Carleman estimate for the Navier-Stokes equations and an application to a lateral Cauchy problem
Mourad Bellassoued
Department of Mathematics
Faculty of Sciences of Bizerte
University of Carthage
7021 Jarzouna Bizerte
Tunisia
e-mail: {\tt mourad.bellassoued@fsb.rnu.tn
We consider the nonstationary linearized Navier-Stokes equations in a bounded domain and first we prove a Carleman estimate with a regular weight function. Second we apply the Carleman estimate to a lateral Cauchy problem for the Navier-Stokes equations and prove the H\"older stability in determining the velocity and pressure field in an interior domain.
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\title{Carleman estimate for the Navier-Stokes equations and
an application to a lateral Cauchy problem}

\author{Mourad Bellassoued\\
Department of Mathematics, Faculty of Sciences of Bizerte\\
University of  Carthage, 7021 Jarzouna Bizerte, Tunisia\\
e-mail: {\tt mourad.bellassoued@fsb.rnu.tn}\\
Oleg Imanuvilov\\
Department of Mathematics, Colorado State University\\
101 Weber Building, Fort Collins, CO 80523-1874, USA\\
e-mail: {\tt oleg@math.colostate.edu}\thanks{Partially supported
by NSF grant DMS 1312900}\\
Masahiro Yamamoto\\
Graduate School of Mathematical Sciences, the University of Tokyo\\
3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan.\\
e-mail: {\tt myama@ms.u-tokyo.ac.jp}\thanks
{Partially supported by 
Grant-in-Aid for Scientific Research (S) 15H05740 
of Japan Society for the Promotion of Science}
}

\date{}

\begin{document}
 \maketitle

%\begin{center}
%$^1$ Department of Mathematics, Faculty of Sciences of Bizerte\\
%University of  Carthage %7021 Jarzouna Bizerte, Tunisia\\
%e-mail: {\tt mourad.bellassoued@fsb.rnu.tn}
%\\
%$^2$ Department of Mathematics, Colorado State University\\
%101 Weber Building, Fort Collins, CO 80523-1874, USA\\
%e-mail: {\tt oleg@math.colostate.edu}\\
%$^3$ Graduate School of Mathematical Sciences, the University of Tokyo\\
%3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan.\\
%e-mail: {\tt myama@ms.u-tokyo.ac.jp}
%\end{center}



\begin{abstract}
We consider the nonstationary linearized
Navier-Stokes equations in a bounded domain and
first we prove a Carleman estimate with a regular weight function.
Second we apply the Carleman estimate to a lateral Cauchy problem
for the Navier-Stokes equations and prove the H\"older stability
in determining the velocity and pressure field in an interior
domain.
\end{abstract}

\baselineskip 18pt
\section{Introduction}

Let $\Omega \subset \R^n$, $n=2,3$, be a bounded domain
with smooth boundary $\ppp\Omega$ (e.g., of $C^2$-class),
and let $\nu=\nu(x)$ be the outward unit
normal vector on $\ppp\Omega$ at $x$.
We set $Q := \Omega \times (0,T)$.

We consider the linearized Navier-Stokes equations for
an incompressible viscous fluid:
$$
\ppp_t v(x, t) - \kappa\Delta v(x, t) + (A\cdot\nabla)v
+ (v\cdot \nabla)B + \nabla p = F(x, t) \quad
\mbox{in $Q$},                                           \eqno{(1.1)}
$$
and
$$
\ddd v(x, t) = 0 \quad \mbox{in $Q$}.      \eqno{(1.2)}
$$
Here $v = (v_1, \cdots, v_n)^T$, $n=2,3$, $\cdot^T$ denotes the transpose
of matrices, $\kappa > 0$ is a constant describing the viscosity,
and for simplicity we assume that the density is one.
Let $\ppp_t = \frac{\ppp}{\ppp t}$, $\ppp_j = \frac{\ppp}{\ppp x_j}$,
$1\le j\le n$, $\Delta = \sum_{j=1}^n \ppp_j^2$, $\nabla =
(\ppp_1,\cdots,\ppp_n)^T$, $\nabla_{x,t} = (\nabla, \ppp_t)^T$,
$\ppp_x^{\beta} = \ppp_1^{\beta_1}\cdots \ppp_n^{\beta_n}$ with
$\beta= (\beta_1, \cdots, \beta_n) \in (\N \cup \{0\})^n$,
$\vert \beta\vert = \beta_1 + \cdots + \beta_n$,
$$
(w\cdot \nabla)v = \left( \sum^n_{j=1} w_j\ppp_jv_1, \cdots,
\sum^n_{j=1} w_j\ppp_jv_n\right)
^T
$$
for $v = (v_1,\cdots,v_n)^T$ and $w = (w_1,\cdots,w_n)^T$.
Throughout this paper, we assume
$$
A \in W^{2,\infty}(Q), \quad
\nabla B \in L^{\infty}(Q).                          \eqno{(1.3)}
$$

In this paper, we establish a Carleman estimate with a regular
weight function and apply it to a lateral Cauchy problem for the 
Navier-Stokes equations and prove the H\"older stability in 
an arbitrarily given interior domain.
For stating the main results, we introduce notations.
Let $I_n$ be the $n\times n$ identity matrix and let the
stress tensor $\sigma(v,p)$ be defined by the $n\times n$ matrix
$$
\sigma(v,p) := \kappa (\nabla v + (\nabla v)^T) - pI_n,
$$ where $\kappa$ is some positive constant.
We assume
$$
d \in C^2(\overline{\Omega}), \quad \vert \nabla d(x)\vert > 0\quad
\mbox{on $\overline{\Omega}$}                  \eqno{(1.4)}
$$
and we arbitrarily choose $t_0 \in (0,T)$ and $\beta > 0$.  We set
$$
\psi(x,t) = d(x) - \beta(t-t_0)^2, \quad
\va(x,t) = e^{\lambda\psi(x,t)}
$$
with a sufficiently fixed large constant $\lambda>0$.
We choose a non-empty relatively open subboundary $\Gamma \subset \ppp\Omega$
arbitrarily.

Let $D \subset Q$ be a bounded domain with smooth boundary $\ppp Q$ such that
$\overline{D\cap (\ppp\Omega \times (0,T))} \subset \Gamma \times (0,T)$.

For $k, \ell \in \N \cup \{0\}$, we set
$$
H^{k,\ell}(D) = \{ v \in L^2(D); \thinspace
\ppp_x^{\beta}v \in L^2(D), \thinspace \vert \beta\vert \le k,
\thinspace \ppp_t^{j}v \in L^2(D) \quad 0\le j\le \ell\}
$$
and
$$
\Vert (v,p)\Vert^2_{ \mathcal X_s(D)}:=
\int_D \left\{ \frac{1}{s^2}\left( \vert \ppp_tv\vert^2
+ \sum_{i,j=1}^n \vert \ppp_i\ppp_jv\vert^2\right)
+ \vert \nabla v\vert^2 + s^2\vert v\vert^2
+ \frac{1}{s}\vert \nabla p\vert^2 + s\vert p\vert^2
\right\} \weight dxdt.
$$

We are ready to state our Carleman estimate.
\\
{\bf Theorem 1}.\\
{\it There exist constants $s_0 > 0$ and $C>0$, independent of $s$, such that
$$
\Vert (v,p)\Vert^2_{\mathcal X_s(D)}
\le C\int_D \vert F\vert^2 \weight dxdt
+ C\int_D (\vert h\vert^2 + \vert \nabla_{x,t} h\vert^2)\weight dxdt
$$
$$
+ Ce^{Cs}(\Vert v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_tv\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))})
                                              \eqno{(1.5)}
$$
for all $s \ge s_0$ and $(v,p) \in H^{2,1}(D) \times H^{1,0}(D)$
satisfying (1.1),
$$
\ddd v = h \quad \mbox{in $D$ with $h \in H^{1,1}(D)$},
$$
and
$$
\left\{
\begin{array}{rl}
& v(\cdot,0) = v(\cdot,T) = 0
\quad \mbox{in $\Omega$},\\
& \vert v\vert = \vert \nabla v\vert = \vert p\vert
= 0 \quad \mbox{on $\ppp D \setminus
(\Gamma \times (0,T))$}.
\end{array}
\right.                                   \eqno{(1.6)}
$$}

This is a Carleman estimate for the linearized Navier-Stokes
equations (1.1) with (1.2) with boundary data on $\Gamma \subset
\ppp\Omega$.  

Boulakia \cite{Bo} proves a Carleman estimate with a weight function
similar to ours for the homogeneous Stokes equations:
$\ppp_tv = \Delta v - \nabla p$ and $\ddd v = 0$ with extra
interior or boundary data.
The Carleman estimate in \cite{Bo} requires a stronger norm of boundary
data than our Carleman estimate if it is applied to
the case of the Stokes equations.

As for other Carleman estimates for the Navier-Stokes equations, we refer to
Choulli, Imanuvilov, Puel and Yamamoto \cite{ChImPuYa},
Fern\'andez-Cara, Guerrero, Imanuvilov and Puel \cite{FeGuImPu},
where the authors use a weight function in the form
$$
\exp\left( \frac{2sw(x)}{t(T-t)}\right)
$$
with some function $w$ and the weight function
decays to $0$ at $t=0,T$ exponentially.  Their Carleman
estimates hold over the whole domain $Q$ for $v$ satisfying
$v=0$ on $\ppp\Omega$ but not necessarily $v(\cdot,0) = v(\cdot,T) = 0$.
Those global Carleman estimate is convenient for proving
the Lipschitz stability for an inverse source problem
(e.g., \cite{ChImPuYa}) and the exact null controllability (\cite{FeGuImPu}),
but is not suitable for proving the unique continuation, and such a
weight function does not admit Carleman estimates for the Navier-Stokes
equations coupled with first-order equation or hyperbolic equation such as
a conservation law.
As for Carleman estimates for the Navier-Stokes equations, see also
Fan, Di Cristo, Jiang and Nakamura \cite{Fan1} and
Fan, Jiang and Nakamura \cite{Fan2} with extra data in a neighborhood
of the whole boundary, which is too much by considering the parabolicity of
the equations.



