# New Approach for Vorticity Estimates of Solutions of the Navier-Stokes
Equations

Gui-Qiang G. Chen and Zhongmin Qian Mathematical Institute, University of
Oxford, Oxford OX2 6GG. Email<EMAIL_ADDRESS>Institute,
University of Oxford, Oxford OX2 6GG, and OSCAR, Suzhou, China. Email:
<EMAIL_ADDRESS>
###### Abstract

We develop a new approach for regularity estimates, especially vorticity
estimates, of solutions of the three-dimensional Navier-Stokes equations with
periodic initial data, by exploiting carefully formulated linearized vorticity
equations. An appealing feature of the linearized vorticity equations is the
inheritance of the divergence-free property of solutions, so that it can
intrinsically be employed to construct and estimate solutions of the Navier-
Stokes equations. New regularity estimates of strong solutions of the three-
dimensional Navier-Stokes equations are obtained by deriving new explicit _a
priori_ estimates for the heat kernel (i.e., the fundamental solution) of the
corresponding heterogeneous drift-diffusion operator. These new _a priori_
estimates are derived by using various functional integral representations of
the heat kernel in terms of the associated diffusion processes and their
conditional laws, including a Bismut-type formula for the gradient of the heat
kernel. Then the _a priori_ estimates of solutions of the linearized vorticity
equations are established by employing a Feynman-Kac-type formula. The
existence of strong solutions and their regularity estimates up to a time
proportional to the reciprocal of the square of the maximum initial vorticity
are established. All the estimates established in this paper contain known
constants that can be explicitly computed.

_Key words_ : Gradient estimates, vorticity estimates, approach, iteration
scheme, _a priori_ estimates, strong solutions, Navier-Stokes equations,
vorticity equations, heat kernel, fundamental solution, stochastic diffusion
process, heterogeneous drift-diffusion operator, conditional laws, Bismut-type
formula, Feynman-Kac-type formula.

_MSC classifications_ : Primary: 35Q30, 35Q35, 35B65, 35B45, 35D35, 76D05;
Secondary: 35K45, 35A08, 35B30, 35Q51

##  1 Introduction

We are concerned with the quantitative regularity estimates of solutions of
the Navier-Stokes equations in ${\mathbb{R}}^{3}$. In this paper, we consider
the Cauchy problem for the Navier-Stokes equations for $x\in\mathbb{R}^{3}$
and $t\geq 0$:

$\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla P=\nu\Delta
u,\\\\[2.84526pt] \nabla\cdot u=0,\end{cases}$ (1.1)

subject to the periodic initial condition:

$u|_{t=0}=u_{0},$ (1.2)

satisfying that $u_{0}(x+L\mathbf{k})=u_{0}(x)$ for all
$\mathbf{k}\in\mathbb{Z}^{3}$ and $\nabla\cdot u_{0}=0$, where $\nu>0$ is the
kinematic constant, and $L>0$ is the period of the initial data. Although
periodic flows are special in nature, the periodic solutions can be considered
as ideal solutions for turbulent motions away from their physical boundaries,
or as models for homogeneous turbulent flows.

The Cauchy problem (1.1)–(1.2) for the Navier-Stokes equations (1.1) with
periodic initial data (1.2) seeks for a velocity vector field $u(x,t)$ and a
scalar pressure $P(x,t)$ that are periodic functions with period $L$
satisfying the initial condition that $u(x,0)=u_{0}(x)$, where $u_{0}(x)$ is
the periodic initial velocity in (1.2), which is divergence-free, i.e.,
$\nabla\cdot u_{0}=0$.

The mathematical study of global solutions of the Navier-Stokes equations
(1.1) was initiated in the work of Leray [23, 24] and Hopf [16, 17], in which
global weak solutions were constructed and investigated. Since then, many
properties and features of the solutions of the Navier-Stokes equations (1.1)
have been understood; see [14, 15, 21, 36, 39, 40] and the references cited
therein. The Navier-Stokes equations (1.1) have been studied by using various
methods; the mathematical analysis of (1.1) has been mainly based on several
functional analysis methods and on the results on certain functional spaces
such as the Sobolev spaces and the Besov spaces. Great progress has been made
in the past decades, the existences of local and global solutions of the
Navier-Stokes equations (1.1) have been studied (cf. [10, 13, 20, 26, 34, 35],
besides the references cited therein and above). The partial regularity of the
weak solutions of the Navier-Stokes equations (1.1) has also received
intensive study; see [5, 25, 30, 31, 32, 33] and the references cited therein.
However, the quantitative regularity of solutions, the global existence of
strong solutions, and the uniqueness of weak solutions of the three-
dimensional (3-D) Navier-Stokes equations remain to be the major open problems
in Mathematics.

###  1.1 Main theorem

Assume that the periodic initial data function $u_{0}$ is smooth and
divergence-free. Then any strong solution $u(x,t)$ of the Cauchy problem
(1.1)–(1.2) must have a constant mean velocity, so that it is assumed without
loss of generality that

$\int_{[0,L]^{3}}u(x,t)\,\textrm{d}x=0,$ (1.3)

due to the Galilean invariance of system (1.1). Denote the vorticity:

$\omega(x,t)=\nabla\wedge u(x,t)$ (1.4)

which is the curl of the velocity. Then $\omega_{0}=\nabla\wedge u_{0}$ is the
initial vorticity.

In this paper, we develop a new approach, i.e., an iteration scheme, to
construct and estimate strong solutions of the Cauchy problem (1.1)–(1.2) more
sharply than the existing results, by developing an array of useful
mathematical tools in Analysis. The main results for the quantitative
estimates of strong solutions can be stated in the following theorem.

######  Theorem 1.1 (Main Theorem).

There are two universal constants $C_{1}>0$ and $C_{2}>0$ such that there
exists a unique strong solution $u(x,t)$ of the Cauchy problem (1.1)–(1.2) for
all $t\in[0,T_{0}]$ with

$T_{0}=C_{1}\nu^{2}L^{-4}\left\|\omega_{0}\right\|_{\infty}^{-2},$ (1.5)

so that the following estimates for $u(x,t)$ hold: For all $0<t\leq T_{0}$ and
$x\in\mathbb{R}^{3}$,

$\displaystyle|u(x,t)|\leq
C_{2}L\left\|\omega_{0}\right\|_{\infty},\qquad|\nabla
u(x,t)|\leq\frac{C_{2}L}{\sqrt{\nu t}}\left\|\omega_{0}\right\|_{\infty},$
(1.6) $\displaystyle|\omega(x,t)|\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad|\nabla\omega(x,t)|\leq\frac{C_{2}}{\sqrt{\nu
t}}\left\|\omega_{0}\right\|_{\infty}$ (1.7)

where $\left\|\omega_{0}\right\|_{\infty}$ is the $L^{\infty}$-norm of the
initial vorticity $\omega_{0}$.

The two positive constants $C_{1}$ and $C_{2}$ in Theorem 1.1 are computable,
which can be worked out by tracing all the universal constants in the proof.

The Cauchy problem (1.1)–(1.2) of the Navier-Stokes equations with periodic
initial data $u_{0}(x)$ has been studied traditionally in the Fourier space,
which is particularly the case in turbulence literature; see, for example, [2,
7, 11]. Our approach is necessary to departure from the well-known methods.
The quantitative regularity estimates, Theorem 1.1, will be proved by
developing a new approach via an array of mathematical tools from the theory
of partial differential equations, stochastic analysis, and the Hodge theory
on the torus.

###  1.2 New approach – An iteration scheme for the construction and estimates
of strong solutions

We now describe briefly the new approach – an iteration scheme – developed in
this paper to prove Theorem 1.1; see §6–§7 for details.

First of all, by using the dimensionless scaling,

$U(x,t)=\frac{L}{2\nu}u(Lx,\frac{L^{2}}{2\nu}t)$

has period $1$ and solves the Navier-Stokes equations (1.1) with
$\nu=\frac{1}{2}$. Thus, without loss of generality, we will assume that the
viscosity constant $\nu=\frac{1}{2}$ and $L=1$ in what follows. Furthermore,
from now on, by a (time-dependent) periodic tensor field $f(x,t)$ on
$\mathbb{R}^{3}$, or equivalently by saying that a tensor field $f(x,t)$ is
periodic, we mean that $f(x,t)$ is a tensor field on $\mathbb{R}^{3}$
depending on the time parameter $t$ and satisfies that
$f(x+\mathbb{k},t)=f(x,t)$ for all $x\in\mathbb{R}^{3}$, $t\geq 0$, and
$\mathbb{k}\in\mathbb{Z}^{3}$.

Our new iteration scheme for the construction of strong solutions of the
Cauchy problem (1.1)–(1.2) is based on the vorticity equation for
$\omega=\nabla\wedge u$:

$\partial_{t}\omega+(u\cdot\nabla)\omega-A(u)\omega-\frac{1}{2}\Delta\omega=0,$
(1.8)

where $A(u)$ is the total derivative of $u$, a tensor field with components
$A(u)_{j}^{i}=\partial_{x^{j}}u^{i}$. Although there are several formulations
of the vorticity equations, one of our main observations is that this version
of formulation serves our aims particularly well.

Suppose that $b(x,t)$ is a periodic, smooth, and divergence-free vector field
such that $b(x,0)=u_{0}(x)$. Then we define a vector field $w(x,t)$, which
should be a candidate of the vorticity (while $w$ is not in general the
vorticity of $b$), by solving the Cauchy problem of the following linear
parabolic equations:

$\begin{cases}\partial_{t}w+(b\cdot\nabla)w-A(b)w-\frac{1}{2}\Delta w=0,\\\
w(\cdot,0)=\omega_{0},\end{cases}$ (1.9)

where $A(b)=(A(b)_{j}^{i})=(\partial_{x^{j}}b^{i})$ is the total derivative of
$b$. The unique solution $w(x,t)$ has two properties that are important to our
approach:

  1. (i)

It can be shown that $\nabla\cdot w=0$ (divergence-free again); this is the
key property that makes the linear parabolic equations (1.9) appealing and
workable to our task.

  2. (ii)

$\int_{[0,1]^{3}}w(x,t)\,\textrm{d}x=0$ for all $t>0$, which is satisfied when
$t=0$.

With these, we define the candidate $v(x,t)$ for the velocity by solving the
Poisson equation:

$\begin{cases}\Delta v=-\nabla\wedge w,\\\
\int_{[0,1]^{3}}v(x,t)\,\textrm{d}x=0\qquad\mbox{for any $t>0$}.\end{cases}$

Then $w=\nabla\wedge v$, according to the Hodge theory on the torus.

In this way, we construct a mapping $V$ that sends $b(x,t)$ to $v(x,t)$. That
is, the iteration for obtaining a strong solution is defined as

$u^{(n)}=V(u^{(n-1)})\qquad\,\,\mbox{for $n=1,2,\ldots$},$ (1.10)

with the initial iteration defined by $u^{(0)}(x,t)=u_{0}(x)$ for all $x$ and
$t\geq 0$.

Let us point out that the computational schemes for simulations of turbulent
flows based on different formulations of the vorticity equations have been
very fruitful in the past, cf. [6, 27] for an overview. Our analysis below
shows that the iteration scheme developed in this paper does converge to the
strong solution with inherent gradient estimates of the iteration solutions
uniformly. This shows that the iteration scheme should also be useful for
developing numerical algorithms to compute turbulent solutions.

###  1.3 Assumptions and notations

In order to describe the technical aspects in our study, we introduce a few
notations and assumptions that will be used throughout the paper. By a
function or a tensor field we mean a function defined on $\mathbb{R}^{d}$ or a
tensor field on the torus $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$; in
the latter case, it is identified with a periodic field on $\mathbb{R}^{d}$
with period $1$ in each coordinate variable.

Suppose that $f(x,t)$ for $x\in\mathbb{R}^{d}$ is a tensor field depending on
a time parameter $t\geq 0$. Then we assume that $f$ is Borel measurable on
$\mathbb{R}^{d}\times[0,\infty)$. For $I\subset[0,\infty)$, the
$L^{\infty}$-norm of $f$ over $\mathbb{R}^{d}\times I$ is defined to be

$\left\|f\right\|_{L^{\infty}(I)}=\sup_{(x,t)\in\mathbb{R}^{d}\times
I}\left|f(x,t)\right|,$

and for the case when $I=[0,\infty)$, the previous norm is simply denoted by
$\left\|f\right\|_{\infty}$.

If $0\leq\tau<T$, the parabolic $L^{\infty}$-norm of $f$ over an interval
$[\tau,T]$ will play an important role, which is defined by

$\left\|f\right\|_{\tau\rightarrow
T}=\sup_{(x,t)\in\mathbb{R}^{d}\times[\tau,T]}\left|\sqrt{t-\tau}f(x,t)\right|=\left\|\sqrt{\cdot-\tau}f\right\|_{L^{\infty}([\tau,T])}.$
(1.11)

In the case when $T=\infty$, the interval $[\tau,T]$ is replaced by
$[\tau,\infty)$. From the definition, it is clear that

$\left\|f\right\|_{\tau\rightarrow
T}\leq\sqrt{T-\tau}\left\|f\right\|_{L^{\infty}([\tau,T])}\leq\sqrt{T-\tau}\left\|f\right\|_{\infty}.$
(1.12)

Throughout the paper, the probability density function (PDF) of a normal
random variable with mean zero and variance $t>0$ is denoted by

$G_{t}(x)=\frac{1}{(2\pi t)^{d/2}}\exp(-\frac{|x|^{2}}{2t})\qquad\mbox{for
$x\in\mathbb{R}^{d}$}.$ (1.13)

Notice that $G_{t-\tau}(y-x)$ is the fundamental solution of the heat operator
$\partial_{t}-\frac{1}{2}\Delta$ in the Euclidean space $\mathbb{R}^{d}$.

As a convention, the Laplacian $\Delta$ and the gradient $\nabla$ (in
particular, the divergence operation $\nabla\cdot$ and the curl
$\nabla\wedge$), when operating on time-dependent tensor fields on
$\mathbb{R}^{d}$, apply to space variable $x$ only.

If $b(x,t)$ is a time-dependent vector field on $\mathbb{R}^{d}$ for $t\geq
0$, the heterogeneous drift-diffusion differential operator of second order:

$L_{b(x,t)}=\frac{1}{2}\Delta+b(x,t)\cdot\nabla$ (1.14)

will play an important role in our study. When no confusion arises,
$L_{b(x,t)}$ is denoted simply by $L_{b}$. Among the technical assumptions on
$b(x,t)$, the most essential one is the assumption that $b(x,t)$ is
_solenoidal_ (i.e., _divergence-free_). That is, for every $t$, the divergence
$\nabla\cdot b(\cdot,t)$ vanishes identically in the distributional sense.
Under this assumption, the formal adjoint operator:

$L_{b}^{\star}=L_{-b},$

which is again an elliptic operator of the same type. The second assumption is
technical for the construction of probabilistic structures. For simplicity, we
assume that $b(x,t)$ is Borel measurable and bounded over any finite interval,
i.e., $\left\|b\right\|_{L^{\infty}(\mathbb{R}^{d}\times[0,T])}<\infty$ for
every $T>0$. The probability density function of the $L_{b}$-diffusion (i.e.,
the fundamental solution or heat kernel to the parabolic operator
$L_{b(x,t)}^{\star}+\partial_{t}$; see §2) is denoted by

$p_{b}(\tau,x,t,y)\qquad\mbox{for $t>\tau\geq 0$ and $x,y\in\mathbb{R}^{d}$}.$

Throughout the paper, universal constants (the constants depending only on the
dimension $d$, or some parameters $\beta$, $\gamma$, etc. introduced in
proofs) are denoted by $C_{1}$, $C_{2}$, etc. which may be different at each
occurrence.

