# Carleman estimate for the Navier-Stokes equations and applications

1 Oleg Y. Imanuvilov, 2 Luca Lorenzi, 3,4,5,6 M. Yamamoto

###### Abstract.

For linearized Navier-Stokes equations, we first derive a Carleman estimate
with a regular weight function. Then we apply it to establish conditional
stability for the lateral Cauchy problem and finally we prove conditional
stability estimates for inverse source problem of determining a spatially
varying divergence-free factor of a source term.

1 Department of Mathematics, Colorado State University
101 Weber Building, Fort Collins, CO 80523-1874, USA
e-mail<EMAIL_ADDRESS>
2 Department of Mathematical, Physical and Computer Sciences, University of
Parma, Parco Area delle Scienze 53/A, I-43124 Parma Italy
<EMAIL_ADDRESS>
3 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,
Meguro, Tokyo 153-8914, Japan
4 Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti,
Romania
5 Correspondence member of Accademia Peloritana dei Pericolanti,
Palazzo Università, Piazza S. Pugliatti 1 98122 Messina Italy
6 Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-
Maklaya St, Moscow, 117198, Russian Federation e-mail<EMAIL_ADDRESS>
##  1\. Introduction

Let $T>0$, and $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth
boundary $\partial\Omega$ and $A$, $B$ be sufficiently smooth. By $a^{T}$ we
denote the transpose of a vector $a$ under consideration. We set

$\partial_{j}=\frac{\partial}{\partial x_{j}},\quad
j=1,2,3,\quad\partial_{t}=\frac{\partial}{\partial
t},\quad\nabla=(\partial_{1},\partial_{2},\partial_{3}),\quad\nabla_{x,t}=(\nabla,\partial_{t})$

and

$\mbox{div}\thinspace v=\sum_{j=1}^{3}\partial_{j}v_{j}\quad\mbox{for
$v=(v_{1},v_{2},v_{3})^{T}$}.$

We consider

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}v(x,t)-\Delta
v+(A(x,t)\cdot\nabla)v+(v\cdot\nabla)B(x,t)+\nabla p=F(x,t),\\\
&\mbox{div}\thinspace v(x,t)=0,\quad x\in\Omega,\,0<t<T.\end{array}\right.$

Let $\Gamma\subset\partial\Omega$ and $\Omega_{0}\subset\Omega$ be given
subboundary and subdomain.

In this article, we consider the following two problems.
Continuation of solution by Cauchy data on $\Gamma\times(0,T)$. Let $(v,p)$
satisfy (1.1) with $F=0$ and let $I\subset(0,T)$. Then determine $(v,p)$ in
$\Omega_{0}\times I$ by $v$ and its derivatives on $\Gamma\times(0,T)$.
Inverse source problem. Let $t_{0}\in(0,T)$ be given and let $(v,p)$ satisfy
(1.1). Then determine $F$ by $v$ and its derivatives on $\Gamma\times(0,T)$
and $v(\cdot,t_{0})$.

A method based on Carleman estimate by Bukhgeim and Klibanov [5] is applicable
for proving stability estimates for these problems. Since [5], there have been
many works for parabolic, hyperbolic and other types of equations. We refer
only to Beilina and Klibanov [1], Bellassoued and Yamamoto [3], Imanuvilov and
Yamamoto [14], [16], [17], Klibanov [19], Klibanov and Timonov [20], Yamamoto
[22]. Here the references are very limited and see also the references in the
above works.

On the other hand, there are not sufficient works on inverse problems by the
same methodology for the Navier-Stokes equations in spite of the importance.
We refer only to Imanuvilov and Yamamoto [15], Bellassoued, Imanuvilov and
Yamamoto [2], Boulakia [4], Choulli, Imanuvilov, Puel and Yamamoto [6], Fabre
[7], Fan, Di Cristo, Jiang and Nakamura [8], Fan, Jiang and Nakamura [9]. See
also Imanuvilov and Yamamoto [17], [18] as for inverse problems for the fluid
equations.

Huang, Imanuvilov and Yamamoto [11] recently simplified the method by [5] and
here we apply it. We remark that we use the same type of Carleman estimate for
the Navier-Stokes equations as [2] but the derivation of a Carleman estimate
is simpler than [2].

Thus, in this article, we aim at improving the results in [2], and the main
achievements are:

  * •

a more feasible Carleman estimate (Theorem 1) for the applications.

  * •

an improvement of the conditional stability estimates for the continuation of
solutions by Cauchy data.

  * •

novel conditional stability estimates in determining force term $F(x,t)$ in
view of the divergence-free component.

Now we show the key Carleman estimate, and to this end, we need some
notations. Let $t_{0}\in(0,T)$ be arbitrarily taken. We choose $\delta>0$ such
that $0<t_{0}-\delta<t_{0}+\delta<T$ and we set

$I=(t_{0}-\delta,\,t_{0}+\delta),\quad Q:=\Omega\times I.$

Let
$\gamma=(\gamma_{1},\gamma_{2},\gamma_{3})\in(\mathbb{N}\setminus\\{0\\})^{3}$
and $|\gamma|=\gamma_{1}+\gamma_{2}+\gamma_{3}$. For
$k,\ell\in\mathbb{N}\cup\\{0\\}$ and $s>0$, we set

$H^{k,\ell}(Q)=\\{v\in L^{2}(Q);\thinspace\partial_{x}^{\gamma}v\in
L^{2}(Q),\thinspace|\gamma|\leq k,\thinspace\partial_{t}^{j}v\in L^{2}(Q)\quad
0\leq j\leq\ell\\}.$

Let $d\in C^{2}(\overline{\Omega})$ satisfy

$None$ $|\nabla d|\neq 0\quad\mbox{on $\overline{\Omega}$}.$

Henceforth we fix a large constant $\lambda>0$, and throughout the article, we
omit the $\lambda$-dependency of the constants. For arbitrarily chosen
constant $\beta>0$, we set

$\varphi(x,t)=e^{\lambda(d(x)-\beta(t-t_{0})^{2})},\quad(x,t)\in Q.$

Henceforth by $[u]_{k}$ with $k=1,2,3$, we denote the $k$-th component of a
vector $u$, and $\nabla(\mbox{rot}\thinspace v)$ means the matrix with the
elements $\partial_{j}[\mbox{rot}\thinspace v]_{k}$, $1\leq j,k\leq 3$.

Then
Theorem 1 (Carleman estimate for the Navier-Stokes equations).
There exist constants $s_{0}>0$ and $C>0$ such that

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s}(|\partial_{t}\mbox{rot}\thinspace
v|^{2}+|\partial_{t}v|^{2}+|\Delta(\mbox{rot}\thinspace v)|^{2}+|\Delta
v|^{2})$ $\displaystyle+$ $\displaystyle s(|\nabla(\mbox{rot}\thinspace
v)|^{2}+|\nabla v|^{2})+s^{3}(|\mbox{rot}\thinspace
v|^{2}+|v|^{2})\biggr{\\}}e^{2s\varphi}dxdt$ $\displaystyle\leq$
$\displaystyle C\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{3}\int_{\partial\Omega\times
I}\sum_{j=0}^{1}(|\nabla_{x,t}^{j}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}v|^{2})e^{2s\varphi}dSdt$ $\displaystyle+$ $\displaystyle
Cs^{3}\sum_{\kappa=0}^{1}\int_{\Omega}\sum_{j=0}^{1}|\nabla^{j}(\mbox{rot}\thinspace
v(x,t_{0}+(-1)^{\kappa}\delta))|^{2}e^{2s\varphi(x,t_{0}+\delta)}dx$

for all $s>s_{0}$, $p\in L^{2}(I;H^{1}(\Omega))$ and $v,\mbox{rot}\thinspace
v\in H^{2,1}(Q)$.

We note that $\varphi(x,t_{0}+\delta)=\varphi(x,t_{0}-\delta)$, $x\in\Omega$.
Remark 1.
We can estimate also $\frac{1}{s}|\nabla p|^{2}e^{2s\varphi}$ on the left-hand
side if we replace the term $\int_{Q}|\mbox{rot}\thinspace F|^{2}dxdt$ by
$\int_{Q}(|F|^{2}+|\mbox{rot}\thinspace F|^{2}dxdt$.

Here we choose a regular weight function
$e^{\lambda(d(x)-\beta(t-t_{0})^{2})}$. A similar Carleman estimate with the
same type of weight function for the Navier-Stokes equations is derived in
[2], but the derivation here is simpler. On the other hand, we can choose
other type of weight function with singularity at $t=0,T$, which was created
by Imanuvilov [12] for a single parabolic equation. For linearized Navier-
Stokes equations, we refer to Choulli, Imanuvilov, Puel and Yamamoto [6], Fan,
Di Cristo, Jiang and Nakamura [8], Fan, Jiang and Nakamura [9], Fernández-
Cara, Guerrero, Imanuvilov and Puel [10].

Next we formulate the main result for the continuation of solution. For small
$\delta>0$ satisfying $0<t_{0}-\delta<t_{0}+\delta$, we choose
$0<\delta_{0}<\delta<T$. We consider

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}v(x,t)-\Delta
v+(A(x,t)\cdot\nabla)v+(v\cdot\nabla)B+\nabla p=0,\\\ &\mbox{div}\thinspace
v=0,\quad(x,t)\in Q:=\Omega\times I.\end{array}\right.$

Then we are requested to estimate $(v,p)$ in some subdomain by data of $v$ on
$\Gamma\times(0,T)$, where $\Gamma$ is an arbitrary subboundary of
$\partial\Omega$.

For an arbitrarily chosen subboundary $\Gamma\subset\partial\Omega$, we choose
an arbitrary subdomain $\Omega_{0}\subset\Omega$ such that

$\overline{\Omega_{0}}\subset\Omega\cup\Gamma,\quad\mbox{$\overline{\Omega_{0}}\cap\partial\Omega$
is included in Int $(\Gamma)$: the interior of $\Gamma$.}$

We are ready to state the main result for the continuation.
Theorem 2 (continuation of solution).
Let $(v,p)\in H^{2,1}(Q)\times L^{2}(0,T;H^{2}(\Omega))$ satisfy (1.3) with
$\mbox{rot}\thinspace v\in H^{2,1}(Q)$ and

$None$ $\sum_{j=0}^{1}(\|\nabla_{x,t}^{j}(\mbox{rot}\thinspace
v)\|_{L^{2}(\partial\Omega\times(0,T))}+\|\nabla^{j}v\|_{L^{2}(\partial\Omega\times(0,T))})+\|\mbox{rot}\thinspace
v\|_{L^{\infty}(0,T;H^{1}(\Omega))}\leq M$

with arbitrarily fixed constant $M>0$. Then for any $\varepsilon>0$, there
exist constants $C>0$ and $\theta\in(0,1)$, which depend on
$M,\varepsilon,\Gamma,\Omega_{0}$ such that

$\sum_{\ell=0}^{1}\|\mbox{rot}\thinspace^{\ell}v\|_{H^{1,1}(\Omega_{0}\times(\varepsilon,\,T-\varepsilon))}+\|\Delta(\mbox{rot}\thinspace^{\ell}v)\|_{L^{2}(\Omega_{0}\times(\varepsilon,\,T-\varepsilon))}+\|\nabla
p\|_{L^{2}(\Omega_{0}\times(\varepsilon,T-\varepsilon))}\\\ \leq
C(D+D^{\theta}),$

where

$D:=\sum_{j=0}^{1}(\|\nabla_{x,t}^{j}(\mbox{rot}\thinspace
v)\|_{L^{2}(\Gamma\times(0,T))}+\|\nabla^{j}v\|_{L^{2}(\Gamma\times(0,T))}).$

This theorem asserts stability in determining a solution $(v,p)$ in some
subdomain by boundary data of $v$ on $\Gamma\times(0,T)$ under a priori bound
(1.4).

A similar conditional stability for the continuation was proved in [2], but
our proof is simplified by adapting the proof of Proposition 2 in Huang,
Imanuvilov and Yamamoto [11] which is concerned with a single parabolic
equation. In [11], differently from the conventional arguments by Carleman
estimates, we do not use any cut-off functions $\chi(x,t)$ which is compactly
supported, and do not need to consider $\chi(x,t)v(x,t)$, but directly
$v(x,t)$. In the case of the Navier-Stokes equations, the cut-off destroys the
original structure of the equations, that is, $\mbox{div}\thinspace\chi v=0$
does not hold. Consequently, if we apply the cut-off, then we have to prove a
Carleman estimate for the Navier-Stokes equations without
$\mbox{div}\thinspace v=0$, which causes a complexity (see [2]). Since we do
not here require any cut-off at all, the proofs of Theorems 1 - 3 are
simplified.

