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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 132, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/132\hfil Regularity of weak solutions]
{Regularity of weak solutions to the magneto-hydrodynamics
equations in terms of the direction of velocity}

\author[Y. Luo\hfil EJDE-2009/132\hfilneg]
{Yuwen Luo}

\address{Yuwen Luo\newline
School of Mathematics \& Physics, Chongqing University of Technology,
Chongqing, 400050, China}
\email{petitevin@gmail.com}

\thanks{Submitted September 28, 2009. Published October 16, 2009.}
\subjclass[2000]{35Q35, 76D03}
\keywords{Magneto-hydrodynamics equation; regularity; Serrin criteria}

\begin{abstract}
 In this article, we study the regularity of weak solutions to
 the 3D incompressible magneto-hydrodynamics equations.
 We obtain a new class of regularity criteria in terms of the
 direction of the velocity. Our result extend some results
 known for incompressible Navier-Stokes equations.
\end{abstract}

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

\section{Introduction}

We consider the 3D incompressible magneto-hydrodynamics (MHD) equation
\begin{equation}
\begin{gathered}
  \frac{\partial u}{\partial t} - \nu \Delta u + u \cdot \nabla u
c = - \nabla    p - \frac{1}{2} \nabla b^2 + b \cdot \nabla b, \\
 \frac{\partial b}{\partial t} - \eta \Delta b + u \cdot \nabla b
= b \cdot    \nabla u ,\\
    \nabla \cdot u = \nabla \cdot b = 0,\\
    u (0, x) = u_0 (x), \quad b (0, x) = b_0 (x)
  \end{gathered} \label{mhdeq}
\end{equation}
Here $u, b$ describe the flow velocity vector and the magnetic field
vector respectively, and $p$ is pressure.  While $u_0, b_0$ are the
given initial velocity and initial magnetic field respectively,
with $\nabla \cdot u_0 = 0$, $\nabla \cdot b_0 = 0$. Without loss
of generality, we set $\nu = \eta = 1$ in the rest of the paper
(it can be achieved by rescaling).
If $\nu = \eta = 0$, \eqref{mhdeq} is called the ideal MHD equations.

It is well known that there exist a global Leray-Hopf weak solution
$(u, b) \in L^{\infty} (0, \infty ; L^2 (\mathbb{R}^3)) \cap L^2 (0, \infty ;
\dot{H} (\mathbb{R}^3))$ if the initial data $(u_0, b_0) \in L^2 (\mathbb{R}^3)$. Using the
standard energy method, it can be easily proved that the solution
satisfies the  energy inequality
\[
\|u (t)\|^2 +\|b (t)\|^2 + \int_0^T (\| \nabla u (s)\|^2_{L^2} +\| \nabla
   b (s)\|_{L^{2}}^2) d s \leq \|u_0 \|_{L^2}^2 +\|b_0 \|_{L^2}^2 ,
\quad T\geq 0.
\]
Up to now, it is unknown whether  solutions of \eqref{mhdeq} on $(0, T)$
will develop finite time singularities even if the initial data is
sufficiently smooth. This problem,  global regularity issue, has been
thoroughly studied for the 3D Navier-Stokes equations and many of
these results can be extended to the 3D MHD equations.
Serrin \cite{serrin} showed that a weak solution of 3D
incompressible Navier-Stokes equations with initial data
$u_0 \in L^2 (\mathbb{R}^3)$
lying $L^p (0, \infty ; L^q (\mathbb{R}^3))$ is smooth in the spatial direction
if $p, q \geq 1$ and  $2/p + 3/q < 1$. He and Xin \cite{HX} extended
this criteria to MHD equations, precisely they showed that under the
condition
\[
u \in L^p (0, T ; L^q (\mathbb{R}^3)) \quad
\text{for $1 / p + 3 / 2 q\leq 1/2$ and $q > 3$},
\]
then the solution remains smooth on $[0, T]$.

