfrak{J}\cdot \tau_t) \na_{\tau_t} v\cdot n_t.\nonumber
\]

By (<ref>), (<ref>), Lemma <ref> and (<ref>), we  show immediately

Now we are ready to have estimate of $I_R$. For the first integral in $I_R$, we  integrate by parts to derive

\[
\begin{split}
\int_{\Gamma_t} \D_{\Gamma_t} D_t \mathfrak{J}^\bot\, C_P ds
&=\na_{\tau_t} D_t \mathfrak{J}^\bot  C_P\big|^{p_l}_{p_r}-\int_{\Gamma_t}  \na_{\tau_t} D_t \mathfrak{J}^\bot \cdot \na_{\tau_t}C_Pds \\
&\leq \f1{8} F_1(t)+P(E(t))\big(1+E_1(t)\big)
\end{split}
\]

where (<ref>), (<ref>), (<ref>), Lemma <ref>  and (<ref>) are used.

Moreover, we have from (<ref>) and (<ref>) that

\[
\begin{split}
\int_{\G_t} D_t h_v \, C_Pds
&=\int_{\G_t}\big[\D_{\G_t} v\cdot D_t n_t  +2D^2v\big(\tau_t, (\na_{\tau_t}v)^\top\big)\cdot n_t\big]C_Pds\\
&\leq P(E(t))\big(1+E_1(t)\big),
\end{split}
\]

where similar analysis as in the proof of the previous lemma is applied.

On the other hand,  the second integral in $I_R$
can also be handled by a similar argument as above.
As a result, the proof is finished.

§.§.§ Right side of (<ref>).
We firstly deal with the integral involving $\cR_2$.
One has for some $\delta_0\in (0,1)$ the following estimate:

\[
\begin{split}
\Big|\int_0^T\int_{\Om_t}\cR_2 \cdot \big(\cD_t^2 \mathfrak{J}-\na\cH(\cP_{t,2})\big)dXdt
\Big|  &\le P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) \\
&\quad+\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt+ \f{1}{16} \int_0^T F_1(t).
\end{split}
\]

Proof. Recalling from (<ref>), we deal with the terms in $\cR_2$ one by one.

- Estimates of $D_t   \big(\tilde{R}_0-\na \cH(  D^2_t P_{v,v}  ) \big)  $.
In fact, one knows directly from (<ref>) that

\[
\tilde{R}_0- \na \cH(  D^2_t P_{v,v}  ) =-\si\na \cH(J\cdot \D_{\G_t}n_t)+\si\na \cH\big([n_t,\D_{\G_t}]\cdot \na P_{v,v}\big)+\si\na\cH(R_1+h_v ) +A_1+A_2+A_3 ,
\]

where recall that $R_1+h_v$ only contains lower-order derivatives like $\pa v, \pa n_t, \kappa$ and $A_1$ to $A_3$ are defined in (<ref>)-(<ref>).

As a result, checking term by term on $\tilde{R}_0- \na \cH(  D^2_t P_{v,v}  ) $, one finds that it contains $\pa D_t v$, $D_tJ$, $\pa v$ and other lower-order terms, so acting $D_t$ on $\tilde{R}_0- \na \cH(  D^2_t P_{v,v}  ) $ and applying lemmas in Section <ref> and Section <ref> lead to the following estimate:

\[
\big\|D_t   \big(\tilde{R}_0-\na \cH(  D^2_t P_{v,v}  ) \big) \big\|_{L^2(\Om_t)}\leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

- Estimates of $[\na, D_t]\cH (\D_{\Gamma_t}\mathfrak{J}^\bot-h_v) $.  It is straightforward to show by (<ref>) and Lemma <ref> that

\[
\big\| [\na, D_t]\cH (\D_{\Gamma_t} \mathfrak{J}^\bot-h_v) \big\|_{L^2(\Om_t)} \leq P(E(t))\|\D_{\Gamma_t}\mathfrak{J}^\bot-h_v \|_{H^{1/2}(\G_t)}\leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

- Estimates of $\na [\cH, D_t](\D_{\Gamma_t}\mathfrak{J}^\bot-h_v) $.  Recalling (<ref>)
and using Lemma <ref>, (<ref>) and applying variational estimates as in page 33 [53]  imply that

\[
\big\|\na [\cH, D_t](\D_{\Gamma_t}\mathfrak{J}^\bot-h_v) \big\|_{L^2(\Om_t)} \leq P(E(t))\|\cH(\D_{\Gamma_t}\mathfrak{J}^\bot-h_v)\|_{H^1(\Om_t)}\leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

- Estimates of $\na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot$.  To estimate this term, we use similar arguments as in the proof of Lemma <ref>.