\section{Proof of Theorem 1}

{\bf First Step.}\\
Let $E \subset \R^n$ be a bounded domain with smooth boundary
$\ppp E$ and let
$E_{\delta} := \{ x\in E;\thinspace
\mbox{dist $(x, E) > \delta$}\}$ with small $\delta > 0$.

We prove\\
{\bf Lemma 1}.\\
{\it Let $p \in H^1(E)$ satisfy
$$
\Delta p = f_0 + \sum_{j=1}^n \ppp_jf_j \quad
\mbox{in $E$}
$$
and $\mbox{supp}\, p \subset E_{\delta}$.  Let $d_0 \in C^2(\overline{E})$
satisfy $d_0(x) > 0$ for $x \in E$ and $\vert \nabla d(x)\vert > 0$ for $x \in
E_{\delta}$.  We set $\va_0(x) = e^{\lambda d_0(x)}$ with large
constant $\lambda>0$.  Then there exist constants $C>0$ and
$s_1>0$ such that
$$
\int_E \left( \frac{1}{s}\vert \nabla p\vert^2
+ s\vert p\vert^2\right) e^{2s\va_0(x)} dx
\le C\int_E \left( \frac{1}{s^2}\vert f_0\vert^2
+ \sum_{j=1}^n \vert f_j\vert^2\right) e^{2s\va_0(x)} dx
$$
for all $s \ge s_1$.  The constants $C$ and $s_1$ are independent
of choices of $p$.}
\\
{\bf Proof.}  Since $\ppp E$ is of $C^3$-class, we choose
a function $\mu \in C^3(\overline{E})$ such that $0 \le \mu \le 1$,
$\mu > 0$ in $E$ and
$\mu = \left\{
\begin{array}{rl}
0, \quad &\mbox{in $\R^n \setminus E$},\\
1, \quad &\mbox{in $E_{\delta/2}$}.
\end{array}\right.$.
We set $\www{d_0}(x) = \mu(x) d_0(x)$ and $\www{\va_0}(x)
= e^{\lambda \www{d_0}(x)}$ for $x \in \overline{E}$.
Then $\www{d_0}(x) = 0$ for $x \in \ppp E$ and $\www{d_0}>0$,
$\vert \nabla\www{d_0}\vert = \vert \mu\nabla d_0 + d_0\nabla \mu\vert
= \vert \mu\nabla d_0\vert > 0$ in $E_{\delta}$.
Hence the $H^{-1}$-Carleman estimate for an elliptic operator
by Imanvilov and Puel \cite{ImPu} yields
$$
\int_E \left( \frac{1}{s}\vert \nabla p\vert^2
+ s\vert p\vert^2\right) e^{2s\www{\va_0}(x)} dx
\le C\int_E \left( \frac{1}{s^2}\vert f_0\vert^2
+ \sum_{j=1}^n \vert f_j\vert^2\right) e^{2s\www{\va_0}(x)} dx
$$
for all $s \ge s_1$.
Here we note that in Theorem 1.2 in \cite{ImPu}, we set $\omega = E\setminus
\overline{E_{\delta}}$ and use $p\vert_{\omega} = 0$.
Since $p=0$ in $E \setminus E_{\delta}$ and
$\www{d_0} = d_0$ in $E_{\delta}$, we complete the proof of Lemma 1.
\\
\vspace{0.2cm}
\\
{\bf Lemma 2.}\\
{\it There exist constants $s_0 > 0$ and $C>0$ such that
$$
\Vert (v,p)\Vert^2_{ \mathcal X_s(Q)}\le  C\int_Q \vert F\vert^2 \weight dxdt
+ C\int_Q (\vert h\vert^2 + \vert \nabla_{x,t}h\vert^2) \weight dxdt
                                                             \eqno{(2.1)}
$$
for all $s \ge s_0$ and $(v,p) \in H^{2,1}(Q) \times H^{1,0}(Q)$
satisfying (1.1),
\begin{align*}
&v(\cdot,0) = v(\cdot,T) = 0 \quad \mbox{in $\Omega$},\\
& \vert v\vert = \vert \nabla v\vert = \vert p\vert = 0
\quad \mbox{in $\ppp\Omega \times (0,T)$},
\end{align*}
and
$$
\ddd v = h \quad \mbox{in $Q$}
$$
with some $h \in H^{1,1}(Q)$.}
\\
{\bf Proof of Lemma 2.}
Thanks to the large parameter $s>0$, in view of (1.3),
it is sufficient to prove Lemma 1 for $B=0$ in (1.1).
In fact, the Carleman estimate with $B\ne 0$ follows from the case
of $B=0$ by replacing $F$ by $F-(v\cdot\nabla)B$
and estimating $\vert (F-(v\cdot\nabla)B)(x,t)\vert
\le \vert F(x,t)\vert + C\vert v(x,t)\vert$ for
$(x,t) \in Q$.  Then, choosing $s_0>0$ large, we can
absorb the term $\int_Q \vert v\vert^2 \weight dxdt$ into the
left-hand side of the Carleman estimate.

By the density argument, it is sufficient to prove the lemma for
$(v,p)$ such that $\mbox{supp} \,v$ and $\mbox{supp}\,p$ are compact in $Q$.
We consider
$$
\ppp_tv = \kappa\Delta v - (A\cdot\nabla)v - \nabla p + F
                                      \eqno{(2.2)}
$$
and
$$
\ddd v = h \qquad \mbox{in $Q$}.             \eqno{(2.3)}
$$
Taking the divergence of (2.2) and using (2.3), we obtain
$$
\Delta p = -\sum_{j,k=1}^n \{ \ppp_j((\ppp_kA_j)v_k)
- (\ppp_j\ppp_kA_j)v_k\}
+ \ddd F - \ppp_th - (A\cdot\nabla)h + \kappa\ddd(\nabla h)
\quad \mbox{in $Q$}.                    \eqno{(2.4)}
$$
Here we used
$$
\ddd ((A\cdot\nabla)v)
= \sum_{j,k=1}^n \ppp_k(A_j\ppp_jv_k)
= \sum_{j=1}^n A_j\ppp_j\left(\sum_{k=1}^n \ppp_kv_k\right)
+ \sum_{j,k=1}^n (\ppp_kA_j)\ppp_jv_k
$$
$$
= A\cdot \nabla(\ddd v)
+ \sum_{j,k=1}^n \{ \ppp_j((\ppp_kA_j)v_k)
- (\ppp_j\ppp_kA_j)v_k\}.                         \eqno{(2.5)}
$$
Moreover on the right-hand side of (2.4), the term
$\kappa\ddd (\nabla h)$ is not in $L^2(Q)$ because we assume only
$h \in H^{1,1}(Q)$.  Thus we cannot apply a usual Carleman estimate
requiring $\Delta p \in L^2(Q)$, and we need the $H^{-1}$-Carleman estimate.

By a usual density argument, we can assume that supp $p \subset Q$. 
By supp $p \subset Q$, fixing $t \in [0,T]$, we apply Lemma 1
to (2.4) and obtain
$$
\int_{\Omega} \left( \frac{1}{s} \vert \nabla p(x,t)\vert^2
+ s\vert p(x,t)\vert^2\right) e^{2s\va(x,t_0)} dx
$$
$$
\le C\int_{\Omega} (\vert F\vert^2 + \vert \ppp_th\vert^2
+ \vert \nabla h\vert^2 + \vert h\vert^2)e^{2s\va(x,t_0)} dx
+ C\int_{\Omega} \vert v(x,t)\vert^2 e^{2s\va(x,t_0)} dx
                                              \eqno{(2.6)}
$$
for $s \ge s_1$ where $s_1>0$ is a sufficiently large constant.

Let $s_0 := s_1e^{\lambda \beta T^2}$.  Then, $s \ge s_0$ implies
$$
se^{-\lambda\beta (t-t_0)^2} \ge se^{-\lambda\beta T^2} \ge s_1
$$
for $0 \le t \le T$, so that for fixed $t \in [0,T]$ by replacing
$s$ by $se^{-\la\beta (t-t_0)^2}$, by (2.5) we can see
\begin{align*}
&\int_{\Omega} \left( \frac{1}{s} \vert \nabla p(x,t)\vert^2
+ s\vert p(x,t)\vert^2\right) \exp( 2(se^{-\la\beta (t-t_0)^2})
\va(x,t_0)) dx \\
\le &C\int_{\Omega} (\vert F\vert^2 + \vert \ppp_th\vert^2
+ \vert \nabla h\vert^2 + \vert h\vert^2)\exp( 2(se^{-\la\beta (t-t_0)^2})
\va(x,t_0)) dx\\
+ & C\int_{\Omega} \vert v(x,t)\vert^2
\exp( 2(se^{-\la\beta (t-t_0)^2})\va(x,t_0))dx,
\end{align*}
that is,
\begin{align*}
& \int_{\Omega} \left( \frac{1}{s} \vert \nabla p(x,t)\vert^2
+ s\vert p(x,t)\vert^2\right) e^{2s\va(x,t)} dx \\
\le& C\int_{\Omega} (\vert F\vert^2 + \vert \ppp_th\vert^2
+ \vert \nabla h\vert^2 + \vert h\vert^2)e^{2s\va(x,t)} dx
+ C\int_{\Omega} \vert v(x,t)\vert^2 e^{2s\va(x,t)} dx
\end{align*}
for $s \ge s_0$ and $0 \le t \le T$.  Integrating this
inequality in $t$ over $(0,T)$,
we have
$$
\int_Q \left( \frac{1}{s} \vert \nabla p\vert^2
+ s\vert p\vert^2\right) e^{2s\va} dxdt
$$
$$
\le C\int_Q (\vert F\vert^2 + \vert \ppp_th\vert^2
+ \vert \nabla h\vert^2 + \vert h\vert^2)e^{2s\va} dxdt
+ C\int_Q \vert v\vert^2 e^{2s\va} dxdt           \eqno{(2.7)}
$$
for all $s \ge s_0$.