Another class of regularity criteria which involves the gradient
of $u$ for the 3D Navier-Stoke equations was introduced by
Beir\~ao de Veiga \cite{bdv} . He showed that any Leray-Hopf
solution is smooth given $\nabla u \in L^p (0, T ; L^q (\mathbb{R}^3))$
with $2 / p + 3 / q = 2$, $3 / 2 < q < \infty$. Beale, Kato and
Majda\cite{bkm} dealt with the vorticity $\omega = \nabla \times
u$ and proved the regularity under the condition $\omega \in L^1
(0, T ; L^{\infty} (\mathbb{R}^3))$. He, Xin \cite{HX} and Zhou \cite{zh2}
respectively extended the result of Beir\~ao de Veiga \cite{bdv}
to the MHD equations, they obtained some condition of $\nabla u$
alone to determine the regularity of the MHD equations. Precisely,
the showed that under the condition
\[
\nabla u \in L^{^p} (0, T ; L^q (\mathbb{R}^3)) \quad
\text{with $ 2 / p +3/ q \leq 1$, $3 / 2 < q \leq \infty$},
\]
then the solution can be extended to $t = T$.
Caflisch, Kapper and Steele
\cite{cks} extended the well known result of \cite{bkm}
to the 3D ideal MHD equations, they showed under the condition
\[
\int_0^T (\| \nabla \times u (t)\|_{L^{\infty}} +\| \nabla \times b
   (t)\|_{L \infty}) d t < \infty
\]
then the solutions remains smooth on $[0, T]$.

 Constantin and Fefferman \cite{CF} used the direction of the vorticity
$\omega / | \omega |$ to describe the regularity criterion to the
Navier-Stokes equations. They showed that under a Lipschitz-like
regularity assumption on $\omega / | \omega |$, the solution is smooth.
Under the framework of Constantin and Fefferman, Zhou \cite{zh3,zh4}
get some more relaxed regularity criterion in terms of the direction
of vorticity.  Inspired by the initial work of \cite{CF},
He and Xin\cite{HX} extended the result to the MHD equations.
They showed that if there exist three positive constant
$K, \rho, \Omega$ such that
\[
| \omega (x + y, t) - \omega (x, t) | \leq K| \omega (x + y, t) |
   |y|^{1 / 2}
\]
holds if both $|y| \leq \rho$ and $| \omega (x, t) | \geq \Omega$
for any $t \in [0, T]$, then the solution is remains smooth on
$[0, T]$, where $\omega = \nabla \times u$ is the vorticity of the
velocity. Zhou \cite{zh5} also studied the regularity criterion for
generalized magneto-hydrodynamics
equations in term of the vorticity field and get similar result.

Of the same spirit in \cite{CF}, Vasseur\cite{Ve} used the direction
of the velocity $u / |u|$ to describe the regularity criterion to
the Navier-Stokes equations. He showed that if the initial value
$u_0 \in L^2 (\mathbb{R}^3)$, and
${\mathop{\rm div}}(u / |u|) \in L^p (0, \infty ; L^q (\mathbb{R}^3))$ with
\[
\frac{2}{p} + \frac{3}{q} \leq \frac{1}{2}, \quad q \geq 6,\;
 p \geq 4
\]
then $u$ is smooth on $(0, \infty) \times \mathbb{R}^3$.

We restricted ourselves to $u / |u|$ to study the regularities of the weak
solutions of \eqref{mhdeq}. We followed the same method of \cite{Ve} to
get our result:

\begin{theorem} \label{thm1}
 Let $u, b$ be a Leray-Hopf  solution to MHD equation
   \eqref{mhdeq} with  the initial value $ u_0, b_0 \in
  H^{1} (\mathbb{R}^3) $. If
  \[
b \in L^{\alpha} (0, T ; L^{\beta} (\mathbb{R}^3)) \quad \text{with }
  \frac{2}{\alpha} +    \frac{3}{\beta} \leq 1, \beta > 3,
\]
  and
  \[
\mathop{\rm div} (u / |u|) \in L^p (0, T ; L^q (\mathbb{R}^3))
\quad with \quad\frac{2}{p} +
     \frac{3}{q} \leq \frac{1}{2},\; q \geq 6, p \geq 4,
 \]
  then $ u, b$ is smooth on $ (0, \infty) \times \mathbb{R}^3$.
\end{theorem}

This result shows that it is sufficient to control the norm of $b$ and
the rate of change the direction of the velocity to get full regularity
 of the solution.
The proof is standard, based on energy methods.

