# Landau-Lifshitz-Bloch equation on Riemannian manifold

Boling Guo, Zonglin Jia

###### Abstract.

In this article, we bring in Landau-Lifshitz-Bloch(LLB) equation on
$m$-dimensional closed Riemannian manifold and prove that it admits a unique
local solution. In addition, if $m\geqslant 3$ and $L^{\infty}-$norm of
initial data is sufficiently small, the solution can be extended globally.
Moreover, if $m=2$, we can prove that the unique solution is global without
assuming small initial data.

##  1\. Introduction

Landau-Lifshitz-Gilbert equation describes physical properties of
micromagnetic at temperatures below the critical temperature. The equation is
as follows:

(1.1) $\frac{\partial m}{\partial t}=\lambda_{1}m\times
H_{eff}-\lambda_{2}m\times(m\times H_{eff})$

where $\times$ denotes the vector cross product in $\mathbb{R}^{3}$ and
$H_{eff}$ is effective field while $\lambda_{1}$ and $\lambda_{2}$ are real
constants.

However, at high temperature, the model must be replaced by following Landau-
Lifshitz-Bloch equation(LLB)

(1.2) $\frac{\partial u}{\partial t}=\gamma u\times
H_{eff}+L_{1}\frac{1}{|u|^{2}}(u\cdot
H_{eff})u-L_{2}\frac{1}{|u|^{2}}u\times(u\times H_{eff})$

where $\gamma$, $L_{1}$, $L_{2}$ are real numbers and $\gamma>0$. $H_{eff}$ is
given by

$H_{eff}=\Delta
u-\frac{1}{\chi||}\Big{(}1+\frac{3T}{5(T-T_{c})}|u|^{2}\Big{)}u.$

where $T>T_{c}>0$ and $\chi||>0$.

Now let us recall some previous results about LLB. In [6], Le consider the
case that $L_{1}=L_{2}=:\kappa_{1}>0$. At that time, he rewrites (1.2) as

(1.3) $\displaystyle\frac{\partial u}{\partial t}=\kappa_{1}\Delta u+\gamma
u\times\Delta u-\kappa_{2}(1+\mu|u|^{2})u$

with $\kappa_{2}:=\frac{\kappa_{1}}{\chi||}$ and $\mu:=\frac{3T}{5(T-T_{c})}$
and assume that $\kappa_{2}$, $\gamma$, $\mu$ is positive. Le has proven that
above equation with Neumann boundary value conditions has global weak
solution(the weak solution here is different from ordinary one). Inspired by
Le, in [5] Jia introduces following equation

(1.4) $\left\\{\begin{array}[]{llll}\partial_{t}u=\kappa_{1}\Delta
u+\gamma\nabla F(u)\times\Delta u-\kappa_{2}(1+\mu\cdot F(u))\nabla
F(u)\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,\Omega\times(0,\infty)\\\ \frac{\partial
u}{\partial\nu}=0\,\,\,\,\,\,\,\,\mbox{on}\,\,\,\,\partial\Omega\times(0,\infty)\\\
u(\cdot,0)=u_{0}\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,\Omega\end{array}\right.$

where $\Omega$ is a regular bounded domain of $\mathbb{R}^{d}(d\leqslant 3)$,
$\nu$ is outer normal direction of $\partial\Omega$ and $F\in
C^{3}(\mathbb{R}^{3})$ is a known function. He calls it Generalized Landau-
Lifshitz-Bloch equation(GLLB) and gets that (1.4) admits a local strong
solution provided $u_{0}\in W^{2,2}(\Omega,\mathbb{R}^{3})$ and
$\frac{\partial u_{0}}{\partial\nu}=0$. In [4], Guo, Li and Zeng consider the
coming LLB equation with initial condition

(1.5) $\displaystyle\left\\{\begin{aligned} &u_{t}=\Delta u+u\times\Delta
u-\lambda(1+\mu|u|^{2})u\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^{d}\times(0,T)\\\
&u(,0)=u_{0}\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^{d},\end{aligned}\right.$

where the constant $\lambda,\mu>0$. They prove the existence of smooth
solutions of (1.5) in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. And a small
initial value condition should be added in the latter case.

In this paper, we would like to introduce a equation similar with (1.5) on
Riemannian manifold. Before getting to this, we should make some preparation.

Let $\pi:(E,h,D)\longrightarrow(M,g,\nabla)$ denote a smooth vector bundle
over an $m-$dimensional smooth closed Riemannian manifold $(M,g,\nabla)$ with
$rank(E)=3$. $g$ means Riemannian metric of $M$ and $\nabla$ is its Levi-
Civita connection. $h$ and $D$ are respectively metric and connection of $E$
such that $Dh=0$. Sometimes we also write $h$ as $\langle\cdot,\cdot\rangle$.

###  1.1. $k-$times continuously differentiable section

Suppose $\Gamma(E)$ is the set of all sections in $E$. Under arbitrary local
frame $\\{e_{\alpha}:1\leqslant\alpha\leqslant 3\\}$, a section
$s\in\Gamma(E)$ can be written in the form of $s=s^{\alpha}\cdot e_{\alpha}$.
If $s^{\alpha}$ is $k-$times continuously differentiable, then we say $s$ is
$k-$times continuously differentiable. Since $E$ is smooth, $k-$times
continuous differentiability is independent of the choice of local frame.
Define

$\Gamma^{k}(E):=\\{s\in\Gamma(E):\mbox{$s$ is $k-$times continuously
differentiable}\\}.$

###  1.2. Orientable vector bundle

$E$ is called orientable if there exists an $\omega\in E^{*}\wedge E^{*}\wedge
E^{*}$ such that $\omega$ is continuous and for all $p\in M$,
$\omega(p)\not=0$, where $E^{*}$ is dual bundle of $E$.

Suppose $\\{e_{1},e_{2},e_{3}\\}$ is a frame of $E$. It is called adapted to
the orientation $\omega$ if

$\omega(e_{1},e_{2},e_{3})>0.$

From now on, we always assume that $E$ is orientable unless otherwise stated.

###  1.3. Cross product on orientable vector bundle

Suppose $\omega$ is an orientation of $E$.
$\\{e_{\alpha}:1\leqslant\alpha\leqslant 3\\}$ is a local frame of $E$ which
is adapted to $\omega$. For any $f_{1},f_{2}\in\Gamma(E)$, we assume that
$f_{1}:=f_{1}^{\alpha}\cdot e_{\alpha}$, $f_{2}:=f_{2}^{\alpha}\cdot
e_{\alpha}$. Their cross product $\times$ is defined as follow

$(f_{1}\times f_{2})(p):=f_{1}(p)\times f_{2}(p),$

where

$\displaystyle f_{1}(p)\times f_{2}(p):$ $\displaystyle=$
$\displaystyle\big{(}f_{1}^{2}(p)\cdot f_{2}^{3}(p)-f_{2}^{2}(p)\cdot
f_{1}^{3}(p)\big{)}\cdot e_{1}(p)$ $\displaystyle+$
$\displaystyle\big{(}f_{2}^{1}(p)\cdot f_{1}^{3}(p)-f_{1}^{1}(p)\cdot
f_{2}^{3}(p)\big{)}\cdot e_{2}(p)$ $\displaystyle+$
$\displaystyle\big{(}f_{1}^{1}(p)\cdot f_{2}^{2}(p)-f_{2}^{1}(p)\cdot
f_{1}^{2}(p)\big{)}\cdot e_{3}(p).$

It is not hard to verify that $f_{1}(p)\times f_{2}(p)$ does not depend upon
the choice of local frames which are adapted to $\omega$.

###  1.4. Laplace operator on vector bundle

Define a functional $Energy$ on $\Gamma^{2}(E)$ which is given in the form of

$Energy(X):=\frac{1}{2}\int_{M}|DX|^{2}\,dM.$

It is not hard to see that the Euler-Lagrange equation of $Energy$ is

$\Delta X:=g^{ij}\cdot(D^{2}X)\Big{(}\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial x^{j}}\Big{)}=0,$

where $g_{ij}:=g\Big{(}\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial x^{j}}\Big{)}$ and $(g^{ij})$ is the inverse
matrix of $(g_{ij})$. Then we say that $\Delta$ is the Laplace operator on
vector bundle $E$.

###  1.5. sections depending on time

A section depending on time is a map

$V:I\longrightarrow\Gamma(E),$

where $I$ is an interval of $\mathbb{R}$. Under arbitrary local frame
$\\{e_{\alpha}:1\leqslant\alpha\leqslant 3\\}$, $V(t,x)$ can be written as
$V(t,x):=V^{\alpha}(t,x)\cdot e_{\alpha}(x)$. If $V^{\alpha}$ is $k-$times
continuously differentiable with respect to $t$, we say $V$ is $k-$times
continuously differentiable with respect to $t$ and use the symbol
$C^{k}(I,\Gamma(E))$ to denote all such $V$. Since $E$ is smooth,
differentiability with respect to time is independent of the choice of local
frame. Moreover, we define

$(\partial_{t}^{k}V)(t,x):=(\partial_{t}^{k}V^{\alpha})(t,x)\cdot
e_{\alpha}(x).$

###  1.6. Sobolev space on vector bundle

Equip $\Gamma^{k}(E)$ with a norm $||\cdot||_{H^{k,p}}(p\geqslant 1)$ which is
defined as follow

$||s||^{p}_{H^{k,p}}:=\sum\limits_{i=0}^{k}\int_{M}|D^{i}s|^{p}\,dM.$

The Sobolev space $H^{k,p}(E)$ is the completion of $\Gamma^{k}(E)$ with
respect to the norm $||\cdot||_{H^{k,p}}$. For convenience, we also denote
$H^{k,2}$ by $H^{k}$ and $||\cdot||_{H^{0,p}}$ by $||\cdot||_{p}$.

Having the above preparation, we will give the definition of Landau-Lifshitz-
Bloch equation(LLB) on Riemannian manifold.

For any $T>0,\lambda>0$ and $\mu>0$, let us consider a section depending on
time $V\in C^{1}([0,T],\Gamma^{2}(E))$. LLB is just the following equation

(1.6) $\displaystyle\left\\{\begin{aligned} &\partial_{t}V=\Delta
V+V\times\Delta
V-\lambda\cdot(1+\mu\cdot|V|^{2})V\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,(0,T]\times
M\\\ &V(0,\cdot)=V_{0}\end{aligned}\right.$

Our main results are as follow:

######  Theorem 1.1.

Let $\pi:(E,h,D)\longrightarrow(M,g,\nabla)$ denote a smooth vector bundle
over an $m-$dimensional smooth closed Riemannian manifold $(M,g,\nabla)$ with
$rank(E)=3$ and $Dh=0$. $E$ is orientable. Given $l\geqslant m_{0}+1$(Here
$m_{0}:=[\frac{m}{2}]+3$ and $[q]$ is the integral part of $q$) and $V_{0}\in
H^{l}(E)$, there is a $T^{*}=T^{*}(||V_{0}||_{H^{m_{0}}})>0$ and a unique
solution $V$ of (1.6) satisfying that for any $0\leqslant
j\leqslant[\frac{l}{\hat{m}}]$($\hat{m}:=\max\\{2,[\frac{m}{2}]+1\\}$) and
$\alpha\leqslant l-\hat{m}j$,

(1.7) $\displaystyle\partial_{t}^{j}D^{\alpha}V\in
L^{\infty}([0,T^{*}],L^{2}(E)).$

Furthermore, if $V_{0}\in\Gamma^{\infty}(E)$, then $V\in
C^{\infty}([0,T^{*}],\Gamma^{\infty}(E))$.

######  Theorem 1.2.

Let $\pi:(E,h,D)\longrightarrow(M,g,\nabla)$ denote a smooth vector bundle
over an $m-$dimensional smooth closed Riemannian manifold $(M,g,\nabla)$ with
$rank(E)=3$, $m\geqslant 3$ and $Dh=0$. $E$ is orientable. For any $T>0$ and
$N\geqslant m_{0}+1$, there exists a $\hat{B}_{N}>0$ such that for all
$V_{0}\in H^{N}(E)$ with $||V_{0}||_{\infty}\leqslant\hat{B}_{N}$, there is a
unique solution of (1.6) satisfying

(1.8) $\displaystyle\partial_{t}^{j}D^{\alpha}V\in
L^{\infty}([0,T],L^{2}(E))\,\,\,\,\forall 0\leqslant
j\leqslant\Big{[}\frac{N}{\hat{m}}\Big{]}\,\,\,\,\forall\alpha\leqslant
N-\hat{m}j$

and

(1.9) $\displaystyle\partial_{t}^{i}D^{\beta}V\in
L^{2}([0,T],L^{2}(E))\,\,\,\,\forall 0\leqslant
i\leqslant\Big{[}\frac{N+1}{\hat{m}+1}\Big{]}\,\,\,\,\forall\beta\leqslant
N+1-(\hat{m}+1)i.$

Furthermore, if $V_{0}\in\Gamma^{\infty}(E)$, then $V\in
C^{\infty}([0,T],\Gamma^{\infty}(E))$.

######  Theorem 1.3.

Let $\pi:(E,h,D)\longrightarrow(M,g,\nabla)$ denote a smooth vector bundle
over an $2-$dimensional smooth closed Riemannian manifold $(M,g,\nabla)$ with
$rank(E)=3$ and $Dh=0$. $E$ is orientable. For any $T>0$, $N\geqslant 5$ and
$V_{0}\in H^{5}(E)$, there is a unique solution of (1.6) satisfying

$\displaystyle\partial_{t}^{j}D^{\alpha}V\in
L^{\infty}([0,T],L^{2}(E))\,\,\,\,\forall 0\leqslant
j\leqslant\Big{[}\frac{N}{2}\Big{]}\,\,\,\,\forall\alpha\leqslant N-2j$

and

$\displaystyle\partial_{t}^{i}D^{\beta}V\in
L^{2}([0,T],L^{2}(E))\,\,\,\,\forall 0\leqslant
i\leqslant\Big{[}\frac{N+1}{3}\Big{]}\,\,\,\,\forall\beta\leqslant N+1-3i.$

Furthermore, if $V_{0}\in\Gamma^{\infty}(E)$, then $V\in
C^{\infty}([0,T],\Gamma^{\infty}(E))$.

##  2\. Notation and Preliminaries

In the paper, we appoint that the same indices appearing twice means summing
it. And $Q_{1}\lesssim Q_{2}$ implies there is a universal constant $C$ such
that $Q_{1}\leqslant C\cdot Q_{2}$.