Next, regarding $F-\nabla p$ in (2.2) as non-homogeneous term,
we apply a Carleman estimate for the parabolic operator
$\ppp_tv - \kappa\Delta v + (A\cdot\nabla)v$ (e.g.,
Theorem 3.1 in Yamamoto \cite{Ya}) to (2.2):
$$
\frac{1}{s}\int_Q \left\{ \frac{1}{s}\left( \vert \ppp_tv\vert^2
+ \sum_{i,j=1}^n \vert \ppp_i\ppp_jv\vert^2\right)
+ s\vert \nabla v\vert^2 + s^3\vert v\vert^2\right\} \weight dxdt
$$
$$
\le C\int_Q \frac{1}{s}\vert \nabla p\vert^2 \weight dxdt
+ C\int_Q \frac{1}{s} \vert F\vert^2 \weight dxdt.
                                               \eqno{(2.8)}
$$

Substituting (2.7) into (2.8), we obtain
\begin{align*}
& \int_Q \left\{ \frac{1}{s^2}\left( \vert \ppp_tv\vert^2
+ \sum_{i,j=1}^n \vert \ppp_i\ppp_jv\vert^2\right)
+ \vert \nabla v\vert^2 + s^2\vert v\vert^2\right\} \weight dxdt \\
\le & C\int_Q \vert F\vert^2 \weight dxdt
+ C\int_Q (\vert \ppp_th\vert^2 + \vert \nabla h\vert^2
+ \vert h\vert^2)\weight dxdt\\
+ &C\int_Q \vert v\vert^2 \weight dxdt
+ \frac{C}{s}\int_Q \vert F\vert^2 \weight dxdt.
\end{align*}
Choosing $s_0 > 0$ large, we can absorb the third term on the right-hand
side into the left-hand side, again with (2.7), we complete the proof of
Lemma 2. $\blacksquare$
\\
\vspace{0.2cm}
\\
{\bf Second Step.}\\
Without loss of generality, we can assume that $d>0$ in $\Omega$ because
we replace $d$ by $d+C_0$ with large constant $C_0 > 0$ if necessary.

In this step, we will prove
\\
{\bf Lemma 3}.\\
{\it There exist constants $s_0>0$ and $C>0$ such that
\begin{align*}
& \Vert (v,p)\Vert^2_{ \mathcal X_s(D)}\\
\le& C\int_D \vert F\vert^2 \weight dxdt
+ C\int_D (\vert h\vert^2 + \vert \nabla_{x,t}h\vert^2)\weight dxdt\\
+ &Ce^{Cs}(\Vert v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_tv\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \ppp_{\nu}v \Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert p\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))})
\end{align*}
for all $s\ge s_0$ and $(v,p)\in H^{2,1}(D)\times H^{1,0}(D)$
satisfying (1.1), (1.6) and
$$
\ddd v = h \quad \mbox{in $D$}.                \eqno{(2.9)}
$$}
\\
{\bf Proof of Lemma 3.}
We take the zero extensions of $v,p, A, F$ to $Q$ from $D$ and
by the same letters we denote them:
$$
v
= \left\{
\begin{array}{rl}
v \quad & \mbox{on $\overline{D}$},\\
0 \quad & \mbox{in $Q \setminus D$},
\end{array}
\right.
\quad
p
= \left\{
\begin{array}{rl}
p \quad & \mbox{on $\overline{D}$},\\
0 \quad & \mbox{in $Q \setminus D$}, \thinspace
\mbox{etc.}\\
\end{array}
\right.
$$
By (1.6) we easily see that
$$
\ppp_i v
= \left\{
\begin{array}{rl}
\ppp_i v \quad & \mbox{on $\overline{D}$},\\
0 \quad & \mbox{in $Q\setminus D$},
\end{array}
\right.
\quad
\ppp_t v
= \left\{
\begin{array}{rl}
\ppp_t v \quad & \mbox{on $\overline{D}$},\\
0 \quad & \mbox{in $Q\setminus D$},
\end{array}
\right.
\quad
\ppp_i\ppp_j v
= \left\{
\begin{array}{rl}
\ppp_i\ppp_j v, \quad & \mbox{on $\overline{D}$},\\
0, \quad & \mbox{in $Q \setminus D$},
\end{array}
\right.
$$
and
$$
\ppp_i p
= \left\{
\begin{array}{rl}
\ppp_i p, \quad & \mbox{on $\overline{D}$},\\
0, \quad & \mbox{in $Q\setminus D$}
\end{array}
\right.
$$
for $1 \le i,j \le n$.
Moreover, since $v=0$ on $\ppp D \setminus
(\Gamma \times (0,T))$ by (1.6), setting
\\
$h = \left\{
\begin{array}{rl}
h \quad & \mbox{on $\overline{D}$},\\
0 \quad & \mbox{in $Q\setminus D$}
\end{array}
\right.
$, we see that $h \in H^{1,1}(Q)$ and
$$
\ddd v = h \quad \mbox{in $Q$}    \eqno{(2.10)}
$$
and
$$
\ppp_tv = \kappa\Delta v + (A\cdot \nabla)v + \nabla p + F
\quad \mbox{in $Q$}.                                \eqno{(2.11)}
$$

By the Sobolev extension theorem, there exist $\www{p}
\in L^2(0,T;H^1(\Omega))$ and $v \in H^{2,1}(Q)$ such that
$$
\left\{
\begin{array}{rl}
&\www{v} = v, \thinspace \ppp_{\nu}\www{v} = \ppp_{\nu} v, \thinspace
\www{p} = p \quad \mbox{on $\ppp\Omega \times (0,T)$},\\
& \mbox{supp $\www{v}(x,\cdot) \subset (0,T)$ for almost all $x \in \Omega$}
\end{array}
\right.                      \eqno{(2.12)}
$$
and
$$
\Vert \www{v}\Vert_{H^{2,1}(Q)}
+ \Vert \ppp_t{\www{v}}\Vert_{L^2(0,T;H^1(\Omega))}
+ \Vert \www{p}\Vert_{L^2(0,T;H^1(\Omega))}
$$
$$
\le C(\Vert v\Vert_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_t v\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \ppp_{\nu} v\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert p\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}).      \eqno{(2.13)}
$$
The last condition in (2.12) can be seen by $v(\cdot, 0) =
v(\cdot,T) = 0$ in $\Omega$ which follows from (1.6).

We set
$$
u = v - \www{v}, \quad q = p-\www{p} \quad \mbox{in $Q$}.
$$
Then, in view of (2.10) - (2.12), we have
$$
\vert u\vert  = \vert \nabla u\vert = \vert q\vert = 0 \quad
\mbox{on $\ppp\Omega\times (0,T)$}          \eqno{(2.14)}
$$
and
$$
\ppp_tu - \kappa\Delta u + \nabla q + (A\cdot\nabla)u
= F - (\ppp_t\www{v} - \kappa\Delta \www{v} + (A\cdot\nabla)\www{v}
+ \nabla\www{p}) =: G \quad \mbox{in $Q$},    \eqno{(2.15)}
$$
$$
\ddd u = h -\ddd \www{v} \in H^{1,1}(Q).              \eqno{(2.16)}
$$

We choose a bounded domain $\www{\Omega}$ with smooth boundary
$\ppp\www{\Omega}$ such that $\www{\Omega} \supset \Omega$,
$\overline{\Gamma} = \ppp\Omega \cap \www{\Omega}$ and
$\ppp\widetilde{\Omega} \cap \overline{\Omega}
= \ppp\Omega \setminus \Gamma$.  In other words, the
domain $\widetilde{\Omega}$ is constructed by expanding $\Omega$ only over
$\Gamma$ to the exterior such that the boundary $\ppp\widetilde{\Omega}$
is smooth.  We set
$$
\www{Q} =\www{\Omega} \times (0,T).
$$

Let us recall that $d$ satisfies (1.4).  Since we can further
choose $\widetilde{\Omega}$ such that
$\widetilde{\Omega}\setminus \Omega$ is included in a sufficiently small
ball, we see that there exists
an extension $\widetilde{d}$ in $\widetilde{\Omega}$ of $d$
satisfying $\vert \nabla\www{d}\vert > 0$ in $\www{\Omega}$.