###  1.4 _A priori_ estimates of solutions of the parabolic equations

In order to prove Theorem 1.1, the main effort is to derive precise _a priori_
estimates of solutions of the Cauchy problem (1.9) for the parabolic
equations. To achieve this, there are two tasks to be carried out.

1\. The main task is to derive explicit _a priori_ estimates for the
fundamental solution (or called the heat kernel) of the parabolic operator
$\partial_{t}-L_{b}$ and its gradient in terms of the bound of $b$ and the
parabolic norm of $\nabla b$, when $b(x,t)$ is a bounded, divergence-free, and
smooth vector field on $\mathbb{R}^{d}$.

Although the regularity theory for linear parabolic equations has been well
established (cf. [12, 22, 38]), our _a priori_ estimates contain the universal
constants depending only on the dimension $d$ and a parameter $\beta>1$ fixed
in our estimates. In addition to its explicit form, for a divergence-free
vector field $b(x,t)$, the gradient estimate for the heat kernel associated
with the parabolic operator $\partial_{t}-L_{b}$, depends only on the bound of
$b$ and its first-order derivative $\nabla b$.

######  Theorem 1.2.

Let $b(x,t)$ be a smooth, divergence-free, and bounded time-dependent vector
field on $\mathbb{R}^{d}$. Then, for every $\beta>1$, there are constants
$C_{1}$ and $C_{2}$ depending only on $\beta$ and dimension $d$ such that

$\displaystyle p_{b}(\tau,x,t,y)\leq
C_{1}\textrm{e}^{C_{2}(t-\tau)\left\|b\right\|_{\infty}^{2}}G_{\beta(t-\tau)}(y-x),$
(1.15)
$\displaystyle\left|\nabla_{y}p_{b}(\tau,x,t,y)\right|\leq\frac{C_{1}}{\sqrt{t-\tau}}\textrm{e}^{C_{2}(t-\tau)\left\|b\right\|_{\infty}^{2}+\frac{1}{2}\sqrt{t-\tau}\left\|\nabla
b\right\|_{\tau\rightarrow t}}G_{\beta(t-\tau)}(y-x),$ (1.16)

for all $t>\tau$ and $x,y\in\mathbb{R}^{d}$.

These estimates are quite delicate to derive: They are obtained by introducing
substantial tools from stochastic analysis, mainly various functional integral
representations for the fundamental solutions (cf. [29, 28]) and a new kind of
Bismut’s formulas (cf. [3, 4]), together with careful and explicit
computations. Estimates (1.15)–(1.16) will be proved in §4 and §5,
respectively.

2\. The second task in our study is to prove the explicit _a priori_ estimates
to the iteration $u^{(n)}=V(u^{(n-1)})$ in (1.10), or equivalently, to derive
the a priori estimates of the solutions of the Cauchy problem (1.9) for the
linear parabolic equations that define the nonlinear mapping $V$, namely the
solution of the Cauchy problem for the parabolic equations:

$\begin{cases}(\partial_{t}-L_{-b})w=A(b)w,\\\
w(\cdot,0)=\omega_{0},\end{cases}$ (1.17)

where, as before, $L_{-b}$ denotes the time-dependent elliptic operator
$\frac{1}{2}\Delta-b\cdot\nabla$.

The crucial observation is that the term on the right-hand side, $A(b)w$, is a
linear zero-order term, which differs fundamentally from the linearized
Navier-Stokes equations. This crucial difference allows us to apply the
Feynman-Kac-type formula, obtained in this context in [28], to derive the
necessary explicit _a priori_ estimates.

The _a priori_ estimates and technical tools are worked out for a general
dimension $d$ and a general vector field $b(x,t)$ that is divergence-free on
$\mathbb{R}^{d}$. Therefore, they have independent interests and are likely
useful for both treating the Navier-Stokes equations with other boundary
conditions and dealing with other linear/nonlinear PDEs.

###  1.5 Organisation of the paper

In §2, several probabilistic structures associated with a time-dependent
vector field $b(x,t)$ are reviewed, and then a functional integration
representation formula for the heat kernel of $\frac{1}{2}\Delta-b\cdot\nabla$
and a Bismut-type formula for the gradient of the heat kernel are established,
which are the tools for deriving the _a priori_ estimates in Theorem 1.2 we
need. In §3, several technical potential estimates are established, which will
be used in the proof of Theorem 1.2 that will be carried out in §4–§5. In §6,
the linearized vorticity equations are carefully analyzed, and the main
regularity results for the strong solutions of the Cauchy problem (1.1)–(1.2)
with periodic initial data (1.2) for the Navier-Stokes equations (1.1) in
${\mathbb{R}}^{3}$ will be proved in §7.

##  2 Probabilistic Tools for the New Approach

In this section, we first introduce several probabilistic structures
associated with a time-dependent vector field $b(x,t)\in\mathbb{R}^{d}$ that
is divergence-free (in the distributional sense), bounded, and Borel
measurable. Then we establish a functional integration representation formula
and a Bismut-type formula for the heat kernel of the corresponding
heterogeneous drift-diffusion operator.

###  2.1 Fundamental solutions and diffusions

Let $\varGamma_{b}(x,t,\xi,\tau)$, for $0\leq\tau<t$ and
$\xi,x\in\mathbb{R}^{d}$, denote the fundamental solution of the forward
parabolic operator $L_{b}-\partial_{t}$, and let
$\varGamma_{b}^{\star}(x,t,\xi,\tau)$, for $0\leq t<\tau$ and
$x,\xi\in\mathbb{R}^{d}$, denote the fundamental solution of the backward
parabolic equation $L_{b}^{\star}+\partial_{t}$; see [12] for their
definitions and basic constructions. Since $b(x,t)$ is bounded,
$\varGamma_{b}(x,t,\xi,\tau)$ and $\varGamma_{b}^{\star}(x,t,\xi,\tau)$ exist
and unique, and

$\varGamma_{b}(x,t,\xi,\tau)=\varGamma_{b}^{\star}(\xi,\tau,x,t)\qquad\mbox{for
any $t>\tau\geq 0$ and $x,\xi\in\mathbb{R}^{d}$}.$ (2.1)

Moreover, $\varGamma_{b}(x,t,\xi,\tau)$ is positive and continuous in $\tau<t$
and $x,\xi\in\mathbb{R}^{d}$, and $\varGamma_{b}^{\star}(x,t,\xi,\tau)$ is
positive and continuous in $t<\tau$ and $x,\xi\in\mathbb{R}^{d}$.

Let $p_{b}(\tau,\xi,t,y)$, for $0\leq\tau<t$ and $\xi,y\in\mathbb{R}^{d}$,
denote the transition probability density function of the $L_{b}$-diffusion
(cf. [18, 37, 38]). Since $\nabla\cdot b=0$,

$p_{b}(\tau,\xi,t,y)=\varGamma_{-b}^{\star}(\xi,\tau,y,t)=\varGamma_{-b}(y,t,\xi,\tau)\qquad\mbox{for
$t>\tau\geq 0$ and any $\xi,y\in\mathbb{R}^{d}$}.$ (2.2)

For $T>0$, $b^{T}(x,t)$ denotes a bounded divergence-free vector field such
that $b^{T}(x,t)$ coincides with $b(x,T-t)$ for all $0\leq t\leq T$ and
$x\in\mathbb{R}^{d}$. Then

$\displaystyle p_{b^{T}}(T-t,y,T-\tau,\xi)=\varGamma_{b}(y,t,\xi,\tau),$ (2.3)
$\displaystyle p_{b}(\tau,\xi,t,y)=p_{-b^{T}}(T-t,y,T-\tau,\xi),$ (2.4)

for all $0\leq\tau<t\leq T$ and $\xi,y\in\mathbb{R}^{d}$.

Let $\varOmega=C([0,\infty),\mathbb{R}^{d})$ be the space of continuous paths
$\varphi:[0,\infty)\rightarrow\mathbb{R}^{d}$, equipped with the natural
filtration denoted by $\mathcal{F}_{t}^{0}$ for $t\geq 0$. For $\tau\geq 0$
and $\xi\in\mathbb{R}^{d}$, there is a unique probability measure
$\mathbb{P}_{b}^{\tau,\xi}$ on $\varOmega$ of all continuous paths such that

$\mathbb{P}_{b}^{\tau,\xi}\left[\varphi\in\varOmega\,:\,\varphi(t)=\xi\,\textrm{
for all }0\leq t\leq\tau\right]=1,$

and the marginal distribution for any finite partition
$\tau=t_{0}<t_{1}<\cdots<t_{k}$:

$\mathbb{P}_{b}^{\tau,\xi}\left[\varphi(t_{1})\in{\rm d\hskip
0.28453pt}x_{1},\cdots,\varphi(t_{k})\in{\rm d\hskip 0.28453pt}x_{k}\right]$

is given by

$p_{b}(\tau,\xi,t_{1},x_{1})p_{b}(t_{1},x_{1},t_{2},x_{2})\cdots
p_{b}(t_{k-1},x_{k-1},t_{k},x_{k}){\rm d\hskip 0.28453pt}x_{1}\cdots{\rm
d\hskip 0.28453pt}x_{k}.$

The collection $\mathbb{P}_{b}^{\tau,\xi}$, for $\tau\geq 0$ and
$\xi\in\mathbb{R}^{d}$, is called the diffusion with infinitesimal generator
$L_{b}$. The construction of $L_{b}$-diffusion, or equivalently of
$p_{b}(s,x,t,y)$, can be based on the Cauchy problem for the stochastic
differential equation (SDE):

$\begin{cases}\textrm{d}X=b(X,t)\textrm{d}t+\textrm{d}B,\\\
X_{\tau}=\xi,\end{cases}$ (2.5)

where $B=(B^{1},\cdots,B^{d})$ is a Brownian motion on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$ (cf. [18, 37] for the details). If
$b(x,t)$ is jointly continuous and global Lipschitz continuous in $x$
(uniformly in $t$), then the Cauchy problem (2.5) has a unique strong
solution, whose distribution gives rise to the probability measure
$\mathbb{P}_{b}^{\tau,\xi}$.

###  2.2 Diffusion bridges and a Feynman-Kac-type formula

Let $\tau\geq 0$, $T>\tau$, and $\xi,\eta\in\mathbb{R}^{d}$ be fixed. The
conditional law $\mathbb{P}_{b}^{\xi,\tau\rightarrow\eta,T}$, also called the
pinned measure or $L_{b}$-diffusion bridge measure, is formally defined to be
$\mathbb{P}_{b}^{\tau,\xi}\left[\,\,\cdot\,\,|\,w(T)=\eta\right]$, which is
again Markovian. According to (14.1) in [8],
$\mathbb{P}_{b}^{\xi,\tau\rightarrow\eta,T}$ is the unique probability measure
on $\varOmega$ with time non-homogeneous transition probability density
function:

$q_{b}(s,x,t,y)=\frac{p_{b}(s,x,t,y)p_{b}(t,y,T,\eta)}{p_{b}(s,x,T,\eta)}\qquad\,\,\mbox{for
$\tau<s<t<T$ and $x,y\in\mathbb{R}^{d}$}.$ (2.6)

It can be shown (cf. [29]) that

$\left.\frac{\textrm{d}\mathbb{P}_{b}^{\xi,\tau\rightarrow\eta,T}}{\textrm{d}\mathbb{P}_{b}^{\xi,\tau}}\right|_{\mathcal{F}_{s}^{0}}=\frac{p_{b}(t,\varphi(t),T,\eta)}{p_{b}(\tau,\xi,T,\eta)}\qquad\,\,\mbox{for
$t\in[\tau,T)$},$ (2.7)

where $\varphi(t)$ denotes the general sample point and the canonical process
on $\Omega$.

######  Theorem 2.1.

The pinned measures satisfy the following duality relation:

$\mathbb{P}_{b}^{\eta,0\rightarrow\zeta,T}=\mathbb{P}_{-b^{T}}^{\zeta,0\rightarrow\eta,T}\circ\tau_{T},$

where $\tau_{T}$ is the time reversal operator at $T$, that is,
$\tau_{T}:\varOmega\rightarrow\varOmega$ which sends $w$ to
$\tau_{T}w(t)=w(T-t)$ for $t\in[0,T]$.

Therefore, if $\\{X_{t}\\}$ is an $L_{b}$-diffusion and $\\{Y_{t}\\}$ is an
$L_{-b^{T}}$-diffusion on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$, then

$\mathbb{P}\left[F(X_{\cdot})\,|\,X_{0}=\eta,X_{T}=\zeta\right]=\mathbb{P}\left[F(Y_{T-\cdot})\,|\,Y_{0}=\zeta,Y_{T}=\eta\right]$

for any bounded or positive Borel measurable function $F$ on $\varOmega$.

The following is a Feynman-Kac-type formula that was established in [28] in
this version:

######  Theorem 2.2.

Suppose that $f(x,t)$ is a strong solution to the parabolic equations:

$\big{(}\partial_{t}-L_{-b(x,t)}\big{)}f^{i}(x,t)=A_{j}^{i}(x,t)f^{j}(x,t)\qquad\mbox{
in $\mathbb{R}^{d}\times[0,\infty)$},$ (2.8)

subject to the initial condition: $f(x,0)=f_{0}(x)$, and $A_{j}^{i}(x,t)$ are
joint continuous in $(x,t)$, Lipschitz continuous in $x$ $($uniformly in $t$
in any finite interval, where $i,j=1,\ldots,d$). Then

$f(x,t)=\int_{\mathbb{R}^{d}}f_{0}(\xi)p_{b}(0,\xi,t,x)\mathbb{P}_{b}^{\xi,0\rightarrow
x,t}\left[Q(0,t)\right]{\rm d\hskip 0.28453pt}\xi,$ (2.9)

where, for every $t>0$, $Q(s,t,\varphi)=\big{(}Q_{j}^{i}(s,t,\varphi)\big{)}$
$($but the sample point $\varphi\in\varOmega$ will be suppressed if no
confusion arises$)$ is the unique solution of the Cauchy problem for the
differential equations:

$\displaystyle\begin{cases}\frac{{\rm d\hskip 0.28453pt}}{{\rm d\hskip
0.28453pt}s}Q_{j}^{i}(s,t,\varphi)=-Q_{k}^{i}(s,t,\varphi)A_{j}^{k}(\varphi(s),s),\\\\[2.84526pt]
Q_{j}^{i}(t,t,\varphi)=\delta_{j}^{i}\qquad\mbox{for $s\leq t$},\end{cases}$

for $i,j=1,\ldots,d$, and $\varphi\in\varOmega$.