###  2.1. Riemannian curvature tensor on vector bundle

Using the connection $D$ on $E$, we can define a tensor $R^{E}$ called
Riemannian curvature tensor. For any $X,Y\in TM$ and $s\in\Gamma^{2}(E)$,

$R^{E}(X,Y)s:=D_{X}D_{Y}s-D_{Y}D_{X}s-D_{[X,Y]}s.$

Let $R^{M}$ be the Riemannian curvature tensor of $M$. Being going to
represent $R^{M}$ and $R^{E}$ in local frame, we appoint
$\frac{\partial}{\partial x^{i}}$ as $\partial_{i}$. Then,

$R^{M}(\partial_{i},\partial_{j})\partial_{r}:=(R^{M})^{h}_{ijr}\cdot\partial_{h}\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,R^{E}(\partial_{i},\partial_{j})e_{\beta}:=(R^{E})^{\alpha}_{ij\beta}\cdot
e_{\alpha}.$

Now we give two tensors

$\mathcal{R}^{M}:=(\mathcal{R}^{M})_{ijkl}\cdot dx^{i}\otimes dx^{j}\otimes
dx^{k}\otimes dx^{l}$

and

$\mathcal{R}^{E}:=(\mathcal{R}^{E})^{\alpha\beta}_{ij}\cdot dx^{i}\otimes
dx^{j}\otimes e_{\alpha}\otimes e_{\beta},$

where

$(\mathcal{R}^{M})_{ijkl}:=(R^{M})^{h}_{ijk}\cdot
g_{hl}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\mathcal{R}^{E})^{\alpha\beta}_{ij}:=(R^{E})^{\alpha}_{ij\theta}\cdot
h^{\theta\beta}.$

$(h_{\alpha\beta})$ is the metric matrix of $h$ and $(h^{\theta\beta})$ is its
inverse matrix.

###  2.2. Cross product of tensors

We also want to introduce cross product between two tensors. Given
$S\in\Gamma(T^{*}M^{\otimes k}\otimes E)$ and $T\in\Gamma(T^{*}M^{\otimes
l}\otimes E)$, let us define

$S\times T:=(S_{i_{1}\cdots i_{k}}\times T_{j_{1}\cdots j_{l}})\otimes
dx^{i_{1}}\otimes\cdots\otimes dx^{i_{k}}\otimes
dx^{j_{1}}\otimes\cdots\otimes dx^{j_{l}},$

where

$S_{i_{1}\cdots
i_{k}}:=S(\partial_{i_{1}},\cdots,\partial_{i_{k}})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{j_{1}\cdots
j_{l}}:=T(\partial_{j_{1}},\cdots,\partial_{j_{l}}).$

It is easy to check

(2.1) $\displaystyle|S\times T|\leqslant|S|\cdot|T|$

###  2.3. Properties of cross product

######  Theorem 2.1.

For any $f_{1},f_{2}\in\Gamma^{1}(E)$, we have

(2.2) $\displaystyle D(f_{1}\times f_{2})=(Df_{1})\times
f_{2}+f_{1}\times(Df_{2}).$

Proof. Take any $p\in M$. Then there exists a neighbourhood $U$ and a positive
number $\delta$ such that the following map

$\exp_{p}:N_{\delta}\triangleq\\{\hat{v}\in
T_{p}M:||\hat{v}||<\delta\\}\longrightarrow U$

is a diffeomorphism. Take $v\in T_{p}M$ such that $||v||=1$. Define
$\gamma_{v}(t):=\exp_{p}(tv)$, where $t\in[0,\delta)$. Now take arbitrary
orthonormal basis $\\{e_{p\alpha}:1\leqslant\alpha\leqslant 3\\}$ in $E_{p}$
which is adapted to $\omega$ and let it move parallelly along $\gamma_{v}$ to
get

$\\{e_{\alpha}(t,v):t\in[0,\delta),1\leqslant\alpha\leqslant 3\\}.$

Clearly,

$w(t):=\omega(e_{1}(t,v),e_{2}(t,v),e_{3}(t,v))>0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,\delta)$

since $w$ is a continuous function with respect to $t$. In the next, let $v$
range all the direction in $T_{p}M$ to obtain

$\\{e_{\alpha}(t,v):t\in[0,\delta),v\in
T_{p}M,||v||=1,1\leqslant\alpha\leqslant 3\\}.$

It is a orthonormal frame on $U$ which is adapted to $\omega$ and

(2.3) $\displaystyle(De_{\alpha})(p)=0.$

Assume that $f_{1}=f_{1}^{\alpha}\cdot e_{\alpha}$ and
$f_{2}=f_{2}^{\beta}\cdot e_{\beta}$. Then, (2.3) yields

$Df_{1}(p)=df_{1}^{\alpha}(p)\otimes
e_{\alpha}(p)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Df_{2}(p)=df_{2}^{\beta}(p)\otimes
e_{\beta}(p).$

Recalling the definition of cross product, we have

$\displaystyle f_{1}\times f_{2}:=\big{(}f_{1}^{2}\cdot
f_{2}^{3}-f_{2}^{2}\cdot f_{1}^{3}\big{)}\cdot e_{1}+\big{(}f_{2}^{1}\cdot
f_{1}^{3}-f_{1}^{1}\cdot f_{2}^{3}\big{)}\cdot e_{2}+\big{(}f_{1}^{1}\cdot
f_{2}^{2}-f_{2}^{1}\cdot f_{1}^{2}\big{)}\cdot e_{3}.$

Therefore, since of (2.3), one can get

$\displaystyle[D(f_{1}\times f_{2})](p):$ $\displaystyle=$
$\displaystyle\big{[}df_{1}^{2}(p)\cdot f_{2}^{3}(p)+f_{1}^{2}(p)\cdot
df_{2}^{3}(p)-df_{2}^{2}(p)\cdot f_{1}^{3}(p)-f_{2}^{2}(p)\cdot
df_{1}^{3}(p)\big{]}\otimes e_{1}(p)$ $\displaystyle+$
$\displaystyle\big{[}df_{2}^{1}(p)\cdot f_{1}^{3}(p)+f_{2}^{1}(p)\cdot
df_{1}^{3}(p)-df_{1}^{1}(p)\cdot f_{2}^{3}(p)-f_{1}^{1}(p)\cdot
df_{2}^{3}(p)\big{]}\otimes e_{2}(p)$ $\displaystyle+$
$\displaystyle\big{[}df_{1}^{1}(p)\cdot f_{2}^{2}(p)+f_{1}^{1}(p)\cdot
df_{2}^{2}(p)-df_{2}^{1}(p)\cdot f_{1}^{2}(p)-f_{2}^{1}(p)\cdot
df_{1}^{2}(p)\big{]}\otimes e_{3}(p),$ $\displaystyle[f_{1}\times(Df_{2})](p)$
$\displaystyle=$ $\displaystyle f_{1}(p)\times(Df_{2})(p)$ $\displaystyle=$
$\displaystyle\big{[}f_{1}^{2}(p)\cdot df_{2}^{3}(p)-df_{2}^{2}(p)\cdot
f_{1}^{3}(p)\big{]}\otimes e_{1}(p)$ $\displaystyle+$
$\displaystyle\big{[}df_{2}^{1}(p)\cdot f_{1}^{3}(p)-f_{1}^{1}(p)\cdot
df_{2}^{3}(p)\big{]}\otimes e_{2}(p)$ $\displaystyle+$
$\displaystyle\big{[}f_{1}^{1}(p)\cdot df_{2}^{2}(p)-df_{2}^{1}(p)\cdot
f_{1}^{2}(p)\big{]}\otimes e_{3}(p),$

and

$\displaystyle[(Df_{1})\times f_{2}](p)$ $\displaystyle=$
$\displaystyle(Df_{1})(p)\times f_{2}(p)$ $\displaystyle=$
$\displaystyle\big{[}df_{1}^{2}(p)\cdot f_{2}^{3}(p)-f_{2}^{2}(p)\cdot
df_{1}^{3}(p)\big{]}\otimes e_{1}(p)$ $\displaystyle+$
$\displaystyle\big{[}f_{2}^{1}(p)\cdot df_{1}^{3}(p)-df_{1}^{1}(p)\cdot
f_{2}^{3}(p)\big{]}\otimes e_{2}(p)$ $\displaystyle+$
$\displaystyle\big{[}df_{1}^{1}(p)\cdot f_{2}^{2}(p)-f_{2}^{1}(p)\cdot
df_{1}^{2}(p)\big{]}\otimes e_{3}(p).$

This theorem follows easily from combining (2.3) with (2.3) and (2.3).
$\hfill\Box$

Because of (2.2), it is easy to verify that

(2.7) $\displaystyle D(S\times T)=(DS)\times T+S\times(DT),$

provided $S\in\Gamma^{1}(T^{*}M^{\otimes k}\otimes E)$ and
$T\in\Gamma^{1}(T^{*}M^{\otimes l}\otimes E)$.

###  2.4. Hamilton’s notation

Suppose $k,l,p,q\in\mathbb{N}$, $S\in T^{*}M^{\otimes k}\otimes E^{\otimes p}$
and $T\in T^{*}M^{\otimes l}\otimes E^{\otimes q}$, where

$E^{\otimes p}:=\underbrace{E\otimes\cdots\otimes
E}\limits_{\mbox{$p-$times}}.$

we will write $S\ast T$, following Hamilton [2], to denote a tensor formed by
contraction on some indices of $S\otimes T$ using the coefficients $g^{ij}$ or
$h_{\alpha\beta}$.

######  Theorem 2.2.

$|S\ast T|\leqslant|S|\cdot|T|$

Proof. We will get the above formula in an orthonormal basis of $M$ and an
orthonormal basis of $E$.

$\displaystyle|S\ast T|^{2}$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}\mbox{free}\\\
\mbox{indices}\end{subarray}}\Bigg{(}\sum\limits_{\begin{subarray}{c}\mbox{contracted}\\\
\mbox{indices}\end{subarray}}S^{\alpha_{1}\cdots\alpha_{p}}_{i_{1}\cdots
i_{k}}\cdot T^{\beta_{1}\cdots\beta_{q}}_{j_{1}\cdots j_{l}}\Bigg{)}^{2}$
$\displaystyle\leqslant$
$\displaystyle\sum\limits_{\begin{subarray}{c}\mbox{free}\\\
\mbox{indices}\end{subarray}}\Bigg{[}\sum\limits_{\begin{subarray}{c}\mbox{contracted}\\\
\mbox{indices}\end{subarray}}\Big{(}S^{\alpha_{1}\cdots\alpha_{p}}_{i_{1}\cdots
i_{k}}\Big{)}^{2}\Bigg{]}\cdot\Bigg{[}\sum\limits_{\begin{subarray}{c}\mbox{contracted}\\\
\mbox{indices}\end{subarray}}\Big{(}T^{\beta_{1}\cdots\beta_{q}}_{j_{1}\cdots
j_{l}}\Big{)}^{2}\Bigg{]}$ $\displaystyle\leqslant$
$\displaystyle\Bigg{[}\sum\limits_{\begin{subarray}{c}\mbox{free}\\\
\mbox{indices}\end{subarray}}\sum\limits_{\begin{subarray}{c}\mbox{contracted}\\\
\mbox{indices}\end{subarray}}\Big{(}S^{\alpha_{1}\cdots\alpha_{p}}_{i_{1}\cdots
i_{k}}\Big{)}^{2}\Bigg{]}\cdot\Bigg{[}\sum\limits_{\begin{subarray}{c}\mbox{free}\\\
\mbox{indices}\end{subarray}}\sum\limits_{\begin{subarray}{c}\mbox{contracted}\\\
\mbox{indices}\end{subarray}}\Big{(}T^{\beta_{1}\cdots\beta_{q}}_{j_{1}\cdots
j_{l}}\Big{)}^{2}\Bigg{]}$ $\displaystyle=$ $\displaystyle|S|^{2}\cdot|T|^{2}$

$\hfill\Box$
Because we do not specifically illustrate which indices are contracted, we
have to appoint that

$S_{1}\ast T_{1}-S_{2}\ast T_{2}:=S_{1}\ast T_{1}+S_{2}\ast T_{2}.$

We will use the symbol $\mathfrak{q}_{s}(T_{1},\cdots,T_{r})$ for a polynomial
in the tensors $T_{1},\cdots,T_{r}$ and their iterated covariant derivatives
with the $\ast$ product like

$\mathfrak{q}_{s}(T_{1},\cdots,T_{r}):=\sum\limits_{j_{1}+\cdots+j_{r}=s}c_{j_{1}\cdots
j_{r}}\cdot D^{j_{1}}T_{1}\ast\cdots\ast D^{j_{r}}T_{r},$

where for $1\leqslant i\leqslant r$, $T_{i}\in\Gamma^{j_{i}}(T^{*}M^{\otimes
t_{i}}\otimes E^{\otimes q_{i}})$ and $c_{j_{1}\cdots j_{r}}$ are some
universal constants.

###  2.5. Ricci identity

Given $s\in\Gamma^{2}(T^{*}M^{\otimes k}\otimes E)$, it is obvious to see that
$s$ can be written as follow

$s:=s^{\alpha}_{i_{1}\cdots i_{k}}\cdot dx^{i_{1}}\otimes\cdots\otimes
dx^{i_{k}}\otimes e_{\alpha}.$

We denote $Ds$ in the form of components

$Ds:=s^{\alpha}_{i_{1}\cdots i_{k},p}\cdot dx^{i_{1}}\otimes\cdots\otimes
dx^{i_{k}}\otimes dx^{p}\otimes e_{\alpha}.$

At some time, we also employ the coming convention

(2.8) $\displaystyle Ds:=s_{i_{1}\cdots i_{k},p}\cdot
dx^{i_{1}}\otimes\cdots\otimes dx^{i_{k}}\otimes dx^{p}.$

Thanks to the above agreement, Ricci identity is conveniently represented in
the next theorem.