First, using Green's Formula, we  obtain

\[
\begin{split}
&\int_{\Om_t} \na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot \cdot \cD_t^2 \mathfrak{J}dX= \int_{\Gamma_t}[\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot \, \cD_t^2 \mathfrak{J}\cdot n_tds\\
&=\int_{\G_t} [\big(\na_{\tau_t}\na_{\tau_t}-(\na_{\tau_t}\tau_t\cdot\tau_t)\na_{\tau_t}\big), D_t]\mJ^\perp \ \cD_t^2\mJ\cdot n_tds\\
&= \int_{\Gamma_t}
2( \na_{\tau_t}v\cdot\tau_t)\na^2_{\tau_t}\mJ^\perp \ \cD_t^2\mJ\cdot n_tds\\
&\quad + \int_{\Gamma_t}\big[\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t)+D_t(\na_{\tau_t}\tau_t\cdot \tau_t)-( \na_{\tau_t}\tau_t\cdot\tau_t)( \na_{\tau_t}v\cdot\tau_t)
\big] \na_{\tau_t}\mJ^\perp \  \cD_t^2 \mathfrak{J}\cdot n_tds.
\end{split}
\]

These two integrals can be handled in a similar way as before, and here we only give the details for the first part in the second integral above. Moreover, the analysis for $\int_{\Om_t} \na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot \cdot  \na\cH(\cP_{t,2})dX$ can be done in a similar and easier way thanks to (<ref>), and  hence estimates for all these remainder parts are omitted.

Recalling (<ref>) and the definition of $C_P$, we rewrite the first part in the second integral above  as follows:

\[
\begin{split}
& \int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, \cD_t^2 \mathfrak{J}\cdot n_t ds\\
&=  \int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t^2 \mathfrak{J}\cdot n_t ds+  \int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, C_P ds \triangleq I_3+I_4.
\end{split}
\]

Similarly as the estimates for $I_1$ in the proof of Lemma <ref>,  one has   for $I_3$ that

\[
\begin{split}
I_3 &=\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t^2 (\mathfrak{J}-\mJ|_c)\cdot n_t ds+\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t^2 (\mathfrak{J}|_c)\cdot n_t ds\\
&\leq   \f1{8} F_1(t)+P(E(t))\big(1+E_1(t)\big)+\f d{dt}\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t (\mathfrak{J}-\mJ|_c)\cdot n_t ds\\
&\quad -\int_{\Gamma_t}D_t\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t (\mathfrak{J}-\mJ|_c)\cdot n_t ds
-\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) D_t\na_{\tau_t} \mathfrak{J}^\bot \, D_t (\mathfrak{J}-\mJ|_c)\cdot n_t ds\\
&\le \f1{8} F_1(t)+P(E(t))\big(1+E_1(t)^{3/2}\big)+\f d{dt}\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) \na_{\tau_t} \mathfrak{J}^\bot \, D_t (\mathfrak{J}-\mJ|_c)\cdot n_t ds,
\end{split}
\]

where we use (<ref>), Lemma <ref> and  Lemma <ref> to have

\[
\begin{split}
&\int_{\Gamma_t}\na_{\tau_t}( \na_{\tau_t}v\cdot\tau_t) D_t\na_{\tau_t} \mathfrak{J}^\bot \, D_t (\mathfrak{J}-\mJ|_c)\cdot n_t ds\\
&\le P(E(t))\Big(1+\sum_i\big\|r^{\delta}(\na^2_{\tau_t} v)\circ T^{-1}_i\big\|_{L^\infty(0, r_0)}
\big\|\|D_t\na_{\tau_t}\mJ^\perp\|_{L^2(\G_t)}\big\|r^{-\delta}D_t (\mathfrak{J}-\mJ|_c)\circ T^{-1}_i\big\|_{L^2(0, r_0)}
\Big)
\end{split}
\]