We take the zero extensions of $u, q, A, G$ and $h-\ddd \www{v}$
to $\www{\Omega}$ and by the same letters we denote them.
Therefore by (2.14) - (2.16), the zero extensions of $u$ and $h - \ddd\www{v}$
satisfies
$$
\ddd u = h - \ddd\widetilde{v} \in H^{1,1}(\www{Q})    \eqno{(2.17)}
$$
and
$$
\ppp_tu - \kappa\Delta u + \nabla q + (A\cdot\nabla)u = G \quad
\mbox{in $\www{Q}$}.                               \eqno{(2.18)}
$$
By the zero extensions and (1.6), we obtain
$$
u(\cdot,0) = u(\cdot, T) = 0 \quad \mbox{in $\www{\Omega}$},
$$
$$
\vert u\vert = \vert \nabla u\vert = \vert q\vert = 0 \quad
\mbox{on $\ppp\www{\Omega} \times (0,T)$}.        \eqno{(2.19)}
$$
Therefore, by noting (2.19), we apply Lemma 2 to (2.17)
and (2.18), and we obtain
\begin{align*}
& \Vert (u,q)\Vert^2_{ \mathcal X_s(\tilde Q)}
\le  C\int_{\www{Q}} \vert G\vert^2 \weight dxdt\\
+ &C\int_{\www{Q}} (\vert h - \ddd\www{v}\vert^2
+ \vert \nabla_{x,t} (h-\ddd\www{v})\vert^2)\weight dxdt
\end{align*}
for $s \ge s_0$.
Hence
\begin{align*}
& \Vert (v-\tilde v,p-\tilde p)\Vert^2_{ \mathcal X_s(Q)} \\
\le & C\int_Q \vert F\vert^2 \weight dxdt\\
+ &C\int_Q \vert \ppp_t\www{v} - \kappa\Delta\www{v}
+ (A\cdot\nabla)\www{v} + \nabla\www{p}\vert^2 \weight dxdt\\
+& C\int_Q \left( \vert h\vert^2 + \vert \nabla_{x,t}h\vert^2
+ \vert \nabla\www{v}\vert^2
+ \sum_{i,j=1}^n \vert \ppp_i\ppp_j\www{v}\vert^2
+ \vert \nabla(\ppp_t\www{v})\vert^2\right)\weight dxdt.
\end{align*}
Using $\vert \ppp_tv\vert^2 \le 2\vert \ppp_t\www{v}\vert^2
+ 2\vert \ppp_t(v-\www{v})\vert^2$, etc. on the left-hand side, we
have
\begin{align*}
& \Vert (v,p)\Vert^2_{ \mathcal X_s(Q)}
\le 2\Vert (\tilde v,\tilde p)\Vert^2_{ \mathcal X_s(Q)} \\
+ &2C\int_Q \vert F\vert^2 \weight dxdt\\
+ &2C\int_Q \vert \ppp_t\www{v} - \kappa\Delta\www{v}
+ (A\cdot\nabla)\www{v} + \nabla\www{p}\vert^2 \weight dxdt\\
+& 2C\int_Q \left( \vert h\vert^2 + \vert \nabla_{x,t}h\vert^2
+ \vert \nabla\www{v}\vert^2
+ \sum_{i,j=1}^n \vert \ppp_i\ppp_j\www{v}\vert^2
+ \vert \nabla(\ppp_t\www{v})\vert^2\right)\weight dxdt\\
\le& C\int_Q \vert F\vert^2 \weight dxdt\\
+& Ce^{Cs}(\Vert \www{v}\Vert^2_{H^{2,1}(Q)}
+ \Vert \www{p}\Vert^2_{H^{1,0}(Q)}
+ \Vert \nabla\ppp_t\www{v}\Vert^2_{L^2(Q)})\\
+& C\int_Q (\vert h\vert^2 + \vert \nabla_{x,t}h\vert^2) \weight dxdt
\end{align*}
for $s \ge s_0$.  Since $F$ and $h$ are zero outside of $D$,
in view of (2.13), the proof of Lemma 3 is completed. $\blacksquare$
\\
\vspace{0.2cm}
\\

{\bf Third Step.}\\
For $r>0$ and $x_0 \in \R^n$, we set $B_r(x_0):= \{x \in \R^n; \thinspace
\vert x-x_0\vert < r\}$.
Then we prove
\\
{\bf Lemma 4.}\\
{\it Let $v \in H^2(\Omega)$ and $p \in H^1(\Omega)$.\\
{\bf (1) Case $n=3$}:  For any $x_0 \in \ppp\Omega$, there exist
$r>0$ and a $10\times 10$ matrix $A \in C^1(\overline{B_r(x_0)})$ such that
$$
\ppp\Omega \cap B_r(x_0) = \{ x(\theta_1,\theta_2);
\thinspace (\theta_1,\theta_2) \in D_1\}
$$
where $x(\theta_1,\theta_2) = (x_1(\theta_1,\theta_2),
x_2(\theta_1,\theta_2), x_3(\theta_1,\theta_2)) \in \R^3$,
$D_1 \subset \R^2$ is a bounded domain and the functions $x_1, x_2, x_3$ 
with respect to $\theta_1, \theta_2$ are in $C^2(\overline{D_1})$ and
$$
\mbox{det}\thinspace A(x(\theta_1,\theta_2)) \ne 0, \quad
(\theta_1, \theta_2) \in \overline{D_1}
$$
and
$$
A(x(\theta_1,\theta_2))
\left(
\begin{array}{cccc}
&(\nabla_xv_1)(x(\theta_1,\theta_2))\\\
&(\nabla_xv_2)(x(\theta_1,\theta_2))\\
&(\nabla_xv_3)(x(\theta_1,\theta_2))\\
&p(x(\theta_1,\theta_2))\\
\end{array}\right)
= \left(
\begin{array}{ccccc}
&\nabla_{\theta_1,\theta_2}(v_1(x(\theta_1,\theta_2)))\\
&\nabla_{\theta_1,\theta_2}(v_2(x(\theta_1,\theta_2)))\\
&\nabla_{\theta_1,\theta_2}(v_3(x(\theta_1,\theta_2)))\\
&(\sigma(v,p)\nu)(x(\theta_1,\theta_2))\\
&(\ddd v)(x(\theta_1,\theta_2))\\
\end{array}\right),
\quad (\theta_1, \theta_2) \in D_1.
$$
\\
{\bf (2) Case $n=2$}:  For any $x_0 \in \ppp\Omega$, there exist
$r>0$ and a $5\times 5$ matrix $A \in C^1(\overline{B_r(x_0)})$ such that
$$
\ppp\Omega \cap B_r(x_0) = \{ x(\theta_1); \thinspace
\theta_1 \in I_1\}
$$
where $x(\theta_1) := (x_1(\theta_1), x_2(\theta_1)) \in \R^2$,
$I_1 \subset \R$ is an open interval, and the functions $x_1, x_2$ are in
$C^2(\overline{I_1})$, and
$$
\mbox{det}\thinspace A(x(\theta_1)) \ne 0, \quad \theta_1 \in \overline{I_1}
$$
and
$$
A(x(\theta_1))
\left(
\begin{array}{ccc}
&(\nabla_xv_1)(x(\theta_1))\\
&(\nabla_xv_2)(x(\theta_1))\\
&p(x(\theta_1))\\
\end{array}\right)
= \left(
\begin{array}{cccc}
&\frac{d}{d\theta_1}v_1(x(\theta_1))\\
&\frac{d}{d\theta_1}v_2(x(\theta_1))\\
&(\sigma(v,p)\nu)(x(\theta_1))\\
&(\ddd v)(x(\theta_1))\\
\end{array}\right),
\quad \theta_1 \in I_1.
$$}
\\
\vspace{0.2cm}
{\bf Remark.}
The lemma guarantees that the boundary data $(v,\ppp_{\nu}v,p)$ and
$(v,\sigma(v,p)\nu)$ are equivalent (e.g., Imanuvilov and Yamamoto
\cite{ImYa2}).  As related papers on inverse boundary value problems
for the Navier-Stokes equations in view of this equivalence, see
Imanuvilov and Yamamoto \cite{ImYa1}, Lai, Uhlmann and
Wang \cite{LUW}.
\\
{\bf Proof of Lemma 4.}
We prove only in the case of $n=3$.  The case of $n=2$ is similar and simpler.
It is sufficient to consider only on
a sufficiently small subboundary $\Gamma_0$ of
$\ppp\Omega$.  Without loss of generality, we can
assume that $\Gamma_0$ is represented by by $(x_1,x_2,\gamma(x_1,x_2))$ where
$\gamma \in C^2(\overline{D_1})$,
$\theta_1 = x_1$, $\theta_2 = x_2$,
$x_3 = \gamma(x_1,x_2)$ for $(x_1,x_2) \in D_1$.  Moreover we assume that
$\Omega$ is located upper $x_3 = \gamma(x_1,x_2)$.

By the density argument, we can assume that $v \in C^1(\overline{\Omega})$ and
$p \in C(\overline{\Omega})$.

We set $\gamma_1;= \ppp_1\gamma$ and $\gamma_2:=
\ppp_2\gamma$.
On $\Gamma_0$, we have
$$
\nu(x) = \frac{1}{1+\gamma_1^2+\gamma_2^2}
\left(
\begin{array}{ccc}
&\gamma_1\\
&\gamma_2\\
&-1\\
\end{array}\right).                 \eqno{(2.20)}
$$