######  Theorem 2.3.

$\displaystyle s^{\alpha}_{i_{1}\cdots i_{k},pq}-s^{\alpha}_{i_{1}\cdots
i_{k},qp}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{k}s^{\alpha}_{i_{1}\cdots
i_{l-1}hi_{l+1}\cdots i_{k}}\cdot(R^{M})^{h}_{pqi_{l}}-s^{\beta}_{i_{1}\cdots
i_{k}}\cdot(R^{E})^{\alpha}_{pq\beta}$ $\displaystyle=$ $\displaystyle k\cdot
s\ast\mathcal{R}^{M}+s\ast\mathcal{R}^{E}.$

Proof. The proof is straightforward if one takes normal coordinates. So we
omit it. $\hfill\Box$

Given $V\in\Gamma^{k+1}(E)$ and $S\in\Gamma^{k+1}(T^{*}M\otimes E)$, by
Theorem 2.3 and induction, the following formulas are easy.
Formula 1.
There exist $a_{ij}\in\mathbb{Z}$ and $b_{rl}\in\mathbb{Z}$ such that

(2.9) $\displaystyle V_{,pi_{1}\cdots i_{k}}-V_{,i_{1}\cdots i_{k}p}$
$\displaystyle=$ $\displaystyle\sum\limits_{i+j=k-1}a_{ij}\cdot D^{i}V\ast
D^{j}\mathcal{R}^{E}+\sum\limits_{r+l=k-2}b_{rl}\cdot
D^{r+1}V\ast\nabla^{l}\mathcal{R}^{M}$ $\displaystyle=$
$\displaystyle\mathfrak{q}_{k-1}(V,\mathcal{R}^{E})+\mathfrak{q}_{k-2}(DV,\mathcal{R}^{M})$

Formula 2.
There exist $a_{ij}\in\mathbb{Z}$ and $b_{rl}\in\mathbb{Z}$ such that

(2.10) $\displaystyle S_{p,qi_{1}\cdots i_{k}}-S_{p,i_{1}\cdots i_{k}q}$
$\displaystyle=$ $\displaystyle\sum\limits_{i+j=k-1}a_{ij}\cdot D^{i}S\ast
D^{j}\mathcal{R}^{E}+\sum\limits_{r+l=k-1}b_{rl}\cdot
D^{r}S\ast\nabla^{l}\mathcal{R}^{M}$ $\displaystyle=$
$\displaystyle\mathfrak{q}_{k-1}(S,\mathcal{R}^{E})+\mathfrak{q}_{k-1}(S,\mathcal{R}^{M})$

###  2.6. Interpolation for sections

We shall prove Gagliardo-Nirenberg inequality of sections on vector bundle.

######  Theorem 2.4.

$(M,g)$ is a $m-$dimensional smooth closed Riemannian manifold. $(E,h,D)$ is a
smooth vector bundle over $M$ with $Dh=0$. $rank(E)$ may not be 3 and $E$ may
not be orientable. Let $T$ be a smooth section of $E$. Given
$s\in\mathbb{R}^{+}$ and $j\in\mathbb{Z}^{+}$, we will have

(2.11) $\displaystyle||D^{j}T||_{\frac{2s}{l}}\leqslant
C(m,s,k,j)\cdot||D^{k}T||^{\frac{j}{k}}_{\frac{2s}{l+k-j}}\cdot||T||^{1-\frac{j}{k}}_{\frac{2s}{l-j}},$

provided $k\in[j,\infty)\cap\mathbb{Z}$,
$l\in[1,s]\cap[j,s+j+1-k]\cap\mathbb{Z}$.

Proof. Apply induction for $j$.
Step 1: When $j=1$, (2.11) is equivalent to

(2.12) $\displaystyle||DT||_{\frac{2s}{l}}\leqslant
C(m,s,k)\cdot||D^{k}T||^{\frac{1}{k}}_{\frac{2s}{l+k-1}}\cdot||T||^{1-\frac{1}{k}}_{\frac{2s}{l-1}},$

for all $l\in[1,s]\cap[1,s+2-k]\cap\mathbb{Z}$. In order to show (2.12), we
use induction for $k$.

When $k=1$, (2.12) holds obviously.

When $k=2$, by 12.1 Theorem of [2] we know (2.12) holds.

Assume that for $2\leqslant\hat{k}\leqslant k$, we obtain

$\displaystyle||DT||_{\frac{2s}{l}}\leqslant
C_{1}(m,s,\hat{k})\cdot||D^{\hat{k}}T||^{\frac{1}{\hat{k}}}_{\frac{2s}{l+\hat{k}-1}}\cdot||T||^{1-\frac{1}{\hat{k}}}_{\frac{2s}{l-1}},$

provided $l\in[1,s]\cap[1,s+2-\hat{k}]\cap\mathbb{Z}$.

When $\hat{k}=k+1$, pick any $l\in[1,s]\cap[1,s+2-(k+1)]\cap\mathbb{Z}$.
Clearly,

$l+1\in[1,s]\cap[1,s+2-k]\cap\mathbb{Z},$

since $k\geqslant 2$. Using induction hypothesis, we get

(2.13) $\displaystyle||D^{2}T||_{\frac{2s}{l+1}}\leqslant
C_{2}(m,s,k)\cdot||D^{k}(DT)||^{\frac{1}{k}}_{\frac{2s}{l+k}}\cdot||DT||^{1-\frac{1}{k}}_{\frac{2s}{l}}.$

Because $1\leqslant l\leqslant s+2-(k+1)<s$, using induction hypothesis for
$k=2$ gives

(2.14) $\displaystyle||DT||_{\frac{2s}{l}}\leqslant
C(m,s)\cdot||D^{2}T||^{\frac{1}{2}}_{\frac{2s}{l+1}}\cdot||T||^{\frac{1}{2}}_{\frac{2s}{l-1}}.$

Combing (2.13) with (2.14) yields

$||DT||_{\frac{2s}{l}}\leqslant
C_{3}(m,s,k)\cdot||D^{k+1}T||^{\frac{1}{2k}}_{\frac{2s}{l+k}}\cdot||DT||^{\frac{1}{2}(1-\frac{1}{k})}_{\frac{2s}{l}}\cdot||T||^{\frac{1}{2}}_{\frac{2s}{l-1}},$

which implies

$||DT||_{\frac{2s}{l}}\leqslant
C(m,s,k+1)\cdot||D^{k+1}T||^{\frac{1}{k+1}}_{\frac{2s}{l+k}}\cdot||T||^{1-\frac{1}{k+1}}_{\frac{2s}{l-1}}.$

Step 2: Suppose that for all the indices not greater than $j$, (2.11) is true.
Now we consider $j+1$. At this moment, we take any
$k\in[j+1,\infty)\cap\mathbb{Z}$ and any
$l\in[1,s]\cap[j+1,s+j+2-k]\cap\mathbb{Z}$. It is easy to deduce that

$k-1\in[j,\infty)\cap\mathbb{Z}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,l\in[1,s]\cap[j,s+j+1-(k-1)]\cap\mathbb{Z}.$

Using induction hypothesis leads to

(2.15) $\displaystyle||D^{j}(DT)||_{\frac{2s}{l}}\leqslant
C_{1}(m,s,k,j)\cdot||D^{k-1}(DT)||^{\frac{j}{k-1}}_{\frac{2s}{l+k-1-j}}\cdot||DT||^{1-\frac{j}{k-1}}_{\frac{2s}{l-j}}.$

Since $l-j\in[1,s]\cap[1,s+2-k]\cap\mathbb{Z}$, by Step 1 we have

(2.16) $\displaystyle||DT||_{\frac{2s}{l-j}}\leqslant
C(m,s,k)\cdot||D^{k}T||^{\frac{1}{k}}_{\frac{2s}{l-j+k-1}}\cdot||T||^{1-\frac{1}{k}}_{\frac{2s}{l-j-1}}.$

Combining (2.15) with (2.16) gives

$\displaystyle||D^{j+1}T||_{\frac{2s}{l}}$ $\displaystyle\leqslant$
$\displaystyle
C(m,s,k,j+1)\cdot||D^{k}T||^{\frac{j}{k-1}}_{\frac{2s}{l+k-1-j}}\cdot||D^{k}T||^{\frac{1}{k}(1-\frac{j}{k-1})}_{\frac{2s}{l+k-j-1}}\cdot||T||^{(1-\frac{1}{k})(1-\frac{j}{k-1})}_{\frac{2s}{l-j-1}}$
$\displaystyle=$ $\displaystyle
C(m,s,k,j+1)\cdot||D^{k}T||^{\frac{j+1}{k}}_{\frac{2s}{l+k-1-j}}\cdot||T||^{1-\frac{j+1}{k}}_{\frac{2s}{l-j-1}}.$

This completes the proof.$\hfill\Box$

######  Theorem 2.5.

$(M,g)$ is a $m-$dimensional smooth closed Riemannian manifold. $(E,h,D)$ is a
smooth vector bundle over $M$ with $Dh=0$. $rank(E)$ may not be 3 and $E$ may
not be orientable. Let $T$ be a smooth section of $E$. If $r,q\geqslant 2$,
then there is a universal constant $C=C(m,r,q,j,k)$ such that

(2.17) $\displaystyle||D^{j}T||_{p}\leqslant
C\cdot||D^{k}T||_{r}^{\frac{j}{k}}\cdot||T||_{q}^{1-\frac{j}{k}},$

provided

$1\leqslant j\leqslant
k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{k}{p}=\frac{j}{r}+\frac{k-j}{q}.$

Proof. We consider 3 cases.
Case 1: When $2\leqslant r<q\leqslant\infty$, there exist $s$ and $l$ such
that

$q=\frac{2s}{l-j}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r=\frac{2s}{l+k-j}.$

Since

$\frac{k}{p}=\frac{j}{r}+\frac{k-j}{q},$

we have $p=\frac{2s}{l}$. From Theorem 2.4 it follows that

$||D^{j}T||_{\frac{2s}{l}}\leqslant
C(m,s,k,j)\cdot||D^{k}T||^{\frac{j}{k}}_{\frac{2s}{l+k-j}}\cdot||T||^{1-\frac{j}{k}}_{\frac{2s}{l-j}},$

which means

$||D^{j}T||_{p}\leqslant
C(m,r,q,j,k)\cdot||D^{k}T||_{r}^{\frac{j}{k}}\cdot||T||_{q}^{1-\frac{j}{k}}.$

Case 2: When $2\leqslant q<r\leqslant\infty$, the proof is similar.
Case 3: When $2\leqslant q=r$, clearly we have $p=q=r$. From 12.1 Theorem in
[2] it follows that

$||DT||_{p}\leqslant
C(m,p)\cdot||D^{2}T||^{\frac{1}{2}}_{p}\cdot||T||_{p}^{\frac{1}{2}},$

which implies

$||D^{j}T||_{p}\leqslant
C(m,p)\cdot||D^{j+1}T||^{\frac{1}{2}}_{p}\cdot||D^{j-1}T||_{p}^{\frac{1}{2}}.$

Let $f(j):=||D^{j}T||_{p}$. It is easy to check that $f$ meets the condition
of 12.5 Corollary in [2]. Then we conclude this theorem.$\hfill\Box$

##  3\. Proof of Theorem 1.1

Given any $T>0$, define an operator

$P:C^{1}([0,T],\Gamma^{2}(E))\longrightarrow C([0,T],\Gamma(E)),$

here

$P(V):=\partial_{t}V-\Delta V-V\times\Delta V-\lambda(1+\mu|V|^{2})V.$

It is not difficult to check that the leading coefficient of the linearised
operator of $P$ meets Legendre-Hadamard condition. By Main Theorem 1 in page 3
of [1] we know (1.6) admits a unique local smooth solution $V$ provided
$V_{0}\in\Gamma^{\infty}(E)$.

In the sequel, we would like to know the lower bound of maximal existence time
$T_{\max}$ of the above smooth solution. Our strategy is to deduce a Gronwall
inequality. That is to say, we shall control
$\frac{d}{dt}||V(t)||^{2}_{H^{l}}$. Before getting to this, it is important to
obtain an upper bound of $||V(t)||_{\infty}$.

Taking inner product with $|V|^{p-2}V(p>2)$ in (1.6), and integrating the
result over $M$, we get

$\displaystyle\int_{M}|V|^{p-2}\langle V,\partial_{t}V\rangle\,dM$
$\displaystyle=$ $\displaystyle\int_{M}|V|^{p-2}\langle V,\Delta
V\rangle\,dM-\lambda\int_{M}(1+\mu|V|^{2})|V|^{p}\,dM$
$\displaystyle\leqslant$
$\displaystyle-\int_{M}|V|^{p-2}\cdot|DV|^{2}\,dM-(p-2)\int_{M}|V|^{p-4}\cdot|\langle
V,DV\rangle|^{2}\,dM\leqslant 0.$

The left hand side of the above inequality is
$\frac{1}{p}\frac{d}{dt}||V(t)||_{p}^{p}$, so this inequality means

$\displaystyle||V(t)||_{p}\leqslant||V_{0}||_{p}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,T_{\max}).$

Taking the limit $p\rightarrow\infty$ leads to

(3.1)
$\displaystyle||V(t)||_{\infty}\leqslant||V_{0}||_{\infty}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,T_{\max}).$

Given $k\geqslant 1$, recalling our appointment (2.8), we have the next
identity

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{M}|D^{k}V|^{2}\,dM$
$\displaystyle=$ $\displaystyle\int_{M}g^{pq}g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\langle V_{,j_{1}\cdots j_{k}},V_{,pqi_{1}\cdots
i_{k}}\rangle\,dM$ $\displaystyle+\int_{M}g^{pq}g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\langle V_{,j_{1}\cdots j_{k}},(V\times V_{,p})_{,qi_{1}\cdots
i_{k}}\rangle\,dM$ $\displaystyle-\lambda\cdot\mu\int_{M}g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\langle V_{,j_{1}\cdots j_{k}},(|V|^{2}V)_{,i_{1}\cdots
i_{k}}\rangle\,dM$ $\displaystyle-\lambda\int_{M}|D^{k}V|^{2}\,dM$

Applying (2.9) and (2.10) to exchange the order of derivatives yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{M}|D^{k}V|^{2}\,dM$
$\displaystyle=$
$\displaystyle-\int_{M}|D^{k+1}V|^{2}\,dM+\int_{M}D^{k}V\ast\mathfrak{q}_{k}(V,\mathcal{R}^{E})\,dM$
$\displaystyle+\int_{M}D^{k}V\ast\mathfrak{q}_{k-1}(DV,\mathcal{R}^{M})\,dM$
$\displaystyle-\int_{M}g^{pq}g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}\langle
V_{,qj_{1}\cdots j_{k}},(V\times V_{,p})_{,i_{1}\cdots i_{k}}\rangle\,dM$
$\displaystyle-\int_{M}D^{k}V\ast\mathfrak{q}_{k-1}(V\times
DV,\mathcal{R}^{M})\,dM$
$\displaystyle-\int_{M}D^{k}V\ast\mathfrak{q}_{k-1}(V\times
DV,\mathcal{R}^{E})\,dM$
$\displaystyle-\lambda\int_{M}|D^{k}V|^{2}\,dM-\lambda\mu\int_{M}|V|^{2}\cdot|D^{k}V|^{2}\,dM$
$\displaystyle-\lambda\mu\sum\limits_{i+j=k-1}b_{ij}\cdot\int_{M}D^{k}V\ast
D^{i+1}(|V|^{2})\ast D^{j}V\,dM$
$\displaystyle+\int_{M}\mathfrak{q}_{k}(V,\mathcal{R}^{E})\ast D^{k-1}(V\times
DV)\,dM$ $\displaystyle+\int_{M}\mathfrak{q}_{k-1}(DV,\mathcal{R}^{M})\ast
D^{k-1}(V\times DV)\,dM,$

here $b_{ij}\in\mathbb{Z}^{+}$ are some universal constants. Note that

$\displaystyle g^{pq}g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}\langle
V_{,qj_{1}\cdots j_{k}},(V\times V_{,p})_{,i_{1}\cdots
i_{k}}\rangle=\sum\limits_{i+j=k-1}a_{ij}\cdot D^{k+1}V\ast(D^{i+1}V\times
D^{j+1}V)$

where $a_{ij}\in\mathbb{Z}^{+}$ are some universal constants. Taking norms on
the right hand side of (3) leads to

