\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 177, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/177\hfil Energy quantization]
{Energy quantization for approximate H-surfaces and applications}

\author[S. Zheng \hfil EJDE-2013/177\hfilneg]
{Shenzhou Zheng}  

\address{Shenzhou Zheng \newline
Department of Mathematics,  Beijing Jiaotong University,
Beijing 100044, China}
\email{shzhzheng@bjtu.edu.cn, Phone +86-10-51688449}

\thanks{Submitted February 2, 2013. Published July 30, 2013.}
\subjclass[2000]{35J50, 35K40, 58D15}
\keywords{Approximate H-surface maps; energy quantization;
H-surface flows; \hfill\break\indent concentration of energy;  
bubbling phenomena}

\begin{abstract}
 We consider weakly convergent sequences of approximate H-surface maps
 defined in the plane with their tension fields bounded in $L^p$ for
 $p> 4/3$, and establish an energy quantization that accounts for
 the loss of their energies by the sum of energies over finitely many
 nontrivial bubbles maps on $\mathbb{R}^2$. As a direct consequence,
 we establish the energy identity at finite singular time to their
 H-surface flows.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The main aim of this study is to discuss the energy quantization
of weakly convergent sequences for the weak solutions of approximate 
H-surface maps. Similar to approximate harmonic or biharmonic maps 
with the controlled tension or bi-tension fields 
 \cite{DT,QT,LW1,WZ1,WZ2}, we consider energy quantization of approximate
 H-surface maps not only its own interest but also an important application 
to H-surface flows. In fact, As a direct consequence we will show energy 
identity to so called  H-surface flows.

Let $\Omega\subset \mathbb{R}^2$ be a bounded smooth domain, and 
$H:\mathbb{R}^3\to \mathbb R$ be a given bounded measurable
function; i.e., $H(\cdot)\in L^{\infty}(\mathbb{R}^3) $. 
First we recall the notion of approximate H-surface maps.

\begin{definition}\rm
 A map $u\in W^{1,2}(\Omega,\mathbb R^3)$ is called an approximate 
H-surfaces, if there exists a tension
field $\tau\in L^p_{\rm{loc}}(\Omega, \mathbb R^{3}), p\ge 1$ such that
\begin{equation}\label{approx_H}
\tau(u)=\Delta u - 2H(u)u_x\wedge u_y,\quad\text{in }\Omega.
\end{equation}
In particular, if $\tau\equiv0$, then the map $u$ satisfies
\begin{equation}\label{H_surfaces}
\Delta u = 2H(u)u_x\wedge u_y ,\quad\text{in }\Omega
\end{equation}
which is called a $H$-surface.
\end{definition}

It is well-known that if $u$ is a conformal representation of a surface 
$\mathcal{S}=u(\Omega)$; i.~e.,
\[ %\label{conformal_repres}
\|u_x\|^2- \|u_y\|^2=u_x  \cdot u_y=0 ,
\]
then $H(u)$ is the mean curvature of the surface $\mathcal{S}$ at the point $u$.

Notice that H-surface is a critical point of the following energy 
functional in $W^{1,2}(\Omega, \mathbb R^3)$.
\begin{equation}\label{H functional}
J_H(u) := \int_{\Omega} \Big(|\nabla u|^2 +\frac 43 Q(u)u_x\wedge u_y\Big)
\end{equation}
with
$$
Q(u)= \Big(\int_{0}^{u_1} H(s, u_2,u_3)ds,\int_{0}^{u_2} H(u_1, s, u_3)ds,
 \int_{0}^{u_3}H(u_1, u_2, s)ds\Big).
$$
From the view of geometrical significance, the system \eqref{H_surfaces} 
can be regarded to be the minimization problem of the standard energy 
$E(u,\Omega):=\int_{\Omega} |\nabla v|^2$ with a constraint of the 
prescribed volume $V(v):=\frac 13 \int_{\Omega} v\cdot v_x\wedge v_y
=\text{Constant}$; that is,
\begin{equation}
\min_{v\in W^{1,2}(\Omega,R^3)}\Big\{\int_{\Omega} |\nabla v|^2: v
=\phi \text{ on }\partial\Omega, V(v)=C\Big\},
\end{equation}
for any given $\phi\in W^{1,2}(\Omega)$, and here $H$ is so-called
Lagrangian multiplier.

Wente  \cite{Wente} and Hildebrandt  \cite{Hil} made fundamental contributions 
on the existence of solutions to the planar Plateau problem or surfaces 
with constant mean curvature, respectively 
(see also Helein's monograph  \cite{Hel}). Later, Brezis-Coron \cite{BC1} 
and Struwe \cite{St} showed existence of multiple solutions of H-surface 
maps in a bounded domain of $\mathbb{R}^2$ for given boundary data. 
As we knew, for variable $H$ there were many significant works by 
Rey \cite{Re}, Bethuel-Rey \cite{BR}, Caldiroli-Musina \cite{CM} and 
Chen-Levine \cite{CL}. Meanwhile, the regularity and bubbling phenomena 
analysis to so-called  H-surface flows in $ W^{1,2}(\Omega, \mathbb{R}^3)$ 
has been shown in various cases such as  $H$ is a constant, H depends only 
on two variables, or $H(u)$ is uniformly Lipschizt continuous 
(see Brezis-Coron\cite {BC} and Hong-Hsu \cite{HH}). In addition, for the 
high dimensional case ($n>2$), Mou-Yang \cite{MY} introduced H-systems in 
a bounded domain of $\mathbb R^n$ and established the existence of multiple 
solutions of H-system for a constant $H$ and given boundary data. 
Furthermore, Duzaar-Grotowski \cite{DG} studied the existence of solutions 
of the H-system with a variable function H from a domain into a higher 
dimensional compact Riemannian manifold. All in all, it is an important 
observation that H-surface maps are invariant under the dilation 
transformations in $\mathbb R^2$. Such a property leads to non-compactness 
of sequences of H-surfaces in $\mathbb R^2$, which prompts studies 
by Brezis-Coron \cite{BC1} concerning the failure of strong convergence 
for weakly convergent H-surfaces.  Roughly speaking, the results in  \cite{BC1} 
assert that the failure of strong convergence occurs at finitely many 
concentration points of its energy, where finitely many bubbles 
(i. e. any nontrivial solutions in $\mathbb R^2$) are generated, and the 
total energies from these bubbles account for the total loss of its energies 
during the process of convergence.

