Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations

TL;DR

This paper establishes the existence of stochastic Lagrangian trajectories for Leray solutions of the 3D Navier-Stokes equations, linking probabilistic methods with classical fluid dynamics solutions.

Contribution

It introduces a novel stochastic framework for Leray solutions, proving existence and convergence of associated stochastic particle paths.

Findings

Existence of weak solutions to the SDE with Leray velocity fields.

Density of solutions belongs to specific Sobolev spaces under certain conditions.

Weak convergence of mollified solutions to the original stochastic trajectories.

Abstract

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $\rho_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2\nu}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.

Equations

$$\mathrm{d}{X}_{s,t}=\mathbf{u}(s,X_{s,t})\mathrm{d}t+\sqrt{2\nu }\mathrm{d}{W}_t,X_{s,s}=x,t⩾s,$$

$${\partial }_t\mathbf{u}=\nu \Delta \mathbf{u}+\mathbf{u}\cdot \nabla \mathbf{u}+\nabla p,\mathrm{div}\mathbf{u}\equiv 0,{\mathbf{u}}_0=\phi ,$$

$$\Vert \mathbf{u}{\Vert }_{L^{\infty }([0,T];L^2({\mathbb{R}}^d))}+\Vert \nabla \mathbf{u}{\Vert }_{L^2([0,T];L^2({\mathbb{R}}^d))}<\infty ,\forall T>0.$$

$$\mathrm{d}{X}_t=\mathbf{u}(t,X_t)\mathrm{d}t+\sqrt{2\nu }\mathrm{d}{W}_t,X_0=x,$$

$$\mathbf{u}(t,x)=\mathcal{P}\mathbb{E}({\nabla }^{\mathrm{t}}{X}_t^{-1}(x)\cdot \phi (X_t^{-1}(x))),$$

$$\mathbf{u}\in {\cap }_{T>0}{L}^q([0,T];L^p({\mathbb{R}}^d)),p,q⩾2,\frac{d}{p}+\frac{2}{q}<1,$$

$$\underset{x\in {\mathbb{R}}^d}{\sup }\mathbf{E}\left(\underset{t\in [0,T]}{\sup }|\nabla X_t(x){|}^p\right)<\infty ,\forall p⩾1,T>0.$$

$$\mathbf{u}\in {\cap }_{T>0}{L}^q([0,T];L^p({\mathbb{R}}^d)),p,q⩾2,\frac{d}{p}+\frac{2}{q}>\frac{d}{2}.$$

$$\mathrm{d}{X}_{s,t}=b(t,X_{s,t})\mathrm{d}t+\sqrt{2}\mathrm{d}{W}_t,t>s,X_{s,s}=x,$$

$${\mathcal{L}}_t^b:=\Delta +b(t,\cdot )\cdot \nabla .$$

$$b\in {\cap }_{T>0}{L}^q([0,T];L^p({\mathbb{R}}^d))=:L_{loc}^q(L^p),p,q⩾2,\frac{d}{p}+\frac{2}{q}<2.$$

$${\int }_s^t|b(r,X_{s,r})|\mathrm{d}r<\infty ,\forall t⩾s,a.s.,$$

$$X_{s,t}=x+{\int }_s^{t}b(r,X_{s,r})\mathrm{d}r+\sqrt{2}(W_t-W_s),\forall t⩾s,a.s.$$

$$\mathbb{E}\exp \left(\frac{1}{2}{\int }_0^T|b{|}^2({\omega }_t)\mathrm{d}t\right)<\infty$$

$${\mathcal{E}}_t^b=\exp \left({\int }_0^{t}b({\omega }_s)\mathrm{d}{\omega }_s-\frac{1}{2}{\int }_0^{t}b({\omega }_s)\mathrm{d}s\right)$$

$$X_t=-c{\int }_0^t{X}_s|X_s{|}^{-2}\mathrm{d}s+W_t,c\in \mathbb{R}.$$

$$\mathrm{div}b(x)=-c(d-2)|x{|}^{-2}\notin L_{loc}^{d/2}.$$

$${\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left({\int }_s^t|b(r,{\omega }_r)|\mathrm{d}r\right)<\infty .$$

$$M_t^f(\omega ):=f({\omega }_t)-f(x)-{\int }_s^t{\mathcal{L}}_r^{b}f({\omega }_r)\mathrm{d}r,t⩾s.$$

$$W_t:=\frac{\sqrt{2}}{2}\left({\omega }_t-{\omega }_s-{\int }_s^{t}b(r,{\omega }_r)\mathrm{d}r\right),t⩾s,$$

$${\omega }_t=x+{\int }_s^{t}b(r,{\omega }_r)\mathrm{d}r+\sqrt{2}{W}_t,t⩾s.$$

$$|\text{mspace}-1.0mu|\text{mspace}-1.0mu|b|\text{mspace}-1.0mu|\text{mspace}-1.0mu{|}_{0,p_1;q_1}+|\text{mspace}-1.0mu|\text{mspace}-1.0mu|(\mathrm{div}b{)}^{-}|\text{mspace}-1.0mu|\text{mspace}-1.0mu{|}_{0,p_2;q_2}<\infty ,$$

$$\displaystyle{\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\int^{t_{1}}_{t_{0}}f(t,\omega_{t}){\mathord{{\rm d}}}t\Bigg{|}{\mathcal{B}}_{t_{0}}\right)\leqslant C(t_{1}-t_{0})^{\theta}|\mspace{-1.0mu}|\mspace{-1.0mu}|f|\mspace{-1.0mu}|\mspace{-1.0mu}|_{-\alpha,p;q}.$$

$$\displaystyle{\mathbb{P}}^{n}_{s,x}\mbox{ weakly converges to ${\mathbb{P}}_{s,x}\in{\mathscr{M}}^{b}_{s,x}$},\mbox{ where ${\mathbb{P}}^{n}_{s,x}\in{\mathscr{M}}^{b_{n}}_{s,x}$}.$$

$${\mathbb{E}}^{{\mathbb{P}}_{s,x}}(f({\omega }_{t_1})|{\mathcal{B}}_{t_0})={\mathbb{E}}^{{\mathbb{P}}_{t_0,{\omega }_{t_0}}}(f({\omega }_{t_1})),{\mathbb{P}}_{s,x}-a.s.$$

$$\Vert {\mathcal{T}}_{s,t}f{\Vert }_p⩽C\Vert f{\Vert }_p.$$

$${\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\int_{\tau}^{\tau+\delta}f(t,\omega_{t}){\mathord{{\rm d}}}t\Big{|}{\mathcal{B}}_{\tau}\right)\leqslant C\delta^{\theta}|\mspace{-1.0mu}|\mspace{-1.0mu}|f|\mspace{-1.0mu}|\mspace{-1.0mu}|_{\alpha,p;q},$$

$$\displaystyle{\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\int_{\tau}^{\tau+\delta}f(t,\omega_{t}){\mathord{{\rm d}}}t\Big{|}{\mathcal{B}}_{\tau}\right)=\lim_{n\to\infty}{\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\int_{\tau_{n}}^{\tau_{n}+\delta}f(t,\omega_{t}){\mathord{{\rm d}}}t\Big{|}{\mathcal{B}}_{\tau_{n}}\right)$$

$$\displaystyle\qquad=\lim_{n\to\infty}{\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\sum_{a\in{\mathbb{T}}}{\bf 1}_{\{\tau_{n}=a\}}\int_{a}^{a+\delta}f(t,\omega_{t}){\mathord{{\rm d}}}t\Big{|}{\mathcal{B}}_{\tau_{n}}\right)$$

$$\displaystyle\qquad=\lim_{n\to\infty}\sum_{a\in{\mathbb{T}}}{\bf 1}_{\{\tau_{n}=a\}}{\mathbb{E}}^{{\mathbb{P}}_{s,x}}\left(\int_{a}^{a+\delta}f(t,\omega_{t}){\mathord{{\rm d}}}t\Big{|}{\mathcal{B}}_{a}\right)$$

$$⩽C{\delta }^{\theta }|\text{mspace}-1.0mu|\text{mspace}-1.0mu|f|\text{mspace}-1.0mu|\text{mspace}-1.0mu{|}_{-\alpha ,p;q}\underset{n\to \infty }{\lim }\sum _{a\in \mathbb{T}}{1}_{\{{\tau }_n=a\}}=C{\delta }^{\theta }|\text{mspace}-1.0mu|\text{mspace}-1.0mu|f|\text{mspace}-1.0mu|\text{mspace}-1.0mu{|}_{-\alpha ,p;q}.$$

$$\left|{\int }_0^T{\int }_{{\mathbb{R}}^d}f(t,y){\mu }_{s,x}(t,\mathrm{d}y)\mathrm{d}t\right|⩽C\Vert f{\Vert }_{-\alpha ,p;q},$$

Full text

[TABLE]

where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^{n}_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf{u}}_{n}$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.

Keywords: Leray’s solution, Navier-Stokes equation, Stochastic differential equation, De-Giorgi’s iteration, Krylov’s estimate.