###  2.3 Functional integral representation for fundamental solutions

Let $B$ be a standard Brownian motion of dimension $d$ on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$. For $\tau\geq 0$ and
$\xi\in\mathbb{R}^{d}$,

$X_{t}^{\tau,\xi}=\begin{cases}B_{t}-B_{\tau}+\xi\quad&\mbox{for
$t\geq\tau$},\\\ \xi\quad&\mbox{for $t\leq\tau$}.\end{cases}$

Then $X^{\tau,\xi}$ is a Brownian motion stated at $\xi$ at the initial time
$\tau$. According to Theorem 6.4.2 in [37], the $L_{b}$-diffusion may also be
constructed by using the Cameron-Martin density:

$U_{b}^{\tau,\xi}(t)=\exp\Big{[}\int_{\tau\wedge
t}^{t}b(X_{s}^{\tau,\xi},s)\cdot\textrm{d}B_{s}-\frac{1}{2}\int_{\tau\wedge
t}^{t}|b|^{2}(X_{s}^{\tau,\xi},s)\,\textrm{d}s\Big{]}.$ (2.10)

######  Theorem 2.3.

The following representation holds for all $t>\tau$:

$p_{b}(\tau,\xi,t,y)=G_{t-\tau}(y-\xi)+\int_{\tau}^{t}\mathbb{P}\Big{[}U_{b}^{\tau,\xi}(s)G_{t-s}(y-X_{s}^{\tau,\xi})b(X_{s}^{\tau,\xi},s)\cdot\frac{y-X_{s}^{\tau,\xi}}{t-s}\Big{]}\,\textrm{\rm
d}s.$ (2.11)

###### Proof.

The theorem was established in [29] for the time homogeneous case, and their
method can be developed to be applied to our case. For the completeness, we
outline the proof here. According to the Cameron-Martin formula,

$\displaystyle\int_{\mathbb{R}^{d}}p_{b}(\tau,\xi,t,x)f(x)\,\textrm{d}x$
$\displaystyle=\mathbb{P}\big{[}U_{b}^{\tau,\xi}(t)f(X_{t}^{\tau,\xi})\big{]}$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathbb{P}\big{[}U_{b}^{\tau,\xi}(t)f(x)\big{|}X_{t}^{\tau,\xi}=x\big{]}\mathbb{P}\big{[}X_{t}^{\tau,\xi}\in\textrm{d}x\big{]}$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathbb{P}\big{[}U_{b}^{\tau,\xi}(t)\big{|}X_{t}^{\tau,\xi}=x\big{]}G_{t-\tau}(x-\xi)f(x)\,\textrm{d}x.$

Choose $f(z){\rm d}z=\delta_{x}(\textrm{d}z)$ to obtain

$\frac{p_{b}(\tau,\xi,t,y)}{G_{t-\tau}(y-\xi)}=\mathbb{P}\big{[}U_{b}^{\tau,\xi}(t)\big{|}X_{t}^{\tau,\xi}=y\big{]}.$
(2.12)

Now we notice that both

$R(s)=\frac{G_{t-s}(y-X_{s}^{\tau,\xi})}{G_{t-\tau}(y-\xi)}$

(see (2.7)) and $U_{b}^{\tau,\xi}(s)$ are exponential martingales so that

$\displaystyle\textrm{d}R(s)=R(s)\nabla\ln
G_{t-s}(y-X_{s}^{\tau,\xi})\cdot\textrm{d}B(s),\qquad R(\tau)=1,$
$\displaystyle\textrm{d}U_{b}^{\tau,\xi}(s)=U_{b}^{\tau,\xi}(s)b(X_{s}^{\tau,\xi},s)\cdot\textrm{d}B(s),\qquad\quad\,\,\,U_{b}^{\tau,\xi}(\tau)=1.$

Therefore, integrating by parts, together with (2.7), yields

$\displaystyle\frac{p_{b}(\tau,\xi,t,y)}{G_{t-\tau}(y-\xi)}$
$\displaystyle=\lim_{s\uparrow
t}\mathbb{P}\big{[}R(s)U_{b}^{\tau,\xi}(s)\big{]}$
$\displaystyle=1+\mathbb{P}\big{[}\big{\langle}R,U_{b}^{\tau,\xi}\big{\rangle}_{t}\big{]}$
$\displaystyle=1+\frac{1}{G_{t-\tau}(y-\xi)}\mathbb{E}\Big{[}\int_{\tau}^{t}U_{b}^{\tau,\xi}(s)b(X_{s}^{\tau,\xi},s)\cdot\nabla
G_{t-s}(y-X_{s}^{\tau,\xi})\,\textrm{d}s\Big{]},$

which leads to the representation formula in (2.11). ∎

###  2.4 A Bismut-type formula

From now on, we make a further assumption that $b(x,t)$ is a time-dependent,
bounded, and divergence-free $C^{1}$-vector field.

Using the idea of Bismut (cf. Chapter 14, §14.1 in Bismut-Lebeau [4], and also
cf. Elworthy-Li [9]), we first establish a Bismut-type formula for the
gradient of the fundamental solution:

$\partial_{x^{j}}\ln p_{b}(\tau,\xi,t,x).$

Let $\tau\geq 0$ and $\xi\in\mathbb{R}^{d}$ be fixed. Bismut’s idea is based
on the following observation: Since $b$ is divergence-free, as a function of
$t>\tau$ and $x\in\mathbb{R}^{d}$, $p_{b}(\tau,\xi,t,x)$ solves the forward
parabolic equation $(L_{-b}-\partial_{t})p_{b}=0$, so that

$\big{(}L_{-b}-\partial_{t}\big{)}\ln p_{b}=-\frac{1}{2}\left|\nabla\ln
p_{b}\right|^{2}\qquad\mbox{on $\mathbb{R}^{d}\times(\tau,\infty)$}.$ (2.13)

Let $T>0$. Consider $f(x,t)=\ln p_{b}(\tau,\xi,\tau+T-t,x)$ for $0\leq t<T$.
Then

$\big{(}L_{-b}+\partial_{t}\big{)}f=-\frac{1}{2}\left|\nabla f\right|^{2},$
(2.14)

where both sides are evaluated at $(\tau,\xi,\tau+T-t,x)$. Now we consider the
following Cauchy problem of the stochastic differential equations:

$\begin{cases}\textrm{d}Y_{t}^{x}=\textrm{d}B_{t}-b(Y_{t}^{x},T+\tau-t)\,\textrm{d}t,\\\
Y_{0}^{x}=x,\end{cases}$ (2.15)

which determines a diffusion process with generator $L_{-b^{T+\tau}}$, where
$B$ is a standard $d$-dimensional Brownian motion on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$, whose filtration is denoted by
$(\mathcal{F}_{t}^{0})_{t\geq 0}$. Then (2.14) implies that

$R_{t}^{x}=\frac{p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})}{p_{b}(\tau,\xi,T+\tau,x)}=\textrm{e}^{f(Y_{t}^{x},t)-f(x,0)}\qquad\mbox{for
$t\in[0,T)$}$ (2.16)

is a positive martingale and

$\begin{cases}\textrm{d}R_{t}^{x}=R_{t}^{x}\nabla
f(Y_{t}^{x},t)\cdot\textrm{d}B_{t},\\\ R_{0}^{x}=1.\end{cases}$ (2.17)

Finally, we set $Z_{j}^{k}(x,t)=\partial_{x^{j}}Y_{t}^{x,k}$ for
$j,k=1,\cdots,d$.

######  Theorem 2.4 (Bismut-Type Formula for the Forward Variable).

Suppose that $b(x,t)$ is bounded and $C^{1}$ with bounded derivative over any
finite interval so that $\nabla\cdot b=0$. Then

$\partial_{x^{j}}\ln
p_{b}(\tau,\xi,T+\tau,x)=\mathbb{Q}\Big{[}\int_{0}^{T}\frac{\rho^{\prime}(t)}{\rho(T)}Z_{j}^{k}(x,t)\,\textrm{d}B_{t}^{k}\Big{]},$
(2.18)

where $\mathbb{Q}$ is the probability measure on
$(\varOmega,\mathcal{F}_{T}^{0})$ such that

$\left.\frac{\textrm{d}\mathbb{Q}}{\textrm{d}\mathbb{P}}\right|_{\mathcal{F}_{t}^{0}}=R_{t}^{x}\qquad\quad\mbox{for
$t\in[0,T)$},$

and $\rho(t)$ is any continuous and piecewise differentiable function with
$\rho(0)=0$ and $\rho(t)>0$ for $t>0$.

###### Proof.

Since $t\rightarrow R_{t}^{x}$ is a martingale so that

$p_{b}(\tau,\xi,T+\tau,x)=\mathbb{P}\left[p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})\right],$

where $\mathbb{P}$ (similarly for $\mathbb{Q}$) also means taking expectation
with respect to $\mathbb{P}$ (resp. $\mathbb{Q}$). By using the previous fact,

$\displaystyle\partial_{x^{j}}\ln p_{b}(\tau,\xi,T+\tau,x)$
$\displaystyle=\frac{\partial_{x^{j}}p_{b}(\tau,\xi,T+\tau,x)}{p_{p}(\tau,\xi,T+\tau,x)}$
$\displaystyle=\frac{\mathbb{P}\left[\partial_{x^{j}}p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})\right]}{p_{p}(\tau,\xi,T+\tau,x)}$
$\displaystyle=\mathbb{P}\Big{[}\frac{p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})}{p_{p}(\tau,\xi,T+\tau,x)}\partial_{x^{j}}\ln
p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})\Big{]}$
$\displaystyle=\mathbb{P}\big{[}R_{t}^{x}Z_{j}^{k}(x,t)\partial_{x^{j}}\ln
p_{b}(\tau,\xi,T+\tau-t,Y_{t}^{x})\big{]}\qquad\,\,\,\mbox{for all $0<t<T$},$

where $Z_{j}^{k}(x,t)=\partial_{x^{j}}Y_{t}^{x,k}$ for $j,k=1,\cdots,d$. Let

$M_{t}^{j}=\int_{0}^{t}\rho^{\prime}(s)Z_{j}^{k}(s)\,\textrm{d}B_{s}^{k}\qquad\,\,\mbox{for
$0<t<T$}.$

Observe that

$\left\langle
R,M^{j}\right\rangle_{t}=\int_{0}^{t}\rho^{\prime}(s)Z_{j}^{k}(s)R_{s}\partial_{x^{k}}\ln
p_{b}(\tau,\xi,T+\tau-s,Y_{s})\,\textrm{d}s.$

Thus we have

$\displaystyle\partial_{x^{j}}\ln p_{b}(\tau,\xi,T+\tau,x)$
$\displaystyle=\frac{1}{\rho(t)}\mathbb{P}\Big{[}\int_{0}^{T}\rho^{\prime}(s)Z_{j}^{k}(s)R_{s}\partial_{x^{k}}\ln
p_{b}(\tau,\xi,T+\tau-s,Y_{s})\,\textrm{d}s\Big{]}$
$\displaystyle=\frac{1}{\rho(T)}\mathbb{P}\left[\left\langle
R^{x},M^{j}\right\rangle_{T}\right]$
$\displaystyle=\frac{1}{\rho(T)}\mathbb{P}\Big{[}R_{T}^{x}\int_{0}^{T}\rho^{\prime}(t)Z_{j}^{k}(t)\,\textrm{d}B_{t}^{k}\Big{]}.$

This completes the proof. ∎

##  3 Basic Estimates for the Heat Kernel

In this section, we establish several estimates for the heat kernel. In
particular, Lemma 3.3 contains the technical estimates needed for the proof of
Theorem 4.1 below.

Recall that $G_{t}(x)=(2\pi t)^{-\frac{d}{2}}\exp(-\frac{|x|^{2}}{2t})$ for
$x\in\mathbb{R}^{d}$ and $t>0$. Clearly,

$\nabla_{x}\ln G_{t-s}(y-x)=\frac{y-x}{t-s}\qquad\,\,\mbox{for $t>s\geq 0$},$
(3.1)

which leads to the following equality:

$|\sqrt{t-s}\nabla_{x}G_{t-s}(y-x)|=\frac{|y-x|}{\sqrt{t-s}}G_{t-s}(y-x).$
(3.2)

Now we notice that, for $\beta>0$,

$G_{t-s}(y-x)=\beta^{\frac{d}{2}}e^{-\frac{\beta-1}{\beta}\frac{|y-x|^{2}}{2(t-s)}}G_{\beta(t-s)}(y-x),$
(3.3)

so that, for $\beta>1$,

$\displaystyle\frac{|y-x|}{\sqrt{t-s}}G_{t-s}(y-x)$
$\displaystyle=\beta^{\frac{d}{2}}\frac{|y-x|}{\sqrt{t-s}}e^{-\frac{\beta-1}{\beta}\frac{|y-x|^{2}}{2(t-s)}}G_{\beta(t-s)}(y-x)$
$\displaystyle=\beta^{\frac{d}{2}}\frac{|y-x|}{\sqrt{t-s}}\frac{1}{1+\frac{\beta-1}{\beta}\frac{|y-x|^{2}}{2(t-s)}+\cdots}G_{\beta(t-s)}(y-x)$
$\displaystyle\leq\frac{\beta^{\frac{d+1}{2}}}{\sqrt{2(\beta-1)}}G_{\beta(t-s)}(y-x).$

Therefore, using (3.2) and the same argument, we have the following lemma
which will be used in our computations later.

######  Lemma 3.1.

For the heat kernel in $\mathbb{\mathbb{R}}^{d}$, the following estimates
hold:

  1. (i)

For $\beta>1$,

$|\sqrt{t-s}\nabla_{x}G_{t-s}(y-x)|\leq\frac{\beta^{\frac{d+1}{2}}}{\sqrt{2(\beta-1)}}G_{\beta(t-s)}(y-x)\qquad\mbox{for
all $t>s\geq 0$ and $x,y\in\mathbb{R}^{d}$}.$ (3.4)

  2. (ii)

For $\alpha\geq 0$ and $\beta>1$,

$\Big{(}\frac{|y-x|}{\sqrt{t-s}}\Big{)}^{\alpha}G_{t-s}(y-x)\leq\beta^{\frac{d}{2}}k!\Big{(}\frac{2\beta}{\beta-1}\Big{)}^{k}G_{\beta(t-s)}(y-x)\qquad\mbox{for
all $t>s\geq 0$ and $x,y\in\mathbb{R}^{d}$},$ (3.5)

where $k=\left[\frac{\alpha}{2}\right]+1$ if $\frac{\alpha}{2}$ is not an
integer, and $k=\frac{\alpha}{2}$ otherwise.

######  Lemma 3.2.

Let $B$ be an $\mathbb{R}^{d}$-Brownian motion on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$. Consider

$I_{\alpha,\beta}(\tau,x,s,t,y)=\mathbb{P}\Big{[}\Big{(}\big{|}\frac{y-X_{s}}{t-s}\big{|}^{\alpha}G_{t-s}(y-X_{s})\Big{)}^{\beta}\Big{]}\qquad\mbox{for
$t>s>\tau$ and $x,y\in\mathbb{R}^{d}$},$ (3.6)

where $\alpha\geq 0$, $\beta>0$ and $X_{s}=B_{s}-B_{\tau}+x$. Then

$I_{\alpha,\beta}(\tau,x,s,t,y)=C_{1}t_{1}^{-(\frac{d(\beta-1)}{2}+\alpha\beta)}G_{t_{1}+t_{2}}(y-x)\int_{\mathbb{R}^{d}}\Big{|}\sqrt{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}z+\frac{t_{1}(x-y)}{t_{1}+t_{2}}\Big{|}^{\alpha\beta}G_{1}(z)\,{\rm
d}z,$ (3.7)

where

$t_{1}=\beta^{-1}(t-s),\quad t_{2}=s-\tau,$ (3.8)

and

$C_{1}=(2\pi)^{-\frac{d(\beta-1)}{2}}\beta^{-(\frac{d}{2}+\alpha)\beta}$ (3.9)

depend only on $\beta>0$ and $d$.