Finally we state the main result for the inverse source problem. In (1.1), we
assume

$None$ $\mbox{div}\thinspace F(x,t_{0})=0,\quad x\in\Omega.$

We introduce two kinds of conditions:

$None$ $|\partial_{t}^{k}\mbox{rot}\thinspace F(x,t)|\leq
C|\mbox{rot}\thinspace F(x,t_{0})|,\quad(x,t)\in Q,\,k=0,1.$ $None$
$|\partial_{t}^{k}\mbox{rot}\thinspace F(x,t)|\leq C(|\nabla
F(x,t_{0})|+|F(x,t_{0})|),\quad(x,t)\in Q,\,k=0,1,2.$

We notice that (1.6) implies (1.7), that is, condition (1.7) is more generous
than (1.7).

We are ready to state different conditional stability estimates according to
(1.6) and (1.7).
Theorem 3 (inverse source problem).
Let $\Gamma\subset\partial\Omega$ be arbitrary and $\Omega_{0}\subset\Omega$
be a subdomain satisfying $\overline{\Omega_{0}}\subset\Omega\cup\Gamma$ and
$\overline{\Omega_{0}}\cap\partial\Omega\subset\mbox{Int}\,(\Gamma)$.
(i) We set

$\mathcal{F}_{1}:=\\{F\in L^{2}(Q);\,\mbox{rot}\thinspace
F,\mbox{div}\thinspace F\in H^{1}(I;L^{2}(\Omega)),\,\mbox{(1.6) is
satisfied}\\}.$

We set

$None$
$D_{1}:=\sum_{j,k=0}^{1}(\|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)\|_{L^{2}(\Gamma\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\Gamma\times
I)})+\|v(\cdot,t_{0})\|_{H^{3}(\Omega)}.$

We assume that a solution $(v,p)\in H^{2,1}(Q)\times H^{1,0}(Q)$ to (1.1)
satisfies $\partial_{t}\mbox{rot}\thinspace v\in H^{2,1}(Q)$,
$\partial_{t}p\in H^{1,0}(Q)$ and

$\sum_{j,k=0}^{1}(\|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)\|_{L^{2}(\partial\Omega\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\partial\Omega\times I)})$ $None$
$+\sum_{\kappa=0}^{1}\sum_{k=0}^{1}\|\partial_{t}^{k}\mbox{rot}\thinspace(\cdot,\,t_{0}+(-1)^{\kappa}\delta)\|_{H^{1}(\Omega)}\leq
M_{1}$

with arbitrarily given constant $M_{1}>0$. Then there exist constants $C>0$
and $\theta\in(0,1)$, dependent on $\Omega_{0},\Gamma,M_{1}$, such that

$\|\mbox{rot}\thinspace F(\cdot,t_{0})\|_{L^{2}(\Omega_{0})}\leq
C(D_{1}^{\theta}+D_{1})$

for each $F\in\mathcal{F}_{1}$.
(ii) We set

$\displaystyle\mathcal{F}_{2}:=\\{F\in
H^{2}(I;H^{2}(\Omega));\,\|F(\cdot,t_{0})\|_{H^{2}(\Omega)}\leq m_{1},\quad
F(\cdot,t_{0})=|\nabla F(\cdot,t_{0})|=0\quad\mbox{on $\Gamma$},$
$\displaystyle\quad\mbox{(1.5) and (1.7) are satisfied}\\}$

with arbitrarily chosen constant $m_{1}>0$. We further set

$D_{2}:=\sum_{j=0}^{1}\sum_{k=0}^{2}(\|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)\|_{L^{2}(\Gamma\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\Gamma\times
I)})+\|v(\cdot,t_{0})\|_{H^{4}(\Omega)}.$

We assume that a solution $(v,p)\in H^{2,1}(Q)\times H^{1,0}(Q)$ to (1.1)
satisfies $\partial_{t}^{k}\mbox{rot}\thinspace v\in H^{2,1}(Q)$,
$\partial_{t}^{k}p\in H^{1,0}(Q)$ with $k=0,1,2$ and

$\displaystyle\sum_{j=0}^{1}\sum_{k=0}^{2}(\|\nabla_{x,t}^{j}\partial_{t}^{k}\mbox{rot}\thinspace
v\|_{L^{2}(\partial\Omega\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\partial\Omega\times I)})$
$\displaystyle+$
$\displaystyle\sum_{\kappa=0}^{1}\sum_{k=0}^{2}\|\partial_{t}^{k}\mbox{rot}\thinspace
v(\cdot,\,t_{0}+(-1)^{\kappa}\delta)\|_{H^{1}(\Omega)}\leq M_{2}$

with arbitrarily given constant $M_{2}>0$. Then there exist constants $C>0$
and $\theta\in(0,1)$, dependent on $\Omega_{0},\Gamma,m_{2},M_{2}$, such that

$\|F(\cdot,t_{0})\|_{H^{1}(\Omega_{0})}\leq C(D_{2}^{\theta}+D_{2})$

for each $F\in\mathcal{F}_{2}$.

In Theorem 3 (i) we aim at the determination of rot $F(x,t_{0})$ for
$x\in\Omega$, not $F$ itself, and we do not need to assume (1.5). Theorem 3
asserts two kinds of conditional stability according to the admissible sets
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$. Theorem 3 (i) determines only the
rotation component of $F(x,t_{0})$ and so it is not necessary to assume (1.5).

In Theorem 3, not assuming the boundary condition on the whole boundary of
$\Omega$, we establish stability in determining $F$ in a subdomain of
$\Omega$. On the other hand, the stability over $\Omega$ is proved with the
boundary condition on the whole $\partial\Omega\times(0,T)$ in Choulli,
Imanuvilov, Puel and Yamamoto [6] for the Navier-Stokes equations, and Fan,
Jiang and Nakamura [9] for the Boussinesq equations.

We will test conditions (1.5) - (1.7). First, as the following example shows,
the condition (1.5) is essential for the uniqueness in determining
$F(x,t_{0})$ in Thoerem 3 (ii), where no data of $p$ are observed.
Obstruction to the uniqueness in determining $F$:
We consider a simple case

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}v(x,t)-\Delta v+\nabla
p=f(x),\\\ &\mbox{div}\thinspace v=0,\quad(x,t)\in Q:=\Omega\times I,\\\
&v(x,t_{0})=0\quad\mbox{for $x\in\Omega$},\quad\mbox{supp
$v\subset\Omega\times(0,T)$}.\end{array}\right.$

Here $f$ is an $\mathbb{R}^{3}$-valued smooth function. It is trivial that
$(v,p)=(0,0)$ satisfies (1.10) with $f=0$. Let $\psi\in
C^{\infty}_{0}(\Omega)$. Then $(v,p):=(0,\,\psi)$ satisfies (1.10) with
$f:=\nabla\psi$. In other words, by the appearance of the pressure field $p$
in the Navier-Stokes equations, there is no possibility for the uniqueness in
determining the component of $f$ given by a scalar potential.

Next we examine the conditions (1.5) - (1.7) by considering examples.
Example 1. We see that (1.5) is replaced in terms of a vector potential:

$F(x,t):=\mbox{rot}\thinspace q(x,t),\quad(x,t)\in Q,$

where $q=q(x,t)$ is smooth. Then (1.5) is automatically satisfied. Moreover,
we see that (1.7) is equivalent to

$|\mbox{rot}\thinspace\mbox{rot}\thinspace(\partial_{t}^{k}q(x,t))|\leq
C(|\nabla\mbox{rot}\thinspace q(x,t_{0})|+|\mbox{rot}\thinspace
q(x,t)|),\quad(x,t)\in Q,\,k=0,1,2.$

Example 2.
Let

$F(x,t)=r(t)f(x),\quad x\in\Omega,\,0<t<T,\quad r(t_{0})\neq
0,\quad\mbox{div}\thinspace f=0\quad\mbox{in $\Omega$},$

where real-valued $r\in C^{2}[0,T]$ and $f=(f_{1},f_{2},f_{3})^{T}$ are
smooth. Then (1.5) and (1.6) are satisfied. Indeed we can directly verify
$\mbox{div}\thinspace F(x,t_{0})=r(t_{0})\mbox{div}\thinspace f(x)=0$ for
$x\in\Omega$, which is (1.5). Moreover, we have

$|\partial_{t}^{k}\mbox{rot}\thinspace
F(x,t)|=\left|\frac{d^{k}r}{dt^{k}}(t)\mbox{rot}\thinspace f(x)\right|\leq
C|\mbox{rot}\thinspace f(x)|,\quad k=0,1$

and

$|\mbox{rot}\thinspace F(x,t_{0})|=|r(t_{0})||\mbox{rot}\thinspace
f(x)|,\quad(x,t)\in Q,$

which is (1.6).
Example 3.
For more general $F$ unlike Example 2, it is not always easy to verify (1.6),
but in the following case the condition (1.7) can be readily verified. Let

$F(x,t)=R(x,t)f(x),$

where $R(x,t)=(r_{jk}(x,t))_{1\leq j,k\leq 3}$ and each component $r_{jk}$ is
smooth in $(x,t)$, and $f(x)=(f_{1}(x),f_{2}(x),f_{3}(x))^{T}$. Then (1.7) is
satisfied if

$|\mbox{det}\,R(x,t_{0})|>0\quad\mbox{for $x\in\overline{\Omega}$}.$

Indeed, we have

$|\partial_{t}^{k}\mbox{rot}\thinspace F(x,t)|\leq C(|\nabla
f(x)|+|f(x)|),\quad(x,t)\in\overline{Q},\quad k=0,1,2.$

Therefore it is sufficient to verify

$None$ $|\nabla f(x)|+|f(x)|\leq C(|\nabla F(x,t_{0})|+|F(x,t_{0})|),\quad
x\in\Omega.$

Verification of (1.11). By $|\mbox{det}\,R(x,t_{0})|\neq 0$ for
$x\in\overline{\Omega}$, we have

$|f(x)|\leq C|F(x,t_{0})|,\quad x\in\overline{\Omega}.$

Next, setting $F=(F_{1},F_{2},F_{3})^{T}$, we have

$\partial_{\ell}F_{k}(x,t_{0})=\sum_{j=1}^{3}r_{kj}(x,t_{0})\partial_{\ell}f_{j}(x)+\sum_{j=1}^{3}(\partial_{\ell}r_{kj})(x,t_{0})f_{j}(x),$

and so

$\sum_{j=1}^{3}r_{kj}(x,t_{0})\partial_{\ell}f_{j}(x)=\partial_{\ell}F_{k}(x,t_{0})-\sum_{j=1}^{3}(\partial_{\ell}r_{kj}(x,t_{0}))f_{j}(x),$

that is,

$R(x,t_{0})\partial_{\ell}f(x)=\partial_{\ell}F(x,t_{0})-(\partial_{\ell}R(x,t_{0}))f(x).$

Therefore, by $|\mbox{det}\,R(x,t_{0})|\neq 0$ for $x\in\overline{\Omega}$, we
see

$|\partial_{\ell}f(x)|\leq C(|\partial_{\ell}F(x,t_{0})|+|F(x,t_{0})|),\quad
x\in\overline{\Omega},\quad 1\leq\ell\leq 3.$

Thus (1.11) is verified.

Here for the form $F(x,t)=R(x,t)f(x)$ with $3\times 3$ matrix $R(x,t)$, we do
not know a convenient sufficient condition for (1.6), but condition (1.7) can
be more directly verified. Therefore, Theorem 3 (ii) is more feasible in
applications.
Remark 2.
A special case $F(x,t)=f(x)r(x,t)$, where $f$ is real-valued and $r$ is
$\mathbb{R}^{3}$-valued, is trivial by (1.5). Because (1.5) means that $f$
satisfies a first-order transport equation $r(x,t_{0})\cdot\nabla
f(x)+f(x)\mbox{div}\thinspace r(x,t_{0})=0$ for $x\in\Omega$, and so $f$ can
be uniquely determined by boundary data of $f$ under a certain condition on
$r(x,t_{0})$.
Remark 3.
Theorem 3 (i) estimates only $\mbox{rot}\thinspace F(x,t_{0})$. However, since
$\mbox{div}\thinspace F(x,t_{0})=0$ for $x\in\Omega$, we have

$-\Delta F(x,t_{0})=\mbox{rot}\thinspace\mbox{rot}\thinspace
F(x,t_{0})-\nabla\mbox{div}\thinspace
F(x,t_{0})=\mbox{rot}\thinspace\mbox{rot}\thinspace F(x,t_{0}),\quad
x\in\Omega.$

If we are given boundary data $F(x,t_{0})$ with $x\in\partial\Omega$ in a
suitable space, then we can further estimate by means of $H^{-1}$-estimate of
the solution to $-\Delta F(x,t_{0})=h$ under condition (1.6). Here we do not
pursue more. In Section 5.4, we will discuss this issue by means of a Carleman
estimate in $H^{-1}$-space under a weaker assumption than (1.7).