AMS 2010 Mathematics Subject Classification: Primary: 60H10, 35Q30; Secondary: 76D05

Research of X. Zhang is partially supported by NNSFC grant of China (No. 11731009). Research of G. Zhao is supported by the German Research Foundation (DFG) through the Collaborative Research Centre(CRC) 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”.

1. Introduction

Throughout the paper we assume $d\geqslant 2$. Consider the following Navier-Stokes equation:

[TABLE]

where ${\mathbf{u}}=(u_{1},\cdots,u_{d})$ is the velocity field of the fluid, $\nu>0$ is the viscosity constant, and $p$ stands for the pressure. It is well known that for any divergence free vector field $\varphi\in L^{2}({\mathbb{R}}^{d})$, there exists a divergence free Leray weak solution to 3D-NSEs in the class

[TABLE]

In a recent remarkable paper, Buckmaster and Vicol [4] showed that there are infinitely many weak solutions ${\mathbf{u}}\in C({\mathbb{R}}_{+};L^{2}({\mathbb{T}}^{3}))$ for 3D-NSEs on the torus. However, it is still not known whether the above Leray solution is unique and smooth, which are in fact famous open problems for a long time.

In this work we are interesting in the following problem: For any Leray solution ${\mathbf{u}}$, is it possible to construct the stochastic Lagrangian particle trajectory $X_{t}=X_{t}(x)$ associated with the velocity field ${\mathbf{u}}$? More precisely, for each starting point $x$, is there a unique solution to the following SDE?

[TABLE]

where $W$ is a $d$-dimensional standard Brownian motion on some probability space $(\Omega,{\mathscr{F}},{\mathbf{P}})$. If ${\mathbf{u}}$ is smooth in $x$, then by Constantin and Iyer’s representation [6] (see also [23, 25]), ${\mathbf{u}}$ can be reconstructed from $X_{t}(x)$ as follows:

[TABLE]

where ${\mathcal{P}}$ is the Leray projection and $X^{-1}_{t}(x)$ is the inverse of stochastic flow $x\mapsto X_{t}(x)$, and $\nabla^{\rm t}$ stands for the transpose of Jacobian matrix. By Krylov and Röckner’s result [14], under the following assumption

[TABLE]

there is a unique strong solution to SDE (1.2) for any starting point $x\in{\mathbb{R}}^{d}$. Moreover, the unique solution $X_{t}(x)$ is weakly differentiable in $x$ and satisfies (see [7, 22, 27]):

[TABLE]

Unfortunately, Leray’s solution does not satisfy (1.3). Indeed, by (1.1) and Sobolev’s embedding (see Lemma 2.1 below),

[TABLE]

Notice that the deterministic Lagrangian particle trajectories associated with ${\mathbf{u}}$ have been studied very well (for example, see [17, Chapter 17] and [5]), which depends on further regularity on Leray’s solution. Here we want to solve SDE (1.2) only basing on (1.4) for $d=3$.

For given $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, we consider the following SDE in ${\mathbb{R}}^{d}$ starting from $x$ at time $s$:

[TABLE]

where $b(t,x):{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is a measurable vector field. The generator associated with the above SDE is given by

[TABLE]

In this paper, we focus on the weak solution of SDE (1.5) with lower regularity $b$, that is,

[TABLE]

Roughly speaking, a weak solution of SDE (1.5) is a semimartingale $(X_{s,t})_{t\geqslant s}$ so that

[TABLE]

and

[TABLE]

When $b\in L^{q}_{loc}(L^{p})$ for some $p,q\in[2,\infty)$ with $\frac{d}{p}+\frac{2}{q}<1$, as mentioned above, by Girsanov’s transformation and $L^{p}$-theory of second order parabolic equations, Krylov and Röckner [14] showed that there is a unique strong solution to SDE (1.5), which extended the main results in [21] and [30]. The strong well-posedness of SDE (1.5) driven by multiplicative Brownian noise was studied in [22, 27] by Zvonkin’s transformation introduced in [30]. Moreover, the flow property and weak differentiability of $X_{s,t}(x)$ in $x$ are also obtained therein. When $b\in H^{-\alpha,p}$ with $\alpha\in(0,\frac{1}{2})$ and $p\in(\frac{d}{1-\alpha},\frac{d}{\alpha})$ is time-independent, Flandoli, Issoglio and Russo [9] showed the existence and uniqueness of “virtual” solutions (a class of special weak solutions) to SDE (1.5). Later, the well-posedness of martingale solutions and weak solutions (which may not be a semimartingale but a Dirichlet process) was established in [28] for $b\in H^{-\alpha,p}$ with $\alpha\in(0,\frac{1}{2}]$ and $p\in(\frac{d}{1-\alpha},\infty)$. We also mention that Bass and Chen in [2] studied the weak well-posedness of SDE (1.5) in the class of semimartingales when $b$ belongs to some generalized Kato’s class ${\bf K}_{d-1}$(see also [29]), in particular, the space $L^{p}$ with $p>d$ is included in that class.

It should be emphasized that even in the weak sense, all the works mentioned above does not cover the borderline case $b\in L^{q}_{loc}(L^{p})$ with $\frac{d}{p}+\frac{2}{q}=1$, not to mention the supercritical case $\frac{d}{p}+\frac{2}{q}>1$. Let us explain the difficulty firstly. In order to get the weak existence of SDE (1.5) with singular drifts, a straightforward way is to use Girsanov’s transform as in [14]. However, this approach does not work in the case when $p\leqslant d$. Let us make a detailed analysis for this point. Let ${\mathbb{C}}$ be the space of all continuous functions from ${\mathbb{R}}_{+}$ to ${\mathbb{R}}^{d}$, which is endowed with the usual Borel $\sigma$-field ${\mathcal{B}}({\mathbb{C}})$. All the probability measures over $({\mathbb{C}},{\mathcal{B}}({\mathbb{C}}))$ is denoted by ${\mathscr{P}}({\mathbb{C}})$. Let $\omega_{t}$ be the canonical process over ${\mathbb{C}}$. For $t\geqslant 0$, let ${\mathcal{B}}_{t}:={\mathcal{B}}_{t}({\mathbb{C}})$ be the natural filtration generated by $\{\omega_{s}:s\leqslant t\}$. Let ${\mathbb{P}}\in{\mathscr{P}}({\mathbb{C}})$ be the classical Wiener measure so that $t\mapsto\omega_{t}$ is a $d$-dimensional standard Brownian motion. For $b\in L^{p}({\mathbb{R}}^{d})$ with $p\leqslant d$, one can check that the Novikov condition

[TABLE]

for the exponential supermartingale

[TABLE]

may not hold. Notice that condition (1.7) is somehow equivalent to say that $b$ belongs to some Kato’s class (see [1]). In fact, without other conditions, if $b$ only belongs to $L^{d-\varepsilon}_{loc}\setminus L^{d}_{loc}$, then the weak existence may be failed. For example, consider the following SDE:

[TABLE]

If $c\geqslant d$, Kinzebulatov and Semenov [13, page 3] explained why the above SDE does not allow a solution (see also [3]). Meanwhile, for $c<c_{d}$, where $c_{d}\in(0,d)$ is some constant only depending on $d$, they proved that there exists a weak solution to the above SDE by utilizing the analytic construction of the semigroup ${\mathrm{e}}^{-t(\Delta+b\cdot\nabla)}$. By direct calculations, for $b(x):=-cx|x|^{-2}$ and $d\geqslant 3$, we have

[TABLE]

Intuitively, if $c\geqslant d$, then the centripetal force is so strong such that the particle can not escape from the origin immediately so that even though a random perturbation is added, there is no solution for SDE (1.8). However, our result below shows that if $b\in L^{d/2+\varepsilon}({\mathbb{R}}^{d})$ for some $\varepsilon>0$, then equation (1.5) has at least one semimartingale solution, provided that the negative part of $\mathord{{\rm div}}b$ satisfies some integrable conditions. We emphasize that Kinzebulatov and Semenov’s result in [13] can not be applied to the case $b\in L^{d-\varepsilon}_{loc}\setminus L^{d}_{loc}$. We believe that the divergence condition is necessary for this case. Moreover, the singular time-dependent drift $b$ is not treated in [13]. If it is not possible, it seems hard to directly construct the two-parameter semigroups associated with time-dependent drifts by analytic method.

Before stating our results, we introduce the following notion of martingale solutions.

Definition 1**.**

For given $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, we call a probability measure ${\mathbb{P}}_{s,x}\in{\mathscr{P}}({\mathbb{C}})$ a martingale solution of SDE (1.5) with starting point $(s,x)$ if

1. (i)

${\mathbb{P}}_{s,x}(\omega_{t}=x,t\leqslant s)=1$, and for each $t>s$,

[TABLE] 2. (ii)

For all $f\in C^{2}_{c}({\mathbb{R}}^{d})$, $M^{f}_{t}$ is a ${\mathcal{B}}_{t}$-martingale under ${\mathbb{P}}_{s,x}$, where

[TABLE]

All the martingale solution ${\mathbb{P}}_{s,x}$ with starting point $(s,x)$ and drift $b$ is denoted by ${\mathscr{M}}^{b}_{s,x}$.