First, we need to introduced some definitions and symbols.

\begin{definition}[\cite{zh1}] \label{def1} \rm
  A measurable vector pair $(u, b)$ is called a weak solution to
the the generalized   magneto-hydrodynamics equations \eqref{mhdeq},
if it satisfies the following  properties
  \begin{enumerate}
    \item $u \in L^{\infty} ([0, T) ; L^2 (\mathbb{R}^3))
\cap L^2 ([0, T) ; H(\mathbb{R}^3))$,\\
$b\in L^{\infty} ([0, T) ; L^2 (\mathbb{R}^3)) \cap L^2 ([0, T) ;
    H (\mathbb{R}^3))$;

 \item $(u, b)$ satisfies  \eqref{mhdeq} in the sense of
    distribution; that is,
\begin{align*}
  & \int_0^T\!\!\!\! \int_{\mathbb{R}^3}\Big( \frac{\partial \phi}
{\partial t}+u\cdot  \nabla\phi \Big) u \,dx\,dt +\int_{\mathbb{R}^3}
u_0 \phi (0, x)\,dx \\
& =\int_0^T \!\!\!\!\int_{\mathbb{R}^3} (\nabla u:\nabla\phi
+ b\cdot\nabla\phi        \cdot b) \,dx\,dt\,,
\\
&\int_0^T\!\!\!\! \int_{\mathbb{R}^3}
\Big( \frac{\partial \phi}{\partial
        t}+u\cdot\nabla\phi\Big)b \,dx\,dt
+ \int_{\mathbb{R}^3}b_0\phi (0,x)\, dx\\
&=\int_0^T\!\!\!\!\int_{\mathbb{R}^3} (\nabla b :\nabla\phi
+ b\cdot\nabla\phi\cdot u) \,dx\,dt
\end{align*}
 for all $\phi \in C_0^{\infty} (\mathbb{R}^3 \times [0, T))$ with
$\nabla \cdot \phi = 0$, and
    \[
\int_0^T\!\!\!\! \int_{\mathbb{R}^3} u \cdot \nabla \phi \,dx\,dt = 0,\quad
\int_0^T\!\!\!\!\int_{\mathbb{R}^3} b \cdot \nabla \phi \,dx\,dt = 0
\]
for every  $\phi \in C_0^{\infty} (\mathbb{R}^3 \times [0, T))$.

\item The energy inequality holds; that is,
\begin{gather*}
 \|u (t)\|_{L^2}^2 + 2 \int_0^t \| \nabla u\|_{L^2}^2 d s
       \  \leq \|u_0 \|_{L^2}^2, \\
 \|b (t)\|_{L^2}^2 + 2 \int_0^t \| \nabla b\|_{L^2}^2 d s
       \  \leq \|b_0 \|_{L^2}^2,
\end{gather*}
 for all $t \in [0, T)$.
  \end{enumerate}
\end{definition}

The space $L^{p, q}$ consists of functions $f$ for which
$\|f\|_{L^{p, q}} < + \infty$, where
\[
\|f\|_{L^{p, q}} = \begin{cases}
  \Big( \int_0^T \|u (\tau, \cdot)\|_{L^q}^p \,d\tau \Big)^{1 / p}, &
     \text{if } 1 \leq p < + \infty,\\
   \mathop{\rm ess\,sup}_{0 < \tau < t} \|u (\tau, \cdot)\|_{L^q}, &
\text{if } p =     + \infty
\end{cases}
\]
where
\[
\|u (\tau, \cdot)\|_{L^q} = \begin{cases}
    \Big( \int_0^T |u (\tau, x) |^q \,dx \Big)^{1 / q}, &
    \text{if } 1 \leq q < + \infty,\\
   \mathop{\rm ess\,sup}_{x \in \mathbb{R}^3} \|u (\tau, \cdot)
 \|_{L^q} &\text{if } q = + \infty
   \end{cases}
\]