By the definition, we have
$$
\sigma(v,p)\nu = \kappa
\left(
\begin{array}{ccc}
2\ppp_1v_1 - \frac{p}{\kappa} & \ppp_1v_2+\ppp_2v_1 &
\ppp_1v_3+\ppp_3v_1 \\
\ppp_1v_2+\ppp_2v_1 & 2\ppp_2v_2-\frac{p}{\kappa} &
\ppp_2v_3+\ppp_3v_2 \\
\ppp_1v_3 + \ppp_3v_1 & \ppp_2v_3+\ppp_3v_2 &
2\ppp_3v_3-\frac{p}{\kappa} \\
\end{array}\right)
\left(
\begin{array}{ccc}
& \nu_1\\
& \nu_2\\
& \nu_3\\
\end{array}\right)                           \eqno{(2.21)}
$$
$$
=:
\left(
\begin{array}{ccc}
& q_1\\
& q_2\\
& q_3\\
\end{array}\right).
$$
We further set
$$
q_4 := (\ddd v)(x_1,x_2,\gamma(x_1,x_2)),
$$
$$
g_k(x_1,x_2) := v_k\XXX, \quad k=1,2,3.
$$
Then
$$
\ppp_1g_k = \ppp_1v_k + \gamma_1(\ppp_3v_k)\XXX,
$$
$$
\ppp_2g_k = \ppp_2v_k + \gamma_2(\ppp_3v_k)\XXX,
$$
that is,
$$
\left\{
\begin{array}{rl}
&\ppp_1v_k\XXX = \ppp_1g_k - \gamma_1(\ppp_3v_k)\XXX, \\
&\ppp_2v_k\XXX = \ppp_2g_k - \gamma_2(\ppp_3v_k)\XXX, \quad k=1,2,3,\\
\end{array}\right.
                                     \eqno{(2.22)}
$$
and
$$
(\ppp_3v_3)\XXX = q_4 - (\ppp_1v_1+\ppp_2v_2)\XXX     \eqno{(2.23)}
$$
for $(x_1,x_2) \in D_1$.
Setting
$$
\left\{
\begin{array}{rl}
& h_1(x_1,x_2) = (\ppp_3v_1)\XXX, \\
& h_2(x_1,x_2) = (\ppp_3v_2)\XXX, \\
\end{array}\right.
                                     \eqno{(2.24)}
$$
by (2.22) and (2.23) we obtain
\begin{align*}
& (\ppp_3v_3)\XXX
= q_4 - (\ppp_1v_1+\ppp_2v_2)\XXX\\
=& (q_4 - \ppp_1g_1 - \ppp_2g_2)(x_1,x_2)
+ (\gamma_1\ppp_3v_1+\gamma_2\ppp_3v_2)\XXX\\
=: & g_0(x_1,x_2) + (\gamma_1\ppp_3v_1+\gamma_2\ppp_3v_2)\XXX
\end{align*}
$$
= (g_0+\gamma_1h_1+\gamma_2h_2)(x_1,x_2)    \eqno{(2.25)}
$$
and so
$$
\left\{
\begin{array}{rl}
& (\ppp_1v_1)\XXX = (\ppp_1g_1 - \gamma_1h_1)\XXX,\\
& (\ppp_2v_1)\XXX = (\ppp_2g_1 - \gamma_2h_1)\XXX,\\
& (\ppp_1v_2)\XXX = (\ppp_1g_2 - \gamma_1h_2)\XXX,\\
& (\ppp_2v_2)\XXX = (\ppp_2g_2 - \gamma_2h_2)\XXX,\\
& (\ppp_1v_3)\XXX = \ppp_1g_3 - \gamma_1g_0 - \gamma_1^2h_1
- \gamma_1\gamma_2h_2,\\
& (\ppp_2v_3)\XXX = \ppp_2g_3 - \gamma_2g_0 - \gamma_1\gamma_2h_1
- \gamma_2^2h_2,\quad (x_1,x_2) \in D_1.
\end{array}\right.
\eqno{(2.26)}
$$
On the other hand, (2.21) yields
$$
\frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_1
= (2\gamma_1\ppp_1v_1+\gamma_2\ppp_1v_2+\gamma_2\ppp_2v_1
- \ppp_1v_3 - \ppp_3v_1)\XXX - \frac{\gamma_1}{\kappa}p,
$$
$$
\frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_2
= (\gamma_1\ppp_1v_2+\gamma_2\ppp_2v_1 + 2\gamma_2\ppp_2v_2
- \ppp_2v_3 - \ppp_3v_2)\XXX - \frac{\gamma_2}{\kappa}p
$$
and
$$
\frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_3
= (\gamma_1\ppp_1v_3+\gamma_1\ppp_3v_1 + \gamma_2\ppp_2v_3
+ \gamma_2\ppp_3v_2 - 2\ppp_3v_3)\XXX + \frac{1}{\kappa}p,
\quad (x_1,x_2) \in D_1.
$$
Substitute (2.25) and (2.26), we have
$$
\left\{
\begin{array}{rl}
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_1
= -(1+\gamma_1^2+\gamma_2^2)h_1 - \frac{\gamma_1}{\kappa}p+ G_1, \\
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_2
= -(1+\gamma_1^2+\gamma_2^2)h_2 - \frac{\gamma_2}{\kappa}p+ G_2, \\
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_3
= -\gamma_1(1+\gamma_1^2+\gamma_2^2)h_1
  -\gamma_2(1+\gamma_1^2+\gamma_2^2)h_2 + \frac{1}{\kappa}p + G_3.
\end{array}
\right.                                  \eqno{(2.27)}
$$
Here $G_k$, $k=1,2,3$, are linear combinations of $\ppp_jg_k, q_1,q_2,
q_3, q_4$, $j=1,2, k=1,2,3$, with coefficients given by $\gamma$ and
its first-order derivatives.
We can uniquely solve (2.27) with respect to $h_1,h_2, p$:
$$
\left(
\begin{array}{ccc}
& h_1(x_1,x_2)\\
& h_2(x_2,x_2)\\
& p\XXX\\
\end{array}\right)
= \widetilde{A}(x_1,x_2)
\left(
\begin{array}{ccc}
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_1 - G_1\\
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_2 - G_2\\
& \frac{1+\gamma_1^2+\gamma_2^2}{\kappa}q_3 - G_3\\
\end{array}\right), \quad (x_1,x_2) \in D_1.
                                    \eqno{(2.28)}
$$
Here $\widetilde{A} \in C^1(\overline{D_1})$ and
 $\mbox{det}\,\widetilde{A} \ne 0$ on $\overline{D_1}$.  The equations
(2.25), (2.26) and (2.28) imply the existence of a $10\times 10$ matrix
$A \in C^1(\overline{D_1})$ satisfying the conditions in the lemma.
Thus the proof of Lemma 4 is completed. $\blacksquare$
\\
\vspace{0.2cm}

Now, in terms of Lemmata 3 and 4, we complete the proof of Theorem 1
as follows.
We consider only the case of $n=3$.  Without loss of generality,
$\Gamma$ is given by
$\Gamma = \{ \XXX;\thinspace x_1,x_2 \in D_1\}$ with
$\gamma \in C^2(\overline{D_1})$.
\\
We set $\nabla_{x_1,x_2}v = (\ppp_1v_1,\ppp_2v_1,
\ppp_1v_2, \ppp_2v_2, \ppp_1v_3, \ppp_2v_3)^T$.
Then, by Lemmata 2 and 3, we have
\begin{align*}
& \left(
\begin{array}{cc}
& \ppp_{\nu}v\XXX\\
& p\XXX\\
\end{array}\right)
= \left(
\begin{array}{cc}
& \frac{1}{1+\gamma_1^2+\gamma_2^2}
((\ppp_1\gamma)\ppp_1v + (\ppp_2\gamma)\ppp_2v - \ppp_3v)\XXX\\
& p\XXX\\
\end{array}\right)\\
=& B_1(x_1,x_2)
\left(
\begin{array}{ccc}
&(\nabla_{x_1,x_2}v)\XXX\\
&(\sigma(v,p)\nu)\XXX\\
\end{array}\right), \quad (x_1,x_2) \in D_1,
\end{align*}
with a $4\times 6$ matrix $B_1 \in C^1(\overline{D_1})$.
Therefore
$$
\Vert \ppp_{\nu}v(\cdot,t)\Vert_{H^1(\Gamma)}
+ \Vert p(\cdot,t)\Vert_{H^1(\Gamma)}
= \left\Vert
B\left(
\begin{array}{ccc}
&\nabla_{x_1,x_2}v\\
&\sigma(v,p)\nu\\
\end{array}\right)
(\cdot,t)\right\Vert_{H^1(\Gamma)}
\le C\left\Vert
\left(
\begin{array}{ccc}
&\nabla_{x_1,x_2}v\\
&\sigma(v,p)\nu\\
\end{array}\right)
(\cdot,t)\right\Vert_{H^1(\Gamma)}
$$
and
$$
\Vert \ppp_{\nu}v(\cdot,t)\Vert_{L^2(\Gamma)}
\le C\left\Vert
\left(
\begin{array}{ccc}
&\nabla_{x_1,x_2}v\\
&\sigma(v,p)\nu\\
\end{array}\right)
(\cdot,t)\right\Vert_{L^2(\Gamma)}
$$
by $B \in C^1(\overline{D_1})$.  Consequently the interpolation
inequality (e.g., Theorem 7.7 (p.36) in Lions and Magenes \cite{LM})
yields
$$
\Vert \ppp_{\nu}v(\cdot,t)\Vert_{H^{\frac{1}{2}}(\Gamma)}
+ \Vert p(\cdot,t)\Vert_{H^{\frac{1}{2}}(\Gamma)}
\le \left\Vert
\left(
\begin{array}{ccc}
&\nabla_{x_1,x_2}v\\
&\sigma(v,p)\nu\\
\end{array}\right)
(\cdot,t)\right\Vert_{H^{\frac{1}{2}}(\Gamma)}
$$
for $0\le t \le T$.
Hence
$$
\Vert \ppp_{\nu}v\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert p(\cdot,t)\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
\le C(\Vert v\Vert_{L^2(0,T;H^1(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}).
$$
With this, Lemma 3 completes the proof of Theorem 1. $\blacksquare$
\\
\section{Conditional stability for the lateral Cauchy problem}

In this section, we discuss \\
{\bf lateral Cauchy problem}\\
We are given a suboundary $\Gamma$ of $\ppp\Omega$ arbitrarily.
Let $(v,p) \in H^{2,1}(Q) \times H^{1,0}(Q)$ satisfy (1.1) and
(1.2).  Determine $(v,p)$ in some subdomain of $Q$ by
$(v, \sigma(v,p)\nu)$ on $\Gamma \times (0,T)$.

In the case of the parabolic equation, there are very many works, and
here we do not list up comprehensively and as restricted references,
see Landis \cite{La}, Mizohata \cite{Mi},
Saut and Scheurer \cite{SS}, Sogge \cite{So}.  See also the monographs
Beilina and Klibanov \cite{BeKl}, Isakov \cite{Is},
Klibanov and Timonov \cite{KlTi}.

Combining a Carleman estimate and a cut-off function, we can 
prove
\\
{\bf Proposition 1.}\\
{\it Let $\va(x,t)$ be given in Theorem 1.  We set
$$
Q(\ep) = \{ (x,t) \in \Omega \times (0,T); \va(x,t) >
\ep\}
$$
with $\ep > 0$.  Moreover we assume that
$$
\oooo{Q(0)} \subset Q \cup (\Gamma \times [0,T])
$$
with subboundary $\Gamma \subset \ppp\Omega$.  Then for any small
$\ep>0$, there exist constants $C>0$ and $\theta\in (0,1)$ such that
$$
\Vert v\Vert_{H^{2,1}(Q(\ep))} + \Vert p\Vert_{H^{1,0}(Q(\ep))}
\le C(\Vert v\Vert^{1-\theta}_{H^{1,1}(Q)}
+ \Vert p\Vert_{L^2(Q)})G^\theta
+ CG,
$$
where we set
$$
G^2 := \Vert F\Vert_{L^2(Q)}^2
+ \Vert v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_tv \Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}.
$$}

As for the proof of Proposition 1, see Theorem 3.2.2 in section 3.2 of
\cite{Is} for example.

Proposition 1 gives an estimate of the solution in $Q(\ep)$ by
data on $\Gamma \times (0,T)$, and $Q(\ep)$ and $\Gamma$ are
determined by an a priori given function $d(x)$.
Therefore the proposition does not give a suitable answer
to our lateral Cauchy problem as stated above, where we are
requested to estimate the solution by data on as a small subboundary
$\Gamma \times (0,T)$ as possible.