Remark 1.1**.**

Let ${\mathbb{P}}_{s,x}\in{\mathscr{M}}^{b}_{s,x}$. By Lévy’s characterization for Brownian motion, one sees that

[TABLE]

is a $d$-dimensional standard Browian motion under ${\mathbb{P}}_{s,x}$ (see [19, Theorem 4.2.1]), so that

[TABLE]

In other words, $({\mathbb{C}},{\mathscr{B}}({\mathbb{C}}),{\mathbb{P}}_{s,x},\omega_{t},W_{t})$ is a weak solution of SDE (1.5).

Our main result is

Theorem 1.1**.**

Suppose that for some $p_{i},q_{i}\in[2,\infty)$ with $\tfrac{d}{p_{i}}+\tfrac{2}{q_{i}}<2$, $i=1,2$,

[TABLE]

where $|\mspace{-1.0mu}|\mspace{-1.0mu}|\cdot|\mspace{-1.0mu}|\mspace{-1.0mu}|_{\alpha,p;q}$ is defined by (2.2) below. For each $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, there exists at least one martingale solution ${\mathbb{P}}_{s,x}\in{\mathscr{M}}^{b}_{s,x}$, which satisfies the following Krylov’s type estimate: for any $\alpha\in[0,1]$ and $p,q\in(1,\infty)$ with $\frac{d}{p}+\frac{2}{q}<2-\alpha$, there exist $\theta=\theta(\alpha,p,q)>0$ and a constant $C>0$ such that for all $s\leqslant t_{0}<t_{1}<\infty$ with $t_{1}-t_{0}\leqslant 1$ and $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$,

[TABLE]

Moreover, we have the following conclusions:

1. (i)

(Weak uniqueness) For any mollifying approximation $b_{n}$ of $b$, there is a Lebesgue-null set ${\mathcal{N}}\subset{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$ such that for all $(s,x)\in{\mathcal{N}}^{c}$,

[TABLE] 2. (ii)

(Almost surely Markov property) For each $(s,x)\in{\mathcal{N}}^{c}$, there is a Lebesgue null set $I_{s,x}\subset[s,\infty)$ such that for all $t_{0}\in(s,\infty)\setminus I_{s,x}$, any $t_{1}>t_{0}$ and $f\in C_{c}({\mathbb{R}}^{d})$,

[TABLE] 3. (iii)

($L^{p}$-semigroup) Let ${\mathcal{T}}_{s,t}f(x):={\mathbb{E}}^{{\mathbb{P}}_{s,x}}f(\omega_{t})$. For any $p\geqslant 1$ and $T>0$, there is a constant $C>0$ such that for Lebesgue almost all $0\leqslant s<t\leqslant T$ and $f\in L^{p}({\mathbb{R}}^{d})$,

[TABLE]

Remark 1.2**.**

By discretization stopping time approximation, Krylov estimate (1.10) is equivalent to say that for any $\delta\in(0,1)$ and stopping time $\tau\in[s,\infty)$,

[TABLE]

where {\mathcal{B}}_{\tau}:=\sigma\big{\{}\omega_{t\wedge\tau},t\geqslant 0\big{\}} is the stopping time $\sigma$-field. In fact, let $\tau_{n}$ be a sequence of decreasing stopping times taking values in ${\mathbb{T}}:=\{k\cdot 2^{-n}:k,n\in{\mathbb{N}}\}$ and converging to $\tau$. For any $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$ and $\delta\in(0,1)$, by the dominated convergence theorem and martingale convergence theorem, we have

[TABLE]

Moreover, let $\mu_{s,x}(t,{\mathord{{\rm d}}}y):={\mathbb{P}}_{s,x}\circ\omega_{t}^{-1}$. For any $\alpha\in[0,1]$ and $p,q\in(1,\infty)$ with $\frac{d}{p}+\frac{2}{q}<2-\alpha$, by (1.10), for any $T>0$ there is a constant $C>0$ such that for all $f\in C^{\infty}_{c}([0,T]\times{\mathbb{R}}^{d})$,

[TABLE]

which in turn implies that $\mu_{s,x}(t,{\mathord{{\rm d}}}y)=\rho_{s,x}(t,y){\mathord{{\rm d}}}y$ with $\rho_{s,x}\in{\mathbb{H}}^{\alpha,p/(p-1)}_{q/(q-1)}$.

Remark 1.3**.**

If $(\mathord{{\rm div}}b)^{-}\equiv 0$, then $\|{\mathcal{T}}_{s,t}f\|_{1}\leqslant\|f\|_{1}$ in (1.13). If $\mathord{{\rm div}}b\equiv 0$, then for any nonnegative $f\in L^{1}({\mathbb{R}}^{d})$, $\|{\mathcal{T}}_{s,t}f\|_{1}=\|f\|_{1}$. By (1.4), we can apply the above theorem to the Leray solution of 3D-NSEs.

Remark 1.4**.**

Let $d\geqslant 3$ and $\alpha<3$. Define

[TABLE]

where for some $M>0$, $\gamma_{z}\in(0,M)$ is a constant and $\phi\in C^{\infty}_{c}({\mathbb{R}}_{+};[0,1])$ with $\phi(r)=1$ for $r\in[0,1]$ and $\phi(r)=0$ for $r>2$. It is easy to see that (1.9) holds.

Remark 1.5**.**

It should be compared with the results in [24, 26]. Therein, under the assumptions

[TABLE]

the existence and uniqueness of almost everywhere stochastic flows are obtained in the framework of DiPerna-Lions’ theory. By the estimate (1.13), we can weaken the assumption on the boundedness of $(\mathord{{\rm div}}b)^{-}$ in [26] when the noise is nondegenerate. On the other hand, in [24, 26], under (1.15), the existence of a solution is only shown for Lebesgue almost all $x\in{\mathbb{R}}^{d}$, while, under (1.9) we can show the existence of a solution for all starting point $x\in{\mathbb{R}}^{d}$.

To prove Theorem 1.1, the key point for us is to establish the maximum principle for the following parabolic equation under (1.9):

[TABLE]

More precisely, for any $\alpha\in[0,1]$ and $q,p\in(1,\infty)$ with $\frac{d}{p}+\frac{2}{q}<2-\alpha$,

[TABLE]

When $f\equiv 0$, under (1.9) the local maximum principle is proved by Nazarov and Ural’tseva in [15] by using Moser’s iteration. We also refer to [11] for the study of elliptic equations with drift $b\equiv 0$ and $f\in L^{p}({\mathbb{R}}^{d})$ for $p>\frac{d}{2}$. Here an open question is that whether we can show (1.11)-(1.13) for all $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, which is closely related to find a continuous solution for PDE (1.16) under (1.9).

This paper is organized as follows: In Section 2, we establish the key maximum estimate (1.17) by De Giorgi’s method. In fact, we shall show a more general result by allowing $b$ and $f$ being in negative Sobolev spaces, which are not treated in [11, 15]. In Section 3, we prove our main result Theorem 1.1. In Appendix, some properties of certain local Sobolev spaces are given. Throughout this paper we shall use the following conventions:

• •

We use $A\lesssim B$ to denote $A\leqslant CB$ for some unimportant constant $C>0$.

• •

For any $\varepsilon\in(0,1)$, we use $A\lesssim\varepsilon B+D$ to denote $A\leqslant\varepsilon B+C_{\varepsilon}D$ for some constant $C_{\varepsilon}>0$.

• •

${\mathbb{N}}_{0}:={\mathbb{N}}\cup\{0\}$, ${\mathbb{R}}_{+}:=[0,\infty)$, $a\vee b:=\max(a,b)$, $a\wedge b:={\mathord{{\rm min}}}(a,b)$, $a^{+}:=a\vee 0$.