with $1>\delta>3-\pi/\om_i$ for $\om_i\in (\pi/3,\pi/2)$  and

\[
\begin{split}
\|D_t\na_{\tau_t}\mJ^\perp\|_{L^2(\G_t)}
&\le \big\|(\na_{\tau_t}v\cdot \tau_t)\na_{\tau_t}\mJ^\perp\big\|_{L^2(\G_t)}+\big\|\na_{\tau_t}D_t(\mJ\cdot n_t)\big\|_{L^2(\G_t)}\\
&\le P(E(t))\big(1+\|\na_{\tau_t}D_t\mJ\|_{L^2(\G_t)}+\|\na^2_{\tau_t}v^\perp\|_{L^2(\G_t)}\big)\le
\end{split}
\]

As a result, we have

\[
\begin{split}
\Big|\int_0^T I_3 dt  \Big|
&\le P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) \\
&\quad+\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt+ \f{1}{64} \int_0^T F_1(t).
\end{split}
\]

Since similar arguments can be applied to $I_4$ and other integrals, we  conclude directly

\[
\begin{split}
\Big|\int_0^T\int_{\Om_t} \na \cH [\D_{\Gamma_t}, D_t]\mathfrak{J}^\bot \cdot \cD_t^2 \mathfrak{J}dXdt \Big|&\le P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) \\
&\quad+\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt+ \f{1}{64} \int_0^T F_1(t).
\end{split}
\]

- Estimates of $D_t( \na P_{\na P_{\mathfrak{J}, v}, v} )$.  Recalling the definition of $P_{\mathfrak{J}, v}$ and $P_{\na P_{\mathfrak{J}, v}, v} $ by (<ref>), one has firstly the estimate (<ref>).
Moreover, one can  take $D_t$ on the system of $P_{\na P_{\mathfrak{J}, v}, v}$ as in (<ref>) to obtain the system for $D_tP_{\na P_{\mathfrak{J}, v}, v}$.

Consequently, checking term by term and applying Lemma <ref>,  Lemma <ref>, one derives

\[
\|D_t( \na \cP_{\na \cP_{\mathfrak{J}, v}, v} )\|_{L^2(\Om_t)} \leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

- Estimates of the last term in $\cR_2$.  Thanks to (<ref>), (<ref>), (<ref>) and Lemma <ref>, we have

\[
\big\|D_t \na \cH \big[D_t\big(v\cdot (\na P_{v,v}|_{c})\big)+ v\cdot (\na \mathcal P_{t,1}|_{c})\big]\big\|_{L^2(\Om_t)} \leq P(E(t))\big(1+E_1(t)^{1/2}\big).
\]

In the end, summing up all these estimates above, the proof is finished.

At this moment,   it remains to  handle the last integral on the right side of (<ref>).   In fact, similar arguments as in  the proof of  Lemma <ref>, we can conclude the following estimate:

\[
\begin{split}
&\int_0^{t'}\int_{\Om_t}\big(  D_t \cD_t\na P_{J, v}+D_t \na P_{\cD_t J, v}   \big)\cdot \big(\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big)dXdt\\
&\le P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) +\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt+ \f{1}{16} \int_0^T F_1(t).
\end{split}
\]

§.§.§ The end of the higher-order energy estimate.

Summing up all the estimates in the previous two subsections, we finally arrive at:

\[
\begin{split}
\sup_{t\in [0, T]}\Big(\big\|\cD^2_t \mathfrak{J} -\na \cH(\cP_{t,2})\big\|^2_{L^2(\Om_t)}+\int_{\Gamma_t}|\na_{\tau_t}D_t \mathfrak{J}^\bot |^2\Big)+\f1{4}\int_0^T F_1(t)\\
&\leq P(E(0))\big(1+E_1(0)^{\delta_0}\big)+\sup_{t\in[0, T]}P(E(t))\big(1+E_1(t)^{\delta_0}\big) +\int_0^T P(E(t))\big(1+E_1(t)^{3/2}\big)dt.
\end{split}
\]

Thanks to (<ref>) and Lemma <ref>, we have for a  number $\delta_1>0$ small enough such  that
sup_t∈[0, T] E_1(t)+ ∫_0^T F_1(t)≤P(E(0))(1+E_1(0)^δ_0)+δ_1sup_t∈[0, T] E_1(t)+∫_0^TP(E(t))(1+E_1(t)^3/2)dt.
Therefore, we've finished the proof of Theorem <ref>.