Based on the above observation, our main purpose is to extend the results 
from  \cite{BC1,Re,HH} to the context of suitable approximate H-surface 
maps due to its more flexible applications. More precisely, we have

\begin{theorem}\label{energy_identity}
Let $\Omega\subset\mathbb{R}^2$ be a bounded smooth domain. Suppose
that $\{u^k\}_{k=1}^{\infty}\subset W^{1,2}(\Omega,\mathbb{R}^3)\cap
L^{\infty}(\Omega,\mathbb{R}^3)$ is a sequence of approximate
H-surface satisfying
\begin{equation}\label{approx_sequence}
\Delta u^k = 2H(u^k)u^k_x\wedge u^k_y +f^k
\end{equation}
with $H(\cdot)\in L^{\infty}(\mathbb{R}^3)$ and $f^k(x)\in
L^p(\Omega)$ with $p>4/3 $. Let
\begin{equation}\label{uniform-bound}
\sup_{k\in \mathbb{N}}\Big(\|\nabla
u^k\|_{L^2(\Omega)}+ \|H(u^k)\|_{L^{\infty}(\mathbb{R}^3)}
+\|f^k\|_{L^p(\Omega)}\Big)\le M <+\infty.
\end{equation}
Then there exist finite points $\{x_1,x_2,\dots,x_L\}$ and an
approximate H-surface $u\in W^{1,2}(\Omega,\mathbb{R}^3)$ so that
\begin{equation}
\Delta u = 2H(u)u_x\wedge u_y +f,\quad \text{in } \Omega,
\end{equation}
where
$u^k\rightharpoonup u$ in $W^{1,2}$ and  
$f^k\rightharpoonup f$  in $L^{p}$.
Moreover, we have:
\begin{itemize}
\item[(1)]  $u^k\to u$  strongly in $ W^{1,2}_{\rm loc}\cap C^0_{\rm loc}
(\Omega\setminus\{x_1,x_2,\dots,x_L\},\mathbb{R}^3)$.
\item[(2)] There exist $L_i\in \mathbb{N}$ and bubbles
$\{\omega_{ij}\}_{j=1}^{L_i}$, which are nontrivial H-surface systems
from $\mathbb{R}^2$ to $S^{2}$, such that
\begin{equation}\label{energy_id1}
\lim_{k\to\infty}\int_{B_{r_i}(x_i)} |\nabla u^k|^2 dx
=\int_{B_{r_i}(x_i)}|\nabla
u|^2dx+\sum_{j=1}^{L_i}\int_{\mathbb{R}^2} |\nabla \omega_{ij}|^2
\end{equation}
where
$$
r_i=\frac 12\min_{1\le j\le L,j\neq
i}\{|x_i-x_j|,d(x_i,\partial\Omega) \}.
$$
\end{itemize}
\end{theorem}

Similar to initially Lin-Wang's argument to deal with harmonic maps \cite{LW1}, 
our main idea is to show no concentration of energy in the neck.  
More precisely, that is done while establishing there is no concentration 
of angular energy in the neck region; then controlling the radial energy 
in the neck region by angular energy and $L^p$-norm of its tension field 
with $p>4/3$ through so called Pohozaev argument \cite{LW1} \cite{WZ1}.
 During using control the radial energy by the angular hessian energy and 
$L^p$-norm of its tension fields by Pohozaev  argument in the neck region, 
the assumption $p>4/3$ seems to be necessary to validate the Pohozaev 
argument, since we need
$\Delta u^k\cdot(x\cdot\nabla u^k) \in L^1$ and 
$f^k\cdot(x\cdot\nabla u^k)\in L^1$.

A typical application of Theorem \ref{energy_identity} is to study asymptotic 
behavior at finite time for H-surface flows in the plane.
 We can directly obtain identity energy at finite time to H-surface 
flows with initial data $u_0$ as follows.
\begin{equation}\label{flow-pro}
\begin{gathered}
u_t=\Delta u - 2H(u)u_x\wedge u_y, \quad  (x,t)\in \Omega\times (0,+\infty)\\
u|_{t=0}=u_0,\quad \quad  x\in \Omega\\
u|_{\partial\Omega}=u_0|_{\partial\Omega}, \quad  t>0, x\in \partial\Omega
\end{gathered}
\end{equation}
where $u_0\in W^{1,2}(\Omega) $ and $H\in L^{\infty}(\mathbb{R}^3)$. 
In particular, note that any $t$-independent solution
$u:\Omega \to \mathbb{R}^3$ of \eqref{flow-pro} is a H-surface system.

We are inspired by Hong-Hsu's energy inequality in \cite[Theorem 3.7]{HH}:
for arbitrary $u_0\in C^2(\Omega,\mathbb{R}^3)$ satisfying 
$\|u_0\|_{L^{\infty}}\|H\|_{L^{\infty}}<1$,
there exists  a time $T_0 > 0$ such that
\begin{equation}\label{energy_ineq}
\|u_t\|_{L^2(\Omega\times(0,T_0))}\le J_H[u_0],
\end{equation}
where $J_H$ is represented by \eqref{H functional}. 
Then we also consider that, for a finite singular time $T_0 < +\infty$, 
energy identity accounting for the $ \delta$ mass by finite many bubbles. 
This observation can be proved by applying the rescaled maps to conformal 
invariance of H-surface flows. Then, from the energy inequality 
\eqref{energy_ineq} there exists a sequence $t_k\uparrow T_0$ such
 that $u^k:=u(\cdot, t_k)\in W^{1,2}(\Omega,\mathbb{R}^3)$ satisfies 
\begin{itemize}
\item[(i)] $ \tau_2(u^k):=\|u_t(t_k)\|_{L^2}\to 0$; and
\item[(ii)] $u^k$ satisfies in the distribution sense
\begin{equation}\label{approx_H-surf-1}
\Delta u^k = 2H(u^k)u^k_x\wedge u^k_y +\tau_2(u^k).
\end{equation}
\end{itemize}
Therefore, from Theorem \ref{energy_identity} we derive that an energy 
identity of the weak limit of H-surface flows are connected together
 without any neck region. In particular, the image of $u_n$ 
converges pointwise to the image of the limit bubble tree maps, 
which is similar to harmonic map flows ( \cite{DT} \cite{QT} \cite{LW1}). 
More precisely, we have the following theorem.

\begin{theorem}\label{flow_identity}
 For some $T_0 < +\infty$, let 
$u\in C^{2+\alpha,1+\alpha}_{\rm loc}(\Omega\times (0, T_0))$ 
be a solution to \eqref{flow-pro} with 
$\|u_0\|_{L^{\infty}(\Omega)}\|H\|_{L^{\infty}(\Omega)}<1$, where 
$T_0$ is a singular time. Then there exist a finite many bubbles 
$\omega_i,i=1,\dots,L $ such
that
\begin{equation}\label{energy_id2}
\lim_{t\to T_0}E(u(\cdot,t),\Omega)=E(u(\cdot,T_0),\Omega)
+\sum_{j=1}^{L}E(\omega_i,\mathbb S^2),
\end{equation}
where $E(\cdot,\mathbb S^2)$ is the energy of finite many bubbles on 
the unit sphere $\mathbb  S^2$.
\end{theorem}

The article is organized as follows. In \S 2, we establish a 
locally H\"oler continuity of weak solutions and the higher 
integrability of their first and second order derivatives, 
strong convergence and  blow-up analysis to any approximate H-surface 
maps with the smallness energy condition and its tension field in $L^p$ 
for some $p>1$. In \S 3, we prove main Theorem \ref{energy_identity}
 by establishing  that there is no concentration of angular energy in 
the neck region;  and then controlling the radial energy in the neck region 
by angular  energy and $L^p$-norm of its tension field with $p>4/3$ 
by the Pohozaev  argument. 
In \S 4, As a consequence of the main Theorem, we set up the 
energy identity at finite singular time $T_0 < +\infty$ to sequences of
 H-surface flows.