2. Maximum principle for parabolic equations by De Giorgi’s method

We first introduce some spaces, functions and notations for later use:

• •

Let $C^{\infty}_{c}({\mathbb{R}}^{d+1})$ be the space of all smooth functions with compact supports and ${\mathscr{D}}^{\prime}$ the dual space of $C^{\infty}_{c}({\mathbb{R}}^{d+1})$, which is also called distribution space. The duality between ${\mathscr{D}}^{\prime}$ and $C^{\infty}_{c}({\mathbb{R}}^{d+1})$ is denoted by ${\langle}\!{\langle}\cdot,\cdot{\rangle}\!{\rangle}$. In particular, if $f(t,x)$ and $g(t,x)$ are two real functions in ${\mathbb{R}}\times{\mathbb{R}}^{d}$, then

[TABLE]

• •

For two distributions $f,g\in{\mathscr{D}}^{\prime}$, one says that $f\leqslant g$ if for any nonnegative $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$,

[TABLE]

• •

For $\alpha\in{\mathbb{R}}$ and $p\in[1,\infty]$, let $H^{\alpha,p}$ be the usual Bessel potential space with norm:

[TABLE]

If $f\in H^{\alpha,p}$, $g\in H^{-\alpha,p^{\prime}}$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, we denote

[TABLE]

• •

For $\alpha\in{\mathbb{R}}$ and $p,q\in[1,\infty]$, let ${\mathbb{H}}^{\alpha,p}_{q}:=L^{q}({\mathbb{R}};H^{\alpha,p})$ be the space of spatial-time functions with norm

[TABLE]

If $f\in{\mathbb{H}}^{\alpha,p}_{q}$, $g\in{\mathbb{H}}^{-\alpha,p^{\prime}}_{q^{\prime}}$ with $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $\frac{1}{q}+\frac{1}{q^{\prime}}=1$, we also denote

[TABLE]

• •

Let ${\mathbb{H}}^{\alpha,p}_{q,loc}$ be the space of all functions $f:{\mathbb{R}}^{d+1}\to{\mathbb{R}}$ with

[TABLE]

• •

For $r>0$, we define

[TABLE]

• •

Fix $\chi\in C^{\infty}({\mathbb{R}}^{d+1};[0,1])$ with $\chi|_{Q_{1}}=1$ and $\chi|_{Q^{c}_{2}}=0$. For $r>0$ and $(s,z)\in{\mathbb{R}}^{d+1}$, define

[TABLE]

• •

Fix $r>0$. Let $\widetilde{\mathbb{H}}^{\alpha,p}_{q}$ be the Banach space of all functions $f\in{\mathbb{H}}^{\alpha,p}_{q,loc}$ with

[TABLE]

• •

We shall simply write

[TABLE]

• •

Let ${\mathscr{V}}:={\mathbb{L}}^{2}_{\infty}\cap{\mathbb{H}}^{1,2}_{2}$, ${\mathscr{V}}_{loc}:={\mathbb{L}}^{2}_{\infty,loc}\cap{\mathbb{H}}^{1,2}_{2,loc}$ and $\widetilde{\mathscr{V}}:=\widetilde{\mathbb{L}}^{2}_{\infty}\cap\widetilde{\mathbb{H}}^{1,2}_{2}$. Define

[TABLE]

• •

Let $\rho\in C^{\infty}_{c}(B_{1})$ with $\int\rho=1$. For $\varepsilon\in(0,1)$ and $n\in{\mathbb{N}}$, we shall use the mollifiers:

[TABLE]

• •

For $d\geqslant 2$, define

[TABLE]

• •

For given $(\alpha,p,q)\in{\mathscr{I}}_{d}$, we define $r,s\in[2,\infty]$ by relation

[TABLE]

Notice that $\frac{d}{p}+\frac{2}{q}<2-\alpha$ implies

[TABLE]

• •

The following Gagliardo-Nirenberge’s interpolation inequality will be used frequently:

[TABLE]

where $\alpha\in[0,1]$, $\theta\in[\alpha,1]$ and $p,q,r\in[1,\infty]$ satisfy

[TABLE]

2.1. Localization estimates

In this subsection we prove an important localization lemma for later use, which is a consequence of Gagliado-Nirenberge’s interpolation inequality and Hölder’s inequality. First of all, we have the following interpolation estimates.

Lemma 2.1**.**

Let $r,s\geqslant 2$ with $\frac{d}{r}+\frac{2}{s}>\frac{d}{2}$. For any $\varepsilon\in(0,1)$, there is a constant $C_{\varepsilon}>0$ such that

[TABLE]

where $\theta:=\frac{d}{2}-\frac{d}{r}\in[0,1)$. In particular, if supp$f\subset Q_{2}$, then for some $C=C(d,r,s)>0$,

[TABLE]

Proof.

For $r\in[2,\infty)$ if $d=2$ or $r\in[2,2d/(d-2)]$ if $d\geqslant 3$, by (2.5) we have

[TABLE]

Since $s\theta<2$, by Hölder’s inequality we further have

[TABLE]

which gives the desired embedding by Young’s inequality. ∎

Lemma 2.2**.**

Let $Q=I\times D\subset{\mathbb{R}}\times{\mathbb{R}}^{d}$ be a bounded domain. For any $p,q,r,s\in[1,\infty]$, there is a constant $C>0$ only depending on $Q,p,q,r,s$ such that for any $A\subset Q$,

[TABLE]

Proof.

Define

[TABLE]

If $r/p\leqslant s/q$, then by Hölder’s inequality,

[TABLE]

If $r/p>s/q$, then by Hölder’s inequality,

[TABLE]

The proof is complete. ∎

The following lemma is the key localization result.

Lemma 2.3**.**

Let $\eta:{\mathbb{R}}^{d+1}\to[0,1]$ be a smooth function with compact support contained in $Q_{2}$. Let $(\alpha,p,q)\in{\mathscr{I}}_{d}$ and $r,s\in[2,\infty]$ be defined by (2.4). Let $\chi_{2}$ be the cutoff function defined by (2.1). For any $\varepsilon\in(0,1)$, there is a constant $C_{\varepsilon}=C_{\varepsilon}(d,\alpha,p,q)>0$ such that for any $c,b,f\in{\mathbb{H}}^{-\alpha,p}_{q,loc}$ and $w\in{\mathscr{V}}_{loc}$,

[TABLE]

[TABLE]

[TABLE]

Proof.

Since $\alpha\in[0,1]$, by relation (2.4), one sees that

[TABLE]

Thus by Hölder’s inequality and Gagliardo-Nirenberge’s inequality (2.5), we have

[TABLE]

(i) Since $\chi_{2}|_{Q_{2}}\equiv 1$ and $\eta|_{Q^{c}_{2}}=0$, by (2.10) with $g=c\chi_{2}$ and $h=\eta^{2}w^{2}$, we have

[TABLE]

where we drop the time variable $t$ and the last step is due to Hölder’s inequality. Integrating both sides in the time variable, and due to $\frac{2-\alpha}{s}+\frac{\alpha}{2}+\frac{1}{q}=1$, by Hölder’s inequality again we get

[TABLE]

which gives (2.7) by Young’s inequality.  

(ii) By (2.10) with $g=b\chi_{2}$ and $h=\nabla\eta^{2}w^{2}$, we have

[TABLE]

Notice that

[TABLE]

and by Hölder’s inequality,

[TABLE]

Hence,

[TABLE]

and by Hölder’s inequality and due to $\frac{1}{q}+\frac{2-\alpha}{s}+\frac{\alpha}{2}=1$, $s\geqslant\frac{2q}{q-1}$,

[TABLE]

The desired estimate (2.8) follows by Young’s inequality and $2q/(q-1)\leqslant s$.  

(iii) By (2.10) with $g=f\eta$ and $h=\eta w$, we have

[TABLE]

Since $\nabla(\eta w)=\nabla(\eta w)^{+}-\nabla(\eta w)^{-}=0$ on $\{\eta w=0\}$ (cf. [10, Lemma 7.6]), we have

[TABLE]

Thus, by Hölder’s inequality, we further have

[TABLE]

and due to $\frac{1}{q}+\frac{1-\alpha}{s}+\frac{\alpha}{2}+\frac{1}{s}=1$, $\frac{d}{r}+\frac{2}{s}>\frac{d}{2}$,

[TABLE]

Notice that by $\eta=\chi_{2}\eta$ and (4.2) below,

[TABLE]

Substituting this into (2.12) and by Young’s inequality, we obtain (2.9). ∎

2.2. Local energy estimate

Throughout this paper we shall always assume

[TABLE]

and consider the following PDE in ${\mathbb{R}}^{d+1}$:

[TABLE]

Definition 2**.**

A function $u\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ is called a weak solution (subsolution or supersolution) of PDE (2.13) with coefficients $(b,f)$ if for any nonnegative smooth function $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$ and almost all $t\in{\mathbb{R}}$,

[TABLE]

Now we prove the following local energy estimate.

Lemma 2.4** (Energy estimate).**

Suppose that for some $(\alpha_{i},p_{i},q_{i})\in{\mathscr{I}}_{d}$, $i=1,2,3$,

[TABLE]

Let $(r_{i},s_{i})\in[2,\infty]$ be defined by (2.4) and $\kappa\geqslant 0$. For any weak subsolution $u\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ of PDE (2.13), there is a constant $C>0$ depending only on $d,\alpha_{i},p_{i},q_{i},i=1,2,3$ and

[TABLE]

where $\chi_{2}$ is defined by (2.1), such that for $w:=(u-\kappa)^{+}$ and any $\eta\in C^{\infty}_{c}(Q_{2})$ and $t\geqslant 0$,

[TABLE]

where ${\mathcal{I}}_{t}(\cdot):={\bf 1}_{(-\infty,t]}(\cdot)$, and

[TABLE]

Proof.