###### Proof.

The proof follows from an elementary computation. First notice that

$G_{t_{1}}(z-y)G_{t_{2}}(z-x)=G_{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}(z-\frac{t_{1}x+t_{2}y}{t_{1}+t_{2}})G_{t_{1}+t_{2}}(y-x)\qquad\mbox{for
$t_{1}>0$ and $t_{2}>0$}.$ (3.10)

Now, for $s>\tau$, the law of $X_{s}$ is normal with PDF $G_{s-\tau}(z-x)$,
and

$\big{(}G_{t-s}(y-z)\big{)}^{\beta}=\beta^{-\frac{d\beta}{2}}(2\pi)^{-\frac{d(\beta-1)}{2}}t_{1}^{-\frac{d(\beta-1)}{2}}G_{t_{1}}(z-y),$

where $t_{1}=\beta^{-1}(t-s)$, so that

$I_{\alpha,\beta}(\tau,x,s,t,y)=\beta^{-(\frac{d}{2}+\alpha)\beta}(2\pi)^{-\frac{d(\beta-1)}{2}}t_{1}^{-(\frac{d(\beta-1)}{2}+\alpha\beta)}\int_{\mathbb{R}^{d}}\left|z-y\right|^{\alpha\beta}G_{t_{1}}(z-y)G_{t_{2}}(z-x)\,\textrm{d}z.$

Then

$\displaystyle I_{\alpha,\beta}(\tau,x,s,t,y)$
$\displaystyle=\frac{G_{t_{1}+t_{2}}(y-x)}{(2\pi)^{\frac{d(\beta-1)}{2}}\beta^{(\frac{d}{2}+\alpha)\beta}t_{1}^{\frac{d\beta-d}{2}+\alpha\beta}}\int_{\mathbb{R}^{d}}\left|z-y\right|^{\alpha\beta}G_{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}(z-\frac{t_{1}x+t_{2}y}{t_{1}+t_{2}})\,\textrm{d}z$
$\displaystyle=\frac{G_{t_{1}+t_{2}}(y-x)}{(2\pi)^{\frac{d(\beta-1)}{2}}\beta^{\frac{d\beta}{2}+\alpha\beta}t_{1}^{\frac{d(\beta-1)}{2}+\alpha\beta}}\int_{\mathbb{R}^{d}}\Big{|}\sqrt{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}z+\frac{t_{1}(x-y)}{t_{1}+t_{2}}\Big{|}^{\alpha\beta}G_{1}(z)\,\textrm{d}z,$

where the second step follows by changing the variable in the integral, which
leads to (3.7). ∎

######  Lemma 3.3.

Let $\alpha\geq 0$ and $\beta\geq 1$.

  1. (i)

For $t>s>\tau\geq 0$,

$\displaystyle\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}\leq
C_{3}(t-\tau)^{\frac{d(\beta-1)}{2\beta}-\frac{\alpha}{2}}(t-s)^{-\frac{d(\beta-1)}{2\beta}}\Big{[}\kappa_{1}\big{(}\frac{s-\tau}{t-s}\big{)}^{\frac{\alpha}{2}}+\big{(}\frac{|x-y|}{\sqrt{t-\tau}}\big{)}^{\alpha}\Big{]}G_{\beta(t-\tau)}(y-x),$
(3.11)

where

$\kappa_{1}=\sqrt[\beta]{\int_{\mathbb{R}^{d}}\left|z\right|^{\alpha\beta}G_{1}\left(z\right)\,\textrm{d}z},\qquad
C_{3}=\beta^{\frac{2d+\alpha}{2}+\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2}}(2\pi)^{\frac{d(\beta-1)}{2\beta}}C_{1}^{\frac{1}{\beta}},$
(3.12)

where $C_{1}$ given in (3.9) depends only on $\alpha\geq 0$, $\beta\geq 1$,
and $d$.

  2. (ii)

If $\alpha\geq 0$ and $\beta\geq 1$ such that
$\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2}<1$, then

$\displaystyle\int_{\tau}^{t}\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}\,\textrm{d}s\leq
C_{4}(t-\tau)^{1-\frac{\alpha}{2}}G_{\beta(t_{1}+t_{2})}(y-x)\Big{(}1+\frac{|x-y|^{\alpha}}{(t-\tau)^{\frac{\alpha}{2}}}\Big{)}\qquad\mbox{for
$t>\tau\geq 0$},$ (3.13)

where $C_{4}=\max\left(\kappa_{3},\kappa_{0}\right)C_{3}$ depends only on
$\beta,\alpha$, and $d$ $($ note that $\kappa_{3},\kappa_{0}$ and $C_{3}$ can
be traced in the proof below$)$.

###### Proof.

By (3.7) and the triangle inequality, we obtain

$\displaystyle\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}$
$\displaystyle=C_{1}^{\frac{1}{\beta}}t_{1}^{-(\frac{d(\beta-1)}{2\beta}+\alpha)}\Big{(}\int_{\mathbb{R}^{d}}\Big{|}\sqrt{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}z+\frac{t_{1}(x-y)}{t_{1}+t_{2}}\Big{|}^{\alpha\beta}G_{1}(z)\,\textrm{d}z\Big{)}^{\frac{1}{\beta}}\sqrt[\beta]{G_{t_{1}+t_{2}}(y-x)}.$
$\displaystyle\leq
C_{1}^{\frac{1}{\beta}}t_{1}^{-(\frac{d(\beta-1)}{2\beta}+\alpha)}\Big{(}\kappa_{1}\Big{(}\frac{t_{1}t_{2}}{t_{1}+t_{2}}\Big{)}^{\frac{\alpha}{2}}+\Big{(}\frac{t_{1}|x-y|}{t_{1}+t_{2}}\Big{)}^{\alpha}\Big{)}\sqrt[\beta]{G_{t_{1}+t_{2}}(y-x)}.$

Using the identity

$\sqrt[\beta]{G_{t_{1}+t_{2}}(y-x)}=\beta^{\frac{d}{2}}(2\pi)^{\frac{d(\beta-1)}{2\beta}}(t_{1}+t_{2})^{\frac{d(\beta-1)}{2\beta}}G_{\beta(t_{1}+t_{2})}(y-x),$

and substituting this into the previous inequality, we therefore obtain

$\displaystyle\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}\leq$
$\displaystyle\,C_{2}(t_{1}+t_{2})^{\frac{d(\beta-1)}{2\beta}}(t_{1}+t_{2})^{-\frac{\alpha}{2}}G_{\beta(t_{1}+t_{2})}(y-x)$
$\displaystyle\times\Big{(}\kappa_{1}t_{1}^{-(\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2})}t_{2}^{\frac{\alpha}{2}}+t_{1}^{-\frac{d(\beta-1)}{2\beta}}\frac{|x-y|^{\alpha}}{(t_{1}+t_{2})^{\frac{\alpha}{2}}}\Big{)},$

where

$C_{2}=\beta^{\frac{d}{2}}(2\pi)^{\frac{d(\beta-1)}{2\beta}}C_{1}^{\frac{1}{\beta}}.$
(3.14)

Since

$\displaystyle t_{1}=\beta^{-1}(t-s),\quad t_{2}=s-\tau,$ $\displaystyle
t-\tau\geq t_{1}+t_{2}=\beta^{-1}(t-s)+s-\tau\geq\beta^{-1}(t-\tau),$

and

$G_{\beta(t_{1}+t_{2})}(y-x)\leq\beta^{\frac{d}{2}}G_{\beta(t-\tau)}(y-x),$

substituting these relations into the previous inequality, we thus deduce

$\displaystyle\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}$ $\displaystyle\leq
C_{3}(t-\tau)^{\frac{d(\beta-1)}{2\beta}-\frac{\alpha}{2}}G_{\beta(t-\tau)}(y-x)$
$\displaystyle\quad\,\times\Big{(}\kappa_{1}(s-\tau)^{\frac{\alpha}{2}}(t-s)^{-(\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2})}+(t-s)^{-\frac{d(\beta-1)}{2\beta}}\frac{|x-y|^{\alpha}}{(t-\tau)^{\frac{\alpha}{2}}}\Big{)},$
(3.15)

where

$C_{3}=C_{2}\beta^{\frac{d+\alpha}{2}+\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2}}=\beta^{\frac{2d+\alpha}{2}+\frac{d(\beta-1)}{2\beta}+\frac{\alpha}{2}}(2\pi)^{\frac{d(\beta-1)}{2\beta}}C_{1}^{\frac{1}{\beta}},$
(3.16)

which implies (i).

To prove (ii), we observe that the integral of the first term in the bracket
in (3.15) can be written as

$\kappa_{1}\int_{\tau}^{t}\frac{(s-\tau)^{\frac{\alpha}{2}}}{(t-s)^{\frac{d\beta-d}{2\beta}+\frac{\alpha}{2}}}\,\textrm{d}s\\\
=\kappa_{1}\int_{\tau}^{t}\frac{(t-\tau-(t-s))^{\frac{\alpha}{2}}}{(t-s)^{\frac{d\beta-d}{2\beta}+\frac{\alpha}{2}}}\,\textrm{d}s\\\
=\kappa_{0}(t-\tau)^{1-\frac{d(\beta-1)}{2\beta}},$

where

$\kappa_{0}=\kappa_{1}\int_{0}^{1}(1-s)^{\frac{\alpha}{2}}s^{-\frac{d\beta-d}{2\beta}-\frac{\alpha}{2}}\,\textrm{d}s,$

which is finite when

$\frac{d\beta-d}{2\beta}+\frac{\alpha}{2}<1,\quad\frac{\alpha}{2}>-1.$

Similarly, the second integral can be written as

$\int_{\tau}^{t}(t-s)^{-\frac{d(\beta-1)}{2\beta}}\,\textrm{d}s=\kappa_{3}(t-\tau)^{1-\frac{d(\beta-1)}{2\beta}},$

where $\kappa_{3}=\int_{0}^{1}s^{-\frac{d(\beta-1)}{2\beta}}\,\textrm{d}s$ is
finite when $\frac{d(\beta-1)}{2\beta}<1$. Then

$\displaystyle\int_{\tau}^{t}\sqrt[\beta]{I_{\alpha,\beta}(\tau,x,s,t,y)}\,\textrm{d}s\leq
C_{3}(t-\tau)^{1-\frac{\alpha}{2}}G_{\beta(t-\tau)}(y-x)\Big{(}\kappa_{0}+\kappa_{3}\frac{|x-y|^{\alpha}}{(t-\tau)^{\frac{\alpha}{2}}}\Big{)},$

and (3.13) follows immediately. ∎

##  4 Explicit Estimates for the Fundamental Solution

We retain the basic assumption on the vector field $b(x,t)$ that is bounded
and Borel measurable. Notice that the divergence-free assumption on $b$ in the
following theorem, Theorem 4.1, is not needed.

######  Theorem 4.1.

For every $\beta>1$, there are constants $C_{1}$ and $C_{2}$ depending only on
$\beta$ and the dimension $d$ such that

$p_{b}(\tau,x,t,y)\leq
C_{1}\textrm{e}^{C_{2}(t-\tau)\left\|b\right\|_{\infty}^{2}}G_{\beta(t-\tau)}(y-x)\qquad\,\,\,\mbox{for
$t>\tau\geq 0$ and $x,y\in\mathbb{R}^{d}$}.$ (4.1)

###### Proof.

According to Theorem 2.3,

$p_{b}(\tau,x,t,y)=G_{t-\tau}(y-x)+\mathbb{E}\Big{[}\int_{\tau}^{t}U_{s}b(X_{s},s)\cdot\frac{y-X_{s}}{t-s}\,G_{t-s}(y-X_{s})\,\textrm{d}s\Big{]},$

where $X_{s}=B_{t}-B_{\tau}+x$, $B$ is the standard Brownian motion of
dimension $d$, and $U$ is the Cameron-Martin density of $L_{b}$-diffusion with
respect to the Brownian motion. Consider the second term on the right-hand
side:

$J\equiv\mathbb{E}\Big{[}\int_{\tau}^{t}U_{s}b(X_{s},s)\cdot\frac{y-X_{s}}{t-s}\,G_{t-s}(y-X_{s})\,\textrm{d}s\Big{]}.$

By the Hölder inequality, we have

$|J|\leq\left\|b\right\|_{\infty}\int_{\tau}^{t}\sqrt[\gamma]{\mathbb{P}\left[U_{s}^{\gamma}\right]}\sqrt[\beta]{I_{1,\beta}(\tau,x,s,t,y)}\,\textrm{d}s,$

where $\frac{1}{\gamma}+\frac{1}{\beta}=1$ and

$U_{s}=\exp\Big{[}\int_{\tau\wedge
s}^{s}b(X_{r},r)\cdot\textrm{d}B_{r}-\frac{1}{2}\int_{\tau\wedge
s}^{s}|b|^{2}(X_{r},r)\,\textrm{d}r\Big{]}.$

Since $U$ is an exponential martingale, then

$\mathbb{P}\left[U_{s}^{\gamma}\right]\leq\textrm{e}^{\frac{1}{2}\gamma(\gamma-1)\left\|b\right\|_{\infty}^{2}(s-\tau)},$

so that

$|J|\leq\left\|b\right\|_{\infty}\textrm{e}^{\frac{1}{2}(\gamma-1)\left\|b\right\|_{\infty}^{2}(t-\tau)}\int_{\tau}^{t}\sqrt[\beta]{I_{1,\beta}(\tau,x,s,t,y)}\,\textrm{d}s,$

where

$I_{1,\beta}(\tau,x,s,t,y)=\mathbb{P}\Big{[}\Big{(}\frac{|y-X_{s}|}{t-s}h(s,X_{s},t,y)\Big{)}^{\beta}\Big{]}$
(4.2)

and $X_{s}=B_{s}-B_{\tau}+x$. Thus, by Lemma 3.3,

$|J|\leq
C_{3}\left\|b\right\|_{\infty}\textrm{e}^{\frac{1}{2}(\gamma-1)\left\|b\right\|_{\infty}^{2}(t-\tau)}\left(\sqrt{t-\tau}+|y-x|\right)G_{\beta(t_{1}+t_{2})}(y-x),$

where $C_{3}>1$ depends only on $d$ and $\beta$, and

$t_{1}=\beta^{-1}(t-s),\quad t_{2}=s-\tau.$

The estimate follows from the following inequality: For every $\gamma>1$,
there are constants $C_{5}$ and $C_{6}$ depending only on $\gamma,\beta$, and
$d$ such that

$\sqrt{s}\left\|b\right\|_{\infty}\textrm{e}^{\frac{1}{2(\beta-1)}\left\|b\right\|_{\infty}^{2}s}\big{(}1+\frac{|z|}{\sqrt{s}}\big{)}G_{\beta
s}(z)\leq C_{5}\textrm{e}^{C_{6}\left\|b\right\|_{\infty}^{2}s}G_{\gamma\beta
s}(z).$

∎

As a consequence, we may deduce the following estimate that will be used in
the proof of the gradient estimate for $p_{b}(s,x,t,y)$.