\section{A priori estimates of approximate H-surfaces}

This section is mainly devoted to a locally H\"older continuity of
weak solutions and the higher integrality of the first and second-order 
derivatives under the smallness energy. To this end, we need to use 
Riesz potential estimates in the Morrey spaces due to Adams \cite{A}. 
For an open set $U\subset\mathbb R^n$, $1\le p<+\infty$, $0<\lambda\le n$, 
the Morrey space $M^{p,\lambda}(U)$ is defined by
\begin{equation}\label{Morrey}
M^{p,\lambda}(U)
=\Big\{f\in L^p(U):
\|f\|_{M^{p,\lambda}}^p
=\sup_{B_r\subset U} r^{\lambda-n}\int_{B_r}|f|^p<+\infty \Big\}.
\end{equation}
Note that the weak $L^p$ space is denote by $L^{p,*}(\Omega)$: for any $ t>0$,
which satisfies
$$
\|f\|^p_{L^{p,*}(\Omega)}:= \sup_{t>0}t^p|\{x\in \Omega|
|f(x)|>t\}|<\infty.
$$
Therefore, the weak Morrey space $M^{p,\lambda}_*(\Omega)$ is defined to be the
set of functions $f\in L^p_*(\Omega)$ satisfying
\begin{equation}\label{weak Morrey}
\|f\|^p_{M^{p,\lambda}_*(\Omega)}:= \sup_{x\in\Omega,0<\rho\leq d}
\{\rho^{\lambda-n}\|f\|^p_{L^p_*(\Omega\cap
B_{\rho}(x))}\}<\infty
\end{equation}
with  $0\le \lambda\le n $ and $d=diam(\Omega)$. Let $I_\beta(f)$ 
be the Riesz potential of order $\beta$ ($0<\beta\le n$) defined by
\begin{equation} \label{reisz}
I_{\beta}(f)(x):=\int_{\mathbb{R}^n}\frac
{f(y)}{|x-y|^{n-\beta}} \,dy, \quad x\in \mathbb{R}^n.
\end{equation}
Then we have the following Riesz potential estimates between Morrey 
spaces due to Adams  \cite{A}.

\begin{lemma}\label{reisz_morrey}
\begin{itemize} 
\item[(1)]  For any $\beta>0, 0 <\lambda\le n, 1 < p <\frac
{\lambda}{\beta}$, if $f\in M^{p,\lambda}(\mathbb{R}^n)$, then 
$I_{\beta}(f)\in M^{\tilde{p},\lambda}(\mathbb{R}^n)$, where
 $\tilde{p}=\frac {\lambda p}{\lambda-p\beta} $. Moreover,
\begin{equation}\label{Riesz-Morrey}
\|I_{\beta}(f)\|_{M^{\tilde{p},\lambda}(\mathbb{R}^n)}\le
C\|f\|_{M^{p,\lambda}(\mathbb{R}^n)}.
\end{equation}
\item[(2)] For $0<\beta<\lambda\le n$, if $f\in M^{1,\lambda}(\mathbb{R}^n)$, 
then $I_{\beta}(f)\in M_*^{\frac{\lambda}{\lambda-\beta},\lambda}(\mathbb{R}^n)$. 
Moreover,
\begin{equation}\label{Riesz-weak-Morrey}
\|I_{\beta}(f)\|_{M_*^{\frac{\lambda}{\lambda-\beta},\lambda}(\mathbb{R}^n)}\le
C\|f\|_{M^{1,\lambda}(\mathbb{R}^n)}.
\end{equation}
\end{itemize}
\end{lemma}

In terms of conformal invariance of the H-surface maps, we can do it in the
 unit disc $B_1$ in order to obtain so called $\varepsilon_0$-strong convergence 
for uniformly bounded sequences.
First, we establish the higher integrability of the first and second-order 
derivatives to approximate H-surface maps \eqref{approx_sequence}.

\begin{lemma}\label{integ_smallness}
Suppose $u^k\in W^{1,2}(B_1,\mathbb{R}^3)$ is a sequence of approximate
 H-surface maps; i.e.,
\begin{equation}\label{approx_H-surf}
\Delta u^k = 2H(u^k)u^k_x\wedge u^k_y +f^k,\quad x\in B_1,
\end{equation}
where $H\in L^{\infty}(R^3) $ and $f^k\in L^p(B_1,R^3)$ with any
$1<p<2$. If, for some sufficiently small constant $\varepsilon_0$,
such that
\begin{equation}\label{smallness}
\int_{B_1}|\nabla u^k|^2dx \le \varepsilon_0^2.
\end{equation}
Then: 
\begin{itemize}
\item[(1)] we have $u\in C^{\alpha}(B_{1/2},\mathbb{R}^3)$ for
 $\alpha\in (0,1-\frac 1p)$  and
\begin{equation}\label{holder_est}
[u^k]_{C^{\alpha}(B_{1/2})}
\le C\Big(\varepsilon_0+\|f^k\|_{L^p(B_1)}\Big).
\end{equation}
\item[(2)] For any $p'\in (2,\frac{2p}{2-p})$, we have  
$u\in W^{1,p'}(B_{1/2})$ and
\begin{equation}\label{integ-improve}
\|\nabla u^k\|_{L^{p'}(B_{1/2})}\le
C\left(\varepsilon_0+\|f^k\|_{L^p(B_1)}\right).
\end{equation}
\item[(3)] Furthermore, $u^k\in W^{2,p}(B_{1/2})$ and
\begin{equation}\label{second-est}
\|\nabla^2 u^k\|_{L^{p}(B_{1/2})}
\le C\Big(\varepsilon^2_0+\|f^k\|_{L^p(B_1)}\Big).
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
 (1)  Observe that
\begin{equation}\label{G_1-eqn}
\begin{gathered} 
\Delta G^k_1=2H(u^k)u^k_x\wedge u^k_y,\quad \text{in }  B_1\\
G^k_1=0,\quad \text{on}\ \partial B_1.
\end{gathered}
\end{equation}
and
\begin{equation}\label{G_2-eqn}
\begin{gathered}
 \Delta G^k_2=f^k,\quad \text{in }  B_1\\
G^k_2=0,\quad \text{on }  \partial B_1.
\end{gathered}
\end{equation}
Set $h^k=u^k-G^k_1-G^k_2$, we obtain
\begin{equation}\label{harm-term}
\begin{gathered} 
\Delta h^k=0,\quad \text{in } B_1\\
h^k=u^k,\quad \text{on }  \partial B_1.
\end{gathered}
\end{equation}
To estimate $G^k_1$, noting that 
$u^k_x\wedge u^k_y\in \mathcal{H}^1(B_1)$
(Hardy spaces),  by \cite[Theorem 3.2.9]{Hel}, for $0<\theta<1$, we have
\begin{equation}
\|\nabla G^k_1\|_{L^2(B_{\theta})}\le C\|2H(u^k)u^k_x\wedge
u^k_y\|_{\mathcal{H}^1(B_1)}.
\end{equation}
Due to the boundedness of  $H(u^k)$ and Wente's inequality 
(see \cite[Theorem 3.1.2]{Hel}), 
\begin{equation}\label{G_1-esti}
\|\nabla G^k_1\|_{L^2(B_{\theta})}\le C \|\nabla
u^k\|^2_{L^2(B_1)}\le C\varepsilon_0 \|\nabla u^k\|_{L^2(B_1)}.
\end{equation}
For $G^k_2$, by $L^p$-theory of Laplace operator $\Delta$ we get
$$
\|\nabla^2 G^k_2\|_{L^p(B_{\theta})}\le C\|f^k\|_{L^p(B_1)},
$$
it follows, from the Sobolev's theorem,  that
\begin{equation}\label{G_2-esti}
\begin{aligned}
\|\nabla G^k_2\|_{L^2(B_{\theta})}
&\le  C \|\nabla^2 G^k_2\|_{L^1(B_{\theta})} \\
&\le  C\theta^{2(1-\frac 1p)} \|\nabla^2
G^k_2\|_{L^p(B_{\theta})}\le C\theta^{2(1-\frac
1p)}\|f^k\|_{L^p(B_1)}.
\end{aligned}
\end{equation}
Moreover, in accordance with the standard estimates of harmonic functions, 
one obtains
\begin{equation}\label{h-esti}
\|\nabla h^k\|_{L^2(B_{\theta})}\le C \|\nabla u^k\|_{L^2(B_1)}\le
C\varepsilon_0.
\end{equation}
Now, we put all estimates \eqref{G_1-esti},\eqref{G_2-esti} and 
\eqref{h-esti} together. It yields
\begin{equation}\label{decay-esti}
\|\nabla u^k\|_{L^2(B_{\theta})}\le C\varepsilon_0 \|\nabla
u^k\|_{L^2(B_1)}+C(\varepsilon_0+\theta^{2(1-\frac
1p)})\|f^k\|_{L^p(B_1)}.
\end{equation}
By iterating \cite[Lemma 3.4]{HL},  we obtain
 $u^k \in C^{\alpha}(B_{\vartheta})$ with 
$0<\alpha<1-\frac 1p, 0<\vartheta<1$, and
\begin{equation}\label{Morrey-esti}
\int_{B_{\vartheta}}|\nabla u^k|^2dx\le C \vartheta^{2\alpha}.
\end{equation}