§ WELL-POSEDNESS OF  SYSTEM $\MBOX{(WW)}$

In this section, we use  Picard iteration to prove the existence of solutions to $\mbox{(WW)}$. The main idea is the same as [54], although some necessary modifications are needed and presented here. As a result, we only show the skecth of the proof. For more details, see [54].

§.§ Definitions of surfaces and domains

Following [62, 54], we introduce a map $\Phi_{S_t}$ on the boundary $S_t=\G_t\cup \G_b$ to fix the moving domain $\Om_t$. To start with, we choose a reference domain $\Om_*$ with upper surface $\G_{t*}$ and bottom $\G_{b*}$, which can be taken as the initial domain $\Om_0$ without loss of generality. The contact points of $\Om_*$ are denoted by $p_{i*}$ ($i=l, r$) and the other notations follow similarly.

we define a unit upward vector field $\mu\in H^s(\G_{*}, \cS^1)$ with some large $s$ satisfying

\[
\mu\cdot n_{t*}\ge c_0\quad\hbox{on}\quad \G_{t*},\qquad\hbox{and}\quad\mu|_{p_{l*}}=-\tau_{b*}|_{p_{l*}},
\  \mu|_{p_{r*}}=\tau_{b*}|_{p_{r*}}
\]

for some fixed constant $c_0\in(0,1)$. Here we notice that the conditions above hold at $p_{l*}, p_{r*}$.

Applying the Implicit Function Theorem, there exists a small constant $d_0>0$  such that the map

\[
\Phi: \G_{t*}\times[-d_0,d_0]\rightarrow \R^2\quad\hbox{with}\quad
\Phi(p, d) \triangleq p+d\,\mu(p)
\]

is an $H^s$ diffeomorphism  from its domain to a neighborhood of $\G_*$.

Consequently, this map identifies each upper surface $\G_t$ near $\G_{t*}$ with a unique function

\[
d_{\G_t}: \G_{t*}\rightarrow  \R
\]

and we can define the following map

\[
\Phi_{S_t}: \ \G_{t*}\rightarrow \G_t\subset \R^2\quad\hbox{with}\quad \Phi_{S_t}(p)=p+d_{\G_t}(p)\mu(p).
\]

Meanwhile,  we can use the function $d_{\G_t}(p)$ as the expression of the upper surface $\G_t$, and we have at the corner points that

\[
d_{\G_t}(p_{i*})=p_i,\quad i=l, r.
\]

Moreover, $\Phi_{S_t}$ can be extended to  the entire boundary $S_*=\G_{t*}\cup \G_{b*}$.
Consequently,  we obtain the map on $S_*$:

\[
\Phi_{S_t}: \ \ S_* \rightarrow  {S_t}.
\]

Using the harmonic extension, we define the following map on $\Om_*$:

\[
\cT_{S_t}:\ \Om_*\rightarrow \Om\quad\hbox{with}\quad \cT_{S_t}=\cH_*\big(\Phi_{S_t}-Id_{S_*}\big)+Id.
\]

Here $\cH_*\big(\Phi_{S_t}-Id_{S_*}\big)$ is the harmonic extension of $\Phi_{S_t}-Id_{S_*}$ satisfying

\[
\left\{\begin{array}{ll}
\D \cH_*\big(\Phi_{S_t}-Id_{S_*}\big)=0\qquad\hbox{in}\quad \Om_*,\\
\cH_*\big(\Phi_{S_t}-Id_{S_*}\big)\big|_{\G_{t*}}=d_{\G_t}\mu,
\quad \cH_*\big(\Phi_{S_t}-Id_{S_*}\big)\big|_{\G_{b*}}=\Phi_{S_t}|_{\G_{b*}}-Id_{\G_{b*}}.
\end{array}\right.
\]

§.§ Recovery of the velocity

When the domain $\Om_t$ is defined by $\cT_S$, we can define the velocity $v$ by the free surface function $d_{\G_t}$.
To begin with,  the kinematic condition on $\G_t$ in $\mbox{(WW)}$ is rewritten into

\[
\pa_t \Phi_{S_t}\cdot (n_t\circ \Phi_{S_t})=(v\cdot N_t)\circ \Phi_{S_t}\quad\hbox{with}\quad\pa_t\Phi_{S_t}=(\pa_t d_{\G_t})\mu\quad\hbox{on}\quad \G_*.
\]

So we obtain

_t d__t=(v·n_t)∘Φ_S_tμ·(n_t∘Φ_S_t)  i.e.   v·n_t=(_t d__tμ)∘Φ^-1_S_t·n_t.