By taking the Steklov mean of $u$, without loss of generality we may assume $\partial_{t}u\in{\mathbb{L}}^{2}_{2,loc}$. By (2.14) and smoothing approximation, for any nonnegative $\varphi\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ with compact support in $Q_{2}$, we have

[TABLE]

Let $\eta\in C^{\infty}_{c}(Q_{2})$ and $w:=(u-\kappa)^{+}$ for some $\kappa\geqslant 0$. Taking the test function $\varphi=\eta^{2}w\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ and integrating in time variable from $-\infty$ to $t$, by the integration by parts formula, we have

[TABLE]

where $\Gamma_{t}:=(-\infty,t]\times{\mathbb{R}}^{d}$ and we have used that

[TABLE]

Noticing that

[TABLE]

by the integration by parts formula again, we have

[TABLE]

and

[TABLE]

Therefore, by $-\mathord{{\rm div}}b\leqslant\Theta_{b}$ and smoothing approximation for $w$, we further have

[TABLE]

which yields by definition (2.3) that

[TABLE]

For $I_{1}$, notice that $r_{1},s_{1}\geqslant 2$, we have

[TABLE]

For $I_{2},I_{3}$ and $I_{4}$, by (2.7), (2.8) and (2.9), we have

[TABLE]

and

[TABLE]

[TABLE]

Combining the above calculations and letting $\varepsilon$ be small enough, we obtain

[TABLE]

where $\Xi_{\eta}$ is define by (2.16). From this, we derive (2.15). ∎

Remark 2.1**.**

If $\alpha_{1}=0$ and $\frac{d}{p_{1}}+\frac{2}{q_{1}}=1$ or $b(t,x)=b(x)\in L^{d}_{loc}({\mathbb{R}}^{d})$, then we can remove the assumption on the divergence of $b$. In fact, in this case, we can give a direct treatment for the term $b$ in (2.17) as follows: For any $\varepsilon>0$, let

[TABLE]

Let $\frac{1}{r_{1}}+\frac{1}{p_{1}}=\frac{1}{s_{1}}+\frac{1}{q_{1}}=\frac{1}{2}$, which satisfy $\frac{d}{r_{1}}+\frac{2}{s_{1}}=\frac{d}{2}$ due to $\frac{d}{p_{1}}+\frac{2}{q_{1}}=1$. Since $\chi_{2}\eta=\eta$, by (2.18), Hölder’s inequality and Lemma 2.1, we have

[TABLE]

where $\lim_{\varepsilon\to 0}c_{\varepsilon}=0$ and $\lim_{\varepsilon\to 0}C_{\varepsilon}=\infty$. Using this estimate to replace the corresponding estimate about $b$ and taking $\varepsilon$ small enough, we still have (2.15). Here the reason that for $p=d$ we assume $b$ being time-independent is that in general

[TABLE]

2.3. Maximum principle

The following De Giorgi’s iteration lemma is well known [11].

Lemma 2.5**.**

Let $(a_{n})_{n\in{\mathbb{N}}}$ be a sequence of nonnegative numbers. Suppose that for some $C_{0},\lambda>1$ and $\varepsilon>0$,

[TABLE]

If $a_{1}\leqslant C_{0}^{-1/\varepsilon}\lambda^{-1/\varepsilon^{2}}$, then

[TABLE]

Now we can show the following local maximum principle for PDE (2.13).

Theorem 2.1** (Local maximum estimate).**

Suppose that for some $(\alpha_{i},p_{i},q_{i})\in{\mathscr{I}}_{d}$, $i=1,2,3$,

[TABLE]

For any weak subsolution $u\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ of PDE (2.13), there is a constant $C>0$ depending only on $d,\alpha_{i},p_{i},q_{i},i=1,2,3$ and the quantities

[TABLE]

where $\chi_{2}$ is defined by (2.1), such that

[TABLE]

Proof.

Let $\kappa>0$, which will be determined below. For $n\in{\mathbb{N}}$, define

[TABLE]

and

[TABLE]

Let $\zeta^{\rm t}_{n}\in C^{\infty}_{c}(-2,2)$ be a time-cutoff function so that for some $C>0$ and any $n\in{\mathbb{N}}$,

[TABLE]

Let $\zeta^{\rm x}_{n}\in C^{\infty}_{c}(B_{2})$ be a spatial-cutoff function so that for some $C>0$ and any $n\in{\mathbb{N}}$,

[TABLE]

Now let us define

[TABLE]

Let $\Xi_{\eta_{n}}$ be defined by (2.16). It is easy to see that for some $C>0$ and all $n\in{\mathbb{N}}$,

[TABLE]

Let

[TABLE]

Notice that

[TABLE]

For $i=1,2,3$, due to $\eta_{n}|_{\Gamma^{c}_{n}}=0$, we have

[TABLE]

which means that

[TABLE]

Since $\frac{2}{r_{i}}+\frac{d}{s_{i}}>\frac{d}{2}$, we can choose $\gamma_{i},\beta_{i}>r,\theta_{i},\tau_{i}>s$ so that

[TABLE]

Thus, by $\eta_{n}|_{\Gamma_{n+1}}=1$, Hölder’s inequality, Lemmas 2.1 and 2.2, we have

[TABLE]

Notice $\Gamma_{1}=Q_{2}$. By (2.15) with $\eta=\eta_{n}$ and $w=w_{n+1}$, for $\kappa\geqslant\|f\chi_{2}\|_{-\alpha_{3},p_{3};q_{3}}$, we obtain

[TABLE]

Now we put

[TABLE]

By (2.21) and (2.22), we obtain that for some $C_{0},\varepsilon>0$ and $\lambda>1$,

[TABLE]

provided $\kappa\geqslant\|f\chi_{2}\|_{-\alpha_{3},p_{3};q_{3}}$. Notice that by $\chi_{2}|_{\Gamma_{1}}=1$ and Lemma 2.1,

[TABLE]

If $\kappa\geqslant(C_{1}C_{0}^{1/\varepsilon}\lambda^{1/\varepsilon^{2}}\|u^{+}\chi_{2}\|_{{\mathscr{V}}})\vee\|f\chi_{2}\|_{-\alpha_{3},p_{3};q_{3}}$ so that $a_{1}\leqslant C_{0}^{-1/\varepsilon}\lambda^{-1/\varepsilon^{2}}$, then by Fatou’s lemma and Lemma 2.5,

[TABLE]

which implies that for Lebesgue almost all $(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{d}$,

[TABLE]

The proof is complete. ∎

Remark 2.2**.**

If $\alpha_{1}=0$ and $\frac{d}{p_{1}}+\frac{2}{q_{1}}=1$ or $b(t,x)=b(x)\in L^{d}_{loc}({\mathbb{R}}^{d})$, then by Remark 2.1, we can drop the condition on the divergence of $b$.

Now we aim to prove the following crucial result.

Theorem 2.2**.**

(Global maximum estimate) Suppose that for some $(\alpha_{i},p_{i},q_{i})\in{\mathscr{I}}_{d}$, $i=1,2,3$,

[TABLE]

Let $u\in{\mathscr{V}}_{loc}\cap{\mathbb{L}}^{\infty}_{loc}$ be a weak solution of PDE (2.13) with initial value $u(0)=0$. For any $T>0$, there exists a constant $C>0$ depending only on $T,d,\alpha_{i},p_{i},q_{i}$ and the quantity

[TABLE]

such that

[TABLE]

Proof.

Without loss of generality, we assume $T=1$ and

[TABLE]

Let $\chi_{1}$ be as in (2.1) and define for $z\in{\mathbb{R}}^{d}$,

[TABLE]

By translation and (2.15) with $\eta=\eta_{z}$ and $w=u^{+},u^{-}$, there is a constant $C>0$ depending only on $T,d$, $\alpha_{i},p_{i},q_{i}$, $|\mspace{-1.0mu}|\mspace{-1.0mu}|b|\mspace{-1.0mu}|\mspace{-1.0mu}|_{-\alpha_{1},p_{1};q_{1}}$, $|\mspace{-1.0mu}|\mspace{-1.0mu}|(\mathord{{\rm div}}b)^{-}|\mspace{-1.0mu}|\mspace{-1.0mu}|_{-\alpha_{2},p_{2};q_{2}}$ such that for all $t\in[0,1]$,

[TABLE]

where $\chi^{0,z}_{2}$ is the same as in (2.1). Taking supremum in $z\in{\mathbb{R}}^{d}$ for both sides, we obtain

[TABLE]

Since for each $z\in{\mathbb{R}}^{d}$, there are at most $N$-points $z_{1},\cdots,z_{N}\in{\mathbb{R}}^{d}$ such that

[TABLE]

where $N=N(d)$, we have for $t\in[0,1]$,

[TABLE]

where the last step is due to $\eta_{z_{j}}|_{[0,t]\times B_{1}(z_{j})}=1$. Hence, by (2.25), (2.26) and (4.3) in appendix,

[TABLE]

Let

[TABLE]

Since $\frac{d}{r_{i}}+\frac{2}{s_{i}}>\frac{d}{2}$, by Lemma 2.1, we have for any $\varepsilon\in(0,1)$,

[TABLE]

Combining (2.27) and (2.28), we arrive at

[TABLE]

By choosing $\varepsilon$ small enough, we obtain

[TABLE]

Since $s_{1}^{\prime},s_{2}^{\prime}<\infty$ and $u(t)\equiv 0$ for $t\leqslant 0$, the above inequality implies that for any $t\in[0,1]$,

[TABLE]

By Gronwall’s inequality we obtain

[TABLE]

which together with (2.29) yields

[TABLE]

Finally, by (2.19) and (2.24), we also have

[TABLE]

where $B^{z}_{1}:=\{x:|x-z|<1\}$. The proof is complete. ∎

2.4. Existence-uniqueness and stability

In this subsection we prove the existence-uniqueness and stability of weak solutions for PDE (2.13) by using the apriori estimate (2.24). For $T>0$ and a function $f$ in ${\mathbb{R}}^{d+1}$, we denote

[TABLE]

Theorem 2.3**.**

(Existence-uniqueness) Under (2.23), for any $T>0$, there exists a unique weak solution $u\in\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T}$ to PDE (2.13) with initial value $u(0)=0$.