\section{Proof of Theorem \ref{thm1}}
\begin{proof}
  Multiplying the first equation by $|u|^2 u$, and the second equation by
  $|b|^2 b$. Integrate the first equation over $\mathbb{R}^3$,
 and after suitable  integration by parts, we obtain
\begin{equation}
\begin{aligned}
 &\frac{d}{d t} \int_{\mathbb{R}^3} \frac{|u|^4}{4}\,dx
+ \int_{\mathbb{R}^3}  (|
    \nabla u|^2 |u|^2 + 2| u|^2 | \nabla |u\|^2)\,dx  \\
&= \int_{\mathbb{R}^3} 2 p u|u| \cdot \nabla |u|d x - \int_{\mathbb{R}^3} b \cdot \nabla
    (|u|^2 u) \cdot b\,dx\,.
\end{aligned}\label{NSeq}
\end{equation}
  Integrate the second equation over $\mathbb{R}^3$, and after suitable
integration by   parts, we obtain
  \begin{equation}
    \frac{d}{d t} \int_{\mathbb{R}^3} \frac{|b|^4}{4}\,dx + \int_{\mathbb{R}^3}  (|b|^2
    | \nabla b|^2 + 2| b|^2 | \nabla |b \|^2)\,dx
= \int_{\mathbb{R}^3} (b \cdot \nabla
    u) \cdot |b|^2 b dx \label{MAeq}
  \end{equation}
  Adding (\ref{NSeq}) and (\ref{MAeq}) yields
  \begin{equation}
     \begin{aligned}
& \frac{d}{d t} \int_{\mathbb{R}^3} \frac{|u|^4  + |b|^4}{4}\,dx
 + \int_{\mathbb{R}^3} ( | \nabla u|^2 |u|^2
 + 2| u|^2 | \nabla |u\|^2)dx \\
&\quad +\int_{\mathbb{R}^{3}}(|b|^2 | \nabla
    b|^2 + 2| b|^2 | \nabla |b\|^2)\,dx  \\
&= \int_{\mathbb{R}^3} (  2 p u|u| \cdot \nabla |u| -
 b \cdot \nabla (|u|^2   u) \cdot b  - (b \cdot \nabla |b|^2 b) \cdot u)
\,dx
\end{aligned}\label{addeq}
  \end{equation}
Next we estimate the  right  hand terms one by one.
Because
  \[
- \Delta p = \sum_{i, j = 1}^3 \partial_i \partial_j (u_i u_j - b_i b_j) .
 \]
  The  Calderon-Zygmund inequality tells
   us  that  there exists a  absolute
   constant $C$  such  that
  \[
\|p\|_{L^q} \leq C (\|u\|_{L^{2 q}}^{2} +\|b\|_{L^{2 q}}^{2}),\quad
\text{for }  1 < q < \infty .
\]
By  generalized H\"older's inequality and Young's inequality,
we get
\begin{equation}
  \begin{aligned}
  2{  \int_{\mathbb{R}^3} p u |u| \cdot \nabla |u|}
 & \leq   2
 {  \int_{\mathbb{R}^3}} |p\|u|^2 | \frac{u}{|u|} \cdot \nabla |u| |\,dx \\
  & \leq  2\|p\|_{L^r} \|u\|^2_{L^{2 r}} \| \frac{u}{|u|} \cdot \nabla
  |u|\|_{L^{\bar{q}}} \\
  & \leq  2 (\|u\|_{L^{2 r}}^4 +\|u\|_{L^{2 r}}^2 \|b\|_{L^{2 r}}^2)\|
  \frac{u}{|u|} \cdot \nabla |u|\|_{L^{\bar{q}}} \\
  & \leq  C | | \frac{u}{|u|} \cdot \nabla |u| | | _{L^{\bar{p}}}
  (\|u\|_{L^{2 r}}^4 +\|b\|_{L^{2 r}}^4) .
\end{aligned}\label{preeq1}
\end{equation}
Where  $2 / r + 1 / \bar{q} = 1, 2 \leq r < 6$, and here we use  the fact
  \[
|u|  {\mathop{\rm div}} (u / |u|) = - \frac{u}{|u|} \cdot \nabla |u|.