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{M}|D^{k}V|^{2}\,dM$
$\displaystyle\leqslant$
$\displaystyle-\int_{M}|D^{k+1}V|^{2}\,dM+\sum\limits_{i=0}^{k}C_{i}\int_{M}|D^{k}V|\cdot|D^{i}V|\,dM$
$\displaystyle+\sum\limits_{i+j=k-1}a_{ij}\int_{M}|D^{k+1}V|\cdot|D^{i+1}V|\cdot|D^{j+1}V|\,dM$
$\displaystyle+\sum\limits_{i=0}^{k-1}\bar{C}_{i}\int_{M}|D^{k}V|\cdot|D^{i}(V\times
DV)|\,dM$
$\displaystyle+\lambda\mu\sum\limits_{i+j=k-1}b_{ij}\int_{M}|D^{k}V|\cdot|D^{i+1}(|V|^{2})|\cdot|D^{j}V|\,dM$
$\displaystyle+\sum\limits_{i=0}^{k}C_{i}\int_{M}|D^{i}V|\cdot|D^{k-1}(V\times
DV)|\,dM,$

where $C_{i}$ and $\bar{C}_{i}$ depend upon $\mathcal{R}^{M}$,
$\mathcal{R}^{E}$ and their covariant differentiations. Applying (2.1) and
(2.7) yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{M}|D^{k}V|^{2}\,dM$
$\displaystyle\leqslant$
$\displaystyle-\int_{M}|D^{k+1}V|^{2}\,dM+\sum\limits_{i+j=k-1}a_{ij}\int_{M}|D^{k+1}V|\cdot|D^{i+1}V|\cdot|D^{j+1}V|\,dM$
$\displaystyle+\tilde{C}_{k}\Big{\\{}||D^{k}V||_{2}\cdot||V||_{H^{k}}+\sum\limits_{0\leqslant
r+q\leqslant k-1}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM$
$\displaystyle+\sum\limits_{r+q+j=k}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle+\sum\limits_{i=0}^{k}\sum\limits_{r+q=k-1}\int_{M}|D^{i}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM\Big{\\}},$

where $\tilde{C}_{k}$ depends upon $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant differentiations.

######  Lemma 3.1.

There is a $C^{\prime}_{m_{0}}>0$ depending on $\mathcal{R}^{M}$,
$\mathcal{R}^{E}$ and their covariant differentiations such that, for any
$t\in[0,T_{\max})$, we have

$\displaystyle\frac{d}{dt}||V(t)||^{2}_{H^{m_{0}}}\leqslant
C^{\prime}_{m_{0}}\cdot(1+||V_{0}||^{2}_{H^{m_{0}}})\cdot\\{||V(t)||^{2}_{H^{m_{0}}}+||V(t)||^{4}_{H^{m_{0}}}\\}.$

Proof. Given $1\leqslant k\leqslant m_{0}$, we consider

$\displaystyle\sum\limits_{r+q+j=k}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle=$ $\displaystyle\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}=k\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle+\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}\leqslant
k-1\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM.$

Clearly,

$\displaystyle\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}=k\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM\lesssim||D^{k}V||_{2}^{2}\cdot||V||^{2}_{\infty}\leqslant||D^{k}V||_{2}^{2}\cdot||V_{0}||^{2}_{\infty}.$

And we want to derive the following

$\displaystyle\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}\leqslant
k-1\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle\leqslant$
$\displaystyle\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}\leqslant
k-1\end{subarray}}||D^{k}V||_{2}\cdot||D^{r}V||_{p_{r}}\cdot||D^{q}V||_{p_{q}}\cdot||D^{j}V||_{p_{j}}.$

where $p_{r}$, $p_{q}$ and $p_{j}$, belonging to $[1,\infty]$, will be
determined later and satisfy

(3.4)
$\displaystyle\frac{1}{p_{r}}+\frac{1}{p_{q}}+\frac{1}{p_{j}}=\frac{1}{2}.$

And then we employ Theorem 2.1 due to [3] to obtain

$\displaystyle||D^{r}V||_{p_{r}}\lesssim||V||^{a_{r}}_{H^{m_{0}}}\cdot||V||_{2}^{1-a_{r}}\leqslant||V||_{H^{m_{0}}},$
$\displaystyle||D^{q}V||_{p_{q}}\lesssim||V||^{a_{q}}_{H^{m_{0}}}\cdot||V||_{2}^{1-a_{q}}\leqslant||V||_{H^{m_{0}}},$

and

$\displaystyle||D^{j}V||_{p_{j}}\lesssim||V||^{a_{j}}_{H^{m_{0}}}\cdot||V||_{2}^{1-a_{j}}\leqslant||V||_{H^{m_{0}}}.$

We hope $p_{r}$, $p_{q}$ and $p_{j}$ meet the next conditions:
Condition 1.

$\displaystyle\frac{1}{p_{r}}=\frac{r}{m}+\frac{1}{2}-a_{r}\cdot\frac{m_{0}}{m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{r}\in\Big{[}\frac{r}{m_{0}},1\Big{)},$

which is equivalent to

(3.5)
$\displaystyle\frac{1}{p_{r}}\in\Big{(}\frac{r-m_{0}}{m}+\frac{1}{2},\frac{1}{2}\Big{]}.$

Condition 2.

$\displaystyle\frac{1}{p_{q}}=\frac{q}{m}+\frac{1}{2}-a_{q}\cdot\frac{m_{0}}{m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{q}\in\Big{[}\frac{q}{m_{0}},1\Big{)},$

which is equivalent to

(3.6)
$\displaystyle\frac{1}{p_{q}}\in\Big{(}\frac{q-m_{0}}{m}+\frac{1}{2},\frac{1}{2}\Big{]}.$

Condition 3.

$\displaystyle\frac{1}{p_{j}}=\frac{j}{m}+\frac{1}{2}-a_{j}\cdot\frac{m_{0}}{m}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{j}\in\Big{[}\frac{j}{m_{0}},1\Big{)},$

which is equivalent to

(3.7)
$\displaystyle\frac{1}{p_{j}}\in\Big{(}\frac{j-m_{0}}{m}+\frac{1}{2},\frac{1}{2}\Big{]}.$

We claim there exist $p_{r}$, $p_{q}$ and $p_{j}$ which are in $[1,\infty]$
and satisfy (3.4), (3.5), (3.6) and (3.7). Obviously, that this claim holds is
equivalent to

(3.8)
$\displaystyle\Big{(}\frac{r-m_{0}}{m}+\frac{1}{2}\Big{)}+\Big{(}\frac{q-m_{0}}{m}+\frac{1}{2}\Big{)}+\Big{(}\frac{j-m_{0}}{m}+\frac{1}{2}\Big{)}<\frac{1}{2}\Longleftrightarrow
k<3m_{0}-m.$

Since $k\leqslant m_{0}$ and $m_{0}>\frac{m}{2}$, (3.8) is true. In other
words,

$\displaystyle\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}\leqslant
k-1\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM\lesssim||D^{k}V||_{2}\cdot||V||^{3}_{H^{m_{0}}}$

In conclusion,

$\displaystyle\sum\limits_{r+q+j=k}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle||D^{k}V||_{2}\cdot||V||^{3}_{H^{m_{0}}}+||D^{k}V||_{2}^{2}\cdot||V_{0}||^{2}_{\infty}.$

For the other terms of (3), using the same methods, we get similar
estimations:
Estimation 1.

$\displaystyle\sum\limits_{i+j=k-1}a_{ij}\int_{M}|D^{k+1}V|\cdot|D^{i+1}V|\cdot|D^{j+1}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||DV||_{\infty}+\sum\limits_{\begin{subarray}{c}i+j=k-1\\\
\max\\{i,j\\}\leqslant
k-2\end{subarray}}a_{ij}\cdot||D^{k+1}V||_{2}\cdot||D^{i+1}V||_{p_{i}}\cdot||D^{j+1}V||_{p_{j}}$
$\displaystyle\lesssim$
$\displaystyle||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||DV||_{\infty}+||D^{k+1}V||_{2}\cdot||V||^{2}_{H^{m_{0}}}$
$\displaystyle\lesssim$
$\displaystyle||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||V||_{H^{m_{0}}}+||D^{k+1}V||_{2}\cdot||V||^{2}_{H^{m_{0}}},$

Estimation 2.

$\displaystyle\sum\limits_{0\leqslant r+q\leqslant
k-1}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM\lesssim||V_{0}||_{\infty}\cdot||D^{k}V||^{2}_{2}+||D^{k}V||_{2}\cdot||V||^{2}_{H^{m_{0}}},$

Estimation 3.

$\displaystyle\sum\limits_{i=0}^{k}\sum\limits_{r+q=k-1}\int_{M}|D^{i}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM\lesssim||V||_{H^{k}}\cdot||D^{k}V||_{2}\cdot||V_{0}||_{\infty}+||V||_{H^{k}}\cdot||V||^{2}_{H^{m_{0}}}.$

Summing $k$ from $0$ to $m_{0}$ gives

$\displaystyle\frac{1}{2}\frac{d}{dt}||V||^{2}_{H^{m_{0}}}$
$\displaystyle\leqslant$
$\displaystyle-||V||^{2}_{H^{m_{0}+1}}+L_{m_{0}}\cdot||V||_{H^{m_{0}+1}}\cdot||V||^{2}_{H^{m_{0}}}$
$\displaystyle+\tilde{L}_{m_{0}}\cdot\Big{\\{}||V||^{2}_{H^{m_{0}}}+||V_{0}||_{\infty}\cdot||V||^{2}_{H^{m_{0}}}$
$\displaystyle+||V||^{3}_{H^{m_{0}}}+||V_{0}||^{2}_{\infty}\cdot||V||^{2}_{H^{m_{0}}}+||V||^{4}_{H^{m_{0}}}\Big{\\}},$

where $L_{m_{0}}$ is universal and $\tilde{L}_{m_{0}}$ depends on
$\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their covariant differentiations.
Then the result follows easily from Young’s inequality and Sobolev embedding

$||V_{0}||_{\infty}\lesssim||V_{0}||_{H^{m_{0}}}.$

This completes the proof. $\hfill\Box$

Consider an ODE

(3.10) $\displaystyle\left\\{\begin{aligned}
\frac{df}{dt}&=C^{\prime}_{m_{0}}\cdot(1+||V_{0}||^{2}_{H^{m_{0}}})\cdot(f+f^{2})\\\
f(0)&=||V_{0}||^{2}_{H^{m_{0}}}.\end{aligned}\right.$

Solving the above equation to get an expression of $f$, we know that the
maximal existence time of the solution $f(\cdot,||V_{0}||_{H^{m_{0}}})$ to
(3.10) is not smaller than

$T^{*}:=\frac{1}{C^{\prime}_{m_{0}}\cdot(1+||V_{0}||^{2}_{H^{m_{0}}})}\log\Big{(}\frac{1+2||V_{0}||^{2}_{H^{m_{0}}}}{2||V_{0}||^{2}_{H^{m_{0}}}}\Big{)}.$

And $f(t,||V_{0}||_{H^{m_{0}}})$ is monotone increasing with respect to $t$.
In other words, for all $t\in[0,T^{*}]$,

$f(t,||V_{0}||_{H^{m_{0}}})\leqslant
f(T^{*},||V_{0}||_{H^{m_{0}}})=1+2||V_{0}||^{2}_{H^{m_{0}}}.$

By comparison principle of ODE, we know that for any
$t\in[0,\min\\{T_{\max},T^{*}\\})$,

$||V(t)||_{H^{m_{0}}}\leqslant\sqrt{1+2||V_{0}||^{2}_{H^{m_{0}}}}.$

In the sequel, we focus on the case that $k$ is sufficiently big.

######  Lemma 3.2.

When $k\geqslant m_{0}+1$, there is a $Q_{k}>0$ depending on
$\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their covariant differentiations such
that, for any $t\in[0,T_{\max})$, we have

$\displaystyle\frac{d}{dt}||V(t)||^{2}_{H^{k}}+||V(t)||^{2}_{H^{k+1}}$
$\displaystyle\leqslant$ $\displaystyle
Q_{k}\cdot\Big{[}||V(t)||^{2}_{H^{k}}\cdot||V(t)||^{2}_{H^{m_{0}}}+||V(t)||^{4}_{H^{k-1}}+||V(t)||^{2}_{H^{k}}\cdot(1+||V_{0}||_{\infty}+||V_{0}||^{2}_{\infty})$
$\displaystyle+||V(t)||^{2}_{H^{k-1}}\cdot(||V(t)||^{2}_{H^{m_{0}}}+||V(t)||^{4}_{H^{m_{0}}})+||V(t)||^{6}_{H^{k-1}}\Big{]}.$

Proof. Firstly, let us calculate one term of (3). Applying the same method of
(3), one can see easily that there are $p_{i}$ belonging to $[1,\infty]$ such
that the following inequalities hold

$\displaystyle\sum\limits_{i+j=k-1}a_{ij}\int_{M}|D^{k+1}V|\cdot|D^{i+1}V|\cdot|D^{j+1}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||DV||_{\infty}+||D^{k+1}V||_{2}\cdot||D^{k-1}V||_{2}\cdot||D^{2}V||_{\infty}$
$\displaystyle+\sum\limits_{\begin{subarray}{c}i+j=k-1\\\
\max\\{i,j\\}\leqslant
k-3\end{subarray}}a_{ij}\cdot||D^{k+1}V||_{2}\cdot||D^{i+1}V||_{p_{i}}\cdot||D^{j+1}V||_{p_{j}}$
$\displaystyle\lesssim$
$\displaystyle||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||V||_{H^{m_{0}}}+||D^{k+1}V||_{2}\cdot||D^{k-1}V||_{2}\cdot||V||_{H^{m_{0}}}$
$\displaystyle+||D^{k+1}V||_{2}\cdot||V||^{2}_{H^{k-1}}.$

By the same procedure, we get the next estimations:
Estimation 4.

$\displaystyle\sum\limits_{0\leqslant r+q\leqslant
k-1}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM$ $\displaystyle\lesssim$
$\displaystyle||D^{k}V||_{2}^{2}\cdot||V_{0}||_{\infty}+||D^{k}V||_{2}\cdot||D^{k-1}V||_{2}\cdot||V||_{H^{m_{0}}}$
$\displaystyle+||D^{k}V||_{2}\cdot||D^{k-2}V||_{2}\cdot||V||_{H^{m_{0}}}+||D^{k}V||_{2}\cdot||V||^{2}_{H^{k-1}}.$

Estimation 5.