######  Lemma 4.2.

Let $b$ be a bounded time-dependent vector field with bound
$\left\|b\right\|_{\infty}$, let $x,\xi\in\mathbb{R}^{d}$ and $t>0$ be fixed,
and let $\gamma\geq 1$. Consider the following integral:

$J(\varepsilon)=\sqrt[\gamma]{\int_{\mathbb{R}^{d}}\big{(}p_{b}(\tau,\xi,\tau+\varepsilon,z)\big{)}^{\gamma}p_{-b}(0,x,t-\varepsilon,z)\,{\rm
d}z}.$

Then, for every $\beta>\gamma$, there are $C_{1}$ and $C_{2}$ depending only
on $\beta,\gamma$, and the dimension $d$ such that

$J(\varepsilon)\leq
C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}}\left(\frac{t}{\varepsilon}\right)^{\frac{d(\gamma-1)}{2\gamma}}G_{\gamma\beta
t}(\xi-x)\qquad\,\,\mbox{for any $0<\varepsilon<t$ and
$x,\xi\in\mathbb{R}^{d}$.}$

###### Proof.

Without loss of generality, we may assume that $\tau=0$. Let $\lambda_{1}\geq
1$, $\lambda_{2}\geq 1$, and

$t_{1}=\gamma^{-1}\lambda_{1}\varepsilon,\quad
t_{2}=\lambda_{2}(t-\varepsilon).$

Then

$\frac{|z-\xi|^{2}}{t_{1}}+\frac{|z-x|^{2}}{t_{2}}=\frac{\big{|}z-\big{(}\frac{t_{2}}{t_{1}+t_{2}}\xi+\frac{t_{1}}{t_{1}+t_{2}}x\big{)}\big{|}^{2}}{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}+\frac{|\xi-x|^{2}}{(t_{1}+t_{2})},$

so that the product

$\big{(}G_{\lambda_{1}\varepsilon}(z-\xi)\big{)}^{\gamma}G_{\lambda_{2}(t-\varepsilon)}(z-x)$

equals

$\big{(}\frac{t_{1}+t_{2}}{t_{1}}\big{)}^{\frac{d}{2}(\gamma-1)}\big{(}G_{\gamma(t_{1}+t_{2})}(\xi-x)\big{)}^{\gamma}G_{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}(z-\frac{t_{2}\xi}{t_{1}+t_{2}}-\frac{t_{1}x}{t_{1}+t_{2}}).$

Let $\beta>1$ and $\gamma\geq 1$ be fixed. Then, by (4.1),

$p_{b}(\tau,\xi,\tau+\varepsilon,z)\leq
C_{1}\textrm{e}^{C_{2}\varepsilon\left\|b\right\|_{\infty}^{2}}G_{\beta\varepsilon}(z-\xi)$

and

$p_{-b}(0,x,t-\varepsilon,z)\leq
C_{1}\textrm{e}^{C_{2}(t-\varepsilon)\left\|b\right\|_{\infty}^{2}}G_{\beta(t-\varepsilon)}(z-x),$

so that

$\displaystyle E$
$\displaystyle\equiv\big{(}p_{b}(\tau,\xi,\tau+\varepsilon,z)\big{)}^{\gamma}p_{-b}(0,x,t-\varepsilon,z)$
$\displaystyle\leq
C_{1}^{\gamma+1}\textrm{e}^{C_{2}(\gamma\varepsilon+t-\varepsilon)\left\|b\right\|_{\infty}^{2}}\big{(}G_{\beta\varepsilon}(z-\xi)\big{)}^{\gamma}G_{\beta(t-\varepsilon)}d(z-x)$
$\displaystyle=C_{1}^{\gamma+1}\textrm{e}^{C_{2}(\gamma\varepsilon+t-\varepsilon)\left\|b\right\|_{\infty}^{2}}\big{(}\frac{t_{1}+t_{2}}{t_{1}}\big{)}^{\frac{d}{2}(\gamma-1)}$
$\displaystyle\quad\times\big{(}G_{\gamma(t_{1}+t_{2})}(\xi-x)\big{)}^{\gamma}G_{\frac{t_{1}t_{2}}{t_{1}+t_{2}}}(z-\frac{t_{2}\xi}{t_{1}+t_{2}}-\frac{t_{1}x}{t_{1}+t_{2}}),$

where

$t_{1}=\gamma^{-1}\beta\varepsilon,\quad t_{2}=\beta(t-\varepsilon).$

Thus, we have

$J(\varepsilon)\leq
C_{1}^{1+\frac{1}{\gamma}}\textrm{e}^{\frac{C_{2}}{\gamma}(\gamma\varepsilon+t-\varepsilon)\left\|b\right\|_{\infty}^{2}}\big{(}\frac{t_{1}+t_{2}}{t_{1}}\big{)}^{\frac{d}{2}(1-\frac{1}{\gamma})}G_{\gamma(t_{1}+t_{2})}(\xi-x).$

Since

$\gamma^{-1}\beta t\leq
t_{1}+t_{2}=\gamma^{-1}\beta\varepsilon+\beta(t-\varepsilon)\leq\beta t,$

then

$J(\varepsilon)\leq\gamma^{\frac{d}{2}(1-\frac{1}{\gamma})}C_{1}^{1+\frac{1}{\gamma}}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}}\big{(}\frac{t}{\varepsilon}\big{)}^{\frac{d}{2}(1-\frac{1}{\gamma})}G_{\gamma\beta
t}(\xi-x),$

which yields the required estimate. ∎

##  5 An Explicit Gradient Estimate for the Fundamental Solution

In this section, we assume that the vector field $b(x,t)$ is smooth,
divergence-free, and bounded, and its derivative is also bounded. Under these
assumptions, the $L_{b}$-diffusion may be constructed by solving Itô’s
stochastic differential equations. The goal of this section is to establish an
explicit gradient estimate for $p_{b}(\tau,\xi,t+\tau,x)$ with respect to $x$.
Recall that, for the heat kernel on $\mathbb{R}^{d}$,

$|\nabla_{x}G_{t}(x-\xi)|\leq\frac{\beta^{\frac{d+1}{2}}}{\sqrt{2(\beta-1)}}\frac{1}{\sqrt{t}}G_{\beta
t}(x-\xi)\qquad\,\,\mbox{for any $t>0$},$ (5.1)

where $\beta>1$ is any constant. We aim to achieve a similar bound for
$p_{b}(\tau,\xi,t+\tau,x)$. To this end, we first prove the following
estimate.

######  Lemma 5.1.

Let $T>\tau\geq 0$ be fixed, and let $Y$ be the strong solution of the Cauchy
problem of the stochastic differential equations:

$\begin{cases}\textrm{\rm d}Y^{i}=\textrm{\rm
d}B_{t}^{i}-b^{i}(Y,T+\tau-t)\textrm{\rm d}t,\\\\[2.84526pt]
Y^{i}(0)=x^{i},\end{cases}\qquad\quad\,\mbox{for $i=1,\cdots,d$},$ (5.2)

where $B=(B^{1}_{t},B^{2}_{t},\dots,B^{d}_{t})$ is the standard Brownian
motion of dimension $d$ on a probability space
$(\varOmega,\mathcal{F},\mathbb{P})$. Then
$Z_{j}^{i}=\partial_{x^{j}}Y^{i},i,j=1,\dots,n$, form the solution of the
Cauchy problem of the differential equations:

$\begin{cases}\textrm{\rm d}Z_{j}^{i}=-Z_{j}^{k}\frac{\partial b^{i}}{\partial
y^{k}}(Y,T+\tau-t)\textrm{\rm d}t,\\\\[2.84526pt]
Z_{j}^{i}(0)=\delta_{j}^{i},\end{cases}$ (5.3)

and satisfy

$|Z(t)|\leq|Z(0)|e^{2\left(\sqrt{T}-\sqrt{T-t}\right)\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}}\qquad\,\textrm{ for }t\in[0,T].$ (5.4)

###### Proof.

From (5.3), we deduce

$\begin{cases}\textrm{d}|Z|^{2}=-2Z_{i}^{j}Z_{j}^{k}\frac{\partial
b^{i}}{\partial y^{k}}(Y,T+\tau-t)\,\textrm{d}t,\\\\[2.84526pt]
Z_{j}^{i}(0)=\delta_{j}^{i}.\end{cases}$ (5.5)

By definition, we have

$|\sqrt{s-\tau}\nabla b(x,s)|\leq\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\qquad\textrm{ for }s\in[\tau,\tau+T],$

which yields that

$|\nabla b(x,T+\tau-s)|\leq\frac{1}{\sqrt{T-s}}\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\qquad\textrm{ for }s\in[0,T).$

Therefore, we have

$\displaystyle|Z(t)|^{2}$
$\displaystyle=d^{2}-2\int_{0}^{t}Z_{i}^{j}(s)Z_{j}^{k}(s)\frac{\partial
b^{i}}{\partial y^{k}}(Y,T+\tau-s)\,\textrm{d}s$ $\displaystyle\leq
d^{2}+2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\int_{0}^{t}\frac{|Z(s)|^{2}}{\sqrt{T-s}}\,\textrm{d}s\qquad\,\,\mbox{for
$0\leq t\leq T$}.$

Let

$f(t)=\int_{0}^{t}\frac{|Z(s)|^{2}}{\sqrt{T-s}}\,\textrm{d}s.$

Then the previous integral inequality may be written as

$|Z(t)|^{2}=f^{\prime}(t)\sqrt{T-t}\leq d^{2}+2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}f(t)\qquad\mbox{for $0\leq t\leq T$},$

so that

$f^{\prime}(t)\leq\frac{d^{2}}{\sqrt{T-t}}+\frac{2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}}{\sqrt{T-t}}f(t)\qquad\mbox{for all $0\leq
t<T$}.$

Define

$\displaystyle q(t):=\exp\left(-2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\int_{0}^{t}\frac{1}{\sqrt{T-s}}\,\textrm{d}s\right)=\exp\left(-4\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\big{(}\sqrt{T}-\sqrt{T-t}\big{)}\right).$

Then $q(t)$ satisfies

$\begin{cases}q^{\prime}(t)=-\frac{2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}}{\sqrt{T-t}}q(t),\\\ q(0)=1.\end{cases}$

This implies that

$(fq)^{\prime}(t)\leq\frac{d^{2}}{\sqrt{T-t}}q(t).$

After integrating from $0$ to $t\leq T$, we have

$\displaystyle f(t)\leq d^{2}\textrm{e}^{-4\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\sqrt{T-t}}\int_{0}^{t}\frac{1}{\sqrt{T-s}}\textrm{e}^{4\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\sqrt{T-s}}\,\textrm{d}s=d^{2}\frac{\textrm{e}^{4\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}(\sqrt{T}-\sqrt{T-t})}-1}{2\left\|b\right\|_{\tau\rightarrow\tau+T}}.$

Then

$\displaystyle|Z(t)|^{2}\leq d^{2}+2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}f(t)=d^{2}\textrm{e}^{4\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}(\sqrt{T}-\sqrt{T-t})},$

that is,

$|Z(t)|\leq|Z(0)|\textrm{e}^{2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}(\sqrt{T}-\sqrt{T-t})}\qquad\mbox{for all
$t\in[0,T]$},$

where $|Z(0)|=d$ is the dimension, or the norm of the identity matrix. ∎

######  Theorem 5.2 (Explicit Gradient Estimate).

For any $\beta>1$, there are constants $C_{1}$ and $C_{2}$ depending only on
$\beta$ and the dimension $d$ such that

$\left|\nabla
p_{b}\right|(\tau,\xi,t+\tau,x)\leq\frac{C_{1}}{\sqrt{t}}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+\frac{1}{2}\sqrt{t}\left\|\nabla
b\right\|_{\tau\rightarrow\tau+t}}G_{\beta t}(\xi-x)$ (5.6)

for all $t>0$, $\tau\geq 0$, and $x,\xi\in\mathbb{R}^{d}$.

###### Proof.

According to the forward Bismut-type formula (see Theorem 2.4),

$\partial_{x^{j}}\ln
p_{b}(\tau,\xi,T+\tau,x)=\mathbb{Q}\Big{[}\int_{0}^{T}\frac{\rho^{\prime}(t)}{\rho(T)}Z_{j}^{k}(t)\,\textrm{d}B^{k}(t)\Big{]},$

where $\mathbb{Q}$ is the conditional law that $Y_{\tau}=\xi$ and
$Y_{T+\tau}=x$, and $\rho(t)$ can be any continuous and piecewise
differentiable function with $\rho(0)=0$ and $\rho(t)>0$ for $t>0$. Let
$\varepsilon>0$ be small, and let $\rho$ be the function such that $\rho(t)=t$
for $t\in[0,T-\varepsilon]$ and $\rho(t)=T-\varepsilon$ for $t>T-\varepsilon$.
Then

$\displaystyle\partial_{x^{j}}\ln p_{b}(\tau,\xi,T+\tau,x)$
$\displaystyle=\frac{1}{T-\varepsilon}\mathbb{Q}\Big{[}\int_{0}^{T-\varepsilon}Z_{j}^{k}(t)\,\textrm{d}B^{k}(t)\Big{]}=\frac{1}{T-\varepsilon}\mathbb{P}\Big{[}R_{T-\varepsilon}\int_{0}^{T-\varepsilon}Z_{j}^{k}(t)\,\textrm{d}B^{k}(t)\Big{]},$

where

$R_{T-\varepsilon}=\frac{p_{b}(\tau,\xi,\tau+\varepsilon,Y_{T-\varepsilon})}{p_{b}(\tau,\xi,T+\tau,x)}$

according to (2.7), and $Y$ is the solution of the Cauchy problem (5.2) for
the stochastic differential equations. Therefore, we have

$\displaystyle\left|\nabla\ln p_{b}\right|(\tau,\xi,T+\tau,x)$
$\displaystyle\leq\frac{C_{q}}{T-\varepsilon}\sqrt[p]{\mathbb{E}\Big{[}\Big{(}\frac{p_{b}(\tau,\xi,\tau+\varepsilon,Y_{T-\varepsilon})}{p_{b}(\tau,\xi,T+\tau,x)}\Big{)}^{p}\Big{]}}\sqrt[q]{\mathbb{E}\Big{[}\Big{(}\int_{0}^{T-\varepsilon}|Z(t)|^{2}\,\textrm{d}t\Big{)}^{\frac{q}{2}}\Big{]}},$

where $C_{q}$ is the constant in the Burkholder-Davis-Gundy inequality (cf.
[18]) that is applied to handle the Itô integral with
$\frac{1}{p}+\frac{1}{q}=1$. Thanks to (5.4),

$\displaystyle|Z(t)|\leq|Z(0)|\textrm{e}^{2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}(\sqrt{T}-\sqrt{T-t})}\leq|Z(0)|\textrm{e}^{2\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}\frac{t}{\sqrt{T}}}\qquad\mbox{for all
$t\in[0,T]$},$

where $|Z(0)|=d$ is the dimension, or the norm of the identity matrix. It
follows that

$\displaystyle\sqrt[q]{\mathbb{E}\Big{[}\Big{(}\int_{0}^{T-\varepsilon}|Z(t)|^{2}\,\textrm{d}t\Big{)}^{\frac{q}{2}}\Big{]}}\leq|Z(0)|\sqrt[q]{\mathbb{E}\Big{[}\Big{(}\int_{0}^{T-\varepsilon}\textrm{e}^{At}\,\textrm{d}t\Big{)}^{\frac{q}{2}}\Big{]}}\leq|Z(0)|\sqrt{\frac{\textrm{e}^{A(T-\varepsilon)}-1}{A(T-\varepsilon)}},$

where $A:=\frac{2\left\|\nabla b\right\|_{\tau\rightarrow\tau+T}}{\sqrt{T}}$.

