Local estimates on two linear parabolic equations with singular coefficients

TL;DR

This paper extends regularity and boundedness results for solutions of linear parabolic equations with singular coefficients, including the linearized Navier-Stokes system, revealing new gradient estimates and solution properties.

Contribution

It provides novel boundedness and regularity results for solutions with highly singular and mildly singular data, extending classical theories to more complex equations like the linearized Navier-Stokes system.

Findings

Boundedness of solutions with supercritical singular data

Regularity results for solutions with Kato class data

New gradient estimate for linearized Navier-Stokes equations

Abstract

We treat the heat equation with singular drift terms and its generalization: the linearized Navier-Stokes system. In the first case, we obtain boundedness of weak solutions for highly singular, "supercritical" data. In the second case, we obtain regularity result for weak solutions with mildly singular data ( those in the Kato class). This not only extends some of the classical regularity theory in [AS], [CrZ] and others from the case of elliptic and heat equations to that of linearized Navier-Stokes equations but also proves an unexpected gradient estimate, which extends the recent interesting boundedness result [O]. In the addendum in May 2019, a missing term in Theorem 1.7 and Lemma 3.3 is added. This is due to the use of a formula in a cited reference, which omitted a term. The main conclusion that local solutions of certain linearized Navier-Stokes equation have bounded spatial gradient is intact. This includes bounded local Leray-Hopf solutions to the Navier Stokes equation without condition on the pressure.

Equations

$$\Delta u+b\nabla u=0$$

$${\int }_{{\mathbf{R}}^n}|b{|}^2(x){\varphi }^2(x)dx\le ϵ{\int }_{{\mathbf{R}}^n}|\nabla \varphi {|}^2(x)dx,\varphi \in C_0^{\infty }({\mathbf{R}}^n).$$

$$\int {\int }_{{\mathbf{R}}^n}|b{|}^m{\varphi }^{2}dxdt\le k\int {\int }_{{\mathbf{R}}^n}|\nabla \varphi {|}^{2}dxdt$$

$${\int }_0^T{\int }_D(u{\partial }_t\varphi -\nabla u\nabla \varphi )dxdt-{\int }_0^T{\int }_{D}b\nabla u\varphi dxdt=-{\int }_D{u}_0(x)\varphi (x,0)dx.$$

$$\{\begin{array}{l}\Delta u-b\nabla u-u_t=0,\text{in}\text{D}\times (0,\infty ),\\ u(x,t)=0,(x,t)\in \partial D\times (0,\infty )\\ u(x,0)=u_0(x).\end{array}$$

$$\frac{d}{dt}\int_{D}G(x,t;y,t)^{2}dy=-2\int_{D}|\nabla G(x,t;y,t)|^{2}dy\leq-c\frac{\big{(}\int_{D}G(x,t;y,t)^{2}dy\big{)}^{1+(2/n)}}{\big{(}\int_{D}G(x,t;y,0)dy\big{)}^{4/n}}.$$

$$G(x,t;y,0)\le c/t^{n/2}.$$

$$u(x,t)={\int }_{D}G(x,t;y)u_0(y)dy$$

$${\int }_0^T{\int }_{{\mathbf{R}}^3}$$

$$\displaystyle=\int^{T}_{0}\bigg{(}\int_{{\bf R}^{3}}|b|^{2}dx\bigg{)}^{2/3}\ \bigg{(}\int_{{\bf R}^{3}}\phi^{6}dx\bigg{)}^{1/3}dt$$

$$\displaystyle\leq\sup_{t\in[0,T]}\bigg{(}\int_{{\bf R}^{3}}|b|^{2}(x,t)dx\bigg{)}^{2/3}\ \int^{T}_{0}\bigg{(}\int_{{\bf R}^{3}}\phi^{6}dx\bigg{)}^{1/3}dt$$

$$\displaystyle\leq C\sup_{t\in[0,T]}\bigg{(}\int_{{\bf R}^{3}}|b|^{2}(x,t)dx\bigg{)}^{2/3}\ \int^{T}_{0}\int_{{\bf R}^{3}}|\nabla\phi|^{2}dxdt.$$

$${\int }_{{\mathbf{R}}^3}{u}^2(x,t)dx$$

$$u(x,t)={\int }_{{\mathbf{R}}^3}{G}_0(x,t;y,t_0)u(y,t_0)dy-{\int }_{t_0}^t{\int }_{{\mathbf{R}}^3}{G}_0(x,t;y,s)b\nabla u(y,s)dyds.$$