(2) To get a higher integrability,  we assume that $\widetilde{u}^k$ 
and $\widetilde{f}^k$ defined in $\mathbb{R}^2$ are
extensions of $u^k$ and $f^k$ from $B_1$ respectively, such that
$\|\nabla \widetilde{u}^k\|_{L^2(\mathbb{R}^2)}\le C\|\nabla
\widetilde{u}^k\|_{L^2(B_1)}$ and 
$\|\widetilde{f}^k\|_{L^p(\mathbb{R}^2)}\le
C\|\widetilde{f}^k\|_{L^p(B_1)}$. Let $\Gamma(x)$ be the fundamental 
solution of Laplacian operator
$\Delta$ in $\mathbb{R}^2$, then
$$
\widetilde{u}^k(x)=\int_{\mathbb{R}^2}\Gamma(x-y)\Delta \widetilde{u}^k(y)dy.
$$
Therefore, for $x\in B_{1/2}$, we have
\begin{equation}
\begin{aligned}
\nabla {u}^k(x)
&= \int_{\mathbb{R}^2}\nabla
G(x-y)\left(2H(\widetilde{u}^k)\widetilde{u}^k_x\wedge \widetilde{u}^k_y 
 +\widetilde{f}^k\right)(y)dy\\
&\leq  C \Big(\Big|\int_{\mathbb{R}^2}\nabla G(x-y)|\nabla
\widetilde{u}^k|^2dy\Big|
+ \Big|\int_{\mathbb{R}^2}\nabla G(x-y)|\widetilde{f}^k|dy\Big|\Big)\\
&=  I_1(|\nabla \widetilde{u}^k|^2)+I_1(|\widetilde{f}^k|).
\end{aligned}
\end{equation}
Note that $|\nabla\widetilde{u}|^2\in M^{1,2-2\alpha}(\mathbb{R}^2)$ and 
$\widetilde{f}^k\in L^p(\mathbb{R}^2)$,  by Lemma \ref{Riesz-Morrey} 
it implies $\nabla {u}^k\in L^{p_0,*}(B_{1/2})$ with 
$2<p_0=\min\{\frac{2-\alpha}{1-\alpha},\frac {2p}{2-p}\} $ for
any $1<p<2$; and
\begin{equation}
\begin{aligned}
\|\nabla u^k\|_{L^{p_0,*}(B_{1/2})}
&\leq  \|I_1(|\nabla \widetilde{u}^k|^2)\|_{L^{p_0,*}(B_{1/2})}
 +\|I_1(|\widetilde{f}^k|)\|_{L^{p_0,*}(B_{1/2})}\\
&\leq  C(\|\nabla
\widetilde{u}^k\|_{L^2(\mathbb{R}^2)}+\|\widetilde{f}^k\|_{L^p(\mathbb{R}^2)})\\
&\leq  C(\|\nabla u^k\|_{L^2(B_1)}+\|f^k\|_{L^p(B_1)})
\end{aligned}
\end{equation}
Thanks to  $L^{p_0,*}(B_1)\subset L^{p'}(B_1)$ with $2<p'<p_0$, 
we obtain that $\nabla u^k\in L^{p'}(B_{1/2})$, and
\begin{equation}\label{integrality}
\|\nabla u^k\|_{L^{p'}(B_{1/2})}\le C(\|\nabla
u^k\|_{L^2(B_1)}+\|f^k\|_{L^p(B_1)}).
\end{equation}
According to the assumption of smallness energy, it clearly yields 
\eqref{integ-improve}.