This article is composed of six sections. In Sections 2-4, we prove Theorems 1
-3 respectively. Section 5 is devoted to concluding remarks and in Section 6,
we derive Lemma 2 which is used for the proof of Theorem 1.

##  2\. Proof of Theorem 1

Thanks to the large parameter $s>0$, it suffices to prove Theorem 1 for $B=0$.

We set

$z:=\mbox{rot}\thinspace v.$

Then

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}z(x,t)-\Delta
z+(A(x,t)\cdot\nabla)z=\mbox{rot}\thinspace F-\sum_{j=1}^{3}\nabla
A_{j}\times\partial_{j}v,\\\ &\Delta v=-\mbox{rot}\thinspace z,\quad
x\in\Omega,\,0<t<T.\end{array}\right.$

The system (2.1) provides a decomposition of the Navier-Stokes equations into
a parabolic equation in $z:=\mbox{rot}\thinspace v$ and an elliptic equation
in $v$, where $p$ is eliminated.

First we show a Carleman estimate for a simple parabolic equation with
constant coefficients:

$\partial_{t}u-\Delta u=G\quad\mbox{in $Q$}.$

Henceforth $C>0$ denotes generic constants which are independent of $s>0$.
Lemma 1.
There exist constants $C>0$ and $s_{0}>0$ such that

$\displaystyle\int_{Q}\left\\{\frac{1}{s}(|\partial_{t}u|^{2}+|\Delta
u|^{2})+s|\nabla u|^{2}+s^{3}|u|^{2}\right\\}e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}|G|^{2}e^{2s\varphi}dxdt+Cs^{3}\int_{\partial\Omega\times
I}(|\nabla_{x,t}u|^{2}+|u|^{2})e^{2s\varphi}dSdt$ $\displaystyle+$
$\displaystyle Cs^{3}\sum_{\kappa=0}^{1}\int_{\Omega}(|\nabla
u(x,t_{0}+(-1)^{\kappa}\delta)|^{2}+|u(x,t_{0}+(-1)^{\kappa}\delta)|^{2})e^{2s\varphi(x,t_{0}+\delta)}dx$

for all $s>s_{0}$ and all $u\in H^{2,1}(Q)$.

Lemma 1 is a standard Carleman estimate for a single parabolic equation, and a
direct proof is found for example in Bellassoued and Yamamoto [3] (Lemma 7.1),
Yamamoto [22] (Theorems 3.1 and 3.2).

Next we show two Carleman estimates for the Laplace operator.
Lemma 2.
Let $r\in L^{2}(I;H^{2}(\Omega))$ satisfy

$-\Delta r(x,t)=g(x,t)\quad\mbox{in $Q$.}$

Then

$\displaystyle\int_{Q}(s|\nabla
r(x,t)|^{2}+s^{3}|r(x,t)|^{2})e^{2s\varphi(x,t)}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}|g(x,t)|^{2}e^{2s\varphi(x,t)}dxdt+Cs^{3}\int_{\partial\Omega\times
I}(|\nabla r|^{2}+|r|^{2})e^{2s\varphi(x,t)}dSdt$

for all $s>s_{0}$.
Lemma 3.
Let $w\in H^{2}(\Omega)$ satisfy

$-\Delta w(x)=h(x),\quad x\in\Omega.$

Then

$\displaystyle\int_{\Omega}(s|\nabla
w(x)|^{2}+s^{3}|w(x)|^{2})e^{2s\varphi(x,t_{0})}dx$ $\displaystyle\leq$
$\displaystyle
C\int_{\Omega}|h(x)|^{2}e^{2s\varphi(x,t_{0})}dx+Cs^{3}\int_{\partial\Omega}(|\nabla
w(x)|^{2}+|w(x)|^{2})e^{2s\varphi(x,t_{0})}dS$

for all $s>s_{1}$.

The proof of Lemma 3 is standard and we can prove directly by integration by
parts (e.g., Lemma 7.1 in [3]), and Lemma 3 is used also for the proof of
Theorem 3 in Section 4.

The proof of Lemma 2 is done by integrating the standard elliptic Carleman
estimate in Lemma 3 for $-\Delta$ over the time interval $I$. For
completeness, the derivation of Lemma 2 from Lemma 3 is provided in Appendix.

Now we complete the proof of Theorem 1. We apply Lemmata 1 and 2 to (2.1):

$\displaystyle\int_{Q}\left(\frac{1}{s}(|\partial_{t}z|^{2}+|\Delta
z|^{2})+s|\nabla z|^{2}+s^{3}|z|^{2}\right\\}e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle C\int_{Q}\left|\mbox{rot}\thinspace
F-\sum_{j=1}^{3}A_{j}\times(\partial_{j}v)\right|^{2}e^{2s\varphi}dxdt+CJ_{1}$
$None$ $\leq C\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+C\int_{Q}(|\nabla
v|^{2}+|v|^{2})e^{2s\varphi}dxdt+CJ_{1}$

and

$None$ $\int_{Q}(s|\nabla v|^{2}+s^{3}|v|^{2})e^{2s\varphi}dxdt\leq
C\int_{Q}|\mbox{rot}\thinspace z|^{2}e^{2s\varphi}dxdt+CJ_{2}.$

Here and henceforth we set

$\displaystyle J_{1}:=s^{3}\int_{\partial\Omega\times
I}(|\nabla_{x,t}(\mbox{rot}\thinspace v)|^{2}+|\mbox{rot}\thinspace
v|^{2})e^{2s\varphi}dSdt$ $\displaystyle+$ $\displaystyle
s^{3}\sum_{\kappa=0}^{1}\int_{\Omega}(|\nabla(\mbox{rot}\thinspace
v(x,t_{0}+(-1)^{\kappa}\delta))|^{2}+|\mbox{rot}\thinspace
v(x,t_{0}+(-1)^{\kappa}\delta)|^{2})e^{2s\varphi(x,t_{0}+\delta)}dx$

and

$J_{2}:=s^{3}\int_{\partial\Omega\times I}(|\nabla
v|^{2}+|v|^{2})e^{2s\varphi}dSdt.$

Hence, substituting (2.3) into the second term on the right-hand side of
(2.1):

$\displaystyle\int_{Q}\left\\{\frac{1}{s}(|\partial_{t}z|^{2}+|\Delta
z|^{2})+s|\nabla z|^{2}+s^{3}|z|^{2}\right\\}e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle C\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+C\int_{Q}|\mbox{rot}\thinspace
z|^{2}e^{2s\varphi}dxdt+C(J_{1}+J_{2}).$

Absorbing the second term on the right-hand side into the left-hand side, we
complete the proof of Theorem 1.

##  3\. Proof of Theorem 2

Theorem 1 holds with general $d(x)$ satisfying only (1.2) in the weight
function, but in order to apply for the proof of Theorems 2 and 3, we have to
choose $d(x)$ closely related to the geometry of $\Gamma$ as follows. First we
construct some domain $\Omega_{1}$. For $\Gamma\subset\partial\Omega$, we
choose a bounded domain $\Omega_{1}$ with smooth boundary such that

$\Omega\subsetneqq\Omega_{1},\quad\overline{\Gamma}=\overline{\partial\Omega\cap\Omega_{1}},\quad\partial\Omega\setminus\Gamma\subset\partial\Omega_{1}.$

In particular, $\Omega_{1}\setminus\overline{\Omega}$ contains some non-empty
open subset. We note that $\Omega_{1}$ can be constructed as the interior of a
union of $\overline{\Omega}$ and the closure of a non-empty domain
$\widehat{\Omega}$ satisfying
$\widehat{\Omega}\subset\mathbb{R}^{n}\setminus\overline{\Omega}$ and
$\partial\widehat{\Omega}\cap\partial\Omega=\Gamma$.

We choose a domain $\omega$ such that
$\overline{\omega}\subset\Omega_{1}\setminus\overline{\Omega}$. Then, by
Imanuvilov [12], we can find $d\in C^{2}(\overline{\Omega_{1}})$ such that

$None$ $d>0\quad\mbox{in $\Omega_{1}$},\quad|\nabla d|>0\quad\mbox{on
$\overline{\Omega_{1}\setminus\omega}$},\quad d=0\quad\mbox{on
$\partial\Omega_{1}$}.$

In particular, by $\overline{\Omega_{0}}\subset\Omega_{1}$, we see

$None$ $d>0\quad\mbox{on $\overline{\Omega_{0}}$},\quad d=0\quad\mbox{on
$\partial\Omega\setminus\Gamma$}.$

We recall that we choose a domain $\Omega_{0}\subset\Omega$ satisfying
$\overline{\partial{\Omega_{0}}\cap\partial\Omega}\subset\mbox{Int}(\Gamma)$
and $\overline{\Omega_{0}}\subset\Omega\cup\Gamma$.

Henceforth we use this $d$ and a sufficiently large constants $\lambda>0$.
Later by (3.10), we choose a constant $\beta>0$.

For given $t_{0}>0$ and $\delta_{2}>0$ satisfying
$0<t_{0}-\delta_{2}<t_{0}+\delta_{2}<T$. We apply the Carleman estimate
Theorem 1 in $\Omega\times(t_{0}-\delta_{2},t_{0}+\delta_{2})$ to obtain

$\displaystyle\int_{\Omega\times(t_{0}-\delta_{2},t_{0}+\delta_{2})}\left(\sum_{j,\ell=0}^{1}|\nabla_{x,t}^{j}(\mbox{rot}\thinspace^{\ell}v)|^{2}+\sum_{\ell=0}^{1}|\Delta(\mbox{rot}\thinspace^{\ell}v)|^{2}\right)e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle
Cs^{4}\int_{\partial\Omega\times(t_{0}-\delta_{2},t_{0}+\delta_{2})}\sum_{j=0}^{1}(|\nabla_{x,t}^{j}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}v|^{2})e^{2s\varphi}dSdt$ $None$
$+Cs^{4}\sum_{\kappa=0}^{1}\int_{\Omega}\sum_{j=0}^{1}|\nabla^{j}(\mbox{rot}\thinspace
v(x,t_{0}+(-1)^{\kappa}\delta_{2}))|^{2}e^{2s\varphi(x,t_{0}+\delta_{2})}dx$

for all $s>s_{0}$.

Here we note that the constants $C>0$ and $s_{0}>0$ are independent of $t_{0}$
because the Carleman estimate is invariant by the translation in time provided
that the translated time interval is included in $(0,T)$.