$\displaystyle\sum\limits_{r+q+j=k}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle\int_{M}|D^{k}V|^{2}\cdot|V|^{2}\,dM+\int_{M}|D^{k}V|\cdot|D^{k-1}V|\cdot|DV|\cdot|V|\,dM$
$\displaystyle+\sum\limits_{\begin{subarray}{c}r+q+j=k\\\
\max\\{r,q,j\\}\leqslant
k-2\end{subarray}}\int_{M}|D^{k}V|\cdot|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle||D^{k}V||_{2}^{2}\cdot||V||^{2}_{\infty}+||D^{k}V||_{2}\cdot||D^{k-1}V||_{2}\cdot||DV||_{\infty}\cdot||V||_{\infty}+||D^{k}V||_{2}\cdot||V||^{3}_{H^{k-1}}$
$\displaystyle\lesssim$
$\displaystyle||D^{k}V||_{2}^{2}\cdot||V_{0}||^{2}_{\infty}+||D^{k}V||_{2}\cdot||D^{k-1}V||_{2}\cdot||V||^{2}_{H^{m_{0}}}+||D^{k}V||_{2}\cdot||V||^{3}_{H^{k-1}}.$

Estimation 6.

$\displaystyle\sum\limits_{i=0}^{k}\sum\limits_{r+q=k-1}\int_{M}|D^{i}V|\cdot|D^{r}V|\cdot|D^{q+1}V|\,dM$
$\displaystyle\lesssim$
$\displaystyle||V||_{H^{k}}\cdot||V||_{\infty}\cdot||D^{k}V||_{2}+||V||_{H^{k}}\cdot||D^{k-1}V||_{2}\cdot||DV||_{\infty}$
$\displaystyle+||V||_{H^{k}}\cdot||D^{k-2}V||_{2}\cdot||D^{2}V||_{\infty}+||V||_{H^{k}}\cdot||V||^{2}_{H^{k-1}}$
$\displaystyle\lesssim$
$\displaystyle||V||_{H^{k}}\cdot||V_{0}||_{\infty}\cdot||D^{k}V||_{2}+||V||_{H^{k}}\cdot||D^{k-1}V||_{2}\cdot||V||_{H^{m_{0}}}$
$\displaystyle+||V||_{H^{k}}\cdot||D^{k-2}V||_{2}\cdot||V||_{H^{m_{0}}}+||V||_{H^{k}}\cdot||V||^{2}_{H^{k-1}}$

Substituting (3), (3), (3) and (3) into (3) and then summing $k$ lead to

$\displaystyle\frac{1}{2}\frac{d}{dt}||V||^{2}_{H^{k}}$
$\displaystyle\leqslant$
$\displaystyle-||V||^{2}_{H^{k+1}}+\hat{Q}_{k}\cdot(||V||_{H^{k+1}}\cdot||V||_{H^{k}}\cdot||V||_{H^{m_{0}}}+||V||_{H^{k+1}}\cdot||V||^{2}_{H^{k-1}})$
$\displaystyle+\tilde{Q}_{k}\cdot(||V||^{2}_{H^{k}}+||V||^{2}_{H^{k}}\cdot||V_{0}||_{\infty}+||V||_{H^{k}}\cdot||V||_{H^{k-1}}\cdot||V||_{H^{m_{0}}}$
$\displaystyle+||V||_{H^{k}}\cdot||V||^{2}_{H^{k-1}}+||V||^{2}_{H^{k}}\cdot||V_{0}||^{2}_{\infty}+||V||_{H^{k}}\cdot||V||_{H^{k-1}}\cdot||V||^{2}_{H^{m_{0}}}$
$\displaystyle+||V||_{H^{k}}\cdot||V||^{3}_{H^{k-1}}),$

where $\tilde{Q}_{k}>0$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant differentiations. Using Young’s inequality, we conclude this
theorem. $\hfill\Box$

Note that (3.2) is linear for $||V||^{2}_{H^{k}}$. It is now clear that
inductively using (3.2) one can show the existence of
$N_{k}=N(||V_{0}||_{H^{k}},Q_{k},Q_{k-1},\cdots,Q_{m_{0}+1})$ for any
$k\geqslant m_{0}+1$ such that

$||V(t)||_{H^{k}}\leqslant N_{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,\min\\{T_{\max},T^{*}\\}),$

which implies

$T_{\max}\geqslant T^{*}.$

Now we return to prove Theorem 1.1. Define

$h(x):=\frac{1}{C^{\prime}_{m_{0}}\cdot(1+x)}\log\Big{(}\frac{1+2x}{2x}\Big{)}$

and we observe that it is a monotone decreasing function. Given $l\geqslant
m_{0}+1$ and $V_{0}\in H^{l}(E)$, there are $V_{i0}\in\Gamma^{\infty}(E)$ such
that as $i\rightarrow\infty$,

$V_{i0}\longrightarrow V_{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{strongly in
$H^{l}(E)$}.$

By the above discussion we know there exist

$T^{*}_{i}\geqslant
h(||V_{i0}||^{2}_{H^{m_{0}}})>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,V_{i}\in
C^{\infty}([0,T^{*}_{i}),\Gamma^{\infty}(E))$

such that

(3.16) $\displaystyle\left\\{\begin{aligned} &\partial_{t}V_{i}=\Delta
V_{i}+V_{i}\times\Delta V_{i}-\lambda\cdot(1+\mu|V_{i}|^{2})V_{i}\\\
&V_{i}(0,\cdot)=V_{i0},\end{aligned}\right.$

here $T^{*}_{i}$ is the maximal existence time of $V_{i}$. Obviously, when $i$
is enough large,

$||V_{i0}||^{2}_{H^{m_{0}}}\leqslant||V_{0}||^{2}_{H^{m_{0}}}+1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,||V_{i0}||_{H^{l}}\leqslant||V_{0}||_{H^{l}}+1,$

which imply

$T^{*}_{i}\geqslant h(||V_{0}||^{2}_{H^{m_{0}}}+1):=2\tilde{\delta}>0$

and

$||V_{i}(t)||_{H^{l}}\leqslant
N(||V_{0}||_{H^{l}}+1,Q_{k},Q_{k-1},\cdots,Q_{m_{0}+1})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,\tilde{\delta}].$

Then $V_{i}$ is a bounded sequence in
$L^{\infty}([0,\tilde{\delta}],H^{l}(E))$. It is not hard to verify that
$\partial_{t}V_{i}$ is a bounded sequence in
$L^{\infty}([0,\tilde{\delta}],L^{2}(E))$. So there exists a $V\in
L^{\infty}([0,\tilde{\delta}],H^{l}(E))$ and a subsequence which is still
denoted by $\\{V_{i}\\}$ such that

$V_{i}\rightharpoonup V\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{weakly $*$ in
$L^{\infty}([0,\tilde{\delta}],H^{l}(E))$}.$

By Aubin-Lions lemma, one can find a subsequence still denoted by
$\\{V_{i}\\}$ such that

$V_{i}\longrightarrow V\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{strongly in
$L^{\infty}([0,\tilde{\delta}],H^{l-1}(E))$}.$

Because $l-1\geqslant m_{0}$, $H^{l-1}(E)$ can be embedded into
$\Gamma^{2}(E)$. In other words, $V$ is a solution to (1.6). Using LLB to
transform time derivatives into spatial derivatives gives that for all
$0\leqslant j\leqslant\big{[}\frac{l}{\hat{m}}\big{]}$ and all
$\alpha\leqslant l-\hat{m}j$, we have

(3.17) $\displaystyle\partial^{j}_{t}D^{\alpha}V\in
L^{\infty}([0,\tilde{\delta}],L^{2}(E)).$

######  Remark 3.3.

The proof of (3.17) is easy if one employs induction for $j$.

At last, since $l\geqslant[\frac{m}{2}]+4$, by the same method of Theorem 3 in
[4] it is not difficult to know that the solution of (1.6) with initial data
$V_{0}\in H^{l}(E)$ is unique. This completes the proof. $\hfill\Box$

##  4\. Proof of Theorem 1.2

Now we focus on global existence of LLB. Suppose that $V$ is the local smooth
solution of (1.6). Our trick is to deduce a uniform estimation for
$||V||_{H^{k}}$. To this goal, firstly we should get a linear Gronwall
inequality.

By (3) and Hölder inequality, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}||D^{k}V||_{2}^{2}$
$\displaystyle\leqslant$
$\displaystyle-||D^{k+1}V||_{2}^{2}+\sum\limits_{i+j=k-1}a_{ij}\cdot||D^{k+1}V||_{2}\cdot\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}$
$\displaystyle+\tilde{C}_{k}\cdot\Big{\\{}||V||^{2}_{H^{k}}+||D^{k}V||_{2}\sum\limits_{0\leqslant
r+q\leqslant k-1}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q+1}V|\Big{|}\Big{|}_{2}$
$\displaystyle||D^{k}V||_{2}\sum\limits_{r+q+j=k}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\Big{|}\Big{|}_{2}$
$\displaystyle+||V||_{H^{k}}\sum\limits_{r+q=k-1}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q+1}V|\Big{|}\Big{|}_{2}\Big{\\}}.$

For the second term on the right hand side of (4),

$\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}\leqslant||D^{i+1}V||_{\frac{2k+2}{i+1}}\cdot||D^{k-i}V||_{\frac{2k+2}{k-i}},$

since $i+j=k-1$. Theorem 2.4 implies

$||D^{i+1}V||_{\frac{2k+2}{i+1}}\lesssim||D^{k+1}V||_{2}^{\frac{i+1}{k+1}}\cdot||V||_{\infty}^{\frac{k-i}{k+1}}$

and

$||D^{k-i}V||_{\frac{2k+2}{k-i}}\lesssim||D^{k+1}V||_{2}^{\frac{k-i}{k+1}}\cdot||V||_{\infty}^{\frac{i+1}{k+1}}.$

So

$\displaystyle\sum\limits_{i+j=k-1}a_{ij}\cdot||D^{k+1}V||_{2}\cdot\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}$
$\displaystyle\leqslant$ $\displaystyle
B_{k}\cdot||V||_{\infty}\cdot||D^{k+1}V||_{2}^{2}\leqslant
B_{k}\cdot||V_{0}||_{\infty}\cdot||D^{k+1}V||_{2}^{2},$

where $B_{k}$ is a universal constant. By the same way, we will get

$\displaystyle\sum\limits_{0\leqslant r+q\leqslant
k-1}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q+1}V|\Big{|}\Big{|}_{2}\leqslant\sum\limits_{0\leqslant
r+q\leqslant
k-1}||D^{r}V||_{\frac{2r+2q+2}{r}}\cdot||D^{q+1}V||_{\frac{2r+2q+2}{q+1}}$
$\displaystyle\lesssim$ $\displaystyle\sum\limits_{0\leqslant r+q\leqslant
k-1}\Big{[}||D^{r+q+1}V||_{2}^{\frac{r}{r+q+1}}\cdot||V||^{\frac{q+1}{r+q+1}}_{\infty}\Big{]}\cdot\Big{[}||D^{r+q+1}V||_{2}^{\frac{q+1}{r+q+1}}\cdot||V||^{\frac{r}{r+q+1}}_{\infty}\Big{]}$
$\displaystyle=$ $\displaystyle\sum\limits_{0\leqslant r+q\leqslant
k-1}||D^{r+q+1}V||_{2}\cdot||V||_{\infty}\lesssim||V||_{H^{k}}\cdot||V_{0}||_{\infty}$

and

(4.4)
$\displaystyle\sum\limits_{r+q=k-1}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q+1}V|\Big{|}\Big{|}_{2}\lesssim||D^{k}V||_{2}\cdot||V_{0}||_{\infty}.$

Moreover, Theorem 2.4 yields

$\displaystyle\sum\limits_{r+q+j=k}\Big{|}\Big{|}|D^{r}V|\cdot|D^{q}V|\cdot|D^{j}V|\Big{|}\Big{|}_{2}\leqslant\sum\limits_{r+q+j=k}||D^{r}V||_{\frac{2k}{r}}\cdot||D^{q}V||_{\frac{2k}{q}}\cdot||D^{j}V||_{\frac{2k}{j}}$
$\displaystyle\lesssim$
$\displaystyle\sum\limits_{r+q+j=k}\Big{(}||D^{k}V||^{\frac{r}{k}}_{2}\cdot||V||^{\frac{q+j}{k}}_{\infty}\Big{)}\cdot\Big{(}||D^{k}V||^{\frac{q}{k}}_{2}\cdot||V||^{\frac{r+j}{k}}_{\infty}\Big{)}\cdot\Big{(}||D^{k}V||^{\frac{j}{k}}_{2}\cdot||V||^{\frac{q+r}{k}}_{\infty}\Big{)}$
$\displaystyle\lesssim$
$\displaystyle||D^{k}V||_{2}\cdot||V||^{2}_{\infty}\leqslant||D^{k}V||_{2}\cdot||V_{0}||^{2}_{\infty}.$

Substituting (4), (4), (4.4) and (4) into (4) leads to

$\displaystyle\frac{1}{2}\frac{d}{dt}||D^{k}V||_{2}^{2}+(1-B_{k}\cdot||V_{0}||_{\infty})\cdot||D^{k+1}V||_{2}^{2}$
$\displaystyle\leqslant$ $\displaystyle
G_{k}\cdot\\{||V||^{2}_{H^{k}}+||D^{k}V||_{2}\cdot||V||_{H^{k}}\cdot||V_{0}||_{\infty}+||D^{k}V||_{2}^{2}\cdot||V_{0}||^{2}_{\infty}\\}$
$\displaystyle\leqslant$ $\displaystyle
G_{k}\cdot(1+||V_{0}||_{\infty}+||V_{0}||^{2}_{\infty})\cdot||V||^{2}_{H^{k}},$

where $G_{k}$ depends upon $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant differentiations. In the sequel, using Gronwall inequality gives the
following theorem.

######  Theorem 4.1.

Given $N\in\mathbb{N}$, there exists an $\tilde{B}_{N}>0$ such that if
$||V_{0}||_{\infty}\leqslant\tilde{B}_{N}$, we will obtain

(4.7)
$\displaystyle||D^{k}V(t)||_{2}^{2}+\int^{t}_{0}||D^{k+1}V(s)||_{2}^{2}\,ds\leqslant
C_{k}(||V_{0}||_{H^{k}},\tilde{B}_{N},t),$

provided $0\leqslant k\leqslant N$ and $t\in[0,T_{\max})$. Here $C_{k}(x,y,t)$
is monotone increasing with respect to $x$ and $t$.

Proof. Employ induction for $N$.