(3) On the basis of  Calderon-Zygmund's $L^p$-theory and 
\eqref{approx_H-surf}, we have
\begin{equation}\label{second-integrality}
\begin{aligned}
\|\nabla^2 u^k\|_{L^{p}(B_{1/2})} 
&\leq  C\|\Delta u^k\|_{L^p(B_{\frac 34})}\\
&\leq  C\Big( \| |\nabla
u^k|^2\|_{L^p(B_{\frac 34})}+\|f^k\|_{L^p(B_{\frac 34})}\Big)\\
&\leq  C\Big( \|\nabla u^k\|_{L^2(B_{\frac 34})}\|\nabla u^k\|_{L^{p'}
(B_{\frac 34})}+\|f^k\|_{L^p(B_1)}\Big),
\end{aligned}
\end{equation}
with $2<p'<p_0$. Thanks to the smallness assumption \eqref{smallness} 
and the higher integrability of derivative \eqref{integ-improve}, 
we obtain $u^k\in W^{2,p}(B_{1/2})$ and
\begin{equation}
\begin{aligned}
\|\nabla^2 u^k\|_{L^{p}(B_{1/2})}
&\leq  C \Big(\varepsilon_0\|\nabla u^k\|_{L^{p'}(B_{\frac 34})}
+\|f^k\|_{L^p(B_1)}\Big)\\
&\leq  C \Big(\varepsilon^2_0+\|f^k\|_{L^p(B_1)}\Big).
\end{aligned}
\end{equation}
This completes the proof.
\end{proof}

It is well known that the energy concentration leads to both the failure 
of strong convergence and the
formation of singularity for sequences of approximate H-surface maps. 
With the help of the higher integrability of the first and second
derivatives for weak solutions, we further consider the blow-up 
analysis and $\varepsilon_0$-strong convergence to a sequence of 
approximate H-surface maps \eqref{approx_sequence}.

\subsection*{Proof of Theorem \ref{energy_identity} (1)}
Since
$u^k\rightharpoonup u$ weakly in $W^{1,2}(\Omega,R^2)$, we have that
$\mu_k=|\nabla u^k|^2dx$ is a family of nonnegative Radon measures
such that $N=\sup_k \mu_k(\Omega) <\infty$. Therefore, after taking
possible subsequences, we may assume that there is a nonnegative
Radon measure $\mu$ such that $\mu_k\to \mu $ as convergence
of Radon measures. Moreover, by Fatou's Lemma, we have that there is
a nonnegative Radon measure $\nu $, called as the defect measure,
such that $\mu=|\nabla u|^2dx+\nu$. Denote by $\Sigma $ the support
of $\nu $. Then we have
\begin{equation}\label{singular-set}
\Sigma=\cap_{r>0}\Big\{x\in \Omega: \lim\inf_{k}\int_{B_r(x)}|\nabla
u^k|^2dy\ge \varepsilon^2_0 \Big\}.
\end{equation}
Let $\Sigma_1 = \{x_1,x_2,\dots,\}$ be any discrete points of
$\Sigma$, and $\{B_{\delta_0} (x_i)\}_{i=1}^\infty $ be mutually
disjoint balls for small $\delta_0$. Then we have
$$
\lim\inf_{k}\int_{B_r(x_i)}|\nabla u^k|^2dy\ge
\varepsilon^2_0,\quad \forall\ 1\le i\le \infty.
$$
Therefore, there exists a natural number $K$ such that for $k\ge K$
we have
$$
\int_{B_r(x_i)}|\nabla u^k|^2dy\ge \varepsilon^2_0,\quad \forall\
1\le i\le \infty.
$$
Let $\mathcal{H}_0$ denote the 0-dimensional Hausdorff measure, then
\begin{equation}\label{finite-sigma}
\begin{aligned}
\varepsilon^2_0 \mathcal{H}_0(\Sigma)
&\leq \sum_{i=0}^{\infty}\int_{B_r(x_i)}|\nabla u^k|^2dy\\
&=  H_0(\Sigma)\int_{\cup_{i=0}^{\infty} B_r(x_i)}|\nabla u^k|^2dy\\
&\leq  \int_{\Omega}|\nabla u^k|^2dy\le N<\infty;
\end{aligned}
\end{equation}
this implies $\mathcal{H}_0(\Sigma)\le L:= \frac {N}{\varepsilon^2_0 } $.
By a compact embedding: 
$W^{1,p'}(B_1,\mathbb{R}^3)\hookrightarrow W^{1,2}(B_1,\mathbb{R}^3) (p'>2) $ due
to Lemma \ref{integ_smallness}, therefore, for any compact subset 
$K\subset \Omega\setminus \Sigma$ it
follows from a simple covering argument that $\nu(K) = 0$ and
$u^k\to u$ strongly in $W^{1,2}(K,\mathbb{R}^3)$. Moreover, for any $x_0\in
K$ there is a $r_{0}> 0$ such that
$$
\lim_k \int_{B_{r_0} (x_0)} |\nabla u^k|^2\le \varepsilon^2_0.
$$
By the standard diagonal process we can extract a subsequence of
$u^k$, still denoted as itself, such that $u^k\to u$ in
$W^{1,2}(\Omega\setminus \{x_1,\dots,x_L\}, \mathbb{R}^3)\cap C^{0}(\Omega\setminus
\{x_1,\dots,x_L\}), \mathbb{R}^3)$. Hence, it is easy to see that 
the expression \eqref{energy_id1} holds with ``$=$'' replaced by ``$\ge$''.

To prove ``$\le$''  of \eqref{energy_id1},
we need to show that the $L^{2}$-norm of $\nabla u^k$ over any
neck region is arbitrarily small. This will mainly be done in the next
sections. Therefore, we will return to the proof of 
Theorem \ref{energy_identity} (2) in the next section.


\section{No concentration of energy in the neck region}


In this section, we  show that there is no concentration of
$\|\nabla u^k\|_{L^{2}}$ in the neck region. This will be done in two steps: 
the first step is to show that there is no angular energy concentration 
in the neck region by comparing with radial harmonic functions over 
dyadic annulus. The second step is to control the radial component of
 energy by the angular component of energy by way of the Pohozaev argument.

\subsection*{Proof of Theorem \ref{energy_identity} (2)}
 Without loss of generality, we suppose that 
$\{u^k \}\subset W^{1,2}(B_1,\mathbb{R}^3)$ is a sequence of approximate 
H-surface maps with
\begin{equation}\label{uniform-bound2}
\sup_{k\in \mathbb{N}}\Big(\|\nabla
u^k\|_{L^2(B_1)}+ \|H(u^k)\|_{L^{\infty}(\mathbb{R}^3)}+\|f^k\|_{L^p(B_1)}\Big)
\le M,
\end{equation}
which satisfy $u^k\rightharpoonup u$ in $W^{1,2}(B_1)$, $f^k\rightharpoonup f$ 
in $L^p(B_1)$, and $u^k\to u $ in $W^{1,2}_{\rm loc}(B_1\setminus \{0\})$ 
but not in $W^{1,2}(B_1)$. In according of Ding and Tian  \cite{DT}, we may 
assume that  the total number of bubbles generated at $0$ is $L=1$. 
Then, for any $\epsilon>0$, there is $r_k\downarrow 0, R>1$ large enough 
and $0<\delta<\epsilon$ such that for $k$ sufficiently large there holds
$$
\int_{B_{2\rho}\setminus B_{\rho}}|\nabla u^k|^2\le \epsilon^2,\quad 
\forall \frac 12 Rr_k\le \rho\le 2\delta.
$$
Let us consider it in two steps.