Plugging into the previous inequality, we obtain

$\displaystyle\left|\nabla\ln
p_{b}\right|(\tau,\xi,T+\tau,x)\leq\frac{C_{q}|Z(0)|}{T-\varepsilon}\sqrt{\frac{\textrm{e}^{A(T-\varepsilon)}-1}{A(T-\varepsilon)}}\sqrt[p]{\mathbb{E}\Big{[}\Big{(}\frac{p_{b}(\tau,\xi,\tau+\varepsilon,Y_{T-\varepsilon})}{p_{b}(\tau,\xi,T+\tau,x)}\Big{)}^{p}\Big{]}}.$

The estimate is true for any $p>1$, so is for $1\leq q<\infty$. It follows
that

$\displaystyle\left|\nabla_{x}p_{b}\right|(\tau,\xi,T+\tau,x)\leq\frac{C_{q}|Z(0)|}{\sqrt{T-\varepsilon}}\sqrt{\frac{\textrm{e}^{A(T-\varepsilon)}-1}{A(T-\varepsilon)}}\sqrt[p]{\int_{\mathbb{R}^{d}}\big{(}p_{b}(\tau,\xi,\tau+\varepsilon,z)\big{)}^{p}p_{-b}(0,x,T-\varepsilon,z)\,\textrm{d}z}$

for any $p>1$. By choosing $\varepsilon=\frac{T}{2}$ and using the fact that
$\frac{\textrm{e}^{x}-1}{x}\leq\textrm{e}^{x}$ for $x>0$, the previous
inequality yields that

$\displaystyle\left|\nabla
p_{b}\right|(\tau,\xi,T+\tau,x)\leq\sqrt{2}C_{q}|Z(0)|\frac{1}{\sqrt{T}}e^{\frac{\sqrt{T}\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}}{2}}\sqrt[p]{\int_{\mathbb{R}^{d}}\big{(}p_{b}(\tau,\xi,\tau+\frac{T}{2},z)\big{)}^{p}p_{-b}(0,x,\frac{T}{2},z)\,\textrm{d}z}$

for any $p>1$. Finally, according to Lemma 4.2, for any $\beta>p$,

$\displaystyle
J(\varepsilon)=\sqrt[p]{\int_{\mathbb{R}^{d}}\big{(}p_{b}(\tau,\xi,\tau+\varepsilon,z)\big{)}^{p}p_{-b}(0,x,T-\varepsilon,z)\,\textrm{d}z}\leq
K_{1}\textrm{e}^{K_{2}T\left\|b\right\|_{\infty}^{2}}\Big{(}\frac{T}{\varepsilon}\Big{)}^{\frac{d}{2}(1-\frac{1}{\gamma})}G_{\gamma\beta
T}(\xi-x).$

Therefore, we conclude that

$\left|\nabla
p_{b}\right|(\tau,\xi,T+\tau,x)\leq\frac{K_{3}}{\sqrt{T}}\textrm{e}^{K_{2}T\left\|b\right\|_{\infty}^{2}+\frac{1}{2}\sqrt{T}\left\|\nabla
b\right\|_{\tau\rightarrow\tau+T}}G_{\gamma\beta T}(\xi-x),$

which yields the estimate. ∎

##  6 Linearized Vorticity Equations

In this section, we exploit the explicit estimates in §3 –§5 to the study of
the linearized vorticity equations for the the Navier-Stokes equations (1.1)
with the viscosity constant $\nu=\frac{1}{2}$ (without loss of generality).

We make the following identification: Any tensor field $F$ defined on
$[0,1]^{3}$ satisfying the periodic condition that $F(x+\mathbb{k})=F(x)$, for
$i=1,2,3$, $x\in[0,1]^{3}$, and $\mathbb{k}\in\mathbb{Z}^{3}$, will be
identified with its periodic extension (with period $1$) on $\mathbb{R}^{3}$.

###  6.1 Linear parabolic equations arising from the vorticity equations

Our construction of strong solutions of the Cauchy problem (1.1)–(1.2) of the
Navier-Stokes equations (1.1) with $\nu=\frac{1}{2}$ will be based on the
vorticity equation for $\omega=\nabla\wedge u$:

$\partial_{t}\omega+(u\cdot\nabla)\omega-A(u)\omega-\frac{1}{2}\Delta\omega=0,$
(6.1)

where $A(u)$ is the total derivative of $u$, which is a tensor with its
components $A(u)_{j}^{i}=\partial_{x^{j}}u^{i}$. Although there are several
formulations of the vorticity equations, we will explain that this version
serves our aims well. The crucial observation is based on an elementary
identity:

$\nabla\wedge\left((u\cdot\nabla)u\right)=(u\cdot\nabla)\omega-A(u)\omega$
(6.2)

so that the vorticity equations are equivalent to

$\nabla\wedge\big{(}\partial_{t}u+(u\cdot\nabla)u-\frac{1}{2}\Delta
u\big{)}=0,$ (6.3)

which is not surprising though. One can even argue that this is from where the
vorticity equations come. However, this formulation allows us to define an
iteration for the construction of strong solutions of the Navier-Stokes
equations.

More precisely, we start with a periodic, smooth, and divergence-free vector
field $b(x,t)$ such that $b(x,0)=u_{0}(x)$, and we then want to construct a
nonlinear mapping that sends $b(x,t)$ to $v(x,t)$, which will be denoted by
$V$, so that $v=V(b)$. This is achieved by the two steps:

Step 1\. For given $b(x,t)$, we define a vector field $w(x,t)$ by solving the
following Cauchy problem of the linear parabolic equations:

$\begin{cases}\partial_{t}w+(b\cdot\nabla)w-A(b)w-\frac{1}{2}\Delta w=0,\\\
w(\cdot,0)=\omega_{0},\end{cases}$ (6.4)

where $A(b)=(\partial_{x^{j}}b^{i})_{1\leq i,j\leq 3}$ is the total derivative
of $b$; this $w$ should be a candidate of the vorticity, while it is not the
vorticity of $b$. There are two properties of the unique solution $w(x,t)$
which are important to our approach.

First, $w$ is again divergence-free. In fact, the divergence of $w(x,t)$,
denoted by $f(x,t)$, i.e., $f=\nabla\cdot w$, satisfies the following
parabolic equation:

$\begin{cases}\partial_{t}f+(b\cdot\nabla)f-\frac{1}{2}\Delta f=0,\\\
f(\cdot,0)=0.\end{cases}$ (6.5)

This equation may be obtained by differentiating (6.4) and using the following
elementary vector identity:

$\nabla\cdot\left((b\cdot\nabla)w\right)=b\cdot\nabla(\nabla\cdot
w)+\nabla\cdot\left(A(b)w\right),$ (6.6)

which holds for any vector field $w$ and any $b$ such that $\nabla\cdot b=0$.
By the uniqueness of linear parabolic equation, we conclude that $f=0$, so
that $w(x,t)$ is divergence-free.

Second, we show that the mean: $m(t)=\int_{[0,1]^{3}}w(x,t)\,\textrm{d}x=0$.
In fact, since both $b$ and $w$ are divergence-free and periodic, then

$\int_{[0,1]^{3}}(b\cdot\nabla)w\,\textrm{d}x=\int_{[0,1]^{3}}A(b)w\,\textrm{d}x=\frac{1}{2}\int_{[0,1]^{3}}\Delta
w\,\textrm{d}x=0,$

so that $m(t)=m(0)=0$.

Step 2\. We define the candidate for the velocity $v(x,t)$ by solving the
Poisson equation:

$\Delta v=-\nabla\wedge w$ (6.7)

at any instance such that $\int_{[0,1]^{3}}v(x,\cdot)\,\textrm{d}x=0$, which
has a unique solution satisfying the periodic condition. Applying
$\nabla\cdot$ to both sides of the Poisson equation, we obtain that
$\Delta\left(\nabla\cdot v\right)=0$, so that $\nabla\cdot v$ must be constant
for every $t$. Since $\int_{[0,1]^{3}}\nabla\cdot v\,{\rm d}x=0$, then
$\nabla\cdot\nu=0$. Therefore, $\nabla\cdot v=0$ and $\nabla\cdot w=0$. The
Poisson equation for $v$ implies that

$\displaystyle\nabla\wedge\left(\nabla\wedge v-w\right)=-\Delta v-\nabla\wedge
w=0,$ $\displaystyle\nabla\cdot\left(\nabla\wedge v-w\right)=-\nabla\cdot
w=0.$

Then, according to the Hodge theory, $\nabla\wedge v-w$ vanishes identically,
so it follows that $w=\nabla\wedge v$. In particular, $v(x,0)=u_{0}(x)$.

Therefore, in this way, we are able to construct an iteration via the
nonlinear mapping $V$ so that $v=V(b)$.

The advantage for using this iteration via the nonlinear mapping $b\rightarrow
v=V(b)$ can now be put forward in the following: Observe that the parabolic
equation (6.4) can be rewritten as

$\begin{cases}\left(\partial_{t}-L_{-b}\right)w=A(b)w,\\\
w(\cdot,0)=\omega_{0},\end{cases}$

where, as before, $L_{-b}$ denotes the time-dependent elliptic operator
$\frac{1}{2}\Delta-b\cdot\nabla$. The crucial difference from the linearised
Navier-Stokes equations lies in the fact that the term on the right-hand side,
$A(b)w$, is a linear zero-order term. This crucial difference allows us to
apply the Feynman-Kac-type formula to obtain the necessary _a priori_
estimates, which will be derived in the remainder of this section.

###  6.2 Brownian motion on $\mathbb{T}^{3}$

Let $h(\tau,x,t,y)$ denote the heat kernel on $\mathbb{T}^{3}$, which is the
transition probability density function of the Brownian motion on
$\mathbb{T}^{3}$, a diffusion on $\mathbb{T}^{3}$ with its infinitesimal
generator $\frac{1}{2}\Delta$. Then

$h(\tau,x,t,y)=\sum_{\boldsymbol{k}\in\mathbb{Z}^{3}}G_{t-\tau}(y-x+\boldsymbol{k})\qquad\mbox{for
$x,y\in[0,1]^{3}$},$ (6.8)

where the series on the right-hand side and its derivative series indeed
converge uniformly for $(x,y)\in[0,1]^{3}$.

Since $\mathbb{T}^{3}$ is a compact manifold without boundary, there is a
Green function $C(x,y)$ associated with the Laplacian $\Delta$, denoted by
$C(x,y)$ (cf. [1, page 108]), which possesses the following properties:

  1. (i)

$C(x,y)$ is smooth out off the diagonal, periodic in $x,y\in[0,1]^{3}$, and

$|\nabla_{x}^{k}C(x,y)|\leq\frac{C_{1}}{|x-y|^{k}}\qquad\mbox{for
$x,y\in[0,1]^{3}$},$

where $k=0,1,2$, $C_{1}$ is a constant, and
$\int_{[0,1]^{3}}C(x,y)\,\textrm{d}y=0$. In particular,

$\int_{[0,1]^{3}}|\nabla_{x}^{k}C(x,y)|\,\textrm{d}y\leq
C_{2}\qquad\,\,\mbox{for all $x\in\mathbb{R}^{3}$ and $k=0,1,2$},$

where $C_{2}$ is a universal constant.

  2. (ii)

For every periodic $C^{2}$ function $\psi$, the Green formula holds:

$\psi(x)=\int_{[0,1]^{3}}\psi(y)\,\textrm{d}y+\int_{[0,1]^{3}}C(x,y)\Delta\psi(y)\,\textrm{d}y.$

It follows that, for any periodic function $f$ with mean zero:
$\int_{[0,1]^{3}}f(y)\,\textrm{d}y=0$, the Poisson equation:

$\Delta\psi=f$

has a unique periodic solution with mean zero, given by the Green formula:

$\psi(x)=\int_{[0,1]^{3}}C(x,y)f(y)\,\textrm{d}y\qquad\,\,\mbox{for
$x\in\mathbb{R}^{3}$}.$

Let $b(\cdot,t)$ be a vector field on $[0,1]^{3}$ depending on $t\geq 0$,
which is identified with a vector field

$b(x,t)=(b^{1}(x,t),b^{2}(x,t),b^{3}(x,t))\qquad\mbox{ on $\mathbb{R}^{3}$}$

with period $1$ along each coordinate. Let
$\mathcal{L}_{b}=\frac{1}{2}\Delta+b(\cdot,t)\cdot\nabla$ be an elliptic
operator on $[0,1]^{3}$. Let $h_{b}(\tau,x,t,y)$ denote the transition
probability density function of the $\mathcal{L}_{b}$-diffusion on
$[0,1]^{3}$. Then

$h_{b}(\tau,x,t,y)=\sum_{\boldsymbol{k}\in\mathbb{Z}^{3}}p_{b}(\tau,x,t,y+\boldsymbol{k})\qquad\mbox{for
$x,y\in[0,1]^{3}$},$

where $p_{b}(\tau,x,t,y)$ is the transition probability density function on
$\mathbb{R}^{3}$ with infinitesimal generator
$L_{b}=\frac{1}{2}\Delta+b(\cdot,t)\cdot\nabla$ on $\mathbb{R}^{3}$.

###  6.3 _A priori_ estimates for the linearized vorticity equations

In this subsection, we establish several a priori estimates for the first step
of the iteration for solving the vorticity equation, defined in the following:
Let $u_{0}$ be a given smooth, periodic initial velocity vector field with
divergence-free, and $\omega_{0}=\nabla\wedge u_{0}$, and let $b(x,t)$ be a
smooth time-dependent divergence-free periodic vector field on
$\mathbb{R}^{3}$ with $b(x,0)=u_{0}(x)$. Hence, $u_{0}(x+\mathbb{k})=u_{0}(x)$
and $b(x+\mathbb{k},t)=b(x,t)$ for $\mathbb{k}\in\mathbb{Z}^{3}$ and
$x\in\mathbb{R}^{3}$. Define the periodic vorticity field $w(x,t)$ to be the
unique solution of the Cauchy problem (6.4) for the linear vorticity
equations.

The vector field $v(x,t)$ is the unique solution with mean zero to the Poisson
equation (6.7) on $\mathbb{T}^{3}$, which may be given by the Green formula:

$v^{i}(x,t)=\int_{[0,1]^{3}}\varepsilon^{ijk}C(x,y)\partial_{y^{j}}w^{k}(y,t)\,\textrm{d}y\qquad\mbox{for
$t\geq 0$ and $x\in\mathbb{T}^{3}$}.$

We have shown that both $v$ and $w$ are divergence-free, and $w=\nabla\times
v$.