Due to the  assumption that the velocity $v$ is  irrotational,  we define $v$ by

v=ϕwith $\phi$ satisfying

Δϕ=ξγ    in  _t,

_n_t ϕ|_Γ_t= (_t d__t  μ)∘Φ^-1_S_t·n_t,   _n_b ϕ|_Γ_b= 0,

\[
\gamma=|\Om_t|^{-1}\quad\hbox{ and }\quad \xi=\int_{\Gamma_t} v\cdot n_t\,ds=\int_{\G_t}(\pa_t d_{\G_t}\mu)\circ \Phi^{-1}_{S_t}\cdot n_tds.
\]

Moreover, we define as in [54] that

\[
D_{t^*}= \pa_t+\na_{v^*},
\]

where $v^{*}= D\Phi_{S_t}^{-1}(v^\top\circ \Phi_{S_t} -\pa_t d_{\G_t} \mu^{\top} )$. A direct computation shows that

\[
(D_t f)\circ \Phi_{S_t}=D_{t^*}(f\circ \Phi_{S_t}) ,
\]

for a function $f$ on $\G_t$.

§.§ The modified formulation  and the precise form of the main theorem

Before we construct the approximate solutions, we derive a new equation modified from the previous sections, which turns out to be more convenient in this part.

First, we define $\mK_a$ based on  the definition of $\mathfrak{K}$:

𝔎_a=+a  d__t∘Φ_S_t^-1= (κ+a  d__t∘Φ_S_t^-1 )-P_v, v   on   _t
for some constant $a>0$, where $\kappa$ can be expressed by $d_{\G_t}$ and $P_{v,v}$ is defined by (<ref>).
So we have

\[
\mJ_a= \na \cH (\mathfrak{K}_a)=\mJ+\si a\na \cH(d_{\G_t}\circ \Phi_{S_t}^{-1}).
\]

Applying  (<ref>) and using $\mK_a$ instead of $\mK$, we obtain

\[
D_t^2 \mathfrak{K}_a+\sigma (a-\D_{\G_t})\cN(\mathfrak{K}_a)=R_{a, 0},
\]

where $R_{a,0}$ is defined by

\[
\begin{split}
R_{a, 0}&=\si R_1 -D_t^2P_{v,v}+\si[n_t, \D_{\G_t}]\cdot \mJ+\si \D_{\G_t}\na P_{v,v}\cdot n_t+2\si\na_{\tau_t}n_t\cdot \na_{\tau_t}\mJ+\si a D^2_t(d_{\G_t}\circ \Phi_{S_t}^{-1})\\
&\quad -\si^2 a \D_{\G_t}\cN(d_{\G_t}\circ \Phi_{S_t}^{-1})+\si a \cN\mK_a.
\end{split}
\]

Acting $\na \cH$ on both sides of the above equation, we  get the equation of $\mJ_a$:

\[
D^2_t\mJ_a+ \sigma\na \cH\big[(a-\D_{\G_t})\mJ_a^\perp  +h_v\big]=R_a,
\]

\[
R_a= \na \cH(R_{a, 0}+\si h_v)-[\na\cH, D_t^2]\mathfrak{K}_a,
\]

and recall that $h_v$ is defined in (<ref>) and comes from $R_1$.