Proof.

First of all, the uniqueness is a direct consequence of (2.24). We prove the existence by weak convergence method. Let $b_{n}(t,x):=b(t,\cdot)*\rho_{n}(x)$ and $f_{n}(t,x):=f(t,\cdot)*\rho_{n}(x)$. By (ii) of Proposition 4.1 in Appendix, we have

[TABLE]

and

[TABLE]

It is well known that the following PDE has a unique smooth solution $u_{n}\in C({\mathbb{R}}_{+};C^{\infty}_{b}({\mathbb{R}}^{d}))$:

[TABLE]

By (2.30) and Theorem 2.2, we have

[TABLE]

Hence, by the fact that every bounded subset of $\widetilde{\mathbb{H}}^{1,2}_{2}$ is relatively compact, there is a subsequence $n_{k}$ and $\bar{u}\in\cap_{T>0}(\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T})$ such that for any $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$ and $g\in{\mathbb{H}}^{-1,2}_{2;loc}$,

[TABLE]

by taking weak limits for equation (2.31), one finds that $\bar{u}$ is a weak solution of PDE (2.13). Indeed, it suffices to prove that for any $\varphi\in C_{c}^{\infty}({\mathbb{R}}^{d+1})$,

[TABLE]

Let the support of $\varphi$ be contained in $Q_{R}$ for some $R>0$. Since $b\in{\mathbb{L}}^{2}_{2;loc}$, by (2.32) and Hölder’s inequality, we have for some $C>0$ independent of $k$,

[TABLE]

On the other hand, since $\mathord{{\rm div}}(b\varphi)\in{\mathbb{H}}^{-1,2}_{2}$ has compact support, by (2.33) we also have

[TABLE]

Thus we obtain the first limit in (2.34). The second limit in (2.34) is direct. ∎

Theorem 2.4**.**

(Stability) Let $(p_{i},q_{i})\in[2,\infty)$ with $\frac{d}{p_{i}}+\frac{2}{q_{i}}<2$, where $i=1,2,3$. For any $n\in{\mathbb{N}}\cup\{\infty\}=:{\mathbb{N}}_{\infty}$, let $b_{n},f_{n}\in{\mathscr{D}}^{\prime}$ satisfy

[TABLE]

For $n\in{\mathbb{N}}_{\infty}$, let $u_{n}\in\widetilde{\mathscr{V}}\cap{\mathbb{L}}^{\infty}$ be the unique weak solutions of PDE (2.13) associated with coefficients $(b_{n},f_{n})$ with initial value $u(0)=0$. Assume that for any $\varphi\in C_{c}({\mathbb{R}}^{d+1})$,

[TABLE]

Then it holds that for Lebesgue almost all $(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$,

[TABLE]

Proof.

Notice that equation

[TABLE]

holds in the distributional sense (see (2.14)). Letting $r:=\frac{2p_{1}}{p_{1}+2}$ and $s:=\frac{2q_{1}}{q_{1}+2}$, by Proposition 4.1 in Appendix, we have

[TABLE]

By (2.35) and Theorem 2.2, we get

[TABLE]

Thus by Aubin-Lions’ lemma (cf. [18]), there is a subsequence $n_{k}$ and $\bar{u}\in\cap_{T>0}(\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T})$ such that

[TABLE]

By selecting a subsubsequence $n^{\prime}_{k}$, it holds that for Lebesgue almost all $(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$,

[TABLE]

As in showing (2.34), one can show that $\bar{u}$ is a weak solution of PDE (2.13). By the uniqueness, $\bar{u}=u$, and by a contradiction method, the whole sequence converges almost everywhere. ∎

3. Proof of Theorem 1.1

Below we always assume that for some $p_{i},q_{i}\in[2,\infty)$ with $\frac{d}{p_{i}}+\frac{2}{q_{i}}<2$, $i=1,2$,

[TABLE]

Let $b_{n}(t,x)=b(t,\cdot)*\rho_{n}(x)$ be the mollifying approximation of $b(t,\cdot)$. By (ii) of Proposition 4.1 in Appendix, we have

[TABLE]

and

[TABLE]

For $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, consider the following SDE:

[TABLE]

where $W$ is a $d$-dimensional standard Brownian motion on some complete filtered probability space $(\Omega,{\mathscr{F}},({\mathscr{F}}_{t})_{t\geqslant 0},{\mathbf{P}})$. It is well known that there is a unique strong solution $X^{n}_{s,t}(x)$ to the above SDE (cf. [12]).

Now we are in the position to prove our main result and we divide our proof into three parts.

3.1. Existence of martingale solutions

First of all, we prove the following crucial estimate of Krylov’s type.

Lemma 3.1**.**

For any $(\alpha,p,q)\in{\mathscr{I}}_{d}$, there are constants $\theta=\theta(\alpha,p,q)>0$ and $C>0$ depending on $\kappa,d$, $\alpha,p,q$, $p_{i},q_{i}$ such that for any $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$ and $0\leqslant s\leqslant t_{0}<t_{1}<\infty$ with $t_{0}-t_{1}\leqslant 1$,

[TABLE]

In particular, we have the following Khasminskii’s estimate: for any $\lambda\in{\mathbb{R}}$,

[TABLE]

Proof.

Fix $0\leqslant s\leqslant t_{0}<t_{1}<\infty$ with $t_{0}-t_{1}\leqslant 1$ and $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$. Let $u_{n}$ be the smooth solution of the following backward PDE:

[TABLE]

By Itô’s formula we have

[TABLE]

By (3.5) and taking conditional expectation with respect to ${\mathscr{F}}_{t_{0}}$, we obtain

[TABLE]

Since $\frac{d}{p}+\frac{2}{q}<2-\alpha$, we can choose $q^{\prime}<q$ so that $\frac{d}{p}+\frac{2}{q^{\prime}}<2-\alpha$. Thus by Theorem 2.2 and Hölder’s inequality, there is constant $C=C(\kappa,d,\alpha,p,q,p_{i},q_{i})>0$ such that

[TABLE]

Thus we obtain (3.3). As for (3.4), it is a direct consequence of (3.3) and [16, Lemma 1.1] (or see [22]). ∎

Lemma 3.2**.**

For any $T>0$, there is a constant $C>0$ such that for any $f\in L^{1}({\mathbb{R}}^{d})$ and $n\in{\mathbb{N}}$,

[TABLE]

where ${\mathcal{T}}^{n}_{s,t}f(x):={\mathbf{E}}f(X^{n}_{s,t}(x))$. Moreover, if $(\mathord{{\rm div}}b)^{-}\equiv 0$, then the above $C$ can be $1$.

Proof.

Let $Y^{n}_{s,t}:=Y^{n}_{s,t}(x)$ be the inverse flow of $x\mapsto X^{n}_{s,t}(x)$. Notice that $s\mapsto Y^{n}_{s,t}$ solves the following backward SDE:

[TABLE]

Letting $J^{n}_{s,t}:=J^{n}_{s,t}(x):=\nabla Y^{n}_{s,t}(x)$ be the Jacobian matrix, we have

[TABLE]

Hence,

[TABLE]

Thus, by Khasminskii’s estimate (3.4) and (3.1), we have

[TABLE]

Now by the change of variables, for any nonnagative $f\in L^{1}({\mathbb{R}}^{d})$, we have

[TABLE]

Moreover, if $(\mathord{{\rm div}}b)^{-}\equiv 0$, then $\det(J^{n}_{s,t})\leqslant 1$ and the above $C\equiv 1$. ∎

Lemma 3.3**.**

For each $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, let ${\mathbb{P}}^{n}_{s,x}$ be the law of $X^{n}_{s,\cdot}(x)$ in ${\mathbb{C}}$. Then $({\mathbb{P}}^{n}_{s,x})_{n\in{\mathbb{N}}}$ is tight.

Proof.