Moreover, the solution $w(x,t)$ of the Cauchy problem (6.4) satisfies the
following implicit equation:

$w(x,t)=\int_{[0,1]^{3}}h_{-b}(\tau,\xi,t,x)w(\xi,\tau)\,\textrm{d}\xi+\int_{\tau}^{t}\int_{[0,1]^{3}}h_{-b}(s,\xi,t,x)A(b)(\xi,s)w(\xi,s)\,\textrm{d}\xi\textrm{d}s$
(6.9)

for $t>\tau\geq 0$ and $x\in[0,1]^{3}$. On the other hand, by the forward
Feynman-Kac’s formula (2.9), we have the following nonlinear representation:

$w^{k}(x,t)=\int_{[0,1]^{3}}\omega_{0}^{j}(\xi)h_{b}(0,\xi,t,x)\mathbb{P}^{\xi}\big{[}Q_{j}^{k}(0,t)|X_{t}=x\big{]}\,\textrm{d}\xi\qquad\mbox{for
$k=1,2,3$},$ (6.10)

where $X$ is the canonical process on the path space
$\varOmega=C([0,\infty),\mathbb{T}^{3})$,
$\mathbb{P}^{\xi}=\mathbb{P}^{0,\xi}$ is the diffusion family on $\varOmega$
with infinitesimal generator ${\mathcal{L}}_{b}$, and $Q(s)=Q(s,t)$ is the
solution to

$\begin{cases}\frac{\textrm{d}}{\textrm{d}s}Q_{j}^{i}(s)=-Q_{k}^{i}(s)A(b)_{j}^{k}(X_{s},s)\qquad\,\,\mbox{
for $s\leq t$},\\\\[2.84526pt] Q_{j}^{i}(t)=\delta_{j}^{i}.\end{cases}$ (6.11)

We now derive several _a priori_ estimates.

######  Lemma 6.1.

Let $\eta=(\eta(s))_{s\geq 0}$ be any continuous curve in $\mathbb{R}^{3}$,
and let $G(s)=(G_{j}^{i}(s))_{s\leq t}$ be the unique solution of the Cauchy
problem for the ordinary differential equations:

$\begin{cases}\frac{\textrm{\rm d}}{\textrm{\rm
d}s}G_{j}^{i}(s)=-G_{k}^{i}(s)A(b)_{j}^{k}(\eta(s),s)\qquad\,\mbox{ for $0\leq
s\leq t$},\\\\[2.84526pt] G_{j}^{i}(t)=\delta_{j}^{i},\end{cases}$ (6.12)

for $i,j=1,2,3$. Then

$\sum_{i,j}G_{j}^{i}(0)G_{j}^{i}(0)\leq 9\textrm{e}^{4\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}.$ (6.13)

###### Proof.

For simplicity of notation, we set

$f(s):=\sum_{i,j}G_{j}^{i}(s)G_{j}^{i}(s),$

which is the squared Hilbert-Schmidt’s norm of $(G_{j}^{i}(s))$. Then

$\displaystyle\frac{\textrm{d}f(s)}{\textrm{d}s}$
$\displaystyle=2G_{j}^{i}(s)\frac{\textrm{d}}{\textrm{d}s}G_{j}^{i}(s)$
$\displaystyle=-2G_{k}^{i}(s)G_{j}^{i}(s)A(b)_{j}^{k}(\eta(s),s)$
$\displaystyle\geq-2|\nabla b(\eta(s),s)|f(s)$
$\displaystyle\geq-\frac{2}{\sqrt{s}}\left\|\nabla b\right\|_{0\rightarrow
t}f(s)\qquad\mbox{ for $s\in[0,t]$}.$

After integration, we obtain

$\displaystyle\ln f(t)-\ln f(0)\geq-2\left\|\nabla b\right\|_{0\rightarrow
t}\int_{0}^{t}\frac{1}{\sqrt{s}}\,\textrm{d}s=-4\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}.$

Then

$G(0)\leq 9\textrm{e}^{4\sqrt{t}\left\|\nabla b\right\|_{0\rightarrow
t}}\qquad\mbox{for all $t\geq 0$}.$

∎

######  Lemma 6.2.

For every $\beta>1$, there are universal constants $C_{1}$ and $C_{2}$
depending only on $\beta$ such that

$|w(x,t)|\leq
C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}G_{\beta t}(|\omega_{0}|)(x)\qquad\,\mbox{for all
$t>0$ and $x\in\mathbb{T}^{3}$}.$ (6.14)

###### Proof.

The estimate in Lemma 6.1 allows us to control the conditional expectation in
(6.10) and to obtain

$\displaystyle|w(x,t)|$ $\displaystyle\leq 3\textrm{e}^{2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}}\int_{[0,1]^{3}}|\omega_{0}(\xi)|h_{b}(0,\xi,t,x)\,\textrm{d}\xi$
$\displaystyle=3\textrm{e}^{2\sqrt{t}\left\|\nabla b\right\|_{0\rightarrow
t}}\int_{[0,1]^{3}}|\omega_{0}(\xi)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}p_{b}(0,\xi,t,x+\mathbb{k})\,\textrm{d}\xi.$

Using the uniform estimate (cf. Theorem 4.1):

$p_{b}(0,\xi,t,x)\leq
C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}}G_{\beta t}(x-\xi),$

we obtain

$\displaystyle|w(x,t)|$ $\displaystyle\leq 3\textrm{e}^{2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}}\int_{[0,1]^{3}}|\omega_{0}(\xi)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}G_{\beta
t}(x+\mathbb{k}-\xi)\,\textrm{d}\xi$
$\displaystyle=3C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}}\int_{[0,1]^{3}}|\omega_{0}(\xi)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}G_{\beta
t}(x+\mathbb{k}-\xi)\,\textrm{d}\xi$
$\displaystyle=3C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}\int_{\mathbb{R}^{3}}|\omega_{0}(\xi)|G_{\beta
t}(x-\xi)\,\textrm{d}\xi$
$\displaystyle=3C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}G_{\beta t}(|\omega_{0}|)(x).$

∎

With Lemmas 6.1–6.2, we now establish the following three theorems.

######  Theorem 6.3.

For every $\beta>1$, there are two positive constants $C_{1}$ and $C_{2}$
depending only on $\beta$ such that

$|\nabla
w(x,t)|\leq\frac{C_{1}}{\sqrt{t}}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+3\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}G_{\beta t}(|\omega_{0}|)(x)\qquad\,\mbox{for all
$t>0$ and $x\in\mathbb{T}^{3}$.}$

###### Proof.

Recall that $w$ satisfies the following equality:

$w(x,t)=\int_{[0,1]^{3}}h_{-b}(0,\xi,t,x)\omega_{0}(\xi)\,\textrm{d}\xi+\int_{0}^{t}\int_{[0,1]^{3}}h_{-b}(s,\xi,t,x)A(b)(\xi,s)w(\xi,s)\,\textrm{d}\xi\textrm{d}s$

for $t>0$ and $x\in[0,1]^{3}$. Differentiating both sides in $x$ to obtain

$\displaystyle\nabla w(x,t)$
$\displaystyle=\int_{[0,1]^{3}}\nabla_{x}h_{-b}(0,\xi,t,x)\omega_{0}(\xi)\,\textrm{d}\xi$
$\displaystyle\quad+\int_{0}^{t}\int_{[0,1]^{3}}\nabla_{x}h_{-b}(s,\xi,t,x)A(b)(\xi,s)w(\xi,s)\,\textrm{d}\xi\textrm{d}s$
$\displaystyle=\int_{[0,1]^{3}}\sum_{\mathbb{k}\in\mathbb{Z}^{3}}\nabla_{x}p_{-b}(0,\xi,t,x+\mathbb{k})\omega_{0}(\xi)\,\textrm{d}\xi$
$\displaystyle\quad+\int_{0}^{t}\int_{[0,1]^{3}}\sum_{\mathbb{k}\in\mathbb{Z}^{3}}\nabla_{x}p_{-b}(s,\xi,t,x+\mathbb{k})A(b)(\xi,s)w(\xi,s)\,\textrm{d}\xi\textrm{d}s.$

According to Theorem 5.2, for every $\beta>1$, there are two universal
constants $C_{2}$ and $C_{3}$ such that

$\left|\nabla_{x}p_{b}\right|(s,\xi,t,x)\leq
C_{3}\frac{\textrm{e}^{C_{2}(t-s)\left\|b\right\|_{\infty}^{2}+\frac{1}{2}\sqrt{t-s}\left\|\nabla
b\right\|_{s\rightarrow
t}}}{\sqrt{t-s}}G_{\beta(t-s)}(x-\xi)\qquad\,\,\mbox{for any $t>s\geq 0$}.$
(6.15)

Thanks to this estimate, using the triangle inequality, we obtain

$\displaystyle|\nabla w|(x,t)$
$\displaystyle\leq\int_{[0,1]^{3}}|\omega_{0}(\xi)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}|\nabla_{x}p_{-b}|(0,\xi,t,x+\mathbb{k})\,\textrm{d}\xi$
$\displaystyle\quad+\int_{0}^{t}\int_{[0,1]^{3}}|A(b)(\xi,s)||w(\xi,s)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}|\nabla_{x}p_{-b}|(s,\xi,t,x+\mathbb{k})\,\textrm{d}\xi\textrm{d}s$
$\displaystyle\leq C_{3}\frac{\textrm{e}^{\frac{\sqrt{t}}{2}\left\|\nabla
b\right\|_{0\rightarrow
t}+C_{2}t\left\|b\right\|_{\infty}^{2}}}{\sqrt{t}}\int_{[0,1]^{3}}|\omega_{0}(\xi)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}G_{\beta
t}(\xi-x-\mathbb{k})\,\textrm{d}\xi$
$\displaystyle\quad+C_{3}\int_{0}^{t}\frac{\textrm{e}^{\frac{\sqrt{t-s}}{2}\left\|\nabla
b\right\|_{s\rightarrow
t}+C_{2}(t-s)\left\|b\right\|_{\infty}^{2}}}{\sqrt{t-s}}\int_{[0,1]^{3}}|A(b)(\xi,s)||w(\xi,s)|\sum_{\mathbb{k}\in\mathbb{Z}^{3}}G_{\beta(t-s)}(\xi-x-\mathbb{k})\,\textrm{d}\xi\textrm{d}s$
$\displaystyle=C_{3}\frac{\textrm{e}^{\frac{\sqrt{t}}{2}\left\|\nabla
b\right\|_{0\rightarrow
t}+C_{2}t\left\|b\right\|_{\infty}^{2}}}{\sqrt{t}}\int_{\mathbb{R}^{3}}|\omega_{0}(\xi)|G_{\beta
t}(\xi-x)\,\textrm{d}\xi$
$\displaystyle\quad+C_{3}\int_{0}^{t}\frac{\textrm{e}^{\frac{1}{2}\sqrt{t-s}\left\|\nabla
b\right\|_{s\rightarrow
t}+C_{2}\left\|b\right\|_{\infty}^{2}(t-s)}}{\sqrt{t-s}}\int_{\mathbb{R}^{3}}|A(b)(\xi,s)||w(\xi,s)|G_{\beta(t-s)}(\xi-x)\,\textrm{d}\xi\textrm{d}s.$

By definition of $S=S(b)$, we have

$\left|A(b)(\xi,s)w(\xi,s)\right|\leq\frac{1}{\sqrt{s}}\left\|\nabla
b\right\|_{0\rightarrow t}|w(\xi,s)|.$

Together with estimate (6.14), we then deduce

$\left|A(b)(\xi,s)w(\xi,s)\right|\leq\frac{3C_{3}}{\sqrt{s}}\left\|\nabla
b\right\|_{0\rightarrow
t}\textrm{e}^{C_{2}s\left\|b\right\|_{\infty}^{2}+2\sqrt{s}\left\|\nabla
b\right\|_{0\rightarrow s}}G_{\beta s}(|\omega_{0}|)(\xi)\qquad\,\mbox{for all
$t>s>0$}.$

It follows that

$\displaystyle I_{2}(x,t)$
$\displaystyle=\int_{0}^{t}\frac{\textrm{e}^{\frac{1}{2}\sqrt{t-s}\left\|\nabla
b\right\|_{s\rightarrow
t}+C_{2}\left\|b\right\|_{\infty}^{2}(t-s)}}{\sqrt{t-s}}\Big{(}\int_{\mathbb{R}^{3}}|A(b)(\xi,s)||w(\xi,s)|G_{\beta(t-s)}(\xi-x)\,\textrm{d}\xi\Big{)}\textrm{d}s$
$\displaystyle\leq 3C_{3}\left\|\nabla b\right\|_{0\rightarrow t}G_{\beta
t}(|\omega_{0}|)(x)\int_{0}^{t}\frac{\textrm{e}^{\frac{1}{2}\sqrt{t-s}\left\|\nabla
b\right\|_{s\rightarrow t}+2\sqrt{s}\left\|\nabla b\right\|_{0\rightarrow
s}+C_{2}t\left\|b\right\|_{\infty}^{2}}}{\sqrt{t-s}\sqrt{s}}\,\textrm{d}s$
$\displaystyle\leq 3C_{3}\left\|\nabla b\right\|_{0\rightarrow t}G_{\beta
t}(|\omega_{0}|)(x)\textrm{e}^{C_{2}\left\|b\right\|_{\infty}^{2}t+3\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}}\int_{0}^{t}\frac{1}{\sqrt{t-s}}\frac{1}{\sqrt{s}}\,\textrm{d}s$
$\displaystyle\leq C_{4}G_{\beta t}(|\omega_{0}|)(x)\left\|\nabla
b\right\|_{0\rightarrow
t}\textrm{e}^{C_{2}\left\|b\right\|_{\infty}^{2}t+\frac{5}{2}\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}$
$\displaystyle\leq\frac{C_{5}}{\sqrt{t}}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+3\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}G_{\beta t}(|\omega_{0}|)(x).$

Similarly, we have

$\displaystyle I_{1}(x,t)=\int_{\mathbb{R}^{d}}\left|\nabla
h_{-b}\right|(0,\xi,t,x)|\omega_{0}(\xi)|\,\textrm{d}\xi\leq\frac{C_{1}}{\sqrt{t}}\textrm{e}^{\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}+C_{2}t\left\|b\right\|_{\infty}^{2}}G_{\beta
t}(|\omega_{0}|)(x).$

The claimed estimate follows immediately. ∎

######  Theorem 6.4.

Let $v$ be the unique solution with mean zero of the Poisson equation:

$\Delta v=-\nabla\wedge w\qquad\,\,\mbox{in $\mathbb{R}^{3}$},$

which is also periodic. Then there are constants $C_{1}$ and $C_{2}$,
depending only on $\beta$, such that

$\displaystyle|v(x,t)|\leq
C_{1}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}\left\|\omega_{0}\right\|_{\infty},$
$\displaystyle|\nabla
v(x,t)|\leq\frac{C_{1}}{\sqrt{t}}\textrm{e}^{C_{2}t\left\|b\right\|_{\infty}^{2}+3\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow t}}\left\|\omega_{0}\right\|_{\infty},$

for all $t\geq 0$ and $x\in[0,1]^{3}$.