$$u(x,t)={\int }_{{\mathbf{R}}^3}{G}_0(x,t;y,t_0)u(y,t_0)dy+{\int }_{t_0}^t{\int }_{{\mathbf{R}}^3}{\nabla }_y{G}_0(x,t;y,s)bu(y,s)dyds.$$

$${\nabla }_{x}u(x,t)$$

$$\equiv I_1(x,t)+I_2(x,t).$$

$${\nabla }_x{\nabla }_y{G}_0(x,t;y,t_0)=-{\nabla }_x{\nabla }_x{G}_0(x,t;y,t_0)$$

$$|I_2(\cdot ,t)|\in L^q({\mathbf{R}}^3),q>3.$$

$${\int }_{t_1}^{t_2}{\int }_{{\mathbf{R}}^n}<u,{\partial }_t\varphi +\Delta \varphi >dxdt-{\int }_{t_1}^{t_2}{\int }_{{\mathbf{R}}^n}<b\nabla u,\varphi >dxdt=-{\int }_{{\mathbf{R}}^n}<u(x,t),\varphi (x,t)>{|}_{t_1}^{t_2}dx.$$

$$|\nabla u(x,t)|$$

$${\int }_{{\mathbf{R}}^n}G(x,t;y,s)dx\le 1.$$

$$\underset{Q_r}{\sup }{u}^2\le C(r,b)\frac{1}{|Q_{{\sigma }_{0}r}|}{\int }_{Q_{{\sigma }_{0}r}}{u}^{2}dyds.$$

Full text

1. 1 Introduction
2. 2 Proof of Theorem 1.1
3. 3 Proof of Theorem 1.2
4. 4 A regularity condition for Navier-Stokes equations not involving absolute values
5. 5 Addendum May 2019

1. Introduction

The goal of the paper is to prove local boundedness and other regularity of weak solutions to the next two parabolic equations.

[TABLE]

[TABLE]

Here $\Delta$ is the standard Laplacian and $b=b(x,t)$ is a given $L^{2}_{loc}$ singular vector field to be specified later. $\Omega$ is a domain.

There has been a mature theory of existence and regularity for equation (1.1) (see [LSU], [Lieb] e.g.). For instance when $b=b(x)$ and $|b|\in L^{p}_{loc}({\bf R}^{n})$, $p>n$, weak solutions to (1.1) are locally bounded and Hölder continuous. This condition is sharp in general. Here is an example (see [HL] p108). The function $u=\ln\ln|x|^{-1}-\ln\ln R^{-1}$ is an unbounded weak solution of

[TABLE]

in the ball $B(0,R)$ in ${\bf R}^{2}$, $R<1$. Here $b=\nabla u=-\frac{\nabla|x|}{|x|\ln|x|^{-1}}$ and hence $b\in L^{2}_{loc}$ with $n=2$.

The first goal of the paper is to show that the simple condition $divb\leq 0$ will ensure that weak solutions of (1.1) are locally bounded when the data $b$ is almost twice as singular as before. This will be made precise in Theorem 1.1 and Remark 1.1 below. Thus one has achieved a leap in boundedness condition rather than a marginal improvement.

Clearly a strong impetus still exists for the study of parabolic equations with very singular coefficients. In the study of nonlinear equations with gradient structure such as the Navier-Stokes equations and harmonic maps, highly singular functions occur naturally. So, it is very important investigate a possible gain of regularity in the presence of singular drift term $b$. This line of research has been followed in the papers [St], [KS], [Os], [CrZ], [ChZ], [CS] and [Se]. Under the condition $|b|\in L^{n}({\bf R}^{n})$, Stampacchia [St] proved that bounded solutions of $\Delta u+b\nabla u=0$ are Hölder continuous. In the paper [CrZ], Cranston and Zhao proved that solutions to this equation are continuous when $b$ is in a suitable Kato class i.e. $\lim_{r\to 0}\sup_{x}\int_{|x-y|\leq r}\frac{|b(y)|}{|x-y|^{n-1}}dy=0$. In the paper [KS] Kovalenko and Semenov proved the Hölder continuity of solutions to (1.1), when $|b|^{2}$ is independent of time and is sufficiently small in the form sense, i.e., for a sufficiently small $\epsilon>0$,

[TABLE]