\]
Write $\|u\|_{L^{2 r}}^4 =\||u|^2 \|_{L^r}$, then interpolation
inequality of $L^p$ spaces and Sobolev imbedding theorem gives
\[
\|u|^2 \|_{L^r}  \leq  \|| u|^2 \|_{L^2}^{2 \theta}
  \|| u|^2 \|_{L^6}^{2 (1 - \theta)}
 \leq  \|| u|^2 \|_{L^2}^{2 \theta} \| \nabla |u|^2
  \|_{L^2}^{2 (1 - \theta)} .
\]
Where $\theta / 2 + (1 - \theta) / 6 = 1 / r$.
Using Young's inequality again, we obtain
\begin{equation}
  \begin{aligned}
&\| \frac{u}{|u|} \cdot \nabla |u|\|_{L^{\bar{p}}} (\|u\|_{L^{2 r}}^4
  +\|b\|_{L^{2 r}}^4)\\
&\leq  C\| \frac{u}{|u|} \cdot \nabla
  |u|\|_{L^{\bar{p}}}^{1 / \theta} (\|u\|_{L^4}^4 +\|b\|_{L^4}^4) +
  \frac{1}{8} (\| \nabla |u|^2 \|_{L^2}^2 +\| \nabla |b|^2 \|_{L^2}^2)\\
& = C\| \frac{u}{|u|} \cdot \nabla
  |u|\|_{L^{\bar{p}}}^{1 / \theta} (\|u\|_{L^4}^4 +\|b\|_{L^4}^4) +
  \frac{1}{2} (\| |u|\nabla |u|\|_{L^2}^2 +\| |b|\nabla |b| \|_{L^2}^2) .
\end{aligned}\label{preeq2}
\end{equation}
Assume that ${\mathop{\rm div} }(u / |u|) \in L^p (L^q)$,
$u \in L^a (L^b)$. Then we
have $|u|{\mathop{\rm div}} (u / |u|)\in L^{\bar{p}} (L^{\bar{q}})$
where
\[
  \frac{1}{\bar{p}} = \frac{1}{a} + \frac{1}{p},\quad
  \frac{1}{\bar{q}} = \frac{1}{b} + \frac{1}{q}
\]
If $1 / \theta \leq \bar{p}$, then
\[
\int_0^T  \| \frac{u}{|u|} \cdot \nabla |u|\|_{L^{\bar{p}}}^{1 /
   \theta}\,dx < \infty .
\]
Now we seek conditions for  $1 / \theta \leq \bar{p}$. By the relation
 \begin{gather*}
      \frac{2}{r} + \frac{1}{\bar{q}} = 1,\\
      \frac{\theta}{2} + \frac{1 - \theta}{6} = \frac{1}{r},\\
      \frac{1}{\theta} \leq \bar{p} ,
\end{gather*}
we get $2/\bar{p} + 3/\bar{q}\leq 2$. From the definition of weak
solution, we know $2/a + 3/b = 3/2$. So
\[
   \frac{2}{\bar{p}} + \frac{3}{\bar{q}}
 =  { \frac{2}{a} + \frac{3}{b} +   \frac{2}{p} + \frac{3}{q}}
 =  { \frac{3}{2} + \frac{2}{p} + \frac{3}{q}}
 \leq  2;
\]
that is,
\[
\frac{2}{p} + \frac{3}{q} \leq \frac{1}{2} .
 \]
 Next we estimate the second term of \eqref{addeq}.
As in \cite{zh1}, we have
\[
 - \int_{\mathbb{R}^3} b \cdot \nabla (|u|^2 u) \cdot b dx
 \leq  \int_{\mathbb{R}^3} |b|^2 |u| | \nabla |u|^2  |\,dx \\
\]
Using Cauchy's inequality with $\varepsilon$, we obtain
\begin{align*}
\int_{\mathbb{R}^3} |b|^2 |u| | \nabla |u|^2  |\,dx
&\leq  C \int_{\mathbb{R}^3}  |b|^4 |u|^2\,dx + \frac{1}{8} \|
    \nabla |u|^2 \|_{L^2}^2\\
& =  C \int_{\mathbb{R}^3}  |b|^4 |u|^2\,dx + \frac{1}{2} \||u| \nabla
    |u|\|^2_{L^{2}}\,.
\end{align*}
By  generalized H\"older inequality, interpolation inequality