We set

$d_{0}=\min_{x\in\overline{\Omega_{0}}}d(x),\quad
d_{1}=\max_{x\in\overline{\Omega}}d(x).$

Let $\widetilde{\varepsilon}>0$ be given such that
$0<\widetilde{\varepsilon}<\delta_{2}$. Then, by (3.2), we have

$None$
$\max_{x\in\overline{\partial\Omega\setminus\Gamma},t_{0}-\delta_{2}\leq t\leq
t_{0}+\delta_{2}}\varphi(x,t)\leq
1,\quad\max_{x\in\overline{\Omega}}\varphi(x,t_{0}-\delta_{2})=\max_{x\in\overline{\Omega}}\varphi(x,t_{0}+\delta_{2})\leq
e^{\lambda(d_{1}^{2}-\beta\delta_{2}^{2})}$

and

$None$ $\min_{x\in\overline{\Omega_{0}},t_{0}-\widetilde{\varepsilon}\leq
t\leq t_{0}+\widetilde{\varepsilon}}\varphi(x,t)\geq
e^{\lambda(d_{0}^{2}-\beta\widetilde{\varepsilon}^{2})}.$

For concise descriptions, we set

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon}):=\sum_{\ell=0}^{1}(\|\mbox{rot}\thinspace^{\ell}v\|^{2}_{H^{1,1}(\Omega_{0}\times(t_{0}-\widetilde{\varepsilon},\,t_{0}+\widetilde{\varepsilon}))}+\|\Delta(\mbox{rot}\thinspace^{\ell}v)\|^{2}_{L^{2}(\Omega_{0}\times(t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})}).$

Therefore, shrinking the integral domain
$\Omega\times(t_{0}-\delta_{2},t_{0}+\delta_{2})$ to
$\Omega_{0}\times(t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})$
on the left-hand side of (3.3), applying the trace theorem and the embedding
$H^{1}(0,T;H^{1}(\Omega))\subset C([0,T];H^{1}(\Omega))$, in terms of (1.4),
(3.4) and (3.5), we obtain

$None$
$\exp(2se^{\lambda(d_{0}^{2}-\beta\widetilde{\varepsilon}^{2})})N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})$
$\displaystyle\leq$ $\displaystyle
Cs^{4}\left(\int_{\Gamma\times(t_{0}-\delta_{2},t_{0}+\delta_{2})}+\int_{(\partial\Omega\setminus\Gamma)\times(t_{0}-\delta_{2},t_{0}+\delta_{2})}\right)\sum_{j=0}^{1}(|\nabla_{x,t}^{j}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}v|^{2})e^{2s\varphi}dSdt$ $\displaystyle+$ $\displaystyle
Cs^{4}\sum_{\kappa=0}^{1}\int_{\Omega}\sum_{j=0}^{1}|\nabla^{j}\mbox{rot}\thinspace(x,t_{0}+(-1)^{\kappa}\delta_{2})|^{2}e^{2s\varphi(x,t_{0}+\delta_{2})}dx$
$\displaystyle\leq$ $\displaystyle
Cs^{4}e^{Cs}D^{2}+Cs^{4}e^{2s}M^{2}+Cs^{4}M^{2}\exp(2se^{\lambda(d_{1}^{2}-\beta\delta_{2}^{2})})$

for all $s\geq s_{0}$.

Now we will make a specific choice of $\beta>0$, $\widetilde{\varepsilon}>0$
and $\delta_{2}>0$ such that $0<\widetilde{\varepsilon}<\delta_{2}$. First fix
sufficiently large $N>1$ such that

$None$ $N-1>\frac{d_{1}^{2}-d_{0}^{2}}{d_{0}^{2}},\quad\mbox{that
is,}\quad\frac{d_{1}^{2}}{d_{0}^{2}}<N.$

For given $\varepsilon>0$ in the statement of the theorem, we set

$None$
$\widetilde{\varepsilon}:=\frac{\varepsilon}{N-1},\quad\delta_{2}:=N\widetilde{\varepsilon}=\frac{N\varepsilon}{N-1}>\widetilde{\varepsilon}.$

Since it suffices to consider the case where $\varepsilon>0$ is sufficiently
small, we can assume that $\delta_{2}>0$ is small, in particular,
$0<\delta_{2}<T-\delta_{2}<T$. Then, we can prove

$None$
$\frac{d_{1}^{2}-d_{0}^{2}}{\delta_{2}^{2}-\widetilde{\varepsilon}^{2}}<\frac{d_{0}^{2}}{\widetilde{\varepsilon}^{2}}.$

Indeed, by (3.8) and $N>1$ we see

$\frac{\delta_{2}^{2}}{\widetilde{\varepsilon}^{2}}=N^{2}>N.$

Using (3.7), we have

$\frac{d_{1}^{2}}{d_{0}^{2}}<N<\frac{\delta_{2}^{2}}{\widetilde{\varepsilon}^{2}},\quad\mbox{that
is,}\quad\frac{d_{1}^{2}-d_{0}^{2}}{d_{0}^{2}}<\frac{\delta_{2}^{2}-\widetilde{\varepsilon}^{2}}{\widetilde{\varepsilon}^{2}},$

which yields (3.9) by multiplying with
$\frac{d_{0}^{2}}{\delta_{2}^{2}-\widetilde{\varepsilon}^{2}}>0$.

Therefore, we can choose $\beta>0$ such that

$None$
$\frac{d_{1}^{2}-d_{0}^{2}}{\delta_{2}^{2}-\widetilde{\varepsilon}^{2}}<\beta<\frac{d_{0}^{2}}{\widetilde{\varepsilon}^{2}},$

which implies

$d_{0}^{2}-\beta\widetilde{\varepsilon}^{2}>0,\quad
d_{0}^{2}-\beta\widetilde{\varepsilon}^{2}>d_{1}^{2}-\beta\delta_{2}^{2},$

that is,

$\mu_{1}:=e^{\lambda(d_{0}^{2}-\beta\widetilde{\varepsilon}^{2})}>\mu_{2}:=\max\\{1,\,e^{\lambda(d_{1}^{2}-\beta\delta_{2}^{2})}\\}.$

We arbitrarily choose $t_{0}\in(\delta_{2},T-\delta_{2})$. We notice that
$(t_{0}-\delta_{2},t_{0}+\delta_{2})\subset(0,T)$. Hence (3.6) yields

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
Cs^{4}M^{2}e^{-2s\mu_{0}}+Cs^{4}e^{Cs}D^{2}$

for all $s\geq s_{0}$. Here we note

$\mu_{0}:=\mu_{1}-\mu_{2}>0.$

Replacing $s$ and $C$ by $s+s_{0}$ and $e^{Cs_{0}}$ respectively, we see

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
Cs^{4}M^{2}e^{-2s\mu_{0}}+Cs^{4}e^{Cs}D^{2}$

for all $s\geq 0$. By $\mu_{0}>0$, we easily verify
$\sup_{s>0}s^{4}e^{-s\mu_{0}}<\infty$ and $s^{4}e^{Cs}\leq C_{1}e^{C_{1}s}$
with large $C_{1}>0$, we obtain

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
CM^{2}e^{-s\mu_{0}}+Ce^{Cs}D^{2}$

for all $s\geq 0$. Here we note that $C>0$ is a generic constant which is
independent of $s>0$. We make the right-hand side small by choosing $s\geq 0$
suitably.
Case 1: $D<M$.
We choose $s>0$ such that

$M^{2}e^{-s\mu_{0}}=D^{2}e^{Cs},$

that is,

$s=\frac{2}{C+\mu_{0}}\log\frac{M}{D}>0.$

Then

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
2CM^{\frac{2C}{C+\mu_{0}}}D^{\frac{2\mu_{0}}{C+\mu_{0}}}.$

Case 2: $D\geq M$.
Then we can directly

$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
CD^{2}(e^{-s\mu_{0}}+e^{Cs}).$

Hence setting $s=1$ for example, we see
$N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
CD^{2}e^{C}$. Thus in both cases, we obtain

$None$ $N(v;t_{0}-\widetilde{\varepsilon},t_{0}+\widetilde{\varepsilon})\leq
C(M)(D^{2}+D^{2\theta}),$

where $\theta=\frac{\mu_{0}}{C+\mu_{0}}\in(0,1)$. Here the constants $C$ and
$\theta$ are dependent on $\widetilde{\varepsilon},\delta_{2}>0$, but
independent of $t_{0}$. Varying $t_{0}$ over $(\delta_{2},\,T-\delta_{2})$, we
reach

$N(v;\delta_{2}-\widetilde{\varepsilon},T-\delta_{2}+\widetilde{\varepsilon})\leq
C(M)(D^{2}+D^{2\theta}).$

Since
$\delta_{2}-\widetilde{\varepsilon}=(N-1)\widetilde{\varepsilon}=\varepsilon$
and and $T-\delta_{2}+\widetilde{\varepsilon}=T-\varepsilon$, by (3.8), we can
complete the proof of Theorem 2. $\blacksquare$

##  4\. Proof of Theorem 3

Section 4.1. Proof of Theorem 3 (i).
First Step.

We set

$\widetilde{D}_{1}:=\sum_{j,k=0}^{1}(\|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)\|_{L^{2}(\Gamma\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\Gamma\times I)})$

and we see

$D_{1}=\widetilde{D}_{1}+\|v(\cdot,t_{0})\|_{H^{3}(\Omega)}.$

We choose $d\in C^{2}(\overline{\Omega})$ satisfying (3.1) and (3.2), and we
set

$\varphi(x,t)=e^{\lambda(d(x)-\beta(t-t_{0})^{2})}$

with sufficiently large $\lambda>0$ and some $\beta>0$. Later we will specify
$\beta>0$. By (3.2) we can choose small $\varepsilon_{1}>0$ such that $d=0$ on
$\partial\Omega\setminus\Gamma$, that is,

$\varphi(x,t)\leq 1\quad\mbox{for
$(x.t)\in(\partial\Omega\setminus\Gamma)\times I$},\quad
d(x)\geq\varepsilon_{1}\quad\mbox{for $x\in\Omega_{0}$}.$

We set

$None$
$\mu_{1}:=e^{\lambda\varepsilon_{1}},\quad\mu_{2}:=\max\\{1,\,e^{\lambda(\|d\|_{C(\overline{\Omega})}-\beta\delta^{2})}\\}.$

For $\varepsilon_{1}>0$, we choose $\beta>0$ large such that

$None$ $\varepsilon_{1}>\|d\|_{C(\overline{\Omega})}-\beta\delta^{2}.$

Therefore, (4.1) yields

$None$
$\left\\{\begin{array}[]{rl}&\int_{(\partial\Omega\setminus\Gamma)\times
I}\sum_{j,k=0}^{1}(|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}\partial_{t}^{k}v|^{2})e^{2s\varphi}dSdt\leq
CM_{1}^{2}e^{2s\mu_{2}},\cr\\\ &\int_{\Gamma\times
I}\sum_{j,k=0}^{1}(|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}\partial_{t}^{k}v|^{2}e^{2s\varphi})dSdt\leq
Ce^{Cs}\widetilde{D}_{1}^{2}.\end{array}\right.$

Moreover,

$C\sum_{\kappa=0}^{1}\int_{\Omega}\sum_{k=0}^{1}(|\nabla(\mbox{rot}\thinspace\partial_{t}^{k}v)(x,t_{0}+(-1)^{\kappa}\delta)|^{2}+|\mbox{rot}\thinspace\partial_{t}^{k}v(x,t_{0}+(-1)^{\kappa}\delta)|^{2})e^{2s\varphi(x,t_{0}+\delta)}dx$
$None$ $\leq
CM_{1}^{2}e^{2se^{\lambda(\|d\|_{C(\overline{\Omega})}-\beta\delta^{2})}}\leq
CM_{1}^{2}e^{2s\mu_{2}}.$

We apply Theorem 1 to $v$ to obtain

$\int_{Q}(s|\nabla v|^{2}+s|\nabla(\mbox{rot}\thinspace
v)|^{2}+s^{3}|v|^{2})e^{2s\varphi}dxdt\leq C\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt$ $None$
$+Cs^{3}M_{1}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{1}^{2}$

by (4.2) - (4.4).

Setting $v_{1}:=\partial_{t}v$, we have

$\left\\{\begin{array}[]{rl}&\partial_{t}v_{1}(x,t)-\Delta
v_{1}+(A(x,t)\cdot\nabla)v_{1}+(v_{1}\cdot\nabla)B(x,t)\cr\\\
&+(\partial_{t}A\cdot\nabla)v+(v\cdot\nabla)\partial_{t}B+\nabla\partial_{t}p=\partial_{t}F(x,t),\cr\\\
&\mbox{div}\thinspace v_{1}(x,t)=0,\quad
x\in\Omega,\,0<t<T.\end{array}\right.$

Noting that

$None$
$\mbox{rot}\thinspace((\partial_{t}A\cdot\nabla)v)=(\partial_{t}A\cdot\nabla)\mbox{rot}\thinspace
v$
$+\left(\sum_{j=1}^{3}(\partial_{2}a_{j})\partial_{j}v_{3}-(\partial_{3}a_{j})\partial_{j}v_{2},\,\sum_{j=1}^{3}(\partial_{3}a_{j})\partial_{j}v_{1}-(\partial_{1}a_{j})\partial_{j}v_{3},\,\sum_{j=1}^{3}(\partial_{1}a_{j})\partial_{j}v_{2}-(\partial_{2}a_{j})\partial_{j}v_{1}\right)^{T},$

where we set $\partial_{t}A:=(a_{1},a_{2},a_{3})^{T}$, by (4.5) we have

$\displaystyle\int_{Q}(|\mbox{rot}\thinspace((\partial_{t}A\cdot\nabla)v)|^{2}+|\mbox{rot}\thinspace((v\cdot\nabla)\partial_{t}B)|^{2})e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle C\int_{Q}(|\nabla(\mbox{rot}\thinspace
v)|^{2}+|\nabla v|^{2}+|v|^{2})e^{2s\varphi}dxdt$ $None$
$\leq\frac{C}{s}\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}\widetilde{D}_{1}^{2}.$

Therefore, we apply Theorem 1 to $v_{1}$ to obtain

$\int_{Q}\left(\frac{1}{s}|\partial_{t}^{2}\mbox{rot}\thinspace
v|^{2}+s^{3}|\partial_{t}\mbox{rot}\thinspace
v|^{2}+s(|\nabla(\mbox{rot}\thinspace\partial_{t}v)|^{2}+|\nabla\partial_{t}v|^{2})+s^{3}|\partial_{t}v|^{2}\right)e^{2s\varphi}dxdt$
$None$ $\leq C\int_{Q}\sum_{k=0}^{1}|\partial_{t}^{k}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{3}M_{1}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{1}^{2}$

for all large $s>0$.
Second Step.