###### Proof.

By Green’s formula,

$\displaystyle v(x,t)=-\int_{[0,1]^{3}}C(x,y)\nabla\wedge w(y,t)\,{\rm d}t,$
$\displaystyle\nabla_{x}v(x,t)=\int_{[0,1]^{3}}\nabla_{x}C(x,y)\nabla\wedge
w(y,t)\,{\rm d}t,$

where $C(x,y)$ is the Green function of $\mathbb{T}^{3}$. Hence, we conclude
that there exist universal constants $C_{4}$ and $C_{5}$ such that

$\displaystyle|v(x,t)|\leq C_{4}\textrm{e}^{2\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}+C_{2}t\left\|b\right\|_{\infty}^{2}}\left\|\omega_{0}\right\|_{\infty},$
$\displaystyle|\nabla
v(x,t)|\leq\frac{C_{5}}{\sqrt{t}}C_{5}\textrm{e}^{3\sqrt{t}\left\|\nabla
b\right\|_{0\rightarrow
t}+C_{2}t\left\|b\right\|_{\infty}^{2}}\left\|\omega_{0}\right\|_{\infty},$

as $G_{\beta t}(|\omega_{0}|)(x)\leq\left\|\omega_{0}\right\|_{\infty}$. ∎

######  Theorem 6.5.

There are two universal constants $C_{1},C_{2}>0$ such that, if

$\displaystyle\left\|b\right\|_{L^{\infty}([0,T]\times\mathbb{R}^{3})}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad\left\|\nabla
b\right\|_{0\rightarrow T}\leq C_{2}\left\|\omega_{0}\right\|_{\infty},$

then, for

$T=\frac{C_{1}}{\left\|\omega_{0}\right\|_{\infty}^{2}},$

the following estimates hold:

$\displaystyle\left\|v\right\|_{L^{\infty}([0,T]\times\mathbb{R}^{3})}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad\,\,\,\left\|w\right\|_{L^{\infty}([0,T]\times\mathbb{R}^{3})}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},$ $\displaystyle\left\|\nabla
v\right\|_{0\rightarrow T}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad\,\,\,\qquad\left\|\nabla
w\right\|_{0\rightarrow T}\leq C_{2}\left\|\omega_{0}\right\|_{\infty}.$

###### Proof.

Let us choose $T>0$ such that

$C_{0}\lambda e^{C_{2}T\mu^{2}C_{0}^{2}\lambda^{2}+3\sqrt{T}\mu
C_{0}\lambda}=\mu C_{0}\lambda.$

Then we solve $T$ to obtain

$T\mu^{2}C_{0}^{2}\lambda^{2}+\frac{3}{C_{2}}\sqrt{T}\mu
C_{0}\lambda-\frac{1}{C_{2}}\ln\mu=0,$

so that

$\sqrt{T}=\frac{2\ln\mu}{C_{0}C_{2}\mu\left(\sqrt{9+4\ln\mu}+3\right)\lambda},$

where $C_{0}=C_{4}\vee C_{5}\vee\ldots$ and $\lambda=\max G_{\beta
T}(|\omega_{0}|)(x)$. Choose $\mu=e$. Then

$\sqrt{T}=\frac{2}{C_{0}C_{2}e\left(\sqrt{13}+3\right)\lambda}$

so that $T=\frac{C}{\lambda^{2}}$. ∎

##  7 Navier-Stokes Equations

Thanks to the explicit a priori estimates established in §3–§6, we are now in
a position to study the strong solutions of the Cauchy problem:

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u-\frac{1}{2}\Delta u=-\nabla P,\\\
\nabla\cdot u=0,\end{cases}$ (7.1)

with initial data:

$\quad u(x,0)=u_{0}(x),$ (7.2)

subject to the periodic condition that $u_{0}(x+e_{i})=u_{0}(x)$ for
$x\in[0,1]^{3}$, where $i=1,2,3$, $(e_{i})$ is the standard basis of
$\mathbb{R}^{3}$.

######  Theorem 7.1.

There are universal constants $C_{1}>0$ and $C_{2}>0$ such that, for any
periodic initial data $u_{0}$ with mean zero and $\omega_{0}=\nabla\wedge
u_{0}$, there exists a unique strong solution $u(x,t)$ of the Cauchy problem
(7.1)–(7.2) for the Navier-Stokes equations (7.1) with periodic initial data
(7.2) for $t\leq T$ so that
$T=\frac{C_{1}}{\left\|\omega_{0}\right\|_{\infty}^{2}}$, and

$\displaystyle\left\|u\right\|_{L^{\infty}([0,T]\times\mathbb{R}^{3})}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad\left\|\omega\right\|_{L^{\infty}([0,T]\times\mathbb{R}^{3})}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},$ $\displaystyle\left\|\nabla
u\right\|_{0\rightarrow T}\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad\qquad\left\|\nabla\omega\right\|_{0\rightarrow
T}\leq C_{2}\left\|\omega_{0}\right\|_{\infty},$

where $\left\|V\right\|_{0\rightarrow
T}=\sup_{(x,t)\in\mathbb{R}^{3}\times[0,T]}\left|\sqrt{t}V(x,t)\right|$.

###### Proof.

To construct the strong solution of the Cauchy problem (7.1)–(7.2), we
construct the following iterations:

Set $u^{(0)}(x,t)=u_{0}(x)$ for all $x$ and $t\geq 0$, and define
$u^{(n)}=V(u^{(n-1)})$ inductively for $n\geq 1$. Then $\nabla\cdot u^{(n)}=0$
and $\nabla\cdot w^{(n)}=0$, and

$\omega^{(n)}=\nabla\wedge u^{(n)}=w^{(n)}\qquad\,\mbox{for all
$n=1,2,\cdots$}.$

Hence, for each $n\geq 1$, $\omega^{(n)}$ solves the linear parabolic
equations on the torus

$\begin{cases}\partial_{t}\omega^{(n)}+(u^{(n-1)}\cdot\nabla)\omega^{(n)}-A(u^{(n-1)})\omega^{(n)}-\frac{1}{2}\Delta\omega^{(n)}=0,\\\
\omega^{(n)}(\cdot,0)=\omega_{0},\end{cases}$

where $\omega^{(n)}=\nabla\wedge u^{(n)}$ and $\nabla\cdot u^{(n)}=0$.

Let $T=\frac{C_{1}}{\left\|\omega_{0}\right\|_{\infty}^{2}}$ in Theorem 6.5.

Then

$\displaystyle|u^{(n)}(x,t)|\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad|\nabla
u^{(n)}(x,t)|\leq\frac{C_{2}}{\sqrt{t}}\left\|\omega_{0}\right\|_{\infty},$
$\displaystyle|\omega^{(n)}(x,t)|\leq
C_{2}\left\|\omega_{0}\right\|_{\infty},\qquad|\nabla\omega^{(n)}(x,t)|\leq\frac{C_{2}}{\sqrt{t}}\left\|\omega_{0}\right\|_{\infty},$

where $C_{1}$ and $C_{2}$ are universal positive constants. By the standard
parabolic regularity theory (cf. [22]),

$\sup_{\mathbb{R}^{3}\times[\delta,T]}\big{|}\partial_{t}^{k}\nabla^{l}u^{(n)}\big{|}\leq
C_{k,l}\qquad\mbox{for all $n$},$

where the constant, $C_{k,l}$, depends on $\left\|\omega_{0}\right\|_{\infty}$
and $\delta>0$ only, for every $\delta>0$ and $k,l\in\mathbb{N}$. These _a
priori_ estimates allow us to conclude that, if necessary for a convergent
subsequence, $u^{(n)}\rightarrow u$ and $\omega^{(n)}\rightarrow\omega$ (in a
space with a norm including high derivatives) so that

$\partial_{t}\omega+(u\cdot\nabla)\omega-A(u)\omega-\frac{1}{2}\Delta\omega=0,$

where $\nabla\cdot u=0$ and $\omega=\nabla\wedge u$. Then

$\displaystyle\nabla\wedge\big{(}\partial_{t}u+(u\cdot\nabla)u-\frac{1}{2}\Delta
u\big{)}=\partial_{t}\omega+(u\cdot\nabla)\omega-A(u)\omega-\frac{1}{2}\Delta\omega=0,$

which implies that there is a scalar function $P$ such that

$\partial_{t}u+(u\cdot\nabla)u-\frac{1}{2}\Delta u=-\nabla P.$

Together with $\nabla\cdot u=0$, we conclude that $u$ is the unique strong
solution of the Cauchy problem (7.1)–(7.2) for the Navier-Stokes equations
(7.1). This completes the proof. ∎

Acknowledgements. The research of Gui-Qiang G. Chen was supported in part by
the UK Engineering and Physical Sciences Research Council Award EP/L015811/1,
EP/V008854, and EP/V051121/1. The research of Zhongmin Qian was supported in
part by the EPSRC Centre for Doctoral Training in Mathematics of Random
Systems: Analysis, Modelling and Simulation (EP/S023925/1).

## References

  * [1] Aubin, T., _Some Nonlinear Problems in Riemannian Geometry_ , Springer-Verlag: Berlin Heidelberg, 1998.
  * [2] Batchelor, G. K., _The Theory of Homogeneous Turbulence_ , Cambridge University Press: Cambridge, 1953.
  * [3] Bismut, J.-M., _Large Deviations and the Malliavin Calculus_ , Birkhäuser Boston: Boston, MA, 1984.
  * [4] Bismut, J.-M. and Lebeau, G., _The Hypoelliptic Laplacian and Ray-Singer Metrics_ , Princeton University Press: Princeton, 2008.
  * [5] Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, _Comm. Pure Appl. Math._ 35 (1982), 771–831.
  * [6] Chorin, A. J., Numerical study of slightly viscous flow, _J. Fluid Mech._ 57 (1973), 785–796.
  * [7] Constantin, P. and Foias, C., _Navier-Stokes Equations_ , University of Chicago Press: Chicago, 1988\.
  * [8] Dellacherie, C. and Meyer, P.-A., _Probabilitiés et Potentiel_ , Chapitres XII à XVI, Hermann, 1987.
  * [9] Elworthy, K. D. and Li, X. M., Formulae for the derivatives of heat semigroups. _J. of Funct. Anal._ 125(1): 252-286, 1994.
  * [10] Fabes, E. B., Jones, B. F., and Rivière, N. M., The initial value problem for the Navier-Stokes equations with data in $L^{p}$, _Arch. Ration. Mech. Anal._ 45 (1972), 222–240.
  * [11] Foias, C., Manley, O., Rosa, R. and Temam, R., _Navier-Stokes Equations and Turbulence_ , Cambridge University Press: Cambridge, 2001.
  * [12] Friedman, A., _Partial Differential Equations of Parabolic Type_ , Prentice-Hall: Englewood Cliffs, N.J., 1964.
  * [13] Fujita H. and Kato T., On the Navier-Stokes initial value problem I, _Arch. Ration. Mech. Anal._ 16 (1964), 269–315.
  * [14] Galdi, G. P., _An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems_ , Springer-Verlag: New York, 1994.
  * [15] Galdi, G. P., _An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II: Nonlinear Steady Problems_ , Springer-Verlag: New York, 1994.
  * [16] Hopf, E., Ein allgemeiner Endlichkeitssatz der Hydrodynamik, _Math. Ann_. 117 (1941), 764–775.
  * [17] Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, _Math. Nachr_. 4 (1950), 213–231.
  * [18] Ikeda, N. and S. Watanabe S., _Stochastic Differential Equations and Diffusion Processes_ , Second Edition, North-Holland Pub. Company, 1989.
  * [19] Kac, M., On distributions of certain Wiener funcionals. _Trans. Amer. Math. Soc._ 65 (1949), 1–13.
  * [20] Kato, T., Strong $L^{p}$-solutions of the Navier-Stokes equations in $\mathbb{R}^{m}$, with application to weak solutions, _Math. Z._ 187 (1984), 471–480.
  * [21] Lady$\check{z}$enskaja, O. A., _Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid_ , Fizmatgiz, Moscow, 1961; English transl. Gordon & Nreach, New York, 1963.
  * [22] Lady$\check{z}$enskaja, O. A., Solonnikov, V. A., and N. N. Ural’ceva, _Linear and Quasilinear Equations of Parabolic Type_ , AMS: Providence, 1968.
  * [23] Leray, J., Etude de diverses équations intégrales non linéaires et de quelques problémes que pose l’hydrodynamique, _J. Math. Pures Appl_. 12 (1933), 1–82.
  * [24] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, _Acta Math_. 63 (1934), 193–248.
  * [25] Lin, F. H., A New proof of the Caffarelli-Kohn-Nirenberg theorem, _Comm. Pure Appl. Math_. 51 (1998), 241–257.
  * [26] Lin, F. H. and Lei, Z., Global mild solutions of Navier-Stokes equations, _Comm. Pure Appl. Math_. 64 (2011), 1297–1304.
  * [27] Majda, A. J. and A. L. Bertozzi, _Vorticity and Incompressible Flow_ , Cambridge Texts in Applied Math., Cambridge University Press: Cambridge, 2002.
  * [28] Qian, Z., Süli, E., and Zhang, Y., Random vortex dynamics via functional stochastic differential equations, _Proc. R. Soc. A_ 478: 20220030. https://doi.org/10.1098/rspa.2022.0030.
  * [29] Qian, Z. and Zheng, W. A., A representation formula for transition probability densities of diffusions and applications, _Stochastic Processes and their Applications_ , 111 (2004), 57–76.
  * [30] Scheffer, V., Partial regularity of solutions to the Navier-Stokes equations, _Pacific J. Math_. 66 (1976), 535–552.
  * [31] Scheffer, V., Hausdorff measure and the Navier-Stokes equations, _Commun. Math. Phys_. 55 (1977), 97–112.
  * [32] Scheffer, V., The Navier-Stokes equations in space dimension four, _Commun. Math. Phys._ 61 (1978), 41–68.
  * [33] Scheffer, V., The Navier-Stokes equations on a bounded domain, _Commun. Math. Phys_. 73 (1980), 1–42.
  * [34] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, _Arch. Ration. Mech. Anal_. 9 (1962), 187–196.
  * [35] Serrin, J., The initial value problem for the Navier-Stokes equations, pp. 69–98, In: _Nonlinear Problems (Proceedings of a Symposium, Madison, Wisconsin)_ , R. T. Langer, ed., University of Wisconsin, Madison, 1963.
  * [36] Sohr, H., _The Navier-Stokes Equations – An Elementary Functional Analytic Approach_ , Birhäuser Advanced Texts, Birhäuser-Verlag, 2001.
  * [37] Stroock, D. W. and Varadhan, S. R. S., _Multidimensional Diffusion Processes_ , Springer-Verlag: Berlin-Heidelberg, 1979.
  * [38] Stroock, D. W., _Partial Differential Equations for Probabilists_ , Cambridge University Press: Cambridge, 2008.
  * [39] Temam, R., _Navier-Stokes Equations_ , North-Holland: Amsterdam, 1977.
  * [40] von Wahl, W., _The Equations of Navier-Stokes and Abstract Parabolic Equations_ , Friedr. Vieweg & Sohn, Braunschweig / Wiesbaden, 1985.