{\bf Step 1.}  Angular energy estimate in the neck region:
From \eqref{uniform-bound} and Lemma \ref{integ_smallness}, it follows that for any
 $\alpha\in (0, 1-\frac 1p)$ and $p'\in (2,\frac {2p}{2-p})$ with 
$1<p<2$, $u^k\in C^{\alpha}\cap W^{1,p'}(B_{2\rho}\setminus B_{\rho})$ and
$$
[u^k]_{C^{\alpha}(B_{2\rho}\setminus B_{\rho})}
+\|\nabla u^k\|_{L^{p'}(B_{2\rho}\setminus B_{\rho})}\le C\epsilon, 
\quad \forall \frac 12 Rr_k\le \rho\le 2\delta.
$$
In the sequel, to deal with the boundary terms we need  the following 
property by Fubini's theorem:
$$
r\int_{\partial B_r}|\nabla u^k|^2
\le 8\sup_k\int_{B_{2r}\setminus B_{r}}|\nabla u^k|^2\le C\epsilon^2,
$$
which holds for $r=Rr_k, \delta$. For convenience's sake, we assume 
the above inequality holds for all $k\ge 1$.

For simplicity, we may assume $\frac{\delta}{Rr_k}$ is a positive integer. 
We make a dyadic decomposition to the annulus $ \frac 12 Rr_k\le |x|\le 2\delta$. 
Let $N_k\in \mathbb{N}$ be such that $2^{N_k}=[\frac{\delta}{Rr_k}]$, and set
\begin{equation}\label{define annulus}
\mathcal{A}^i_k:=B_{2^{i+1}Rr_k}\setminus B_{2^iR r_k},\quad
\mathcal{B}^i_k:=B_{2^{i+2}Rr_k}\setminus B_{2^{i-1} Rr_k},
\quad 1\le i\le N_k-1.
\end{equation}
Then,  $ B_{\delta}\setminus B_{Rr_k}=\cup_{i=0}^{N_k-1}\mathcal A^i_k$ and
$ B_{2\delta}\setminus B_{\frac 12 Rr_k}=\cup_{i=0}^{N_k-1}\mathcal B^i_k$.
 We now introduce a radial harmonic function $v^k$ on the annulus
$B_{2\delta}\setminus B_{Rr_k}$ as follows. For $0\le i\le N_k-1$,
$v^k(x)=v^k(|x|)$ satisfies
\begin{equation}
\begin{gathered}
\Delta v^k=0 \quad \text{in }\mathcal{A}^i_k,\\
 v^k(r)= -\hspace{-3.8mm}\int_{\partial B_{2^{i+1}Rr_k}}u^k,\quad  
 \text{if } r=2^{i+1}Rr_k,\\
 v^k(r)= -\hspace{-3.8mm}\int_{\partial B_{2^{i}Rr_k}}u^k,\quad  
  \text{if }  r=2^{i}Rr_k;
\end{gathered}
\end{equation}
where $-\hspace{-3mm}\int$ denotes the average integral. 
By the standard estimate of harmonic functions, we have 
$v^k\in C^{\alpha}(\mathcal A^i_k)\cap W^{1,p'}(\mathcal A^i_k)$ for all 
$ 0\le i\le N_k-1$; and
$$
\big[v_k\big]_{C^{\alpha}(\mathcal A^i_k)}\le C\big[u_k\big]_{C^{\alpha}
(\mathcal A^i_k)}\le C\epsilon;
$$
in particular,
\begin{equation}\label{oscillation-annulus}
\text{osc}_{\mathcal A^i_k}(u^k-v^k)\le C\varepsilon,\quad 
\forall 0\le i\le N_k-1.
\end{equation}
Now we perform the estimate similar to Ding-Tian's argument by  \cite{DT} 
or Lin-Wang  \cite{LW1} on harmonic maps. Applying the Green's identity due 
to $u_k-v_k\in W^{2,p}(\mathcal A_k^i)$ we  get that for $0\le i\le N_k-1$,
\begin{equation}\label{Green identity}
\int_{\mathcal A^i_k} \Delta (u^k-v^k)(u^k-v^k)
=-\int_{\mathcal A^i_k} |\nabla (u^k-v^k)|^2
+\int_{\partial\mathcal{A}^i_k} \frac {\partial(u^k-v^k)}{\partial \nu}(u^k-v^k).
\end{equation}
By summing over $0\le i\le N_k-1$, we derive that
\begin{equation}\label{sum over}
\begin{aligned}
&\int_{B_{\delta}\setminus B_{Rr_k}} |\nabla (u^k-v^k)|^2\\
&= -\sum_{i=0}^{N_k-1}\int_{\mathcal A^i_k} \Delta (u^k-v^k)(u^k-v^k)
+\Big(\int_{\partial B_{\delta}}-\int_{\partial B_{Rr_k}}\Big)
 \frac {\partial(u^k-v^k)}{\partial \nu}(u^k-v^k)\\
&= -\sum_{i=0}^{N_k-1}\int_{\mathcal A^i_k} \Delta u^k (u^k-v^k)
+\Big(\int_{\partial B_{\delta}}-\int_{\partial B_{Rr_k}}\Big)
\frac {\partial u^k}{\partial \nu} (u^k-v^k),
\end{aligned}
\end{equation}
where we used that $\Delta v^k=0$ in $\mathcal A_k^i$ and
$\int_{\partial B_\rho} \frac{\partial v^k}{\partial\nu}(u^k-v^k)=0$
for $\rho=\delta$ and $Rr_k$, which is due to the radial form of $v^k$ 
and the boundary conditions:  $v^k=-\hspace{-3mm}\int_{\partial B_{\rho}}u^k$.

Let us first check the estimates of the integral on the boundary in 
the right hand side of \eqref{sum over}. It follows from H\"older 
inequality and Fubini's Theorem that
\begin{equation}\label{1-boundary-est}
\begin{aligned}
\Big|\int_{\partial B_{\delta}}\frac {\partial u^k}{\partial \nu}(u^k-v^k) \Big|
&\leq \int_{\partial B_{\delta}} |\nabla u^k||u^k-v^k|\\
&\leq  C \max_{\partial B_{\delta}}|u^k-v^k | 
 \Big(\delta\int_{\partial B_{\delta}}|\nabla u^k|^2\Big)^{1/2}\\
&\leq  C \epsilon \Big(\int_{B_{2\delta}\setminus 
 B_{\frac 12 \delta}}|\nabla u^k|^2\Big)^{1/2}\le C\epsilon^2.
\end{aligned}
\end{equation}
Similarly,
\begin{equation}\label{2-boundary-est}
\begin{aligned}
\Big|\int_{\partial B_{Rr_k}}\frac {\partial u^k}{\partial \nu}(u^k-v^k) \Big|
&\leq   \max_{\partial B_{Rr_k}}|u^k-v^k|\int_{\partial B_{Rr_k}} |\nabla u_k|\\
&\leq  C \epsilon \Big(Rr_k\int_{\partial B_{Rr_k}}|\nabla u^k|^2\Big)^{1/2}\\
&\leq  C \epsilon \Big(\int_{B_{2Rr_k}\setminus B_{\frac 12 Rr_k}}
|\nabla u^k|^2\Big)^{1/2}\le C\epsilon^2.
\end{aligned}
\end{equation}