We start to estimate $\mbox{rot}\thinspace\partial_{t}v(x,t_{0})$ as follows:

$\displaystyle\int_{\Omega}|\mbox{rot}\thinspace\partial_{t}v(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx=\int^{t_{0}}_{t_{0}-\delta}\left(\partial_{t}\int_{\Omega}|\mbox{rot}\thinspace\partial_{t}v(x,t)|^{2}e^{2s\varphi(x,t)}dx\right)dt$
$\displaystyle+$
$\displaystyle\int_{\Omega}|\mbox{rot}\thinspace\partial_{t}v(x,t_{0}-\delta)|^{2}e^{2s\varphi(x,t_{0}-\delta)}dx$
$\displaystyle=$
$\displaystyle\int^{t_{0}}_{t_{0}-\delta}\int_{\Omega}\left(2((\mbox{rot}\thinspace\partial_{t}v)\cdot(\mbox{rot}\thinspace\partial_{t}^{2}v))+2s\partial_{t}\varphi|\mbox{rot}\thinspace\partial_{t}v|^{2}\right)e^{2s\varphi(x,t)}dxdt$
$\displaystyle+$
$\displaystyle\int_{\Omega}|\mbox{rot}\thinspace\partial_{t}v(x,t_{0}-\delta)|^{2}e^{2s\varphi(x,t_{0}-\delta)}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}(s|\mbox{rot}\thinspace\partial_{t}v|^{2}+|\mbox{rot}\thinspace\partial_{t}v||\mbox{rot}\thinspace\partial_{t}^{2}v|)e^{2s\varphi(x,t)}dxdt+CM_{1}^{2}e^{2s\mu_{2}}.$

Here we applied also (4.1) and the bound $M_{1}$. We have

$\displaystyle|\mbox{rot}\thinspace\partial_{t}v||\mbox{rot}\thinspace\partial_{t}^{2}v|=\left(\frac{1}{s}|\mbox{rot}\thinspace\partial_{t}^{2}v|\right)\left(s|\mbox{rot}\thinspace\partial_{t}v|\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(\frac{1}{s^{2}}|\mbox{rot}\thinspace\partial_{t}^{2}v|^{2}+s^{2}|\mbox{rot}\thinspace\partial_{t}v|^{2}\right)=\frac{1}{2s}\left(\frac{1}{s}|\mbox{rot}\thinspace\partial_{t}^{2}v|^{2}+s^{3}|\mbox{rot}\thinspace\partial_{t}v|^{2}\right),$

and so (4.8) implies

$\displaystyle\int_{Q}|\mbox{rot}\thinspace\partial_{t}v||\mbox{rot}\thinspace\partial_{t}^{2}v|e^{2s\varphi}dxdt\leq$
$\displaystyle\frac{C}{s}\int_{Q}\sum_{k=0}^{1}|\partial_{t}^{k}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}\widetilde{D}_{1}^{2}$

and

$\int_{Q}s|\mbox{rot}\thinspace\partial_{t}v|^{2}e^{2s\varphi}dxdt\leq\frac{C}{s^{2}}\int_{Q}\sum_{k=0}^{1}|\partial_{t}^{k}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+CsM_{1}^{2}e^{2s\mu_{2}}+Cse^{Cs}\widetilde{D}_{1}^{2}.$

Hence,

$None$
$\int_{\Omega}|\mbox{rot}\thinspace\partial_{t}v(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq\frac{C}{s}\int_{Q}\sum_{k=0}^{1}|\partial_{t}^{k}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}D_{1}^{2}.$

By (1.1) and $\partial_{t}\mbox{rot}\thinspace v\in
H^{1}(I;L^{2}(\Omega))\subset C(\overline{I};L^{2}(\Omega))$, we have

$F(x,t_{0})=\partial_{t}v(x,t_{0})-\Delta
v(x,t_{0})+(A\cdot\nabla)v(x,t_{0})+(v(x,t_{0})\cdot\nabla)B(x,t_{0})+\nabla
p(x,t_{0}),$

and

$None$ $\mbox{rot}\thinspace
F(x,t_{0})=\mbox{rot}\thinspace\partial_{t}v(x,t_{0})+a(x),$

where

$None$ $a(x):=\mbox{rot}\thinspace(-\Delta
v(x,t_{0})+(A\cdot\nabla)v(x,t_{0})+(v(x,t_{0})\cdot\nabla)B(x,t_{0})).$

Consequently,

$\int_{\Omega}|\mbox{rot}\thinspace F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx$
$None$
$\leq\frac{C}{s}\int_{Q}\sum_{k=0}^{1}|\partial_{t}^{k}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}D_{1}^{2}.$

By (1.6) and $\varphi(x,t)\leq\varphi(x,t_{0})$ for $(x,t)\in Q$, we have

$\frac{C}{s}\int_{Q}\sum_{k=0}^{1}|\mbox{rot}\thinspace\partial_{t}^{k}F(x,t)|^{2}e^{2s\varphi(x,t)}dxdt\leq\frac{C}{s}\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx.$

Therefore,

$\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq\frac{C}{s}\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}D_{1}^{2}$

for all large $s>0$.

Choosing $s>0$ large, we can absorb the first term on the right-hand side into
the left-hand side, we obtain

$\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq
Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}D_{1}^{2}$

for all large $s>s_{0}$. By $d(x)\geq\varepsilon_{1}$ for $x\in\Omega_{0}$ and
$\Omega_{0}\subset\Omega$, noting $\varphi(x,t_{0})=e^{\lambda d(x)}$, we see

$e^{2se^{\lambda\varepsilon_{1}}}\int_{\Omega_{0}}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}dx\leq\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx.$

Therefore,

$\int_{\Omega_{0}}|\mbox{rot}\thinspace F(x,t_{0})|^{2}dx\leq
Cs^{2}M_{1}^{2}e^{2s\mu_{2}}e^{-2s\mu_{1}}+Cs^{2}e^{Cs}D_{1}^{2}.$

By (4.2), we see

$\mu_{0}:=e^{\lambda\varepsilon_{1}}-\mu_{2}=\mu_{1}-\mu_{2}>0,$

and so

$\int_{\Omega_{0}}|\mbox{rot}\thinspace F(x,t_{0})|^{2}dx\leq
Cs^{2}M_{1}^{2}e^{-2s\mu_{0}}+Cs^{2}e^{Cs}D_{1}^{2}$

for all $s>s_{0}$. By the same argument in obtaining (3.11), we complete the
proof of Theorem 3 (i).
Section 4.2. Proof of Theorem 3 (ii).
First Step.

We set

$\widetilde{D}_{2}:=\sum_{j=0}^{1}\sum_{k=0}^{2}(\|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)\|_{L^{2}(\Gamma\times
I)}+\|\nabla^{j}\partial_{t}^{k}v\|_{L^{2}(\Gamma\times I)})$

and

$D_{2}:=\widetilde{D}_{2}+\|v(\cdot,t_{0})\|_{H^{4}(\Omega)},\quad\widetilde{M}_{2}:=M_{2}+m_{1}.$

Now we choose $\varphi(x,t)=e^{\lambda(d(x)-\beta(t-t_{0})^{2})}$, where $d\in
C^{2}(\overline{\Omega})$ satisfies (3.1) and (3.2), and $\lambda>0$ is
sufficiently large, and the constant $\beta>0$ satisfies (4.2). Then (3.4) and
(3.5) hold.

Therefore,

$None$
$\left\\{\begin{array}[]{rl}&\int_{(\partial\Omega\setminus\Gamma)\times
I}\sum_{j=0}^{1}\sum_{k=0}^{2}(|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}\partial_{t}^{k}v|^{2})e^{2s\varphi}dSdt\leq
CM_{2}^{2}e^{2s\mu_{2}},\cr\\\ &\int_{\Gamma\times
I}\sum_{j=0}^{1}\sum_{k=0}^{2}(|\nabla_{x,t}^{j}\partial_{t}^{k}(\mbox{rot}\thinspace
v)|^{2}+|\nabla^{j}\partial_{t}^{k}v|^{2})e^{2s\varphi}dSdt\leq
Ce^{Cs}\widetilde{D}_{2}^{2}.\end{array}\right.$

Moreover,

$C\sum_{\kappa=0}^{1}\int_{\Omega}\sum_{j=0}^{1}\sum_{k=0}^{2}(|\nabla(\mbox{rot}\thinspace\partial_{t}^{k}v)(x,t_{0}+(-1)^{\kappa}\delta)|^{2}+|\mbox{rot}\thinspace\partial_{t}^{k}v(x,t_{0}+(-1)^{\kappa}\delta)|^{2})e^{2s\varphi(x,t_{0}+\delta)}dx$
$None$ $\leq
CM_{2}^{2}e^{2se^{\lambda(\|d\|_{C(\overline{\Omega})}-\beta\delta^{2})}}\leq
CM_{2}^{2}e^{2s\mu_{2}}.$

Moreover, setting $v_{2}:=\partial_{t}^{2}v$, we have

$\left\\{\begin{array}[]{rl}&\partial_{t}v_{2}(x,t)-\Delta
v_{2}+(A(x,t)\cdot\nabla)v_{2}+(v_{2}\cdot\nabla)B(x,t)\cr\\\
+&2(\partial_{t}A\cdot\nabla)v_{1}+2(v_{1}\cdot\nabla)\partial_{t}B+(\partial_{t}^{2}A\cdot\nabla)v+(v\cdot\nabla)\partial_{t}^{2}B+\nabla\partial_{t}^{2}p=\partial_{t}^{2}F(x,t),\cr\\\
&\mbox{div}\thinspace v_{2}(x,t)=0,\quad
x\in\Omega,\,0<t<T.\end{array}\right.$

By (4.13) and (4.14), we apply Theorem 1 to $v_{2}$ to obtain

$\displaystyle\int_{Q}s|\nabla(\mbox{rot}\thinspace\partial_{t}^{2}v)|^{2}e^{2s\varphi}dxdt\leq
C\int_{Q}|\mbox{rot}\thinspace\partial_{t}^{2}F|^{2}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle
C\int_{Q}\left|\mbox{rot}\thinspace\left\\{2(\partial_{t}A\cdot\nabla)v_{1}+2(v_{1}\cdot\nabla)\partial_{t}B+(\partial_{t}^{2}A\cdot\nabla)v+(v\cdot\nabla)\partial_{t}^{2}B\right\\}\right|^{2}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle
Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}$ $None$ $\leq
C\int_{Q}|\partial_{t}^{2}\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}$

for all large $s>0$.