In fact, in this section, we prove
\\
%
{\bf Theorem 2 (conditional stability)}\\
{\it Let $\Gamma \subset \ppp\Omega$ be an arbitrary non-empty
subboundary of $\ppp\Omega$.
For any $\ep > 0$ and
an arbitrary bounded domain $\Omega_0$ such that
$\overline{\Omega_0} \subset \Omega \cup \Gamma$, $\ppp\Omega_0
\cap \ppp\Omega$ is a non-empty open subset of $\ppp\Omega$ and
$\ppp\Omega_0 \cap \ppp\Omega \subsetneqq \Gamma$,
there exist constants $C>0$ and $\theta \in (0,1)$ such that
$$
\Vert v\Vert_{H^{2,1}(\Omega_0\times (\ep,T-\ep))}
+ \Vert p\Vert_{H^{1,0}(\Omega_0\times (\ep,T-\ep))}
$$
$$
\le C(\Vert v\Vert_{H^{1,1}(Q)} + \Vert p\Vert_{L^2(Q)})
^{1-\theta}
(\Vert F\Vert_{L^2(Q)}
+ \Vert v\Vert_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert v \Vert_{H^1(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))})^{\theta}
$$
$$
+ C(\Vert F\Vert_{L^2(Q)}
+ \Vert v\Vert_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_tv \Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}).
                                                            \eqno{(3.1)}
$$}
\\

In Theorem 2, in order to estimate $(v,p)$, we have to assume
a priori bounds of $\Vert v\Vert_{H^{1,1}(Q)}$ and
$\Vert p\Vert_{L^2(Q)}$.  Thus estimate (3.1)
is called a conditional stability estimate. %
We note that (3.1) is rewritten as
\begin{align*}
&\Vert v\Vert_{H^{2,1}(\Omega_0\times (\ep,T-\ep))}
+ \Vert p\Vert_{H^{1,0}(\Omega_0\times (\ep,T-\ep))}\\
=& O((\Vert F\Vert_{L^2(Q)}
+ \Vert v\Vert_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert v \Vert_{H^1(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))})^{\theta})
\end{align*}
as $\Vert F\Vert_{L^2(Q)} + \Vert v\Vert_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert v \Vert_{H^1(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
\longrightarrow 0$.
Thus the estimate indicates stability of H\"older type. 

For the homogeneous Stokes equations:
$$
\ppp_tv - \Delta v + \nabla p = 0, \quad
\ddd v = 0 \quad \mbox{in $Q$},
$$
Boulakia \cite{Bo} (Proposition 2) proved the conditional stability
in $\Omega_0 \times (\ep, T-\ep)$ on the basis of a Carleman estimate
in \cite{Bo}.  The norm of boundary data in \cite{Bo} is stronger than our
chosen norm.
%

The theorem does not directly give an estimate when $\Omega_0 = \Omega$, but
we can derive an estimate in $\Omega$ by an argument similar to
Theorem 5.2 in Yamamoto \cite{Ya} and we do not discuss details.
Boulakia \cite{Bo} (Theorem 1) established a conditional stability
estimate up to $\ppp\Omega$ by boundary or interior data.
The argument is based on the interior estimate in $\Omega_0
\times (\ep, T-\ep)$ and an
argument similar to Theorem 5.2 in \cite{Ya}.

Theorem 2 immediately implies the global uniqueness of
the solution:
\\
{\bf Corollary}.\\
{\it Let $\Gamma \subset \ppp\Omega$ be an arbitrarily fixed subboundary.
If $(v,p) \in H^{2,1}(Q)\times H^{1,0}(Q)$ satisfies (1.1) and (1.2), and
$v = \sigma(v,p)\nu = 0$ on $\Gamma \times (0,T)$, then
$\vert v\vert = \sigma(v,p)\nu = 0$ in $\Omega \times (0,T)$.}
\\
\vspace{0.2cm}
\\
{\bf Proof of Theorem 2}.
Once a relevant Carleman estimate for the Navier-Stokes equations is proved,
the proof is similar to Theorem 5.1 in \cite{Ya}.
Thus, according to $\Omega_0$ and
$\Gamma$, we have to choose a suitable weight function $\va$.
For this, we show\\
{\bf Lemma 5.}\\
{\it Let $\omega$ be an arbitrarily fixed subdomain of $\Omega$
such that $\overline{\omega} \subset \Omega$.  Then there exists
a function $d \in C^2(\oooo\Omega)$ such that
$$
d(x) > 0 \quad x \in \Omega, \quad d\vert_{\ppp\Omega}
= 0, \quad \vert \nabla d(x)\vert > 0, \quad
x\in \overline{\Omega\setminus\omega}.
$$}
\\
\vspace{0.1cm}
\\
For the proof, see Fursikov and Imanuvilov \cite{FuIm}, Imanuvilov \cite{Im},
Imanuvilov, Puel and Yamamoto \cite{ImPuYa}.

We choose a bounded domain $\Omega_1$
with smooth boundary such that
$$
\Omega \subsetneqq \Omega_1, \quad
\oooo{\Gamma} = \oooo{\ppp\Omega \cap \Omega_1}, \quad
\ppp\Omega \setminus \Gamma \subset \ppp\Omega_1,      \eqno{(3.2)}
$$
and $\Omega_1\setminus \oooo\Omega$ contains
some non-empty open set.
We note that $\Omega_1$ is constructed by taking a union of
$\Omega$ and a domain $\www{\Omega} \subset \R^n \setminus
\overline{\Omega}$ such that
$\www\Omega \cap \ppp\Omega = \Gamma$.
Choosing $\oooo\omega \subset \Omega_1 \setminus \oooo\Omega$,
and applying Lemma 5 to obtain $d \in C^2(\oooo\Omega_1)$ satisfying
$$
d(x) > 0, \quad x \in \Omega_1, \quad
d(x) = 0, \quad x \in \ppp\Omega_1, \quad
\vert \nabla d(x)\vert > 0, \quad x \in \oooo\Omega.
                                                            \eqno{(3.3)}
$$
Then, since $\overline{\Omega_0} \subset \Omega_1$, we can choose
sufficiently large $N > 1$ such that
$$
\{ x\in \Omega_1; \thinspace d(x) > \frac{4}{N}\Vert d\Vert
_{C(\oooo{\Omega_1})} \} \cap \oooo\Omega \supset \Omega_0.                                                                    \eqno{(3.4)}
$$
Moreover we choose sufficiently large $\beta > 0$ such that
$$
\beta\ep^2 < \Vert d\Vert_{C(\oooo{\Omega_1})} < 2\beta\ep^2.     \eqno{(3.5)}
$$
We arbitrarily fix $t_0 \in [\sqrt{2}\ep, T-\sqrt{2}\ep]$.  We set
$\va(x,t) = e^{\la\psi(x,t)}$ with fixed large parameter $\la>0$ and
$\psi(x,t) = d(x) - \beta (t-t_0)^2$,
$\mu_k = \exp\left( \la\left(\frac{k}{N}\Vert d\Vert_{C(\oooo{\Omega_1})}
- \frac{\beta\ep^2}{N}\right)\right)$, $k=1,2,3,4$, and
$D = \{ (x,t); \thinspace x \in\oooo\Omega, \quad \va(x,t) > \mu_1
\}$.

Then we can verify that
$$
\Omega_0 \times \left( t_0 - \frac{\ep}{\sqrt{N}},
t_0 + \frac{\ep}{\sqrt{N}} \right) \subset D
\subset \oooo\Omega \times (t_0-\sqrt{2}\ep,t_0+\sqrt{2}\ep).
                                      \eqno{(3.6)}
$$
In fact, let $(x,t) \in\Omega_0 \times \left( t_0-\frac{\ep}{\sqrt{N}},
t_0 + \frac{\ep}{\sqrt{N}} \right)$.  Then, by (3.4) we have
$x \in \oooo\Omega$ and $d(x) > \frac{4}{N}
\Vert d\Vert_{C(\oooo{\Omega_1})}$, so that
$$
d(x) - \beta(t-t_0)^2
> \frac{4}{N}\Vert d\Vert_{C(\oooo{\Omega_1})}
- \frac{\beta\ep^2}{N},
$$
that is, $\va(x,t) > \mu_4$, which implies that $(x,t) \in D$
by the definition of $D$.
Next let $(x,t) \in D$.  Then $d(x) - \beta (t-t_0)^2
> \frac{1}{N}\Vert d\Vert_{C(\oooo{\Omega_1})}
- \frac{\beta \ep^2}{N}$.  Therefore
$$
\Vert d\Vert_{C(\oooo{\Omega_1})} - \frac{1}{N}
\Vert d\Vert_{C(\oooo{\Omega_1})} + \frac{\beta\ep^2}{N}
> \beta (t-t_0)^2.
$$
Applying (3.5), we have $2\left(1-\frac{1}{N}\right)\beta\ep^2
+ \frac{\beta\ep^2}{N} > \left(1 - \frac{1}{N}\right)\Vert d\Vert
_{C(\oooo{\Omega_1})} + \frac{\beta \ep^2}{N} > \beta(t-t_0)^2$, that is,
$2\beta \ep^2 > \beta(t-t_0)^2$, which implies that
$t_0 - \sqrt{2}\ep < t < t_0+\sqrt{2}\ep$.
The verification of (3.6) is completed.