Next, we estimate the first term in the right hand side of \eqref{sum over}. 
It yields
\begin{equation}\label{first-term}
\begin{aligned}
\int_{\mathcal A^i_k} \Delta u^k (u^k-v^k)
&= \int_{\mathcal A^i_k} (H(u^k) u^k_x\wedge u^k_y+f^k)(u^k-v^k)\\
&\leq C \sup_{0\le i\le N_k-1}\text{osc}_{\mathcal A^i_k}(u_k-v_k)
\int_{\mathcal A^i_k}(|u^k|^2+|f^k|)\\
&\leq  C\epsilon\int_{\mathcal A^i_k}(|u^k|^2+|f^k|)
\end{aligned}
\end{equation}
Putting theses estimates of \eqref{1-boundary-est} \eqref{2-boundary-est} 
\eqref{first-term} into the inequality \eqref{sum over}, then we conclude that
\begin{equation}\label{derivative}
\int_{B_{\delta}\setminus B_{Rr_k}} |\nabla (u^k-v^k)|^2
\le C\epsilon\int_{B_{\delta}\setminus B_{Rr_k}}(|u^k|^2+|f^k|)+C\epsilon^2
\le C\epsilon.
\end{equation}
Since $v^k$ is radial form and 
$|\nabla u^k|^2=|\frac {\partial u^k}{\partial r}|^2 
+\frac 1{r^2} |\frac {\partial u^k}{\partial \theta}|^2$, 
it follows that for $rR_K\le R\le \delta$,
\begin{equation}\label{tangential}
\begin{aligned}
\int_{B_{\delta}\setminus B_{Rr_k}}\frac 1{r^2} 
\Big|\frac {\partial u^k}{\partial \theta}\Big|^2 
&=  \int_{B_{\delta}\setminus B_{Rr_k}}\frac 1{r^2} 
\Big|\frac {\partial (u^k-v^k)}{\partial \theta}\Big|^2\\
&\leq  \int_{B_{\delta}\setminus B_{Rr_k}} |\nabla (u^k-v^k)|^2\le C\epsilon,
\end{aligned}
\end{equation}
where $\frac {\partial u^k}{\partial \theta}$ and 
$\frac {\partial u^k}{\partial r}$ denote the tangential component and 
the radial component of $\nabla u^k$, respectively.


{\bf Step 2.}  Radial component of energy in the neck region:
 Now we are in a position to employ the Pohozaev argument to control 
$\int_{B_{\delta}\setminus B_{Rr_k}}|\frac {\partial u^k}{\partial r}|^2$ 
by $\int_{B_{\delta}\setminus B_{Rr_k}}\frac 1{r^2} 
|\frac {\partial u^k}{\partial \theta}|^2$ and $\|f^k\|_{L^p(B_{\delta})}$. 
Observe that $x\cdot \nabla u^k\in L^{p'}(B_{\delta}), 
\Delta u^k\in L^p(B_{\delta})$ and $f^k\in  L^{p}(B_{\delta})$ with 
$2<p'<\frac {2p}{2-p}$ for $1<p<2$, we have that 
$f^k\cdot(x\cdot \nabla u^k)\in L^{1}(B_{\delta})$ and 
$\Delta u^k\cdot(x\cdot \nabla u^k)\in  L^{1}(B_{\delta})$ only if 
$p>\frac 43$. That is due to $\frac {2p}{2-p}>\frac p{p-1} $ at this point. 
On the other hand, thanks to Equ.\eqref{approx_sequence} it implies  that
$(\Delta u^k-f^k)=H(u^k)u^k_x\wedge u^k_y\perp T_{u^k(x)}\mathcal{S}$, a. e. 
$x\in B_{\delta}$ with $\mathcal{S}=u^k(B_{\delta})$. Therefore, 
by multiplying \eqref{approx_sequence} by $x\cdot \nabla u^k$ and 
integrating it over $B_{r}$,  for $0<r<\delta$,  yields
$$
\int_{B_{r}}\Delta u^k\cdot(x\cdot \nabla u^k)
=\int_{B_{r}}f^k\cdot(x\cdot \nabla u^k).
$$
In terms of  Green's identity, we have
\begin{equation}\label{Pohozaev}
\begin{aligned}
\int_{B_{r}}\Delta u^k\cdot(x\cdot \nabla u^k) 
&=  -\int_{B_{r}}\nabla u^k\cdot\nabla(x\cdot \nabla u^k)
 +\int_{\partial B_{r}}(x\cdot \nabla u^k)(\nabla u^k\cdot \frac x{|x|}) \\
&=  \frac 12\int_{B_{r}} x\cdot\nabla(|\nabla u^k|^2)
 -\int_{B_{r}}|\nabla u^k|^2+r\int_{\partial B_{r}}
 \Big|\frac {\partial u^k}{\partial r}\Big|^2\\
&=  -\frac 12 r\int_{\partial B_{r}}|\nabla u^k|^2+r\int_{\partial B_{r}}
\Big|\frac {\partial u^k}{\partial r}\Big|^2.
\end{aligned}
\end{equation}
Therefore, 
$$
r\int_{\partial B_{r}}\Big|\frac {\partial u^k}{\partial r}\Big|^2 
-\frac 12 r\int_{\partial B_{r}}|\nabla u^k|^2=\int_{B_{r}}f^k\cdot(x\cdot 
\nabla u^k).
$$
It follows from 
$|\nabla u^k|^2=|\frac {\partial u^k}{\partial r}|^2 +\frac 1{r^2} |
\frac {\partial u^k}{\partial \theta}|^2$ that
\begin{equation}\label{}
\int_{\partial B_{r}}\Big|\frac {\partial u^k}{\partial r}\Big|^2
\le \int_{\partial B_{r}} r^{-2}\Big|\frac {\partial u^k}{\partial 
\theta}\Big|^2+2\int_{B_{r}}\Big|f^k\Big|\Big|\nabla u^k\Big|.
\end{equation}
Integrating it over the interval $[Rr_k,\delta]$, it follows from
 H\"older inequality and the tangential estimate of \eqref{tangential}
 that for $0<\delta<\epsilon$ there hold
\begin{equation}\label{radial}
\begin{aligned}
\int_{B_{\delta}\setminus B_{Rr_k}}\Big|
\frac {\partial u^k}{\partial r}\Big|^2 
&\leq  \int_{B_{\delta}\setminus B_{Rr_k}}\frac 1{r^2}
\Big|\frac {\partial u^k}{\partial \theta}\Big|^2+
2\delta\Big\|f^k\Big\|_{L^p(B_{\delta})}\Big\|\nabla u^k
\Big\|_{L^{p'}(B_{\delta})}\\
&\leq   C(\epsilon+\delta)\le C\epsilon.
\end{aligned}
\end{equation}
 Putting \eqref{tangential} and \eqref{radial} together, it yields
\begin{equation}\label{no-energy}
\int_{B_{\delta}\setminus B_{Rr_k}}\Big|\nabla u^k|^2=
\int_{B_{\delta}\setminus B_{Rr_k}}\Big|
\frac {\partial u^k}{\partial r}\Big|^2
+\int_{B_{\delta}\setminus B_{Rr_k}}\frac 1{r^2}\Big|
\frac {\partial u^k}{\partial \theta}\Big|^2\le C\epsilon.
\end{equation}
This implies that there is no neck formation between any two
bubbles. Hence the proof of Theorem \ref{energy_identity} is  complete. 