Similarly to (4.6) and (4.7), in terms of (4.8), we have

$\displaystyle\int_{Q}|\mbox{rot}\thinspace(2(\partial_{t}A\cdot\nabla)v_{1}+2(v_{1}\cdot\nabla)\partial_{t}B)|^{2}e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle C\int_{Q}(|\nabla(\mbox{rot}\thinspace
v_{1})|^{2}+|\nabla v_{1}|^{2}+|v_{1}|^{2})e^{2s\varphi}dxdt$ $\displaystyle=$
$\displaystyle
C\int_{Q}(|\nabla(\mbox{rot}\thinspace\partial_{t}v)|^{2}+|\nabla\partial_{t}v|^{2}+|\partial_{t}v|^{2})e^{2s\varphi}dxdt$
$None$
$\leq\frac{C}{s}\int_{Q}\sum_{k=0}^{1}|\mbox{rot}\thinspace\partial_{t}^{k}F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{2}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}\widetilde{D}_{2}^{2}.$

Here we used also $M_{1}\leq M_{2}$ and $D_{1}\leq D_{2}$. Similarly we can
verify

$\int_{Q}|\mbox{rot}\thinspace((\partial_{t}^{2}A\cdot\nabla)v+(v\cdot\nabla)\partial_{t}^{2}B)|^{2}e^{2s\varphi}dxdt$
$None$ $\leq\frac{C}{s}\int_{Q}|\mbox{rot}\thinspace
F|^{2}e^{2s\varphi}dxdt+Cs^{2}M_{2}^{2}e^{2s\mu_{2}}+Cs^{2}e^{Cs}\widetilde{D}_{2}^{2}.$

Substituting (4.16) and (4.17) into (4.15), we obtain

$\displaystyle\int_{Q}s|\nabla(\mbox{rot}\thinspace\partial_{t}^{2}v)|^{2}e^{2s\varphi}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}\sum_{k=0}^{2}|\mbox{rot}\thinspace\partial_{t}^{k}F|^{2}e^{2s\varphi}dxdt+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}.$

With (4.8), we see

$\int_{Q}(s|\nabla(\mbox{rot}\thinspace\partial_{t}^{2}v)|^{2}+s|\nabla(\mbox{rot}\thinspace\partial_{t}v)|^{2})e^{2s\varphi}dxdt$
$None$ $\leq
C\int_{Q}\sum_{k=0}^{2}|\mbox{rot}\thinspace\partial_{t}^{k}F|^{2}e^{2s\varphi}dxdt+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}$

for all large $s>0$.
Second Step.

Now, similarly to Second Step of the proof of Theorem 3 (i), using (4.18) and
(4.14), we see

$\displaystyle\int_{\Omega}|\nabla(\mbox{rot}\thinspace\partial_{t}v(x,t_{0}))|^{2}e^{2s\varphi(x,t_{0})}dx=\int^{t_{0}}_{t_{0}-\delta}\left(\partial_{t}\int_{\Omega}|\nabla(\mbox{rot}\thinspace\partial_{t}v(x,t))|^{2}e^{2s\varphi(x,t)}dx\right)dt$
$\displaystyle+$
$\displaystyle\int_{\Omega}|\nabla(\mbox{rot}\thinspace\partial_{t}v)(x,t_{0}-\delta)|^{2}e^{2s\varphi(x,t_{0}-\delta)}dx$
$\displaystyle=$
$\displaystyle\int^{t_{0}}_{t_{0}-\delta}\int_{\Omega}(2(\nabla(\mbox{rot}\thinspace\partial_{t}v)\cdot\nabla(\mbox{rot}\thinspace\partial_{t}^{2}v))+2s\partial_{t}\varphi|\nabla(\mbox{rot}\thinspace\partial_{t}v)|^{2})e^{2s\varphi(x,t)}dxdt$
$\displaystyle+$
$\displaystyle\int_{\Omega}|\nabla(\mbox{rot}\thinspace\partial_{t}v)(x,t_{0}-\delta)|^{2}e^{2s\varphi(x,t_{0}-\delta)}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}(s|\nabla(\mbox{rot}\thinspace\partial_{t}v)|^{2}+|\nabla(\mbox{rot}\thinspace\partial_{t}v)||\nabla(\mbox{rot}\thinspace\partial_{t}^{2}v)|)e^{2s\varphi(x,t)}dxdt+CM_{2}^{2}e^{2s\mu_{2}}$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=0}^{2}\int_{Q}|\mbox{rot}\thinspace\partial_{t}^{k}F|^{2}e^{2s\varphi}dxdt+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}.$

Here $(\nabla(\mbox{rot}\thinspace A)\cdot\nabla(\mbox{rot}\thinspace B))$
means that the sum of the products of all the corresponding elements of the
two matrices $\nabla(\mbox{rot}\thinspace A)$ and $\nabla(\mbox{rot}\thinspace
B)$.

Applying (1.7) to the first term on the right-hand side, we obtain

$\int_{\Omega}|\nabla(\mbox{rot}\thinspace\partial_{t}v(x,t_{0}))|^{2}e^{2s\varphi(x,t_{0})}dx\leq
C\int_{Q}(|F(x,t_{0})|^{2}+|\nabla F(x,t_{0})|^{2})e^{2s\varphi}dxdt$ $None$
$+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}+Cs^{3}e^{Cs}\widetilde{D}_{2}^{2}$

for all large $s>0$.

By (1.5), we have

$\mbox{rot}\thinspace\mbox{rot}\thinspace F(x,t_{0})=-\Delta
F(x,t_{0})+\nabla(\mbox{div}\thinspace F(x,t_{0}))=-\Delta F(x,t_{0}),\quad
x\in\Omega$

and (4.10) yields

$\mbox{rot}\thinspace\mbox{rot}\thinspace
F(x,t_{0})=\mbox{rot}\thinspace\mbox{rot}\thinspace\partial_{t}v(x,t_{0})+\mbox{rot}\thinspace
a(x),$

where $a$ is defined by (4.11). Hence,

$-\Delta
F(x,t_{0})=\mbox{rot}\thinspace\mbox{rot}\thinspace(\partial_{t}v(x,t_{0}))+\mbox{rot}\thinspace
a(x),\quad x\in\Omega.$

Therefore,

$|\Delta F(x,t_{0})|\leq
C(|\nabla(\mbox{rot}\thinspace\partial_{t}v(x,t_{0}))|+|\mbox{rot}\thinspace
a(x)|,\quad x\in\Omega$

and so (4.19) implies

$\displaystyle\int_{\Omega}|\Delta F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq
C\int_{Q}(|F(x,t_{0})|^{2}+|\nabla F(x,t_{0})|^{2})e^{2s\varphi(x,t)}dxdt$
$\displaystyle+$ $\displaystyle
Ce^{Cs}(s^{3}\widetilde{D}_{2}^{2}+\|v(\cdot,t_{0})\|^{2}_{H^{4}(\Omega)})+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}$

for all large $s>0$. Since $\varphi(x,t)\leq\varphi(x,t_{0})$ for $(x,t)\in
Q$, we obtain

$\int_{\Omega}|\Delta F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq
C\int_{\Omega}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx$ $None$
$+Ce^{Cs}s^{3}D_{2}^{2}+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}$

for all large $s>0$
Third Step.
Next we apply Lemma 3 shown already in Section 2, which is an elliptic
Carleman estimate. Noting that $F(x,t_{0})=|\nabla F(x,t_{0})|=0$ for
$x\in\Gamma$, application of Lemma 3 to (4.20) yields

$\displaystyle\int_{\Omega}(s|\nabla
F(x,t_{0})|^{2}+s^{3}|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx$
$\displaystyle\leq$ $\displaystyle C\int_{\Omega}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx+Ce^{Cs}s^{3}D_{2}^{2}+Cs^{3}M_{2}^{2}e^{2s\mu_{2}}$
$None$ $+Cs^{3}\int_{\partial\Omega\setminus\Gamma}(|\nabla
F(x,t_{0})|^{2}+|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dS$

for all large $s>0$. Recalling that $\varphi(x,t)\leq 1$ for
$(x,t)\in(\partial\Omega\setminus\Gamma)\times I$, by
$\|F(\cdot,t_{0})\|_{H^{2}(\Omega)}\leq m_{1}$ and (4.1), we apply the trace
theorem to obtain

$\int_{\partial\Omega\setminus\Gamma}(|\nabla
F(x,t_{0})|^{2}+|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dS\leq
m_{1}^{2}\int_{\partial\Omega\setminus\Gamma}e^{2s}dS\leq Cm_{1}^{2}e^{2s}\leq
Cm_{1}^{2}e^{2s\mu_{2}}.$

Choosing $s>0$ large, we can absorb the first term on the right-hand side of
(4.21) into the left-hand side, and so

$\int_{\Omega}s(|\nabla
F(x,t_{0})|^{2}+|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx\leq
Ce^{Cs}s^{3}D_{2}^{2}+Cs^{3}\widetilde{M}_{2}^{2}e^{2s\mu_{2}}$

for all large $s>0$. Shrinking the integral domain $\Omega$ on the left-hand
side to $\Omega_{0}$ and using (4.1) and $e^{2s\varphi(x,t_{0})}\geq
e^{2se^{\lambda\varepsilon_{1}}}=e^{2s\mu_{1}}$ for $x\in\Omega_{0}$,

$e^{2s\mu_{1}}\|F(\cdot,t_{0})\|^{2}_{H^{1}(\Omega_{0})}\leq
Ce^{Cs}s^{3}D_{2}^{2}+Cs^{3}\widetilde{M}_{2}^{2}e^{2s\mu_{2}}$

for all large $s>0$. By (4.2), we have $\mu_{1}>\mu_{2}$ and so

$\|F(\cdot,t_{0})\|^{2}_{H^{1}(\Omega_{0})}\leq
Ce^{Cs}s^{3}D_{2}^{2}+Cs^{3}\widetilde{M}_{2}^{2}e^{-2s\mu_{0}}$

for all large $s>0$. Here we set $\mu_{0}:=\mu_{1}-\mu_{2}>0$. Therefore, by
the same argument as in reaching (3.10), we can complete the proof of Theorem
3 (ii). $\blacksquare$

##  5\. Concluding Remarks

5.1. Original Navier-Stokes equations

We mainly consider linearized Navier-Stokes equations. The original Navier-
Stokes equations reads as

$None$ $\partial_{t}v(x,t)-\Delta v(x,t)+(v\cdot\nabla)v(x,t)+\nabla
p=F(x,t),\quad x\in\Omega,\,0<t<T.$

Let $v_{1}$ and $v_{2}$ satisfy (5.1) with $F_{1}$ and $F_{2}$ respectively.
Then, setting $v:=v_{1}-v_{2}$, we have

$\partial_{t}v(x,t)-\Delta
v(x,t)+(v_{1}\cdot\nabla)v(x,t)+(v\cdot\nabla)v_{2}(x,t)+\nabla
p=(F_{1}-F_{2})(x,t),\quad x\in\Omega,\,0<t<T,$

which corresponds to (1.1). Although the existence of $v_{1},v_{2}$ to the
initial boundary value problems for the Navier-Stokes equations, is in
general, not completely solved with our assumed regularity, for our inverse
problems we assume the existence of $v_{1},v_{2}$ with such regularity.
Moreover, we see by the proof that $T>0$ can be arbitrarily small, and our
inverse problem requires the existence of the solution $(v,p)$ local in time.
Moreover, we notice that for the inverse problems, we consider the solutions
only for positive time interval $t>t_{0}-\delta>0$, so that there is a
possibility that we rely on the smoothing property in time of solution to the
Navier-Stokes equations, in order to gain the necessary regularity of solution
for the inverse problem.

On the other hand, in our inverse source problems, we cannot choose $t_{0}=0$,
in other words, our problem is not an inverse problem for any initial boundary
value problem. In particular, we remark that we do not assume any initial
conditions, but we need $v(\cdot,t_{0})$ in $\Omega$ with some $t_{0}>0$. In
general, the inverse problem with our formulation for parabolic equations as
well as the Navier-Stokes equations has been a long-standing open problem for
the case of $t_{0}=0$.
5.2. Data of pressure field $p$

In this article, we do not use any data of $p$. Therefore, in Theorem 3 (ii),
we have to assume (1.5): $\mbox{div}\thinspace F(\cdot,t_{0})=0$ in $\Omega$.
With information of $p$, we need not such assumption for unknown sources, but
here we do not pursue this direction.
5.3. Available Carleman estimates

As Carleman estimates for parabolic equations including the Navier-Stokes
equations, we can have two types according to the choices of weight functions
in the following forms.

  * •

Regular weight function:

$None$ $\varphi(x,t):=e^{\lambda(d(x)-\beta(t-t_{0})^{2})}.$

  * •

Singular weight function:

$None$
$\varphi(x,t)=\frac{e^{\lambda\eta(x)}-e^{2\lambda\|\eta\|_{C(\overline{\Omega})}}}{h(t)},$

where $\lim_{t\to t_{0}\pm\delta}h(t)=0$.

Here $d,\eta\in C^{2}(\overline{\Omega})$ are chosen suitably.