of $L^{p}$ spaces, Sobolev imbedding theorem and Young's inequality,
we obtain
  \begin{align*}
    C \int_{\mathbb{R}^3} |b|^4 |u|^2 dx
& \leq  C\|b\|_{L^{\beta}}^2
    \|b\|_{L^{2 r}}^2 \|u\|_{L^{2 r}}^2\\
& \leq  C\|b\|_{L^{\beta}}^2 (\|b\|_{L^{2 r}}^4 +\|u\|_{L^{2 r}}^4)\\
& \leq  C\|b\|_{L^{\beta}}^{2 / \xi} (\|b\|_{L^4}^4 +\|u\|_{L^4}^4)
    + \frac{1}{2} (\||u| \nabla |u|\|_{L^2}^2
+\||b| \nabla |b|\|_{L^2}^2)\,,
  \end{align*}
  where
  \begin{equation}
\begin{gathered}
  \frac{1}{\beta}+\frac{1}{r} = \frac{1}{2}\\
  \frac{\xi}{2}+\frac{1-\xi}{6}=\frac{1}{r}\\
  2\leq r<6
\end{gathered} \label{rel1}
\end{equation}
We need $2/\xi\leq \alpha$ so that
\[
\int_{0}^{T}\|b\|_{L^{\beta}}^{2 / \xi}dx<\infty\,.
\]
By the relation (\ref{rel1}), this is satisfied if
$2/\alpha+3/\beta\leq 1$. And $\beta>3$ implies $2\leq r<6$.
Due to the above argument,
  \begin{equation}
    - \int_{\mathbb{R}^3} b \cdot \nabla (|u|^2 u) \cdot b dx\leq
    C\|b\|_{L^{\beta}}^{2 / \xi} (\|b\|_{L^4}^4 +\|u\|_{L^4}^4) + (\||u|
    \nabla |u|^{} \|_{L^2}^2 + \frac{1}{2} \||b| \nabla |b|\|_{L^2}^2)
    \label{mfeq1}
  \end{equation}
The last term of \eqref{addeq} can be treated in the same way,
  \begin{equation}
    - \int_{\mathbb{R}^3} b \cdot \nabla (|b|^2 b) \cdot u dx\leq
    C\|b\|_{L^{\beta}}^{2 / \xi} (\|b\|_{L^4}^4 +\|u\|_{L^4}^4) + (
    \frac{1}{2} \||u| \nabla |u|^{} \|_{L^2}^2 +\||b| \nabla |b|\|_{L^2}^2)
    \label{mfeq2}
  \end{equation}
Combining \eqref{addeq},  \eqref{preeq1},  \eqref{preeq2},
\eqref{mfeq1},  \eqref{mfeq2}, we obtain
\begin{align*}
&\frac{d}{d t} \int_{\mathbb{R}^3} \frac{|u|^4 + |b|^4}{4}\,dx
 + \int_{\mathbb{R}^{3}}      (  | \nabla u|^2 |u|^2
 + |b|^2 | \nabla b|^2)\,dx \\
&\leq C(\|b\|_{L^{\beta}}^{2 / \xi} + \||u|  {\mathop{\rm div}} (u / |
    u|)\|_{L^q }^{1/ \theta} + 1) (\|b\|_{L^b}^4 +\|u\|_{L^b}^4)  \\
& = C(\|b\|_{L^{\beta}}^{2 / \xi} + \||u|  {\mathop{\rm div}}
 (u / |^{} u|)\|_{L^q }^{1 / \theta} + 1) \int_{\mathbb{R}^3}
 \frac{|u|^4 + |b|^4}{4}\,dx
\end{align*}
 Let $A(t)=C(\|b\|_{L^{\beta}}^{2 / \xi} + \||u|
{\mathop{\rm div}} (u / |^{} u|)\|_{L^q    }^{1 / \theta} + 1)$,
then the Gronwell
  inequality implies that, whenever $T$ is finite,
\[
\|u(T)\|_{L^{4}}^{4}+\|b(T)\|_{L^{4}}^{4}
\leq C(\|u_{0}\|_{L^{4}}^{4}+\|b_{0}\|_{L^{4}}^{4})
\exp(t \sup_{t\in[0,T)}A(t)).
 \]
This shows   that the solution $(u, b)$ can be extended to  $t=T$.
This completes the proof.
\end{proof}

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\end{document}