Fix $(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$ and $T>s$. Let $\tau\geqslant s$ be any stopping time less than $T$. Notice that

[TABLE]

By the strong Markov property and (3.3) with $\alpha=0$, we have

[TABLE]

where $C$ is independent of $n$. Thus by [29, Lemma 2.7], we obtain

[TABLE]

From this, by Chebyshev’s inequality, we derive that for any $\varepsilon>0$,

[TABLE]

Hence, by [19, Theorem 1.3.2], the law of $X^{n}_{s,\cdot}(x)$ is tight in ${\mathbb{C}}$. ∎

Now we can show the existence of martingale solutions.

Lemma 3.4**.**

Any accumulation point ${\mathbb{P}}_{s,x}$ of $({\mathbb{P}}^{n}_{s,x})_{n\in{\mathbb{N}}}$ belongs to ${\mathscr{M}}_{s,x}^{b}$. Moreover, for any $(\alpha,p,q)\in{\mathscr{I}}_{d}$, there are $\theta=\theta(\alpha,p,q)>0$ and constant $C>0$ such that for any $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$ and $0\leqslant s\leqslant t_{0}<t_{1}<\infty$ with $t_{1}-t_{0}\leqslant 1$,

[TABLE]

Proof.

Let $(\alpha,p,q)\in{\mathscr{I}}_{d}$. By (3.3), there are $\theta=\theta(\alpha,p,q)>0$ and constant $C>0$ such that for each $n\in{\mathbb{N}}$, $x\in{\mathbb{R}}^{d}$, and any $f\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$, $0\leqslant s\leqslant t_{0}<t_{1}<\infty$ with $t_{1}-t_{0}\leqslant 1$, and $G\in C_{b}({\mathbb{C}})$ being ${\mathcal{B}}_{t_{0}}$-measurable,

[TABLE]

Let ${\mathbb{P}}_{s,x}$ be any accumulation point of $({\mathbb{P}}^{n}_{s,x})_{n\in{\mathbb{N}}}$, that is, for some subsequence $n_{k}$,

[TABLE]

By taking weak limits for (3.8) and a standard monotone class method, we obtain (3.7). In order to prove ${\mathbb{P}}_{s,x}\in{\mathscr{M}}_{s,x}^{b}$, it suffices to prove that for any $t_{1}>t_{0}\geqslant s$ and $f\in C^{2}_{c}({\mathbb{R}}^{d})$,

[TABLE]

where

[TABLE]

By the standard monotone class method, it is enough to show that for any $G\in C_{b}({\mathbb{C}})$ being ${\mathcal{B}}_{t_{0}}$-measurable,

[TABLE]

Note that for each $n\in{\mathbb{N}}$,

[TABLE]

We want to take weak limits, where the key point is to show

[TABLE]

Assume that supp$(f)\subset Q_{R}$. By (3.3) with $\alpha=0$ and (4.5) in Appendix, we have

[TABLE]

where $\chi_{R}|_{Q_{R}}\equiv 1$ has compact support. On the other hand, for fixed $m\in{\mathbb{N}}$, since

[TABLE]

we also have

[TABLE]

which together with (3.10) yields (3.9). The proof is complete. ∎

3.2. Weak convergence of ${\mathbb{P}}^{n}_{s,x}$

In this subsection we show that for Lebesgue almost all $(s,x)$, the accumulation point of $({\mathbb{P}}^{n}_{s,x})_{n\in{\mathbb{N}}}$ is unique, which in turn implies that

[TABLE]

For fixed $T>0$ and $f\in{\mathbb{L}}^{\infty}_{T}$, by Theorem 2.3, there is a unique weak solution $u=u_{T,f}\in\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T}$ to the following backward PDE:

[TABLE]

Let ${\mathbb{Q}}\subset{\mathbb{R}}$ be the set of all rational numbers and ${\mathscr{G}}_{0}$ be a countable dense subset of $C^{\infty}_{c}({\mathbb{R}}^{d})$. For $m\in{\mathbb{N}}$, we recursively define a countable set ${\mathscr{G}}_{m}$ as follows:

[TABLE]

Clearly,

[TABLE]

Lemma 3.5**.**

For $T>0$, $f\in{\mathbb{L}}^{\infty}_{T}$ and $n\in{\mathbb{N}}$, if we define

[TABLE]

then $u^{n}_{T,f}\in\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T}$ uniquely solves PDE (3.11) with $b=b_{n}$. Moreover, there is a Lebesgue null set ${\mathcal{N}}\subset{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$ such that for all $(s,x)\in{\mathcal{N}}^{c}$, $f\in{\mathscr{A}}$ and $s\leqslant T\in{\mathbb{Q}}$,

[TABLE]

Proof.

For $m\in{\mathbb{N}}$, let $f_{m}(t,x):=f(t,\cdot)*\rho_{m}(x)$ and

[TABLE]

It is well known that $u^{n,m}_{T,f}$ solves PDE (3.11) with $b=b_{n}$ and $f=f_{m}$. By Theorem 2.4, for Lebesgue almost all $(s,x)$, we have

[TABLE]

where $u^{n}_{T,f}\in\widetilde{\mathscr{V}}_{T}\cap{\mathbb{L}}^{\infty}_{T}$ is the unique weak solution of of PDE (3.11) with $b=b_{n}$. On the other hand, by Krylov’s estimate (3.3), for each $s\leqslant T$ and $x\in{\mathbb{R}}^{d}$, we have

[TABLE]

which together with (3.14) gives (3.12). For fixed $T\in{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$, by Theorem 2.4 again, there is a Lebesgue null set ${\mathcal{N}}_{T,f}\subset{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$ such that (3.13) holds for all $(s,x)\in{\mathcal{N}}^{c}_{T,f}$. Finally, we just need to take

[TABLE]

The proof is complete. ∎

Lemma 3.6**.**

Let ${\mathcal{N}}$ be as in Lemma 3.5. For fixed $(s,x)\in{\mathcal{N}}^{c}$ and any two accumulation points ${\mathbb{P}}^{(1)}_{s,x}$ and ${\mathbb{P}}^{(2)}_{s,x}$ of $({\mathbb{P}}^{n}_{s,x})_{n\in{\mathbb{N}}}$,

[TABLE]

Proof.

Fix $(s,x)\in{\mathcal{N}}^{c}$. For $s\leqslant T\in{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$, by (3.13) and taking weak limits for (3.12) along different subsequences for ${\mathbb{P}}^{(i)}_{s,x}$, $i=1,2$, one finds that

[TABLE]

which implies that for all $s\leqslant T\in{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$,

[TABLE]

In particular, for all $T\geqslant s$ and $f\in{\mathscr{G}}_{0}$,

[TABLE]

Claim: Let $g_{m}(t,x)\in{\mathbb{L}}^{\infty}$ be uniformly bounded and converge to $g(t,x)$ for Lebesgue almost all $(t,x)$. For any $(s,x)\in{\mathcal{N}}^{c}$ and $T>s$, it holds that for each $x\in{\mathbb{R}}^{d}$,

[TABLE]

Proof of Claim: For $R>0$, define

[TABLE]

By (3.3) with $(\alpha,p,q)=(0,d,4)$ and the dominated convergence theorem, we have

[TABLE]

On the other hand, by SDE (3.2) and (3.3) again, we also have

[TABLE]

where $C$ is independent of $n$. Hence,

[TABLE]

which together with (3.17) yields the claim.

Next let $s\leqslant T_{1}<T_{2}$ be two rational numbers and $f_{1},f_{2}\in{\mathscr{G}}_{0}$. Let $({\mathbb{P}}^{n_{k}}_{s,x})_{k\in{\mathbb{N}}}$ be a subsequence so that $({\mathbb{P}}^{n_{k}}_{s,x})_{k\in{\mathbb{N}}}$ weakly converges to ${\mathbb{P}}^{(1)}_{s,x}$. By the Markov property, we have for $f_{1},f_{2}\in{\mathscr{G}}_{0}$

[TABLE]

where the last step is due to (3.13) and the above Claim. Notice that

[TABLE]

Hence,

[TABLE]

Since the right hand side does not depend on the choice of the subsequence $n_{k}$, we finally obtain that for any rational numbers $s\leqslant T_{1}<T_{2}$ and $f_{1},f_{2}\in{\mathscr{G}}_{0}$,

[TABLE]

From this, as above we derive that for all $f_{1},f_{2}\in{\mathscr{G}}_{0}$ and $T_{2}>T_{1}\geqslant s$,

[TABLE]

Similarly, we can prove that for any $T_{m}>\cdots>T_{1}\geqslant s$ and $f_{1},\cdots,f_{m}\in{\mathscr{G}}_{0}$,

[TABLE]

Thus we obtain (3.15). ∎

3.3. Almost surely Markov property

Let ${\mathcal{N}}$ be as in Lemma 3.5. We fix $(s,x)\in{\mathcal{N}}^{c}$ so that

[TABLE]

Recalling that ${\mathscr{G}}_{0}$ is a countable dense subset of $C^{\infty}_{c}({\mathbb{R}}^{d})$, to show (1.12), it suffices to prove the following claim:

Claim 1: For fixed $t_{1}\in(s,\infty)\cap{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$, there is a Lebesgue-null set $I^{t_{1},f}_{s,x}\subset(s,t_{1})$ so that for all $t_{0}\in(s,t_{1})\setminus I^{t_{1},f}_{s,x}$,

[TABLE]

Indeed, if this is proven, then we can take

[TABLE]

Thus for any $t_{0}\in(s,\infty)\setminus I_{s,x}$, and all $t_{0}<t_{1}\in{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$,

[TABLE]

By a standard approximation argument, the above equality also holds for all $t_{1}>t_{0}$ and $f\in C_{c}({\mathbb{R}}^{d})$.