Next we have
$$
\left\{
\begin{array}{rl}
&\ppp D \subset \Sigma_1 \cup \Sigma_2, \\
&\Sigma_1 \subset \Gamma \times (0,T), \quad
\Sigma_2 = \{ (x,t); \thinspace x\in\Omega, \thinspace
\va(x,t) = \mu_1\}.
\end{array}
\right.                           \eqno{(3.7)}
$$
In fact, let $(x,t) \in \ppp D$.  Then $x \in \oooo\Omega$ and
$\va(x,t) \ge \mu_1$.  We separately consider the cases
$x \in \Omega$ and $x \in \ppp\Omega$.  First let $x \in \Omega$.
If $\va(x,t) > \mu_1$, then $(x,t)$ is an interior point of
$D$, which is impossible.
Therefore $\va(x,t) = \mu_1$, which implies $(x,t) \in
\Sigma_2$.
Next let $x \in \ppp\Omega$.  Let $x \in \ppp\Omega \setminus
\Gamma$.  Then $x \in \ppp\Omega_1$ by the third condition in
(3.2), and $d(x) = 0$ by the second condition in (3.3).
On the other hand, $\va(x,t) \ge \mu_1$ yields that
$$
d(x) - \beta(t-t_0)^2 = -\beta(t-t_0)^2
\ge \frac{1}{N}\Vert d\Vert_{C(\oooo{\Omega_1})}
- \frac{\beta\ep^2}{N},
$$
that is, $0\le \beta(t-t_0)^2 \le \frac{1}{N}(
-\Vert d\Vert_{C(\oooo{\Omega_1})} + \beta\ep^2)$, which is impossible
by (3.5).  Therefore $x \in \Gamma$.  By (3.6), we see that
$0 < t < T$ and the verification of (3.7) is completed.

We apply Theorem 1 in $D$.
Henceforth $C>0$ denotes generic constants independent of $s$ and
choices of $v, p$.
We need a cut-off function because we have no data on
$\ppp D \setminus (\Gamma \times (0,T))$.
Let $\chi \in C^{\infty}(\R^{n+1})$ satisfying $0 \le \chi \le 1$ and
$$
\chi(x,t) =
\left\{
\begin{array}{rl}
1, \qquad & \va(x,t) > \mu_3, \\
0, \qquad & \va(x,t) < \mu_2.
\end{array}
\right.                             \eqno{(3.8)}
$$
We set $y = \chi v$ and $q = \chi p$.  Then, by (1.1) and (1.2), we have
\begin{align*}
& \ppp_ty - \kappa\Delta y + (A\cdot \nabla)y + (y\cdot \nabla)B
+ \nabla q\\
=& \chi F + v\ppp_t\chi - 2\kappa\nabla\chi\cdot \nabla v
- \kappa(\Delta\chi)v + (A\cdot\nabla \chi)v + p(\nabla\chi)
\quad \mbox{in $D$}
\end{align*}
and
$$
\ddd y = \nabla\chi \cdot v \quad \mbox{in $D$}.
$$
By (3.7) and (3.8), we see that
$$
\vert y\vert  = \vert \nabla y\vert = \vert q\vert = 0
\qquad \text{on $\Sigma_2$}.
$$
Hence Theorem 1 yields
\begin{align*}
&  \Vert (y,q)\Vert^2_{ \mathcal X_s(D)}
\le C\int_D \vert F\vert^2 \weight dxdt \\
+ &C\int_D  \vert v\ppp_t\chi - 2\kappa\nabla\chi\cdot \nabla v
- \kappa(\Delta\chi)v + (A\cdot\nabla \chi)v + p(\nabla\chi)\vert^2
\weight dxdt\\
+ &C\int_D (\vert \nabla\chi\cdot v\vert^2
+ \vert \nabla_{x,t}(\nabla\chi\cdot v)\vert^2)\weight dxdt
\end{align*}
$$
+ Ce^{Cs}(\Vert \chi v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_t(\chi v)\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(\chi v, \chi p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))})
                                                     \eqno{(3.9)}
$$
for $s \ge s_0$.
We can verify $\Vert \chi v\Vert_{H^{\gamma}(\Gamma)}
\le C\Vert v\Vert_{H^{\gamma}(\Gamma)}$ with $\gamma = 0,1,2$, and for
$j=\frac{1}{2}$ and $j=\frac{3}{2}$, the interpolation inequality yields
$$
\Vert \chi v\Vert^2_{L^2(0,T;H^j(\Gamma))}
\le C\Vert v\Vert^2_{L^2(0,T;H^j(\Gamma))}, \quad
\Vert \ppp_t(\chi v)\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
\le C\Vert \ppp_tv\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}.
$$
Therefore, since 
$$
\sigma(\chi v, \chi p)\nu = \chi\sigma(v,p)\nu
+ \kappa((\ppp_i\chi)v_j + (\ppp_j\chi)v_i)_{1\le i,j\le n}\nu,
$$
we have 
$$
\Vert \sigma(\chi v, \chi p)\nu\Vert
_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
\le \Vert \sigma(v,p)\nu\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ C\Vert v\Vert_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
$$
by $\chi \in C^{\infty}(\R^{n+1})$.
Hence
\begin{align*}
&\Vert \chi v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_t(\chi v)\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(\chi v, \chi p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}\\
\le& C(\Vert v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_tv\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v, p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}).
\end{align*}
We recall that
$$
G^2 = \Vert F\Vert^2_{L^2(Q)}
+ \Vert v\Vert^2_{L^2(0,T;H^{\frac{3}{2}}(\Gamma))}
+ \Vert \ppp_t v\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}
+ \Vert \sigma(v, p)\nu\Vert^2_{L^2(0,T;H^{\frac{1}{2}}(\Gamma))}.
$$

The integrands of the second and the third terms on the right-hand
side of (3.9) do not vanish only if $\va(x,t) \le \mu_3$, because
these coefficients include derivatives of $\chi$ as factors and
by (3.8) vanish if $\va(x,t) > \mu_3$.  Therefore
\begin{align*}
&\vert \mbox{[the second and the third terms on the right-hand side of
(3.9)]}
\vert\\
\le &C(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert_{L^2(Q)}^2)e^{2s\mu_3}.
\end{align*}
Consequently (3.9) yields
$$
 \Vert (y,q)\Vert^2_{ \mathcal X_s(D)}
\le C(\Vert v\Vert^2_{H^{1,1}(Q)} + \Vert p\Vert_{L^2(Q)}^2)e^{2s\mu_3}
+ Ce^{Cs}G^2     \quad \forall s\ge s_0.                      \eqno{(3.10)}
$$


By (3.4) and the definition of $D$, we can directly verify that
$(x,t) \in \Omega_0 \times \left(t_0 - \frac{\ep}{\sqrt{N}},
t_0 + \frac{\ep}{\sqrt{N}}\right)$ implies $\va(x,t) > \mu_4$.
Therefore, noting (3.6) and (3.8), we see that
\begin{align*}
&  \Vert (y,q)\Vert^2_{ \mathcal X_s(D)}\ge  \Vert (v,p)\Vert^2_{ \mathcal X_s(\Omega_0\times (t_0 - \frac{\ep}{\sqrt{N}},t_0 + \frac{\ep}{\sqrt{N}}) )}\\
\ge& e^{2s\mu_4}\int^{t_0 + \frac{\ep}{\sqrt{N}}}_{t_0 - \frac{\ep}{\sqrt{N}}}
\int_{\Omega_0}
\left\{ \frac{1}{s^2}\left( \vert \ppp_tv\vert^2
+ \ssss \vert \ppp_i\ppp_jv \vert^2\right)
+ \vert \nabla v\vert^2 + s^2\vert v\vert^2
+ \frac{1}{s}\vert \nabla p\vert^2 + s\vert p\vert^2 \right\} dxdt.
\end{align*}
Hence (3.10) yields
\begin{align*}
& e^{2s\mu_4}\int^{t_0+\frac{\ep}{\sqrt{N}}}_{t_0-\frac{\ep}{\sqrt{N}}}
\int_{\Omega_0}
\left\{ \frac{1}{s^2}\left( \vert \ppp_tv\vert^2
+ \ssss \vert \ppp_i\ppp_jv \vert^2\right)
+ \vert \nabla v\vert^2 + s^2\vert v\vert^2
+ \frac{1}{s}\vert \nabla p\vert^2 + s\vert p\vert^2 \right\} dxdt\\
\le & C(\Vert v\Vert^2_{H^{1,1}(Q)} + \Vert p\Vert_{L^2(Q)}^2)e^{2s\mu_3}
+ Ce^{Cs}G^2.
\end{align*}
Therefore
\begin{align*}
& \int^{t_0+\frac{\ep}{\sqrt{N}}}_{t_0-\frac{\ep}{\sqrt{N}}}
\int_{\Omega_0}
\left\{ \left( \vert \ppp_tv\vert^2
+ \ssss \vert \ppp_i\ppp_jv \vert^2\right)
+ \vert \nabla v\vert^2 + \vert v\vert^2
+ \vert \nabla p\vert^2 + \vert p\vert^2 \right\}dxdt\\
\le & Cs^2e^{-2s(\mu_4-\mu_3)}
(\Vert v\Vert^2_{H^{1,1}(Q)} + \Vert p\Vert_{L^2(Q)}^2)
+ Ce^{Cs}G^2\quad\forall s\ge s_0.
\end{align*}

By $\sup_{s>0} se^{-s(\mu_4-\mu_3)} < \infty$, we
estimate $se^{-2s(\mu_4-\mu_3)}$ by $e^{-s(\mu_4-\mu_3)}$ on
the right-hand side.
Moreover, replacing $C$ by $Ce^{Cs_0}$, we can have
$$
\Vert v\Vert^2_{H^{2,1}\left(\Omega_0\times
\left(t_0-\frac{\ep}{\sqrt{N}},t_0+\frac{\ep}{\sqrt{N}}\right)\right)}
+ \Vert p\Vert^2_{H^{1,0}\left(\Omega_0\times
\left(t_0-\frac{\ep}{\sqrt{N}},t_0+\frac{\ep}{\sqrt{N}}\right)\right)}
$$
$$
\le Ce^{-s(\mu_4-\mu_3)}(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)}) + Ce^{Cs}G^2   \eqno{(3.11)}
$$
for all $s \ge 0$.
Let $m \in \N$ satisfy 
$\sqrt{2}\ep + \frac{m\ep}{\sqrt{N}} \le T - \sqrt{2}\ep
\le \sqrt{2}\ep + \frac{(m+1)\ep}{\sqrt{N}} \le T$.