In the case $N=0$, let $\tilde{B}_{0}:=1$. Taking inner product with $V$ in
(1.6) and then integrating the result over $M$, we get

$\frac{1}{2}\frac{d}{dt}||V(t)||_{2}^{2}+||DV(t)||_{2}^{2}+\lambda\int_{M}(1+\mu|V(t)|^{2})\cdot|V(t)|^{2}\,dM=0$

which is equivalent to

(4.8) $\displaystyle||V(t)||_{2}^{2}+2\int_{0}^{t}||DV(s)||_{2}^{2}\,ds$
$\displaystyle+2\lambda\int_{0}^{t}ds\int_{M}(1+\mu|V(s)|^{2})\cdot|V(s)|^{2}\,dM=||V_{0}||_{2}^{2}.$

Assume that for all the indices not larger than $N$, (4.7) holds. Now we
consider $N+1$.

Take $\tilde{B}_{N+1}:=\min\\{\tilde{B}_{N},\frac{1}{2B_{N+1}}\\}$. If
$||V_{0}||_{\infty}\leqslant\tilde{B}_{N+1}$, (4) gives

$\displaystyle\frac{1}{2}\frac{d}{dt}||D^{N+1}V||_{2}^{2}+\frac{1}{2}||D^{N+2}V||_{2}^{2}$
$\displaystyle\leqslant$ $\displaystyle
G_{N+1}\cdot(1+\tilde{B}_{N+1}+\tilde{B}^{2}_{N+1})\cdot\Big{\\{}||D^{N+1}V||_{2}^{2}+\sum\limits_{k=0}^{N}C_{k}(||V_{0}||_{H^{k}},\tilde{B}_{N},t)\Big{\\}}.$

Then this theorem follows easily from Gronwall inequality. This completes the
proof.$\hfill\Box$

Now we return to prove Theorem 1.2. Given $T>0$ and $N\geqslant
m_{0}+1=[\frac{m}{2}]+4$, we take any $V_{0}\in H^{N}(E)$ with
$||V_{0}||_{\infty}\leqslant\frac{1}{2}\tilde{B}_{N}:=\hat{B}_{N}$(This
$\tilde{B}_{N}$ is from Theorem 4.1). Then there are
$V_{0i}\in\Gamma^{\infty}(E)$ converging to $V_{0}$ strongly in $H^{N}(E)$.

Suppose $V_{i}$ satisfies

(4.10) $\displaystyle\left\\{\begin{aligned} &\partial_{t}V_{i}=\Delta
V_{i}+V_{i}\times\Delta V_{i}-\lambda\cdot(1+\mu\cdot|V_{i}|^{2})V_{i}\\\
&V_{i}(0,\cdot)=V_{0i}\end{aligned}\right.$

and its maximal existence time is $T^{*}_{i}$. As $i$ is large enough, we have

$||V_{0i}||_{\infty}\leqslant
2||V_{0}||_{\infty}\leqslant\tilde{B}_{N},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,||V_{0i}||_{H^{m_{0}}}\leqslant
2||V_{0}||_{H^{m_{0}}}$

and $||V_{0i}||_{H^{N}}\leqslant 2||V_{0}||_{H^{N}}$.

If $T^{*}_{i}<T$, then by Theorem 4.1,

$\displaystyle||V_{i}(t)||_{H^{m_{0}}}^{2}+\int^{t}_{0}||DV_{i}(s)||_{H^{m_{0}}}^{2}\,ds\leqslant
C_{m_{0}}(||V_{0i}||_{H^{m_{0}}},\tilde{B}_{N},t)\leqslant
C_{m_{0}}(2||V_{0}||_{H^{m_{0}}},\tilde{B}_{N},T),$

provided $t\in[0,T^{*}_{i})$. Review that in the proof of Theorem 1.1 we have
defined a monotone decreasing function

$h(x):=\frac{1}{C^{\prime}_{m_{0}}\cdot(1+x)}\log\Big{(}\frac{1+2x}{2x}\Big{)}.$

So for arbitrary $t\in[0,T^{*}_{i})$, it is obvious to see

$h(||V_{i}(t)||^{2}_{H^{m_{0}}})\geqslant
h\big{(}C_{m_{0}}(2||V_{0}||_{H^{m_{0}}},\tilde{B}_{N},T)\big{)}:=\delta_{0}>0.$

Now we bring in a new system

(4.11) $\displaystyle\left\\{\begin{aligned}
&\partial_{t}\hat{V}_{i}=\Delta\hat{V}_{i}+\hat{V}_{i}\times\Delta\hat{V}_{i}-\lambda\cdot(1+\mu\cdot|\hat{V}_{i}|^{2})\hat{V}_{i}\,\,\,\,\,\,\,\,\mbox{in}\,\,\,\,\,\,\,\,\Big{(}T^{*}_{i}-\frac{\delta_{0}}{2},\infty\Big{)}\times
M\\\
&\hat{V}_{i}\Big{(}T^{*}_{i}-\frac{\delta_{0}}{2},\cdot\Big{)}=V_{i}\Big{(}T^{*}_{i}-\frac{\delta_{0}}{2},\cdot\Big{)}\end{aligned}\right.$

The maximal existence time of $\hat{V}_{i}$ is not smaller than
$h(||V_{i}(T^{*}_{i}-\frac{\delta_{0}}{2})||^{2}_{H^{m_{0}}})$ which is not
smaller than $\delta_{0}$. By the uniqueness we know that for any
$t\in[T^{*}_{i}-\frac{\delta_{0}}{2},T^{*}_{i})$, $\hat{V}_{i}(t)=V_{i}(t)$.
It means that $V_{i}$ can be extended to $[0,T^{*}_{i}+\frac{\delta_{0}}{2})$.
Because $T^{*}_{i}$ is maximal, we get a contradiction. So $V_{i}\in
C^{\infty}([0,T],\Gamma^{\infty}(E))$ and for all $t\in[0,T]$,

$||V_{i}(t)||_{H^{N}}^{2}+\int^{t}_{0}||DV_{i}(s)||_{H^{N}}^{2}\,ds\leqslant
C_{N}(2||V_{0}||_{H^{N}},\tilde{B}_{N},T).$

By the same method we prove local well-posedness one can know there is a

$V\in L^{\infty}([0,T],H^{N}(E))\cap L^{2}([0,T],H^{N+1}(E))$

such that $V_{i}$ converges to $V$ strongly in
$L^{\infty}([0,T],H^{N-1}(E))$(in the sense of picking subsequence). It means
$V$ is a solution of LLB.

At last, we claim (1.8) and (1.9) are true. Since (1.8) is easy, we only prove
(1.9).
Proof. Employ induction for $i$.

When $i=0$, (1.9) holds.

Suppose that for all the indices not bigger than $i$, (1.9) is true.

Now we consider $i+1$. Choose any
$\beta\in[0,N+1-(\hat{m}+1)(i+1)]\cap\mathbb{Z}$. Applying
$\partial_{t}^{i}D^{\beta}$ to both sides of (1.6), we get

$\displaystyle\partial_{t}^{i+1}D^{\beta}V=\partial_{t}^{i}D^{\beta}\Delta
V-\partial_{t}^{i}D^{\beta}(V\times\Delta
V)-\lambda\cdot\partial_{t}^{i}D^{\beta}\big{[}(1+\mu|V|^{2})V\big{]},$

which implies

$\displaystyle\int_{0}^{T}||\partial_{t}^{i+1}D^{\beta}V(s)||_{2}^{2}\,ds$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{T}||\partial_{t}^{i}D^{\beta+2}V(s)||_{2}^{2}\,ds+\int_{0}^{T}||\partial_{t}^{i}D^{\beta}V(s)||_{2}^{2}\,ds$
$\displaystyle+\sum\limits_{i^{\prime},\beta^{\prime}}\int_{0}^{T}\Big{|}\Big{|}|\partial_{t}^{i^{\prime}}D^{\beta^{\prime}}V(s)|\cdot|\partial_{t}^{i-i^{\prime}}D^{\beta-\beta^{\prime}+2}V(s)|\Big{|}\Big{|}^{2}_{2}\,ds$
$\displaystyle+\sum\limits_{\begin{subarray}{c}i_{1}+i_{2}+i_{3}=i\\\
\beta_{1}+\beta_{2}+\beta_{3}=\beta\end{subarray}}\int_{0}^{T}\Big{|}\Big{|}|\partial_{t}^{i_{1}}D^{\beta_{1}}V(s)|\cdot|\partial_{t}^{i_{2}}D^{\beta_{2}}V(s)|\cdot|\partial_{t}^{i_{3}}D^{\beta_{3}}V(s)|\Big{|}\Big{|}^{2}_{2}\,ds.$

Because $\hat{m}:=\max\\{2,[\frac{m}{2}]+1\\}$,

$i^{\prime}\leqslant
i\leqslant\Big{[}\frac{N+1}{\hat{m}+1}\Big{]}\leqslant\Big{[}\frac{N}{\hat{m}}\Big{]}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\beta^{\prime}\leqslant\beta\leqslant
N+1-(\hat{m}+1)(i+1)\leqslant N-\hat{m}\cdot i^{\prime}-\hat{m},$

(1.8) yields

$||\partial_{t}^{i^{\prime}}D^{\beta^{\prime}}V(t)||_{\infty}\leqslant||\partial_{t}^{i^{\prime}}D^{\beta^{\prime}}V(t)||_{H^{\hat{m}}}<\infty\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
t\in[0,T].$

And since

$\hat{m}(i-i^{\prime})+(\beta-\beta^{\prime}+2)\leqslant\hat{m}\cdot
i+\beta+2\leqslant\hat{m}\cdot i+N+1-(\hat{m}+1)(i+1)+2=N-i-\hat{m}+2\leqslant
N-i\leqslant N,$

by (1.8) we have

$\int_{0}^{T}||\partial_{t}^{i-i^{\prime}}D^{\beta-\beta^{\prime}+2}V(s)||_{2}^{2}\,ds\leqslant\sup\limits_{t\in[0,T]}\\{||\partial_{t}^{i-i^{\prime}}D^{\beta-\beta^{\prime}+2}V(t)||^{2}_{2}\\}\cdot
T<\infty.$

So

$\displaystyle\int_{0}^{T}\Big{|}\Big{|}|\partial_{t}^{i^{\prime}}D^{\beta^{\prime}}V(s)|\cdot|\partial_{t}^{i-i^{\prime}}D^{\beta-\beta^{\prime}+2}V(s)|\Big{|}\Big{|}^{2}_{2}\,ds$
$\displaystyle\leqslant$
$\displaystyle\sup\limits_{t\in[0,T]}\\{||\partial_{t}^{i^{\prime}}D^{\beta^{\prime}}V(t)||^{2}_{\infty}\\}\cdot\int_{0}^{T}||\partial_{t}^{i-i^{\prime}}D^{\beta-\beta^{\prime}+2}V(s)||_{2}^{2}\,ds<\infty$

For the other terms on the right hand side of (4), using similar method we
know all of them are strictly smaller than $\infty$.

This completes the proof.$\hfill\Box$

##  5\. Proof of Theorem 1.3

In this section, we need some formulas. Their proofs are tedious. So we only
list the results.

Formula 3. Suppose that $V\in\Gamma^{2}(E)$. Then we will obtain

$\displaystyle||\Delta V||_{2}^{2}$ $\displaystyle=$
$\displaystyle||D^{2}V||_{2}^{2}+2\int_{M}\langle
DV,DV\ast\mathcal{R}^{E}\rangle\,dM$ $\displaystyle+\int_{M}\langle DV,V\ast
D\mathcal{R}^{E}\rangle\,dM+\int_{M}\langle
DV,DV\ast\mathcal{R}^{M}\rangle\,dM.$

######  Remark 5.1.

Formula 3 easily implies

(5.1) $\displaystyle||D^{2}V||_{2}^{2}\leqslant||\Delta
V||_{2}^{2}+\eta\cdot(||DV||_{2}^{2}+||V||_{2}^{2}),$

where $\eta$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives.

Formula 4. Given $V\in\Gamma^{3}(E)$,

$\displaystyle||\Delta DV||_{2}^{2}$ $\displaystyle=$
$\displaystyle||D^{3}V||_{2}^{2}+3\int_{M}\langle
D^{2}V,D^{2}V\ast\mathcal{R}^{M}\rangle\,dM+2\int_{M}\langle
D^{2}V,D^{2}V\ast\mathcal{R}^{E}\rangle\,dM$ $\displaystyle+\int_{M}\langle
D^{2}V,DV\ast\nabla\mathcal{R}^{M}\rangle\,dM+\int_{M}\langle D^{2}V,DV\ast
D\mathcal{R}^{E}\rangle\,dM.$

######  Remark 5.2.

From Formula 4 it follows that

(5.2) $\displaystyle||D^{3}V||_{2}^{2}\leqslant||\Delta
DV||_{2}^{2}+\eta_{2}\cdot(||D^{2}V||_{2}^{2}+||DV||_{2}^{2}),$

where $\eta_{2}$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives. Since by (2.9) we have

$\displaystyle\Delta DV=D\Delta
V+\mathfrak{q}_{1}(V,\mathcal{R}^{E})+\mathfrak{q}_{0}(DV,\mathcal{R}^{M}),$

integration by parts and Hölder’s inequality yield

(5.3) $\displaystyle||\Delta DV||_{2}^{2}\leqslant||D\Delta
V||_{2}^{2}+\eta_{3}\cdot(||D^{2}V||_{2}^{2}+||DV||_{2}^{2}+||V||_{2}^{2}),$

where $\eta_{3}$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives. Substituting (5.3) into (5.2) gives

(5.4) $\displaystyle||D^{3}V||_{2}^{2}\leqslant||D\Delta
V||_{2}^{2}+(\eta_{2}+\eta_{3})\cdot(||D^{2}V||_{2}^{2}+||DV||_{2}^{2}+||V||_{2}^{2}).$

Formula 5. If $V\in\Gamma^{4}(E)$, then

$\displaystyle||D^{4}V||_{2}^{2}$ $\displaystyle=$
$\displaystyle||\Delta^{2}V||_{2}^{2}+\int_{M}\langle\mathfrak{q}_{3}(V,\mathcal{R}^{E}),D^{3}V\rangle\,dM+\int_{M}\langle\mathfrak{q}_{2}(DV,\mathcal{R}^{M}),D^{3}V\rangle\,dM$
$\displaystyle+\int_{M}\langle\mathfrak{q}_{1}(DV,\mathcal{R}^{E}),\mathfrak{q}_{1}(DV,\mathcal{R}^{E})\rangle\,dM+\int_{M}\langle\mathfrak{q}_{1}(DV,\mathcal{R}^{E}),\mathfrak{q}_{1}(DV,\mathcal{R}^{M})\rangle\,dM$
$\displaystyle+\int_{M}\langle\mathfrak{q}_{1}(DV,\mathcal{R}^{M}),\mathfrak{q}_{1}(DV,\mathcal{R}^{M})\rangle\,dM.$

######  Remark 5.3.

Formula 5, Hölder’s inequality and Young’s inequality lead to

(5.5) $\displaystyle||D^{4}V||_{2}^{2}$ $\displaystyle\leqslant$
$\displaystyle||\Delta^{2}V||_{2}^{2}+\eta_{5}\cdot(||D^{3}V||_{2}^{2}+||D^{2}V||_{2}^{2}+||DV||_{2}^{2}+||V||_{2}^{2})$
$\displaystyle\leqslant$
$\displaystyle||\Delta^{2}V||_{2}^{2}+\eta_{6}\cdot(||D\Delta
V||_{2}^{2}+||\Delta V||_{2}^{2}+||DV||_{2}^{2}+||V||_{2}^{2}),$

where we have used (5.1), (5.4) and $\eta_{5}$, $\eta_{6}$ depend on
$\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their covariant derivatives.