Meanwhile, the condition (<ref>) at the contact points are rewritten as

\[
D_t\mJ_a = \pm\f{\sigma^2 }{\beta_c}\sin\om_i (\na_{\tau_t} \mJ_a)^\bot
\tau_b+R_{c1, a}\quad\hbox{at}\quad p_i (i=l, r)
\]

\[
R_{c1, a}=R_{c1}+\si aD_t\na \cH(d_{\G_t}\circ \Phi_{S_t}^{-1})+\si a \f{\sigma^2 }{\beta_c}(n_t\cdot \tau_b) \big(\na_{\tau_t} \na \cH(d_{\G_t}\circ \Phi_{S_t}^{-1})\big)^\perp.
\]

Next, we consider how to recover  the free surface and the domain from  $\mathfrak{K}_a$, which is slightly different from [54]. In [54], we use the equation of $\mathfrak N_a=\cN(\kappa+a \,d_{\G_t}\circ \Phi_{S_t}^{-1})$ together with the boundary information $d_i=d_{\G_t}|_{p_{i*}}$ of $d_{\G_t}$ to recover the free surface $d_{\G_t}$, so the system of $(\mathfrak{K}_a, d_l, d_r)$ is needed; In this paper, we use the quantity $\mK_a$ instead of $\mathfrak N_a$, where an extra $P_{v,v}$ is added here in (<ref>).
Moreover, we will need to use the equation and norms of $\mJ_a$ in the  iteration scheme, where  $\mK_a$ can be retrieved. In fact, we have $\mJ_a^\perp=\mJ_a\cdot n_t=\cN \mK_a$ on $\G_t$. Therefore,  to identify $\mK_a$, we look at  the Neumann-boundary elliptic system of $\cH(\mK_a)$ with the compatibility condition $\int_{\G_t}\mJ_a^\perp ds=0$. We know immediately that there exists a unique solution $\cH(\mK_a)$ up to an additive constant to this system.
As a result,  as long as  we have $\mJ_a$, we obtain $\mathfrak{K}_a$. (One can also check Lemma 2.5 in [54].)

Consequently, to recover $d_{\G_t}$ (which is the key to recover the water-waves system), we need the system of $(\mK_a, P_{v,v}, d_l, d_r)$. As long as we have proved the existence of the solution to this system, we obtain immediately the following quantity

\[
\kappa+a \,d_{\G_t}\circ \Phi_{S_t}^{-1}=\si^{-1}(\mK_a+P_{v,v})\quad\hbox{with the boundary information} \ d_l, d_r.
\]

As a result,  the the desired function $d_{\G_t}$ can be solved directly from these quantities above  in a similar way as in Proposition 4.2 [54], and then we can finally recover our water-waves system $\mbox{(WW)}$.

Based on the analysis above,  we need to give the boundary condition of $d_{\G_t}$ which is deduce from (<ref>) (for more details, see (4.27) in [54]).  In fact, one has the following evolution equations for $d_i(t)=d_{\G_t}(p_{i*})$ ($ i=l, r$):

d_i”(t)=𝔅_i,  i=l, r,

\[
\begin{split}
\mathfrak{B}_i&=-\f{1}{\mu\cdot (n_t \circ\Phi_{S_t})} \Big(\mu\cdot (n_t \circ\Phi_{S_t})\na_{v^*}\pa_td_{\G_t}+\na_{v^*}\mu\cdot (n_t\circ\Phi_{S_t})\pa_t d_{\G_t}+\si\mathfrak{K}_a\circ \Phi_{S_t}\\
&\quad -\si^2 a \cN (d_{\G_t}\circ \Phi^{-1}_{S_t})\circ \Phi_{S_t}+(\na P_{v,v}+{\bf g})\circ \Phi_{S_t}\cdot (n_t\circ \Phi_{S_t})\Big)\big|_{p_{i*}}.
\end{split}
\]

We rewrite the equation for $P_{v,v}$ by (<ref>):

\[
D_tP_{v,v}=\D^{-1}_N(h_p, f_p, g_p),
\]

where $\D^{-1}_N$ means solving the Neumann-boundary system (<ref>) with $\int_{\Om_t}D_tP_{v,v}dX=0$, and the right-side functions are

\[
h_p=2tr (\na v\cdot \na v\, \na v)+2tr \big(\na(\na P_{v,v}+\mJ)\na v\big)+ 2tr(\na  v  \na^2  P_{v, v}),\quad  f_p=C'_{v,v}(t)+\na_{n_t} v\cdot \na P_{v, v}\big|_{\G_t}\]

\[
g_p=-(mJ+\na P_{v,v}+{\bf g})\cdot\big (\na n_b+(\na n_b)^*\big)\cdot v+v\cdot D_t(\na n_b)\cdot v+\na_{n_b} v\cdot \na P_{v, v}.
\]