The Carleman estimate with the weight (5.3) was proved firstly by Imanuvilov
[12] for a parabolic equation.

As for related inverse problems for the Navier-Stokes equations, Choulli,
Imanuvilov, Puel and Yamamoto [6], Fan, Di Cristo, Jiang and Nakamura [8],
Fan, Jiang and Nakamura [9] used Carleman estimates with (5.3), while
Bellassoued, Imanuvilov and Yamamoto [2] and the current article rely on
(5.2).

Moreover, in order to derive Carleman estimates for the Navier-Stokes
equations with (5.2) or (5.3), we have two options:

  1. (1)

First take rot to obtain a parabolic equation in $\mbox{rot}\thinspace v$ and
then a Poisson equation in $v$. See [6], [8], [9] with the weight in form of
(5.3).

  2. (2)

First take div to obtain a Poisson equation in $p$ and then a parabolic
equation in $v$. See also [2] with the weight in form of (5.2).

The present article is based on Option (1) with the weight function (5.2),
which is a different combination of the weight function and the option.
5.4. Improvement of Theorem 3 (ii)

So far, we prefer to a use of Carleman estimates of a limited category. Here
we apply an $H^{-1}$-Carleman estimate for $\Delta$ and improve Theoerm 3
(ii).

Indeed we can relax condition (1.7) as

$None$ $|\partial_{t}^{k}\mbox{rot}\thinspace F(x,t)|\leq C(|\nabla
F(x,t_{0})|+|F(x,t_{0})|),\quad(x,t)\in Q,\,k=0,1.$

Compared with (1.7), we need not the second derivative in $t$ on the left-hand
side. Then reducing the time regularity by $1$ in data and a priori bound, we
can improve the conclusion in Theorem 3 (ii) as follows. We recall that
$D_{1}$ is defined by (1.8).

Proposition 1.
There exist constants $C>0$ and $\theta\in(0,1)$, which are dependent on
$\Omega_{0}$, $M_{1}$, $\Gamma$, such that

$None$ $\|F\|_{H^{1}(\Omega_{0})}\leq C(D_{1}+D_{1}^{\theta})$

for each

$F\in\mathcal{F}_{3}:=\\{F\in
H^{1}(I;H^{2}(\Omega));\,\mbox{supp}F(\cdot,t_{0})\subset\Omega,\,\mbox{(1.5)
and (5.4) are satisfied}\\},$

provided that (1.9) holds with arbitrarily chosen constant $M_{1}>0$.

We can relax also the condition supp $F(\cdot,t_{0})\subset\Omega$, but for
simplicity we keep this condition in $\mathcal{F}_{3}$ and here give
Sketch of the proof of (5.5).

Under the assumption, we can follow the argument in the proof of Theorem 3 (i)
and reach (4.12). Application of (5.4) to the first term on the right-hand
side of (4.12) yields

$\int_{\Omega}|\mbox{rot}\thinspace
F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx\leq\frac{C}{s}\int_{Q}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t)}dxdt$ $None$
$+Cs^{2}M_{1}^{2}e^{2s\mu_{2}}+Ce^{Cs}s^{2}D_{1}^{2}$

for all large $s>0$. We estimate the first term on the right-hand side of
(5.6) as follows:

$\displaystyle\int_{Q}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t)}dxdt$ $\displaystyle=$
$\displaystyle\int_{\Omega}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}\left(\int^{t_{0}+\delta}_{t_{0}-\delta}e^{2s(\varphi(x,t)-\varphi(x,t_{0}))}dt\right)dx.$

Since $d\geq 0$ on $\overline{\Omega}$, we see

$\displaystyle\int^{t_{0}+\delta}_{t_{0}-\delta}e^{2s(\varphi(x,t)-\varphi(x,t_{0}))}dt=\int^{t_{0}+\delta}_{t_{0}-\delta}e^{-2se^{\lambda
d(x)}(1-e^{-\lambda\beta(t-t_{0})^{2}})}dt$ $\displaystyle\leq$
$\displaystyle\int^{t_{0}+\delta}_{t_{0}-\delta}e^{-2s(1-e^{-\lambda\beta(t-t_{0})^{2}})}dt=o(1)$

as $s\to\infty$ by the Lebesgue theorem. Therefore,

$\int_{Q}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t)}dxdt=o(1)\int_{\Omega}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx$

and so (5.6) yields

$s\int_{\Omega}|\mbox{rot}\thinspace F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx$
$None$ $\leq o(1)\int_{\Omega}(|F(x,t_{0})|^{2}+|\nabla
F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx+Cs^{3}M_{1}^{2}e^{2s\mu_{2}}+Ce^{Cs}s^{3}D_{1}^{2}$

for all large $s>0$.

Now by (1.5), we have

$None$ $-\Delta F(x,t_{0})=\mbox{rot}\thinspace\mbox{rot}\thinspace
F(x,t_{0}),\quad x\in\Omega,$

and supp $F(\cdot,t_{0})\subset\Omega$. We show an $H^{-1}$-Carleman estimate
for $\Delta$ as follows. Let $E,\widetilde{E}\subset\mathbb{R}^{n}$ be bounded
domains with smooth boundaries $\partial E$ and $\partial\widetilde{E}$ such
that $\overline{E}\subset\widetilde{E}$, and
$\omega\subset\widetilde{E}\setminus\overline{E}$ be an arbitrary non-empty
open set. Then in Imanuvilov [12] it is proved that there exists $\eta\in
C^{2}(\overline{E})$ satisfying

$None$ $\eta=0\quad\mbox{on $\partial\widetilde{E}$},\quad\eta>0\quad\mbox{in
$\widetilde{E}$},\quad|\nabla\eta|>0\quad\mbox{on
$\overline{\widetilde{E}\setminus\omega}$}.$

We set $\varphi_{0}(x):=e^{\lambda\eta(x)}$ with sufficiently large constant
$\lambda>0$. Then
Lemma 4.
There exist constants $C>0$ and $s_{0}>0$ such that

$\int_{E}(|\nabla w(x)|^{2}+s^{2}|w(x)|^{2})e^{2s\varphi_{0}(x)}dx\leq
C\sum_{j=1}^{n}s\int_{E}|g_{j}(x)|^{2}e^{2s\varphi_{0}(x)}dx$

for all $s>s_{0}$ and $w\in H^{2}_{0}(E)$ satisfying

$\Delta w=\sum_{j=1}^{n}\partial_{j}g_{j}(x),\quad x\in E.$

This is a Carleman estimate where the right-hand side is estimated in
$H^{-1}$-space, while Lemma 3 in Section 2 is a Carleman estimate whose right-
hand side is estimated in the $L^{2}$-space.

Lemma 4 is proved as follows: We make the zero extension of $w$ in $E$ to
$\widetilde{E}\supset E$. By $w\in H^{2}_{0}(E)$, we see that $w\in
H^{2}_{0}(\widetilde{E})$ and $w=0$ in $\widetilde{E}\setminus E$. Hence,
applying Theorem 2.2 in Imanuvilov and Puel [13], we have

$\int_{\widetilde{E}}(|\nabla
w(x)|^{2}+s^{2}|w(x)|^{2})e^{2s\varphi_{0}(x)}dx\leq
C\sum_{j=1}^{n}s\int_{\widetilde{E}}|g_{j}(x)|^{2}e^{2s\varphi_{0}(x)}dx$

for all large $s>0$, which is Lemma 4.

Here we fix sufficiently large $\lambda>0$ and so in the Carleman estimate we
can omit the dependence of the constants and the estimate on $\lambda$.

We apply Lemma 4 by setting $\widetilde{E}=\Omega_{1}$ and $E=\Omega$ and
$\eta(x)=d(x)$, where $\Omega_{1}\supsetneqq\overline{\Omega}$ satisfies
(3.1). Moreover $\mbox{rot}\thinspace\mbox{rot}\thinspace F(x,t_{0})$ is given
by a linear combination of $\partial_{j}[\mbox{rot}\thinspace
F(x,t_{0})]_{k}$, $1\leq j,k\leq 3$, where $[\cdot]_{k}$ denotes the $k$-th
component of a vector under consideration. Application of Lemma 4 to (5.8)
implies

$\int_{\Omega}(|\nabla
F(x,t_{0})|^{2}+s^{2}|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx\leq
Cs\int_{\Omega}|\mbox{rot}\thinspace F(x,t_{0})|^{2}e^{2s\varphi(x,t_{0})}dx$

for all large $s>0$. Substituting (5.7), we obtain

$\displaystyle\int_{\Omega}(|\nabla
F(x,t_{0})|^{2}+s^{2}|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx$
$\displaystyle\leq$ $\displaystyle o(1)\int_{\Omega}(|\nabla
F(x,t_{0})|^{2}+|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx+Cs^{3}M_{1}^{2}e^{2s\mu_{2}}+Ce^{Cs}s^{3}D_{1}^{2}$

for all large $s>0$.

Choosing $s>0$ sufficiently large, we can absorb the first term on the right-
hand side into the left-hand side, we obtain

$\int_{\Omega}(|\nabla
F(x,t_{0})|^{2}+|F(x,t_{0})|^{2})e^{2s\varphi(x,t_{0})}dx\\\ \leq
Cs^{3}M_{1}^{2}e^{2s\mu_{2}}+Ce^{Cs}s^{3}D_{1}^{2}$

for all large $s>0$. Then, by the same argument as in the final part of the
proof of Theorem 3 (i), we can complete the proof of (5.5). $\blacksquare$

In place of Lemma 4, one may apply a Carleman estimate (e.g., Vogelsang [21])
for a system $\mbox{rot}\thinspace w=g$ and $\mbox{div}\thinspace w=0$, but we
do not exploit here.

##  6\. Appendix. Derivation of Lemma 2 from Lemma 3

Let $t\in I$ be arbitrarily fixed. Then, applying Lemma 3 to $-\Delta
r(x,t)=g(x,t)$ in $Q$, we have

$\int_{\Omega}(s|\nabla
r(x,t)|^{2}+s^{3}|r(x,t)|^{2})e^{2\widetilde{s}\varphi(x,t_{0})}dx$ $None$
$\leq
C\int_{\Omega}|g(x,t)|^{2}e^{2\widetilde{s}\varphi(x,t_{0})}dx+C\widetilde{s}^{3}\int_{\partial\Omega}(|\nabla
r(x,t)|^{2}+|r(x,t)|^{2})e^{2\widetilde{s}\varphi(x,t_{0})}dS$

for all $\widetilde{s}>s_{1}$: some constant.

We set $s_{*}:=s_{1}e^{\lambda\beta\delta^{2}}$. Let $s>s_{*}$. Then

$se^{-\lambda\beta(t-t_{0})^{2}}\geq se^{-\lambda\beta\delta^{2}}>s_{1}$

for all $t\in\overline{I}$. Then in (6.1), setting
$\widetilde{s}:=se^{-\lambda\beta(t-t_{0})^{2}}$, we see

$\displaystyle\int_{\Omega}(se^{-\lambda\beta(t-t_{0})^{2}}|\nabla
r(x,t)|^{2}+s^{3}e^{-3\lambda\beta(t-t_{0})^{2}}|r(x,t)|^{2})e^{2se^{-\lambda\beta(t-t_{0})^{2}}\varphi(x,t_{0})}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}|g(x,t)|^{2}e^{2se^{-\lambda\beta(t-t_{0})^{2}}\varphi(x,t_{0})}dx+Cs^{3}e^{-3\lambda\beta(t-t_{0})^{2}}\int_{\partial\Omega}(|\nabla
r|^{2}+|r|^{2})e^{2se^{-\lambda\beta(t-t_{0})^{2}}\varphi(x,t_{0})}dS.$

Here, by $e^{-\lambda\beta(t-t_{0})^{2}}\geq e^{-\lambda\beta\delta^{2}}$ for
$t\in I$, we see

$\displaystyle\int_{\Omega}(se^{-\lambda\beta\delta^{2}}|\nabla
r(x,t)|^{2}+s^{3}e^{-3\lambda\beta\delta^{2}}|r(x,t)|^{2})e^{2s\varphi(x,t)}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}|g(x,t)|^{2}e^{2s\varphi(x,t)}dx+Cs^{3}\int_{\partial\Omega}(|\nabla
r|^{2}+|r|^{2})e^{2s\varphi(x,t)}dS.$

Multiplying with $e^{3\lambda\beta\delta^{2}}$ and integrating over $t\in I$,
we complete the derivation of Lemma 2. $\blacksquare$

## Acknowledgment

The first author was supported partly by NSF grant DMS 1312900. The third
author was supported by Grant-in-Aid for Scientific Research (S) 15H05740 and
Grant-in-Aid (A) 20H00117 of Japan Society for the Promotion of Science, The
National Natural Science Foundation of China (no. 11771270, 91730303), and the
RUDN University Strategic Academic Leadership Program. Most of the article was
composed when the first author stayed at University of Parma as visiting
professor and he thanks the university for offering an excellent occasion.