Formula 6. Assume that $V\in\Gamma^{4}(E)$. Then we get

(5.6) $\displaystyle||D^{2}\Delta V||_{2}^{2}$ $\displaystyle=$
$\displaystyle||\Delta^{2}V||_{2}^{2}+\int_{M}\langle\mathfrak{q}_{1}(\Delta
V,\mathcal{R}^{E}),D\Delta
V\rangle\,dM+\int_{M}\langle\mathfrak{q}_{0}(D\Delta
V,\mathcal{R}^{M}),D\Delta V\rangle\,dM$ $\displaystyle\lesssim$
$\displaystyle||\Delta^{2}V||_{2}^{2}+\eta_{8}\cdot(||D\Delta
V||_{2}^{2}+||\Delta V||_{2}^{2}),$

where we have applied Hölder’s inequality, Young’s inequality and $\eta_{8}$
depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their covariant
derivatives.

Now let us go on to prove Theorem 1.3. Suppose that $V\in
C^{\infty}\big{(}[0,T^{*}),\Gamma^{\infty}(E)\big{)}$ is the unique local
smooth solution of (1.6), where $T^{*}$ is its maximal existence time. First
of all, we shall estimate $||DV(t)||_{\infty}$ for all $t\in[0,T^{*})$. By
Sobolev embedding it is easy to see that we only need to get a uniform upper
bound of $||DV(t)||_{H^{2}}$(Note that in this section $m=2$). Combining (5.1)
and (5.4) one can know that we only need to estimate

$||DV(t)||_{2}^{2}+||\Delta V(t)||_{2}^{2}+||D\Delta V(t)||_{2}^{2}.$

Using the same method of (2.2) in [4] we can get

(5.7) $\displaystyle||DV(t)||_{2}^{2}+\int_{0}^{t}||\Delta
V(s)||_{2}^{2}\,ds\leqslant\lambda^{2}\cdot(1+\mu||V_{0}||^{2}_{\infty})^{2}\cdot||V_{0}||_{2}^{2}\cdot
t+||DV_{0}||_{2}^{2}.$

For $||\Delta V(t)||_{2}^{2}$, our trick is to deduce a Gronwall’s inequality.
(1.6) yields

$\displaystyle\frac{1}{2}\frac{d}{dt}||\Delta V(t)||_{2}^{2}+\int_{M}|D\Delta
V(t)|^{2}\,dM+\lambda\cdot||\Delta V(t)||_{2}^{2}$ $\displaystyle=$
$\displaystyle-\int_{M}[DV(t)\times\Delta V(t)]\ast D\Delta
V(t)\,dM-\lambda\mu\cdot\int_{M}\langle\Delta[|V(t)|^{2}\cdot V(t)],\Delta
V(t)\rangle\,dM$ $\displaystyle\leqslant$
$\displaystyle\int_{M}|DV(t)|\cdot|\Delta V(t)|\cdot|D\Delta
V(t)|\,dM+C\cdot||V(t)||^{2}_{\infty}\cdot(||\Delta
V(t)||^{2}_{2}+||DV(t)||_{4}^{2})$ $\displaystyle\leqslant$
$\displaystyle||DV(t)||_{4}\cdot||\Delta V(t)||_{4}\cdot||D\Delta
V(t)||_{2}+C\cdot||V(t)||^{2}_{\infty}\cdot(||\Delta
V(t)||^{2}_{2}+||DV(t)||_{4}^{2}),$

here $C$ is a universal constant. Theorem 2.1 of [3], (5.1) and (5.4) give

(5.9) $\displaystyle||DV(t)||_{4}$ $\displaystyle\lesssim$
$\displaystyle||DV(t)||^{\frac{1}{4}}_{H^{2}}\cdot||DV(t)||_{2}^{\frac{3}{4}}$
$\displaystyle\leqslant$ $\displaystyle\eta_{4}\cdot(||D\Delta
V(t)||_{2}^{\frac{1}{4}}+||\Delta
V(t)||_{2}^{\frac{1}{4}}+||DV(t)||_{2}^{\frac{1}{4}}+||V(t)||_{2}^{\frac{1}{4}})\cdot||DV(t)||_{2}^{\frac{3}{4}}$

and

(5.10) $\displaystyle||\Delta V(t)||_{4}$ $\displaystyle\lesssim$
$\displaystyle||\Delta V(t)||^{\frac{1}{2}}_{H^{1}}\cdot||\Delta
V(t)||_{2}^{\frac{1}{2}}$ $\displaystyle\lesssim$ $\displaystyle(||D\Delta
V(t)||_{2}^{\frac{1}{2}}+||\Delta V(t)||_{2}^{\frac{1}{2}})\cdot||\Delta
V(t)||_{2}^{\frac{1}{2}},$

where $\eta_{4}$ depends upon $\eta_{2}$, $\eta_{3}$ and $\eta$. Thus we
derive

$\displaystyle||DV(t)||_{4}\cdot||\Delta V(t)||_{4}\cdot||D\Delta V(t)||_{2}$
$\displaystyle\lesssim$
$\displaystyle\eta_{4}\cdot||DV(t)||_{2}^{\frac{3}{4}}\cdot||\Delta
V(t)||_{2}^{\frac{1}{2}}\cdot||D\Delta V(t)||_{2}^{\frac{7}{4}}$
$\displaystyle+\eta_{4}\cdot||DV(t)||_{2}^{\frac{3}{4}}\cdot||\Delta
V(t)||_{2}^{\frac{3}{4}}\cdot||D\Delta V(t)||_{2}^{\frac{3}{2}}$
$\displaystyle+\eta_{4}\cdot||DV(t)||_{2}\cdot||\Delta
V(t)||_{2}^{\frac{1}{2}}\cdot||D\Delta V(t)||_{2}^{\frac{3}{2}}$
$\displaystyle+\eta_{4}\cdot||DV(t)||_{2}^{\frac{3}{4}}\cdot||\Delta
V(t)||_{2}\cdot||D\Delta V(t)||_{2}^{\frac{5}{4}}$
$\displaystyle+\eta_{4}\cdot||DV(t)||_{2}^{\frac{3}{4}}\cdot||\Delta
V(t)||_{2}^{\frac{5}{4}}\cdot||D\Delta V(t)||_{2}$
$\displaystyle+\eta_{4}\cdot||DV(t)||_{2}\cdot||\Delta
V(t)||_{2}\cdot||D\Delta V(t)||_{2}$
$\displaystyle+\eta_{4}\cdot||V(t)||_{2}^{\frac{1}{4}}\cdot||DV(t)||^{\frac{3}{4}}_{2}\cdot||\Delta
V(t)||_{2}^{\frac{1}{2}}\cdot||D\Delta V(t)||_{2}^{\frac{3}{2}}$
$\displaystyle+\eta_{4}\cdot||V(t)||_{2}^{\frac{1}{4}}\cdot||DV(t)||^{\frac{3}{4}}_{2}\cdot||\Delta
V(t)||_{2}\cdot||D\Delta V(t)||_{2}$ $\displaystyle\leqslant$
$\displaystyle\eta_{4}\gamma_{1}\cdot(||\Delta
V(t)||_{2}^{\frac{1}{2}}\cdot||D\Delta V(t)||_{2}^{\frac{7}{4}}+||\Delta
V(t)||_{2}^{\frac{3}{4}}\cdot||D\Delta V(t)||_{2}^{\frac{3}{2}}$
$\displaystyle+||\Delta V(t)||_{2}^{\frac{1}{2}}\cdot||D\Delta
V(t)||_{2}^{\frac{3}{2}}+||\Delta V(t)||_{2}\cdot||D\Delta
V(t)||_{2}^{\frac{5}{4}}$ $\displaystyle+||\Delta
V(t)||_{2}^{\frac{5}{4}}\cdot||D\Delta V(t)||_{2}+||\Delta
V(t)||_{2}\cdot||D\Delta V(t)||_{2}),$

where $\gamma_{1}$ relies on $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$
and $t$. From Young’s inequality it follows that

$\displaystyle||DV(t)||_{4}\cdot||\Delta V(t)||_{4}\cdot||D\Delta V(t)||_{2}$
$\displaystyle\leqslant$ $\displaystyle\gamma_{2}\cdot(||\Delta
V(t)||_{2}^{4}+1)+\frac{1}{4}\cdot||D\Delta V(t)||_{2}^{2},$

where $\gamma_{2}$ is dependent of $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant derivatives, $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$
and $t$. By the same way, we have

$\displaystyle||V(t)||^{2}_{\infty}\cdot(||\Delta
V(t)||_{2}^{2}+||DV(t)||_{4}^{2})$ $\displaystyle\lesssim$
$\displaystyle||V(t)||^{2}_{\infty}\cdot||\Delta V(t)||_{2}^{2}$
$\displaystyle+\eta_{4}^{2}\cdot||V_{0}||^{2}_{\infty}\cdot(||D\Delta
V(t)||_{2}^{\frac{1}{2}}+||\Delta
V(t)||_{2}^{\frac{1}{2}}+||DV(t)||_{2}^{\frac{1}{2}}+||V(t)||_{2}^{\frac{1}{2}})\cdot||DV(t)||_{2}^{\frac{3}{2}}$
$\displaystyle\leqslant$ $\displaystyle\gamma_{3}\cdot(||\Delta
V(t)||_{2}^{2}+1)+\frac{1}{4}||D\Delta V(t)||_{2}^{2},$

where $\gamma_{3}$ is dependent of $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant derivatives, $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$
and $t$. Substituting (5) and (5) into (5) and Young’s inequality lead to

$\displaystyle\frac{1}{2}\frac{d}{dt}||\Delta
V(t)||_{2}^{2}+\frac{1}{2}||D\Delta V(t)||_{2}^{2}+\lambda||\Delta
V(t)||_{2}^{2}\leqslant\gamma_{4}\cdot(||\Delta V(t)||_{2}^{4}+1),$

here $\gamma_{4}$ relies on $\gamma_{2}$ and $\gamma_{3}$. The generalized
Gronwall’s inequality says that if

$\frac{df}{dt}\leqslant C\cdot f\cdot g+C,$

then

$f\leqslant C\cdot\exp\Big{(}\int_{0}^{t}g(s)\,ds\Big{)}+C.$

So if we replace $f$ and $g$ by $||\Delta V(t)||_{2}^{2}$ and note that (5.7)
implies the boundedness of $\int_{0}^{t}g(s)\,ds$, then

(5.13) $\displaystyle||\Delta V(t)||_{2}^{2}\leqslant\gamma_{5}$

which implies

(5.14) $\displaystyle\int_{0}^{t}||D\Delta V(s)||_{2}^{2}\,ds\leqslant
2\gamma_{4}\cdot(\gamma_{5}^{2}+1)\cdot t,$

where $\gamma_{5}$ is dependent of $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant derivatives,
$||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$, $||\Delta V_{0}||_{2}$ and
$t$.

In the sequel, we are going to estimate $||D\Delta V(t)||_{2}$ for all
$t\in[0,T^{*})$. (1.6) gives

$\displaystyle\frac{1}{2}\frac{d}{dt}||D\Delta V(t)||_{2}^{2}$
$\displaystyle=$
$\displaystyle-||\Delta^{2}V(t)||_{2}^{2}-\int_{M}\langle\Delta^{2}V(t),\Delta[V(t)\times\Delta
V(t)]\rangle\,dM$
$\displaystyle+\lambda\int_{M}\langle\Delta[(1+\mu|V(t)|^{2})\cdot
V(t)],\Delta^{2}V(t)\rangle\,dM$ $\displaystyle=$
$\displaystyle-||\Delta^{2}V(t)||_{2}^{2}-2\int_{M}\Delta^{2}V(t)\ast[DV(t)\times
D\Delta V(t)]\,dM$
$\displaystyle+\lambda\int_{M}\langle\Delta[(1+\mu|V(t)|^{2})\cdot
V(t)],\Delta^{2}V(t)\rangle\,dM$ $\displaystyle=$
$\displaystyle-||\Delta^{2}V(t)||_{2}^{2}-2\int_{M}\Delta^{2}V(t)\ast[DV(t)\times
D\Delta V(t)]\,dM$
$\displaystyle+\lambda\mu\int_{M}\langle\Delta\big{[}|V(t)|^{2}\cdot
V(t)\big{]},\Delta^{2}V(t)\rangle\,dM-\lambda||D\Delta V(t)||_{2}^{2}.$

On the other hand, Hölder inequality yields

$\displaystyle\Big{|}\int_{M}\Delta^{2}V(t)\ast[DV(t)\times D\Delta
V(t)]\,dM\Big{|}$ $\displaystyle\leqslant$
$\displaystyle||DV(t)||_{\frac{16}{5}}\cdot||D\Delta
V(t)||_{\frac{16}{3}}\cdot||\Delta^{2}V(t)||_{2}$

and

$\displaystyle\Big{|}\int_{M}\langle\Delta\big{[}|V(t)|^{2}\cdot
V(t)\big{]},\Delta^{2}V(t)\rangle\,dM\Big{|}$ $\displaystyle\lesssim$
$\displaystyle||V(t)||^{2}_{\infty}\cdot||\Delta^{2}V(t)||_{2}\cdot||\Delta
V(t)||_{2}+||V(t)||_{\infty}\cdot||\Delta^{2}V(t)||_{2}\cdot||DV(t)||_{4}^{2}.$

By Sobolev Embedding, we have

(5.16)
$\displaystyle||DV(t)||_{\frac{16}{5}}\lesssim||DV(t)||^{\frac{1}{8}}_{H^{3}}\cdot||DV(t)||^{\frac{7}{8}}_{2}.$