As a result, we sum up the system of $(\mK_a, P_{v,v}, d_l, d_r)$ as follows:

D_t^2 𝔎_a +σ(a-__t)(𝔎_a)=R_a, 0,

D_t_a  = ±σ^2 β^-1_csin(_τ_t _a)^τ_b +R_c1, a ,   at   p_i ( i=l, r),

D_t P_v,v=^-1_N(h_p, f_p, g_p),

d^2dt^2d_i(t)=𝔅_i,  i=l, r.

Based on these preparations above, we are finally ready to state our precise form of Theorem <ref>. We start with introducing the space $\Sigma$ for given $T, L>0$:

\[
\Sigma=\big\{(\mK_a, P_{v,v}, d_l, d_r)\big| \|(\mK_a, P_{v,v}, d_l, d_r)\|_\Sigma\le L\big\}
\]

where the  norm

\[
\begin{split}
\|(\mK_a, P_{v,v}, d_l, d_r)\|_\Sigma&\triangleq \|\pa_t(\mJ_a\circ \Phi_{S_t})  \|_{C([0, T]; L^2(\Om_*))}
+\| (\mJ_a\circ \Phi_{S_t})\cdot n_{t*}\|_{C([0, T]; H^1(\Gamma_*))} \\
&\quad +\|P_{v,v}\circ  \Phi_{S_t}\|_{C([0, T]; H^{5/2}(\Om_*))} +\sum_{i=l, r}\Big(|d_i|_{C([0, T])}+| d(d_i)/ dt |_{C([0, T])}\Big).
\end{split}
\]

Meanwhile, according to the higher-order energy $E_h(t)$, we also define the space (for given $L_1>0$)

\[
\Sigma_h=\big\{(\mK_a, P_{v,v}, d_l, d_r)\big| \|(\mK_a, P_{v,v}, d_l, d_r)\|_{\Sigma_h}\le L_1\big\}
\]

by the norm

\[
\begin{split}
\|(\mK_a, P_{v,v}, d_l, d_r)\|_{\Sigma_h}&\triangleq \|(\mK_a, P_{v,v}, d_l, d_r)\|_\Sigma+\|\pa_t\big((\mJ_a\circ \Phi_{S_t}) \cdot n_{t*}\big)  \|_{L^\infty([0, T]; H^1(\G_*))}\\
&\quad +\|\pa^2_t(\mJ_a\circ \Phi_{S_t})  \|_{L^\infty([0, T]; L^2(\Om_*))}.
\end{split}
\]

The initial data is given by

\[
\left\{\begin{array}{ll}
(\mJ_a\circ \Phi_{S_t})\cdot n_{t*}\big|_{t=0}=\bar \mJ_{a, 0},  \quad \pa_t(\mJ_a\circ \Phi_{S_t})\big|_{t=0} =\bar\mJ_{a, 1},\\
P_{v,v}\circ \Phi_{S_t}\big|_{t=0}=\bar P_{v, v, 0}, \quad d_i(0)=d_{i, 0},\quad  \f d{dt}d_i(0)=d_{i, 1}.
\end{array}
\right.
\]

Now we can  present  the local well-posedness theorem.
Assume that the  initial data $\big(\bar \mJ_{a, 0}, \bar\mJ_{a, 1}, \bar P_{v, v, 0}, d_{i,0}, d_{i, 1}\big)\in H^1(\G_*)\times L^2(\Om_*)\times  H^{5/2}(\Om_*)\times \R^2$ and initial contact angles $\om_{i0}\in (0, \pi/2) $ for $i=l, r$.  When the  compatibility conditions  (<ref>) at $t=0$  are satisfied for $k=0,1,2,3$, there exists a unique solution $(\mK_a, P_{v,v}, d_l, d_r)\in \Sigma_h$ to system (<ref>). Moreover, system (<ref>) is locally well-posed with  $(\mK_a, P_{v,v}, d_l, d_r)$ depending continuously on the initial data in  $\Sigma$.

§.§ Iteration scheme