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# Sensitivity Analysis in the Dupire Local Volatility Model with Tensorflow

Francois Belletti, Davis King, James Lottes, Yi-Fan Chen, John Anderson
Google Research
Mountain View CA
USA
<EMAIL_ADDRESS>
###### Abstract

In a recent paper [4], we have demonstrated how the affinity between TPUs and
multi-dimensional financial simulation resulted in fast Monte Carlo
simulations that could be setup in a few lines of python Tensorflow code. We
also presented a major benefit from writing high performance simulations in an
automated differentiation language such as Tensorflow: a single line of code
enabled us to estimate sensitivities, i.e. the rate of change in price of
financial instrument with respect to another input such as the interest rate,
the current price of the underlying, or volatility. Such sensitivities —
otherwise known as the famous financial “Greeks” — are fundamental for risk
assessment and risk mitigation. In the present follow-up short paper, we
extend the developments exposed in [4] about the use of Tensor Processing
Units and Tensorflow for TPUs.

###### Index Terms:

Tensorflow, Financial Monte Carlo, Simulation, Tensor Processing Unit,
Hardware Accelerators, TPU, GPU

##  I Introduction

Our aim here is to delve more into Tensorflow as a tool for sensitivity
analysis. To that end, we reproduce results produced in [11] on the
sensitivity analysis of the local volatility model with Automated Adjoint
Differentiation (AAD) (known in the ML community as back-propagation). Using
Tensorflow [3], (which enables automated differentiation and makes leveraging
GPUs and TPUs [9] extremely simple), we aim to reproduce quantitative results
presented in Section 12.4 (page 424) of [11]. We demonstrate in particular
that on TPUs, in spite of the mixed numerical precision, we are able to
closely reproduce results obtained on CPU with standard IEEE float32
precision. First, we recall the context in which Tensorflow can be used for
Monte Carlo simulation in quantitative finance.

##  II Simulating to assess risk

Most financial assets are subject to frequent and unpredictable changes in
price. Quantifying the potential outcomes associated with fluctuations in
value of the instruments underlying a financial portfolio is of primordial
importance to monitor risk exposure. Monte Carlo simulation is routinely used
throughout the financial sector to estimate the potential changes in value of
financial portfolios over a certain time horizon. In particular, if multiple
instruments in a given portfolio have correlated fluctuations and/or if the
portfolio comprises derivatives that share common underlyings, Monte Carlo
simulation helps unravel the different outcomes that emerge out of such
complex correlation and composition structures. As variance is particularly
large for many financial assets, in general, it is necessary to sample many
times (hundreds of thousands to billions) to obtain converged estimates under
the Law of Large Numbers. The issue is even more pronounced in high
dimensional settings where the use of Quasi Random Numbers [5, 10, 11] to
speed up convergence is less practical. In short, we routinely need to run the
same high dimensional simulation of a stochastic process many times.

Running the same program affected by random perturbations over and over again
is now familiar to most ML researchers. Stochastic Gradient Descent, which is
now a cornerstone of Deep Learning [6], runs the same computation millions to
billions of times with different inputs and subjected to different random
perturbations such as dropout or data augmentation techniques. Recently,
Tensor Processing Units [9, 1, 8] (see Figure 1) have been designed to
accelerate the training of deep neural networks (in particular dense and
convolutional feed-forward networks) which in turns is strikingly similar in
terms of computational patterns to multi-dimensional Monte-Carlo simulations
that are employed to assess financial risk. Indeed, both rely on interleaving
element-wise operators with matrix/matrix products as illustrated in Figure 2.

Figure 1: Hardware architecture and programming model of Tensor Processing
Units (TPUs) [1]. Figure 2: Similarity between a DNN layer and a time step of
a multi-dimensional Geometric Brownian Monte Carlo.

In [4], we have demonstrated the efficiency of Tensorflow and TPUs to price
European options with a single underlying, price European basket options and
compute their “delta”, estimate Value-at-Risk and Conditional Value-at-Risk,
and price American options. In particular, sensitivity analysis was limited to
estimating sensitivities with respect to the current price of the underlyings.
We now investigate the estimation of sensitivities with respect to model
parameters such as the local volatility surface in Dupire’s model as in [11].

##  III Automated differentiation for financial risk assessment

In this follow-up paper, we focus closely on the general use of Tensorflow and
TPUs to estimate financial Greeks [7]. In particular, we show how to estimate
the sensitivity of an estimate for the price of an option with respect to a
parameter of key importance: the local volatility of its underlying asset. In
other words, we want to understand how robust the price estimate we obtain is
with respect to errors in the estimation of the volatility parameter. Such a
procedure is key to understanding the risk associated with model parameter
under or over estimation.

A standard technique to estimate such a sensitivity used to rely on the
computation of an empirical numerical first order derivative through
“bumping”. For each parameter of interest, $p$, a simulation is run with a
value of $p_{0}+\frac{\epsilon}{2}$, another with $p_{0}-\frac{\epsilon}{2}$
and the difference between the two outcomes normalized by $\epsilon$ would
serve as an estimate for the sensitivity of the option price with respect to
$p$ at the value $p=p_{0}$. In the case of local volatility model, where the
volatility surface comprises hundreds of parameters, the procedure requires
twice as many converged simulations as there are parameters to compute
sensitivities with respect to.

Recently, the use of Automated Adjoint Differentiation (AAD), i.e. back-
propagation, has provided risk assessment with faster means of estimation for
sensitivities. Such a first order derivative is practically very easy to
implement with the use of an automated differentiator to program the
simulation. The advantages of AAD over bumping are exposed at length in [11].

##  IV TPUs to estimate first order derivatives of option price estimates

Methods other than AAD, such as computing the tangent process of the
simulation or employing Malliavin calculus [10, 5], can be employed to
estimate sensitivities. In all these cases, a simulation of the adjoint,
tangent or Malliavin weight is needed which is also largely simplified by
automated differentiation as they rely on the calculation of the first order
derivative of the transition operation of the discretized stochastic process
under study.

All the methods above benefit from two major advantages provided by the use of
TPUs. First, simulations are directly written with an automated
differentiation language. Second, many simulations (and first order
derivatives) can be computed fast in an embarrassingly parallel manner on
TPUs.

We now demonstrate such advantages in practice as we reproduce an application
presented in [11] which focuses on a textbook example representative of
simulations that are run pervasively when pricing financial derivatives. As in
this reference, we focus on the estimation of the first order derivative of
the option price with respect to each parameter of the volatility surface
which is of key importance to understand the risk profile of the financial
instrument.

###  IV-A The Dupire local volatility model

The Dupire local volatility model considers a single asset (e.g. a stock
price) and assumes (once discretized by a naive Euler explicit scheme) that
tomorrow’s price equals today’s price affected by a deterministic trend and a
crucially important stochastic Gaussian term whose variance depends on today’s
stock price. Such volatility depends more precisely on the level of the stock
price and the day under consideration. The corresponding Stochastic
Differential Equation can be written as [11]

$\frac{dX_{t}}{X_{t}}=\sigma\left(X_{t},t\right)dW_{t}$

where $X_{t}$ is the price of the underlying asset of interest,
$\sigma(\cdot,\cdot)$ is the local volatility function and $dW_{t}$ the
Brownian motion representing the process driving price fluctuations. The
volatility surface is classically calibrated thanks to the Black-Scholes model
and a root finding procedure for a grid of values of price and time
$\left\\{\left(i\times\Delta x,j\times\Delta t\right)\right\\}_{i=0\dots
I-1,j=0\dots J-1}$. An interpolation procedure is then used to output
volatility values for any price and time value. Here we consider this
calibration has been conducted and we want to understand the sensitivity of
the price of a European call option with respect to the volatility surface
parameters. Our experiments employ the very same parameters as our reference
(Section 12.4 page 424). As we conduct our experiment we want to assess
multiple points: Can we implement the Dupire local volatility model
efficiently on TPU? In spite of the use of mixed precision on TPU, can we
obtain price and sensitivity estimates that closely match our reference? Are
TPU based simulations and AAD fast enough when compared to the CPU reference
wall time given in our reference and with respect to Tensorflow GPU?

###  IV-B Implementing the local volatility model on TPU

We implemented our simulation naively in an interactive notebook using our TF-
Risk library [2]. In the code snipped 5 provided in appendix, it is noteworthy
that we only use the library for three elements:

  * •

European call payoff;

  * •

A wrapper around pseudo random normal number generators as provided by
Tensorflow;

  * •

A 2d interpolation method optimized for TPUs.

###  IV-C Accuracy of TPU-based estimates

As a parameter, we employ a volatility surface which — as in our reference —
comprises 30 discretization points for price values and 60 discretization
points for time values. We want to compute the sensitivity of the estimated
price (which in practice is estimated through Monte Carlo sampling and
averaging) with respect to each of the 1800 parameter values of the volatility
surface. Tensorflow enables us to write such a procedure in a few lines of
code while implicitly conducting automated differentiation through back-
propagation to estimate the average first order derivative of the price
estimate with respect to each volatility parameter.

We can plot each of these 1800 estimated first order derivatives as usually do
for the volatility surface. The corresponding sensitivity estimates are
presented in Figure 3. Furthermore, we can accurately reproduce the results
presented in [11] with 500K sampled trajectories as shown in Figure 4.

Figure 3: Estimated sensitivities of the European call price with respect to
the volatility surface parameters as in [11].  Figure 4:  Estimated
sensitivities match the results presented page 425 in [11] closely (our
results are not smoothed).

###  IV-D TPU based estimation speed

It is reported in [11] that the high performance parallel C++ library built by
the authors produced estimates with $500$K sampled trajectories in $575$ms at
best which gives us a state-of-the-art reference for multi-threaded CPU
implementation. After compilation of our python code for TPU with XLA, we
report a median wall time of $16$ms from an interactive notebook backed by a
full Cloud TPU v3 with computations in mixed precision (float32 for element-
wise operations and bfloat16 for operations involving the MXU). Our wall time
comprises the round-trip necessary for the notebook’s kernel to send
instructions to the TPU and get results back. We also compare against a
Tensorflow GPU implementation and there the best wall time is 110ms on a V100
GPU.

It is noteworthy that we optimized our implementation of the two dimensional
interpolation routine employed in the simulation to compute the value of the
local volatility between spots and times for which it has been estimated (our
price can take any positive value while we have 156 simulated time steps and
only 60 maturities for which the local volatility has been estimated). As in
[11], we employ a bi-linear interpolation which creates a differentiable path
between the estimated parameters and the option price estimate. Bi-linear
interpolation can be implemented in various ways and a naive method consists
in looking up values through a $tf.gather\\_nd$. We found that this
implementation was suboptimal both for V100 GPUs and TPUs. Indeed, in both
cases, it was preferable to employ one-hot encodings and matrix/matrix
multiplies which in turn could help us leverage the TensorCore unit on V100s
and the MXU on TPUs.

##  V Conclusion

Although TPUs have limited numerical precision, they can successfully be
leveraged with Tensorflow to conduct reliable general purpose sensitivity
analysis for financial derivatives which in turn improves quantitative risk
assessment. One line of code is sufficient, once a simulation has been written
in Tensorflow, to compute first order financial “Greeks”. Furthermore,
Tensorflow readily enables the use of GPUs or TPUs in the cloud without
substantial code changes which in turn enables strong wall time improvements
for the computation of simulations and sensitivities when compared to CPUs.

Additionally, Tensorflow provides facilities to scale up simulations to
multiple CPU/GPU machines or entire TPU-pods in the cloud, which we will show
in our upcoming work. We will also demonstrate how to provide support for near
double numerical precision on TPUs.

## Appendix A: code snippet

In Figure 5, we showcase the python Tensorflow implementation of Dupire’s
local volatility model in TF-Risk [2].

Figure 5: Implementation of Dupire’s local volatility model with TF-Risk.

## References

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