\section{Application to H-surface flows}

As an application of Theorem \ref{energy_identity}, we will establish
 the energy identity at a finite time singular point for the sequences 
of H-surface flows.

\subsection{Proof of Theorem \ref{flow_identity}}  
 Without loss of generality, suppose that $\Omega= B_1$ and $(x_0, T_0)$  
is the only singular point at $t=T_0$. Let 
$u^k(x, t) = u(\lambda_kx; t_k + \lambda^2_kt)$, then 
$u^k(x, t)$ still satisfies the same equation as \eqref{flow-pro}, 
and by \eqref{energy_ineq} we have 
\begin{equation}
\int^{2}_{-2}\int_{B_{\lambda^{-1}_k}}\Big|\frac{\partial u^k}{\partial t}  
\Big|^2=\int^{t_k+2\lambda^2_k}_{t_k-2\lambda^2_k}\int_{B_1}
\Big|\frac{\partial u}{\partial t}  \Big|^2\to 0,\quad  \text{as }
k\to\infty.
\end{equation}
By Fubini's theorem, there exists $\eta_k\in (-1,-1/2)$ such that
\begin{equation}\label{decay-1}
\int_{B_{\lambda^{-1}_k}}\Big|\frac{\partial u^k(\cdot,\eta_k)}{\partial t}  
\Big|^2\to 0, \quad \int_{B_{\lambda^{-1}_k}\times (-2,2)}\Big|
\frac{\partial u^k}{\partial t}  \Big|^2\to 0.
\end{equation}
Here, just similar to \cite{HH}, then we have the following energy inequality:
 for any $0<s\le \tau<T$ and $B_{2R}(x)\subset \Omega, x\in \Omega$, there holds
\begin{equation}\label{energy-ineq}
E(u(\tau),B_R(x))\le 5 E(u(s),B_{2R}(x))+C\frac{\tau-s}{R^2}J_H(u_0).
\end{equation}
With the help of the energy inequality \eqref{energy-ineq}, 
it is known from \cite[Lemma 4.1]{LW1} that there exists unique positive 
$m$ such that, in the sense of  Radon measure, we have
\begin{equation}\label{Radon-converg}
|\nabla u|^2(x,t)dx\to m \delta_{x_0}+|\nabla u|^2(x,T_0)dx,\quad \text{as }
 t\to T_0,
\end{equation}
at the only singular point at $t=T_0$, where $\delta_{x_0}$ denotes Dirac 
$\delta$-mass at $x_0$. Also from  \eqref{energy-ineq} it follows that 
$$
\int_{B_{R}}\Big|\nabla u^k(\cdot,\eta_k)\Big|^2\ge \int_{B_1}
\Big|\nabla u^k(\cdot,T_0)\Big|^2-CR^{-2}J_0,
$$
let $R\to \infty$, by \eqref{Radon-converg} which implies
\begin{equation}\label{bubble-number}
\lim_{R\to \infty}\int_{B_{R}}\Big|\nabla u^k(\cdot,\eta_k)\Big|^2= m.
\end{equation}
Therefore, for each $R>0$, we know from \eqref{bubble-number} that 
$u^k(\cdot,\eta_k)\rightharpoonup v$ weakly in $W^{1,2}(B_R, \mathbb{R}^3)$.
We claim that $v$ is a constant map. Indeed, let $|t_k|\le 2\lambda^2_k$ we 
observe that
$$
\int_{B_{R}}\Big|u^k(\cdot,\eta_k)-u^k(\cdot,-t_k\lambda^{-2}_k) \Big|^2
\le 4 \int^{2}_{-2}\int_{B_{\lambda^{-1}_k}}\Big|\frac{\partial u^k}{\partial t} 
 \Big|^2\to 0,
$$
and
$$
\int_{B_{R}}\Big|\nabla u^k(\cdot,-t_k\lambda^{-2}_k) \Big|^2
=\int_{B_{\lambda_kR}}\Big|\nabla u\Big|^2(\cdot,T_0)\to 0.
$$
For each $R>0$, now we apply Theorem \ref{energy_identity} of approximate H-surfaces 
to $u^k(\cdot,\eta_k)$ on the ball $B_R$ to conclude that there exist 
finite number bubbles $\{\omega_{i,R}\}_{i=1}^{L_R}$ such that
\begin{equation}\label{i-sum-bubbles}
\lim_{k\to \infty}\int_{B_{R}}\Big|\nabla u^k\Big|^2(\cdot,\eta_k)
=\sum_{i=1}^{L_R}E(\omega_{i,R},\mathbb{S}^2).
\end{equation}
Further, we know that $1\le L_R\le \Big[\frac {m}{\varepsilon_0}\Big]$ 
because there is a $\varepsilon_0$ such that any bubble 
$\omega:\mathbb{S}^2\to \mathbb{R}^3$ satisfying 
$E(\omega,\mathbb{S}^2)\ge \varepsilon_0$. Hence, there exists a
 $d\in \Big[1,\frac {m}{\varepsilon_0}\Big]$ such that, after possible 
a subsequence, $L_R=d$ and
\begin{equation}\label{m-bubbles}
m=\lim_{R\uparrow\infty}\lim_{k\to \infty}\int_{B_{R}}\Big|
\nabla u^k\Big|^2(\cdot,\eta_k)=\lim_{R\uparrow\infty}
\sum_{i=1}^{d}E(\omega_{i,R},\mathbb{S}^2).
\end{equation}
Note that $\{\omega_{i,R}\}_{i=1}^{d}$ have uniformly boundedness of energies, 
from Brezis-Coron \cite{BC1} one concludes that there exist
$N_i\in \Big[1,\frac {m}{\varepsilon_0}\Big]$ and $N_i$ bubbles
 $\{\omega_{i,j}\}_{j=1}^{N_i}$ such that
\begin{equation}\label{j-sum-bubbles}
\lim_{R\uparrow\infty}E(\omega_{i,R},\mathbb{S}^2)
=\sum_{j=1}^{N_i}E(\omega_{i,j},\mathbb{S}^2).
\end{equation}
Now, putting all \eqref{i-sum-bubbles},\eqref{m-bubbles} and 
\eqref{j-sum-bubbles} together, it follows that
$$
m=\sum_{i=1}^{d}\sum_{j=1}^{N_i}E(\omega_{i,j},\mathbb{S}^2).
$$
The proof of Theorem \ref{flow_identity} is complete. 


\subsection*{Acknowledgements} This research is partially supported
by grant 11071012  from the NSFC, and by the program of visiting Chern Institute 
of Mathematics. The author would like to thank Prof. Changyou Wang 
for the valuable suggestions.


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