Combining (5.1), (5.4) and (5.5) we arrive at

$\displaystyle||DV(t)||^{\frac{1}{8}}_{H^{3}}$ $\displaystyle\leqslant$
$\displaystyle\eta_{7}\cdot(||\Delta^{2}V(t)||_{2}^{\frac{1}{8}}+||D\Delta
V(t)||_{2}^{\frac{1}{8}}$ $\displaystyle+||\Delta
V(t)||_{2}^{\frac{1}{8}}+||DV(t)||_{2}^{\frac{1}{8}}+||V(t)||_{2}^{\frac{1}{8}}),$

where $\eta_{7}$ is dependent of $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant derivatives. Moreover,

$\displaystyle||D\Delta V(t)||_{\frac{16}{3}}\lesssim||D\Delta
V(t)||^{\frac{5}{8}}_{H^{1}}\cdot||D\Delta V(t)||_{2}^{\frac{3}{8}}$

and

$\displaystyle||D\Delta V(t)||_{H^{1}}\lesssim||D^{2}\Delta
V(t)||_{2}+||D\Delta V(t)||_{2}.$

By (5.6) we are led to

$\displaystyle||D\Delta
V(t)||_{H^{1}}\lesssim||\Delta^{2}V(t)||_{2}+(\sqrt{\eta_{8}}+1)\cdot||D\Delta
V(t)||_{2}+\sqrt{\eta_{8}}\cdot||\Delta V(t)||_{2},$

which implies

$\displaystyle||D\Delta V(t)||_{\frac{16}{3}}$ $\displaystyle\lesssim$
$\displaystyle\big{[}||\Delta^{2}V(t)||^{\frac{5}{8}}_{2}+(\eta_{8}^{\frac{5}{16}}+1)\cdot||D\Delta
V(t)||_{2}^{\frac{5}{8}}$ $\displaystyle+\eta_{8}^{\frac{5}{16}}\cdot||\Delta
V(t)||_{2}^{\frac{5}{8}}\big{]}\cdot||D\Delta V(t)||_{2}^{\frac{3}{8}}.$

Furthermore, substituting (5.16), (5) and (5) into (5) we arrive at

$\displaystyle\Big{|}\int_{M}\Delta^{2}V(t)\ast[DV(t)\times D\Delta
V(t)]\,dM\Big{|}$ $\displaystyle\lesssim$
$\displaystyle\eta_{7}\cdot(||\Delta^{2}V(t)||_{2}^{\frac{1}{8}}+||D\Delta
V(t)||_{2}^{\frac{1}{8}}+||\Delta
V(t)||_{2}^{\frac{1}{8}}+||DV(t)||_{2}^{\frac{1}{8}}+||V(t)||_{2}^{\frac{1}{8}})$
$\displaystyle\cdot\big{[}||\Delta^{2}V(t)||^{\frac{5}{8}}_{2}+(\eta_{8}^{\frac{5}{16}}+1)\cdot||D\Delta
V(t)||_{2}^{\frac{5}{8}}+\eta_{8}^{\frac{5}{16}}\cdot||\Delta
V(t)||_{2}^{\frac{5}{8}}\big{]}$ $\displaystyle\cdot||D\Delta
V(t)||_{2}^{\frac{3}{8}}\cdot||\Delta^{2}V(t)||_{2}\cdot||DV(t)||_{2}^{\frac{7}{8}}.$

Substituting the upper bounds of $||V(t)||_{2}$, $||DV(t)||_{2}$ and $||\Delta
V(t)||_{2}$ into (5) leads to

$\displaystyle\Big{|}\int_{M}\Delta^{2}V(t)\ast[DV(t)\times D\Delta
V(t)]\,dM\Big{|}$ $\displaystyle\leqslant$
$\displaystyle\eta_{9}\cdot(||\Delta^{2}V(t)||_{2}^{\frac{1}{8}}+||D\Delta
V(t)||_{2}^{\frac{1}{8}}+1)$
$\displaystyle\cdot(||\Delta^{2}V(t)||_{2}^{\frac{5}{8}}+||D\Delta
V(t)||_{2}^{\frac{5}{8}}+1)\cdot||D\Delta
V(t)||_{2}^{\frac{3}{8}}\cdot||\Delta^{2}V(t)||_{2}$ $\displaystyle\lesssim$
$\displaystyle\eta_{9}\cdot(||\Delta^{2}V(t)||_{2}^{\frac{3}{4}}+||D\Delta
V(t)||_{2}^{\frac{3}{4}}+1)\cdot||D\Delta
V(t)||_{2}^{\frac{3}{8}}\cdot||\Delta^{2}V(t)||_{2}$ $\displaystyle\leqslant$
$\displaystyle\eta_{9}\cdot||\Delta^{2}V(t)||_{2}^{\frac{7}{4}}\cdot||D\Delta
V(t)||_{2}^{\frac{3}{8}}+\eta_{9}\cdot||\Delta^{2}V(t)||_{2}\cdot||D\Delta
V(t)||_{2}^{\frac{9}{8}}$
$\displaystyle+\eta_{9}\cdot||\Delta^{2}V(t)||_{2}\cdot||D\Delta
V(t)||_{2}^{\frac{3}{8}}$ $\displaystyle\lesssim$
$\displaystyle\eta_{9}\cdot\Big{(}\varepsilon\cdot||\Delta^{2}V(t)||_{2}^{2}+\frac{1}{\varepsilon}\cdot||D\Delta
V(t)||_{2}^{3}\Big{)}+\eta_{9}\cdot\Big{(}\varepsilon\cdot||\Delta^{2}V(t)||_{2}^{2}+\frac{1}{\varepsilon}\cdot||D\Delta
V(t)||_{2}^{\frac{9}{4}}\Big{)}$
$\displaystyle+\eta_{9}\cdot\Big{(}\varepsilon\cdot||\Delta^{2}V(t)||_{2}^{2}+\frac{1}{\varepsilon}\cdot||D\Delta
V(t)||_{2}^{\frac{3}{4}}\Big{)}$ $\displaystyle\lesssim$
$\displaystyle\eta_{9}\cdot\Big{(}\varepsilon\cdot||\Delta^{2}V(t)||_{2}^{2}+\frac{1}{\varepsilon}\cdot||D\Delta
V(t)||_{2}^{4}+\frac{1}{\varepsilon}\Big{)},$

where $\eta_{9}$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives, $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$,
$||\Delta V_{0}||_{2}$ and $t$.

Moreover, there is a universal constant $\kappa_{1}$ such that

$\displaystyle\Big{|}\int_{M}\langle\Delta\big{[}|V(t)|^{2}\cdot
V(t)\big{]},\Delta^{2}V(t)\rangle\,dM\Big{|}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{4}||\Delta^{2}V(t)||_{2}^{2}+\kappa_{1}\cdot||V(t)||^{4}_{\infty}\cdot||\Delta
V(t)||_{2}^{2}+\kappa_{1}\cdot||V(t)||^{2}_{\infty}\cdot||DV(t)||_{4}^{4}.$

Recalling (5.9) and (5.13) we obtain

(5.20) $\displaystyle\Big{|}\int_{M}\langle\Delta\big{[}|V(t)|^{2}\cdot
V(t)\big{]},\Delta^{2}V(t)\rangle\,dM\Big{|}\leqslant\frac{1}{4}||\Delta^{2}V(t)||_{2}^{2}+\kappa_{1}\cdot||V_{0}||^{4}_{\infty}\cdot\gamma_{5}$
$\displaystyle+\kappa_{1}\cdot||V_{0}||^{2}_{\infty}\cdot\eta_{4}\cdot(||D\Delta
V(t)||_{2}+||\Delta
V(t)||_{2}+||DV(t)||_{2}+||V(t)||_{2})\cdot||DV(t)||_{2}^{3}.$

Substituting the upper bounds of $||\Delta V(t)||_{2}$, $||DV(t)||_{2}$ and
$||V(t)||_{2}$ into (5.20) gives

$\displaystyle\Big{|}\int_{M}\langle\Delta\big{[}|V(t)|^{2}\cdot
V(t)\big{]},\Delta^{2}V(t)\rangle\,dM\Big{|}$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{4}||\Delta^{2}V(t)||_{2}^{2}+\kappa_{2}\cdot(||D\Delta
V(t)||_{2}+1)$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{4}||\Delta^{2}V(t)||_{2}^{2}+\frac{\kappa_{2}}{2}\cdot(||D\Delta
V(t)||^{2}_{2}+3),$

where $\kappa_{2}$ relies on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives, $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$,
$||\Delta V_{0}||_{2}$ and $t$.

In conclusion,

$\displaystyle\frac{1}{2}\frac{d}{dt}||D\Delta
V(t)||_{2}^{2}+||\Delta^{2}V(t)||_{2}^{2}+\lambda\cdot||D\Delta
V(t)||_{2}^{2}$ $\displaystyle\leqslant$
$\displaystyle\eta^{\prime}_{9}\cdot\Big{(}\varepsilon\cdot||\Delta^{2}V(t)||_{2}^{2}+\frac{1}{\varepsilon}\cdot||D\Delta
V(t)||_{2}^{4}+\frac{1}{\varepsilon}\Big{)}+\frac{1}{4}||\Delta^{2}V(t)||_{2}^{2}+\frac{\kappa_{2}}{2}\cdot(||D\Delta
V(t)||^{2}_{2}+3),$

where $\eta^{\prime}_{9}$ depends on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and
their covariant derivatives,
$||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$, $||\Delta V_{0}||_{2}$ and
$t$. Let $\varepsilon$ be small enough. From Young’s inequality it follows
that

$\displaystyle\frac{1}{2}\frac{d}{dt}||D\Delta
V(t)||_{2}^{2}+\frac{1}{2}||\Delta^{2}V(t)||_{2}^{2}+\lambda\cdot||D\Delta
V(t)||_{2}^{2}\leqslant\eta_{10}\cdot(1+||D\Delta V(t)||_{2}^{4}),$

where $\eta_{10}$ is dependent of $\eta_{9}$ and $\kappa_{2}$. Since of
(5.14), the generalized Gronwall’s inequality implies

$||D\Delta V(t)||_{2}^{2}\leqslant\gamma_{6},$

here $\gamma_{6}$ relies on $||D\Delta V_{0}||_{2}$, $\eta_{10}$,
$\gamma_{4}$, $\gamma_{5}$ and $t$. Substituting the upper bounds of $||\Delta
V(t)||^{2}_{2}$ and $||D\Delta V(t)||_{2}^{2}$ into (5.1) and (5.4) yields

$||DV(t)||_{H^{2}}\leqslant\gamma_{7}$

which implies

$||DV(t)||_{\infty}\lesssim\gamma_{7},$

where $\gamma_{7}$ relies on $\mathcal{R}^{M}$, $\mathcal{R}^{E}$ and their
covariant derivatives, $||V_{0}||_{\infty},||V_{0}||_{2},||DV_{0}||_{2}$,
$||\Delta V_{0}||_{2}$, $||D\Delta V_{0}||_{2}$ and $t$.

Now we return to prove Theorem 1.3. Reviewing (4), we know that the key is to
estimate

$\sum\limits_{i+j=k-1}a_{ij}\cdot||D^{k+1}V||_{2}\cdot\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}.$

Hölder inequality yields

$\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}\leqslant||D^{i+1}V||_{\frac{2k-2}{i}}\cdot||D^{j+1}V||_{\frac{2k-2}{k-1-i}}.$

From Theorem 2.4, it follows that

$||D^{i+1}V||_{\frac{2k-2}{i}}\lesssim||D^{k}V||_{2}^{\frac{i}{k-1}}\cdot||DV||_{\infty}^{\frac{k-1-i}{k-1}}$

and

$||D^{j+1}V||_{\frac{2k-2}{k-1-i}}\lesssim||D^{k}V||_{2}^{\frac{k-1-i}{k-1}}\cdot||DV||^{\frac{i}{k-1}}_{\infty}.$

So one can get

(5.21)
$\displaystyle\Big{|}\Big{|}|D^{i+1}V|\cdot|D^{j+1}V|\Big{|}\Big{|}_{2}\lesssim||D^{k}V||_{2}\cdot||DV||_{\infty}.$

Substituting (5.21), (4), (4.4) and (4) into (4) we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}||D^{k}V||_{2}^{2}+||D^{k+1}V||_{2}^{2}$
$\displaystyle\leqslant$
$\displaystyle\kappa_{3}\cdot||D^{k+1}V||_{2}\cdot||D^{k}V||_{2}\cdot||DV||_{\infty}+G_{k}\cdot(1+||V_{0}||_{\infty}+||V_{0}||^{2}_{\infty})\cdot||V||^{2}_{H^{k}}$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\cdot||D^{k+1}V||_{2}^{2}+\frac{\kappa_{3}^{2}}{2}\cdot||D^{k}V||_{2}^{2}\cdot||DV||^{2}_{\infty}+G_{k}\cdot(1+||V_{0}||_{\infty}+||V_{0}||^{2}_{\infty})\cdot||V||^{2}_{H^{k}},$

where $\kappa_{3}$ is a universal constant. Note the fact

$||DV(t)||_{\infty}\lesssim\gamma_{7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall\,\,\,\,t\in[0,T^{*}).$

Summing $k$ from 0 to $N$ and applying Gronwall’s inequality we are led to

$||V(t)||^{2}_{H^{N}}+\int_{0}^{t}||V(s)||^{2}_{H^{N+1}}\,ds\leqslant
C_{N}(||V_{0}||_{H^{N}},t,\gamma_{7},||V_{0}||_{\infty},\kappa_{3},G_{0},\cdots,G_{N}).$

The remaining part of the proof of Theorem 1.3 is as the same as that of
Theorem 1.2. So we omit it. This completes the proof.$\hfill\Box$

## References

  * [1] Charles Baker, The mean curvature flow of submanifolds of high codimension, Ph.D. thesis, Australian National University (2010); arXiv: 1104.4409.
  * [2] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential. Geometry, 17 (1982) 255-306.
  * [3] Weiyue Ding, Youde Wang; _Local Schrödinger flow into Kähler manifolds_ , Science in China, Series A, 2001, Vol. 44, No. 11.
  * [4] Boling Guo, Qiaoxin Li, Ming Zeng, Global smooth solutions of the Landau-Lifshitz-Bloch equation, preprint.
  * [5] Zonglin Jia, Local strong solution to General Landau-Lifshitz-Bloch equation, arXiv:1802.00144.
  * [6] Kim Ngan Le, Weak solutions of the Landau-Lifshitz Bloch equation, J. Differential Equations 261 (2016) 6699-6717

Boling Guo

Institute of Applied Physics and Computational Mathematics, China Academy of
Engineering Physics, Beijing, 100088, P. R. China

Email<EMAIL_ADDRESS>

Zonglin Jia

Institute of Applied Physics and Computational Mathematics, China Academy of
Engineering Physics, Beijing, 100088, P. R. China

Email<EMAIL_ADDRESS>