We here notice that the constant $C$ in (3.11) is independent
also of $t_0$, provided that $\sqrt{2}\ep \le t_0 \le T-\sqrt{2}\ep$.
In (3.11), taking $t_0 = \sqrt{2}\ep + \frac{j\ep}{\sqrt{N}}$,
$j=0,1,2,..., m$ and summing up over $j$, we have
$$
\Vert v\Vert^2_{H^{2,1}\left(\Omega_0\times
\left(\sqrt{2}\ep- \frac{\ep}{\sqrt{N}}, T-\sqrt{2}\ep - \frac{\ep}{\sqrt{N}}
\right)\right)}
+ \Vert p\Vert^2_{H^{1,0}\left(\Omega_0\times
\left(\sqrt{2}\ep- \frac{\ep}{\sqrt{N}}, T-\sqrt{2}\ep - \frac{\ep}{\sqrt{N}}
\right)\right)}
$$
$$
\le Ce^{-s(\mu_4-\mu_3)}(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)})
+ Ce^{Cs}G^2
$$
for all $s \ge 0$.
Here we note that $T - \sqrt{2}\ep \le \sqrt{2}\ep + \frac{(m+1)\ep}
{\sqrt{N}}$ implies
$T - \sqrt{2}\ep - \frac{m\ep}{\sqrt{N}} \le \sqrt{2}\ep + \frac{1}{\sqrt{N}}
\ep$.  Replacing $\left(\sqrt{2}+\frac{1}{\sqrt{N}}\right)\ep$ by
$\ep$, we have
$$
\Vert v\Vert^2_{H^{2,1}(\Omega_0\times (\ep, T-\ep))}
+ \Vert p\Vert^2_{H^{1,0}(\Omega_0\times (\ep,T-\ep))}
$$
$$
\le Ce^{-s(\mu_4-\mu_3)}(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)})
+ Ce^{Cs}G^2                               \eqno{(3.12)}
$$
for all $s \ge s_0$.

First let $G=0$.  Then letting $s \to \infty$ in (3.12), we see that
$\vert v\vert = \vert p\vert =0$ in $\Omega_0 \times (\ep,T-\ep)$, so
that the conclusion of Theorem 2 holds true.  Next let $G\ne 0$.
First let $G \ge \Vert v\Vert_{H^{1,1}(Q)} +
\Vert p\Vert_{L^2(Q)}$.  Then (3.12) implies
$\Vert v\Vert_{H^{2,1}(\Omega_0\times (\ep,T-\ep))}
+ \Vert p\Vert_{H^{1,0}(\Omega_0\times (\ep,T-\ep))}
\le Ce^{Cs}G$ for $s \ge 0$, which already proves the theorem.
Second let $G < \Vert v\Vert_{H^{1,1}(Q)} + \Vert p\Vert
_{L^2(Q)}$.  In order to make the right-hand side of (3.12) smaller,
we choose $s > 0$ such that
$$
e^{-s(\mu_4-\mu_3)}(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)}) = e^{Cs}G^2.
$$
By $G\ne 0$, we can choose
$$
s = \frac{1}{C+\mu_4-\mu_3} \log\frac{\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)}}{G^2} > 0.
$$
Then (3.12) gives
$$
\Vert v\Vert^2_{H^{2,1}(\Omega_0\times (\ep,T-\ep))}
+ \Vert p\Vert^2_{H^{1,0}(\Omega_0\times (\ep,T-\ep))}
\le 2C(\Vert v\Vert^2_{H^{1,1}(Q)}
+ \Vert p\Vert^2_{L^2(Q)})^{\frac{C}{C+\mu_4-\mu_3}}
G^{\frac{2(\mu_4-\mu_3)}{C+\mu_4-\mu_3}}.
$$
The the proof of Theorem 2 is completed. $\blacksquare$
%
%
\\


\begin{thebibliography}{99} %
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Carleman estimates for parabolic equations with
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\bibitem{LUW} R.Y. Lai, G. Uhlmann and J.-N. Wang,
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\bibitem{SS} J.-C. Saut and B. Scheurer,
Unique continuation for some evolution equations,
J. Diff. Eqns {\bf 66} (1987) 118-139.

\bibitem{So} C.D. Sogge,
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\bibitem{Ya} M. Yamamoto,
Carleman estimates for parabolic equations and applications,
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\end{document}



\ref \no 1 \by A. Amirov and M. Yamamoto
\paper Unique continuation and an inverse problem
for hyperbolic equations across a general hypersurface
\jour J\. Phys\.: Conf\. Ser\.
\vol 12
\yr 2005
\pages 1-12
\endref

\ref \no 2 \by L. Baudouin and J.-P. Puel
\paper Uniqueness and stability in an inverse problem for
the Schr\"odinger equation
\jour Inverse Problems
\vol 18
\yr 2002
\pages 1537 -- 1554
\endref

\ref \no 3 \by M. Bellassoued
\paper Uniqueness and stability in determining the speed of
propagation
of second-order hyperbolic equation with variable coefficients
\jour Appl\. Anal\.
\vol 83
\yr 2004
\pages 983--1014
\endref

\ref \no 4 \by M. Bellassoued
\paper Global logarithmic stability in inverse hyperbolic
problem by arbitrary boundary observation
\jour Inverse Problems
\vol 20
\yr 2004
\pages 1033--1052
\endref





\ref \no 10 \by J. Fan, M. Di Cristo, Y. Jiang and G.Nakamura
\paper Inverse viscosity problem for the Navier-Stokes
equation
\jour J\. Math\. Anal\. Appl\.
\vol 365
\yr 2010
\pages 750--757
\endref


\ref \no 12 \by E. Fern\'andez-Cara, S. Guerrero,
O. Yu. Imanuvilov and J.-P. Puel
\paper Remarks on exact controllability for Stokes and
Navier-Stokes systems
\jour C\. R\. Acad\. Paris, Ser\. I\.
\vol 338
\yr 2004
\pages 375--380
\endref

\ref \no 13  \by E. Fern\'andez-Cara, S. Guerrero,
O. Yu. Imanuvilov and J.-P. Puel
\paper Local exact
controllability of the Navier-Stokes system
\jour J\. Math\. Pures et Appl\.
\vol 83
\year 2004
\pages 1501--1542
\endref

\ref \no 14 \by A. V. Fursikov and O. Yu. Imanuvilov
\book Controllability of Evolution Equations
\publ Seoul National University
\publaddr Seoul
\yr 1996
\endref



\ref \no 16 \by O. Yu. Imanuvilov
\paper Boundary controllability of parabolic equations
\jour Sbornik Math\.
\vol 186
\yr 1995
\pages 879--900
\endref




\ref \no 19 \by O. Yu. Imanuvilov and J.-P. Puel
\paper Global Carleman estimates for weak solutions of
elliptic  nonhomogeneous Dirichlet problems
\jour IMRN
\yr 2003
\pages 883--913
\endref

\ref \no 20 \by O. Yu. Imanuvilov, J.-P. Puel and M. Yamamoto
\paper Carleman estimates for parabolic equations with
nonhomogeneous boundary conditions
\jour Chin\. Ann\. Math\. Ser\. B
\vol 30
\yr 2009
\pages 333--378
\endref


1506.02534.extracted_bib
{BeKl L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer-Verlag, Berlin, 2012.
1506.02534.extracted_bib
{Bo M. Boulakia, Quantification of the unique continuation property for the nonstationary Stokes problem, preprint.
1506.02534.extracted_bib
{ChImPuYa M. Choulli, O. Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Appl. Anal. {92 (2013), 2127-2143.
1506.02534.extracted_bib
{Fan1 J. Fan, M. Di Cristo, Y. Jiang, and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, J. Math. Anal. Appl. {365 (2010), 750-757.
1506.02534.extracted_bib
{Fan2 J. Fan, Y. Jiang, and G. Nakamura, Inverse problems for the Boussinesq system, Inverse Probl. {25 (2009), 085007 (10pp).
1506.02534.extracted_bib
{FeGuImPu E. Fern\'andez-Cara, S. Guerrero S, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. {83 (2004), 1501-1542.
1506.02534.extracted_bib
{FuIm A.V. Fursikov and O.Y. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Korea, 1996.
1506.02534.extracted_bib
{Im O.Y. Imanuvilov, Controllability of parabolic equations, Sbornik Math. {186 (1995), 879-900.
1506.02534.extracted_bib
{ImPu O.Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, International Mathematics Research Notices {16 (2003), 883-913.
1506.02534.extracted_bib
{ImPuYa O.Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. {30B (2009), 333-378.
1506.02534.extracted_bib
{ImYa1 O.Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for Navier-Stokes equations and Lam\'e system in two dimensions, Inverse Problems {31 (2015), 035004.
1506.02534.extracted_bib
{ImYa2 O.Y. Imanuvilov and M. Yamamoto, Equivalence of two inverse boundary value problems for the Navier-Stokes equations, 2015, http://arxiv.org/pdf/1501.02550.pdf
1506.02534.extracted_bib
{Is V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, Berlin, 2006.
1506.02534.extracted_bib
{KlTi M.V. Klibanov and A.A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004.
1506.02534.extracted_bib
{LUW R.Y. Lai, G. Uhlmann and J.-N. Wang, Inverse boundary value problem for the Stokes and the Navier-Stokes equations in the plane, Arch. Rational Mech. Anal., Digital Object Identifier (DOI) 10.1007/s00205-014-0794-1
1506.02534.extracted_bib
{La E.M. Landis, Some questions in the qualitative theory of elliptic and parabolic equations, AMS Transl. Series 2 {20 (1962) 173-238.
1506.02534.extracted_bib
{LM J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.
1506.02534.extracted_bib
{Mi S. Mizohata, Unicit\'e du prolongement des solutions pour quelques op\'erateurs diff\'erentiels paraboliques, Mem. College Sci. Univ. Kyoto {A 31 (1958) 219-239.
1506.02534.extracted_bib
{SS J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Diff. Eqns {66 (1987) 118-139.
1506.02534.extracted_bib
{So C.D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat. {28 (1990) 159-182.
1506.02534.extracted_bib
{Ya M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems {25 (2009) 123013 (75pp).