Furthermore, to prove Claim 1, it suffices to prove the following claim:

Claim 2: Let $t_{1}\in(s,\infty)\cap{\mathbb{Q}}$ and $f\in{\mathscr{G}}_{0}$. For fixed $m\in{\mathbb{N}}$, $s_{1},\cdots,s_{m}\in(s,t_{1})\cap{\mathbb{Q}}$ and $g_{1},\dots,g_{m}\in{\mathscr{G}}_{0}$, there exists a null set $I:=I^{s_{1},\cdots,s_{m}}_{g_{1},\cdots,g_{m}}\subset[s_{m},t_{1}]$ so that for all $t_{0}\in[s_{m},t_{1}]\setminus I$,

[TABLE]

Indeed, if this is proven, then we can define

[TABLE]

Thus for any $t_{0}\in(s,t_{1})\setminus I^{t_{1},f}_{s,x}$, (3.20) holds for all $m$ and $s_{1},\cdots,s_{m}\in(s,t_{0}]\cap{\mathbb{Q}}$, $g_{1},\cdots,g_{m}\in{\mathscr{G}}_{0}$. By a standard monotone class argument, we obtain (3.19) for $t_{0}\in(s,t_{1})\setminus I^{t_{1},f}_{s,x}$ from (3.20).

Proof of Claim 2: For simplicity of notations, we shall write

[TABLE]

By the Lebesgue differential theorem, we only need to prove that for any $t_{0}\in[s_{m},t_{1}]$,

[TABLE]

Clearly, by the Markov property of $({\mathbb{P}}^{n}_{s,x})_{(s,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}}$, we have

[TABLE]

By (3.18) we have

[TABLE]

Define

[TABLE]

Since by (3.18), $H_{n}(r,y)\to H(r,y)$ for Lebesgue almost all $r,y$, by (3.16), we have

[TABLE]

On the other hand, for fixed $n$, since $y\mapsto H_{n}(r,y)$ is continuous, we also have

[TABLE]

Therefore,

[TABLE]

which together with (3.22) and (3.23) gives (3.21). The proof is complete.

Proof of Theorem 1.1.

By Lemma 3.4, we have the existence of ${\mathbb{P}}_{s,x}\in{\mathscr{M}}^{b}_{s,x}$, which satisfies the Krylov estimate (1.10). By Lemma 3.6, we have (i). By Subsection 3.3 we have (ii). By Lemma 3.2 and (i), we have (iii). ∎

4. Appendix: Properties of space $\widetilde{\mathbb{H}}^{\alpha,p}_{q}$

In this appendix we prove some important properties about the space $\widetilde{\mathbb{H}}^{\alpha,p}_{q}$. We need the following lemma, which can be found in [20, p.205] and [28, Lemma 2.2].

Lemma 4.1**.**

1. (i)

For any $\alpha\in{\mathbb{R}}$ and $p\in(1,\infty)$, there is a $C=C(d,\alpha,p)>0$ such that

[TABLE] 2. (ii)

Let $p\in(1,\infty)$ and $\alpha\in(0,1]$ be fixed. For any $p_{1}\in[p,\infty)$ and $p_{2}\in[\frac{p_{1}}{p_{1}-1},\infty)$ with $\frac{1}{p}\leqslant\frac{1}{p_{1}}+\frac{1}{p_{2}}<\frac{1}{p}+\frac{\alpha}{d}$, there is a constant $C>0$ such that for all $f\in H^{-\alpha,p_{1}}$ and $g\in H^{\alpha,p_{2}}$,

[TABLE]

The following proposition tells us that the localized norm $|\mspace{-1.0mu}|\mspace{-1.0mu}|\cdot|\mspace{-1.0mu}|\mspace{-1.0mu}|_{\alpha,p;q}$ enjoys the almost same properties as the global norm $\|\cdot\|_{\alpha,p;q}$.

Proposition 4.1**.**

Let $p,q\in(1,\infty)$ and $\alpha\in{\mathbb{R}}$.

1. (i)

For $r\not=r^{\prime}>0$, there is a constant $C=C(d,\alpha,r,r^{\prime})\geqslant 1$ such that for all $f\in\widetilde{\mathbb{H}}^{\alpha,p}_{q}$,

[TABLE]

In other words, the definition of $\widetilde{\mathbb{H}}^{\alpha,p}_{q}$ does not depend on the choice of $r$. 2. (ii)

Let $(\rho_{n})_{n\in{\mathbb{N}}}$ be a family of mollifiers in ${\mathbb{R}}^{d}$ and $f_{n}(t,x):=f(t,\cdot)*\rho_{n}(x)$. For any $f\in\widetilde{\mathbb{H}}^{\alpha,p}_{q}$, it holds that $f_{n}\in L^{q}_{loc}({\mathbb{R}};C^{\infty}_{b}({\mathbb{R}}^{d}))$ and for some $C=C(d,\alpha,p,q)>0$,

[TABLE]

and for any $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d+1})$,

[TABLE] 3. (iii)

For any $k\in{\mathbb{N}}$, there is a constant $C=C(d,k,\alpha,p,q)\geqslant 1$ such that for all $f\in\widetilde{\mathbb{H}}^{\alpha+k,p}_{q}$,

[TABLE] 4. (iv)

Let $p\in(1,\infty)$ and $\alpha\in(0,1]$, $q\in[1,\infty]$. For any $p_{1}\in[p,\infty)$ and $p_{2}\in[\frac{p_{1}}{p_{1}-1},\infty)$ with $\frac{1}{p}\leqslant\frac{1}{p_{1}}+\frac{1}{p_{2}}<\frac{1}{p}+\frac{\alpha}{d}$, and $\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q}$, there is a constant $C>0$ such that

[TABLE] 5. (v)

${\mathbb{L}}^{p}_{q}+{\mathbb{L}}^{\infty}_{\infty}\subsetneq\widetilde{\mathbb{L}}^{p}_{q}$.

Proof.

(i) Let $r>r^{\prime}$. We first prove the right hand side inequality in (4.3). Fix $(s,z)\in{\mathbb{R}}^{d+1}$. Notice that the support of $\chi^{s,z}_{r}$ is contained in $Q^{s,z}_{2r}$. Clearly, $Q^{s,z}_{2r}$ can be covered by finitely many $Q_{r^{\prime}}^{s_{i},z_{i}},i=1,\cdots,N$, where $N=N(d,r,r^{\prime})$ does not depend on $s,z$. Let $(\varphi_{i})_{i=1}^{N}$ be the partition of unity associated with $\{Q_{r^{\prime}}^{s_{i},z_{i}},i=1,\cdots,N\}$ so that

[TABLE]

Thus, due to $\chi^{s_{i},z_{i}}_{r^{\prime}}|_{Q_{r^{\prime}}^{s_{i},z_{i}}}=1$, by (4.1) we have

[TABLE]

where $C=C(N,\alpha,d,r,r^{\prime})>0$, which yields the right hand side inequality in (4.3). On the other hand, since $\chi^{s,z}_{r^{\prime}}=\chi^{s,z}_{2r}\chi^{s,z}_{r^{\prime}}$, by what we have proved, we have

[TABLE]

where $C$ does not depend on $s,z$, which gives the left hand side inequality.

(ii) By the definition of convolutions, it is easy to see that

[TABLE]

Hence,

[TABLE]

which gives (4.4). As for (4.5), it follows by a finitely covering technique.

(iii) We only prove it for $k=1$. By definition and $\chi^{s,z}_{2}\nabla\chi^{s,z}_{1}=\nabla\chi^{s,z}_{1}$ we have

[TABLE]

which in turn gives the right hand side estimate by (i). The left hand side inequality is similar.

(iv) By (4.2) and $\chi^{s,z}_{2}\chi^{s,z}_{1}=\chi^{s,z}_{1}$, we have

[TABLE]

The desired estimate follows by (i).

(v) Let ${\mathbb{Z}}^{d}$ be the set of all lattice points. Define

[TABLE]

It is easy to see that $f\in\widetilde{\mathbb{L}}^{p}_{q}$, but $f\notin{\mathbb{L}}^{p}_{q}+{\mathbb{L}}^{\infty}_{\infty}$. ∎