It is a well known fact that form boundedness condition provides a more general class of singular functions than corresponding $L^{p}$ class, Morrey-Campanato class and Kato class functions. This result was generalized in [Se] to equations with leading term in divergence form. In [Os], Osada proved, among other things, that the fundamental solution of (1.1) has global Gaussian upper and lower bound when $\bf b$ is the derivative of bounded functions (in distribution sense) and $divb=0$. Recently in the paper [LZ], Hölder continuity of solutions to (1.1) was established when $b=b(x)$, $|b|^{2}$ is form bounded and $div{b}=0$. Most recently, in [Z2], we considered (1.1) with time-dependent functions $b=b(x,t)$. It was proven that weak solutions to (1.1) are locally bounded provided that $divb=0$ and for a fixed $m>1$, $|b|^{m}$ is form bounded. That is for any $\phi\in C^{\infty}({\bf R}^{n}\times(0,\infty))$ with compact support in the spatial direction,

[TABLE]

where $k$ is independent of $\phi$. Note the key improvement over previous result is that the power on $b$ drops from $2$ to any number greater than $1$.

It is interesting to note that this class of data $b$ contains the velocity function in the $3$ dimensional Navier-Stokes equations. As a result we gave a different proof of the local boundedness of velocity in $2$ dimensional case. Moreover assuming a local bound in the pressure, we prove boundedness of velocity in $3$ the dimensional case.

The first goal of the paper is to treat the end point case of the above condition i.e. $m=1$. We will prove that weak solutions to (1.1) are locally bounded provided that $|b|[\ln(1+|b|)]^{2}$ is form bounded and $divb\leq 0$.

Many authors have also studied the regularity property of the related heat equation $\Delta u+Vu-u_{t}=0$. Here $V$ is s singular potential. We refer the reader to the papers by Aizenman and Simon [AS], Simon [S] and the reference therein. The function $V$ is allowed in the Kato class which is a little more singular than the corresponding $L^{p}$ class. It remains a challenging problem is to push this theory to broader class of functions.

In this paper we use the following definition of weak solutions.

Definition 1.1* Let $D\subseteq{\bf R}^{n}$ be a domain and $T\in(0,\infty]$. A function $u$ such that $u,|\nabla u|\in L^{2}_{loc}(D\times[0,T])$ is a weak solution to (1.1) if: for any $\phi\in C^{\infty}_{0}(D\times(-T,T))$, there holds*

[TABLE]

Theorem 1.1**.**

Suppose $divb\leq 0$ in the weak sense, $b\in L^{2}_{loc}$, and that $|b|[\ln(1+|b|)]^{2}$ is form bounded i.e. there exists $k>0$ such that for any $\phi\in C^{\infty}({\bf R}^{n}\times(0,\infty))$ with compact support in the spatial direction,

[TABLE]

Then weak solutions to equation (1.1) are locally bounded.

Remark 1.1. In the special case that $b$ is independent of time and $b\in L^{p}({\bf R}^{n})$ with $p>n/2$, then it is easy to check that (1.3) is satisfied. Recall that the standard theory essentially only allows functions in $L^{p}$ with $p>n$. The strength of the theorem comes from the fact that weak solutions are locally bounded in any domain regardless of its value on the parabolic boundary. If the domain is ${\bf R}^{n}\times(0,\infty)$ or if the initial Dirichlet boundary condition is imposed, then using a Nash type estimate, one can show that solutions are locally bounded when $t>0$ as long as the fundamental solution is well defined. In this case one can choose $b$ to be as singular as any $L^{2}$ functions. Actually the presence of $b$ is totally irrelevant except for the purpose of making the integrals in the definition of a weak solution finite. Here is a sketch of the proof. Let u solves

[TABLE]

Let $G(x,t;y,t)$ be the fundamental solution with initial Dirichlet boundary condition. If $divb=0$, differentiating in time shows that $\int_{D}G(x,t;y,0)dy\leq 1$. By Nash inequality, one has

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

is bounded as soon as $t>0$ and $u_{0}$ is in $L^{1}(D)$.

However this does not imply local boundedness of weak solutions unconditionally. It would be interesting to establish existence and more regularity result for (1.1) with the singular data in Theorem 1.1.

We should mention that in the time independent elliptic case, there was already strong indication that standard regularity theory can be improved in the presence of divergence free data. In the important papers [FR1-2], local boundedness of Green’s function (away from the singularity) of the operator $-\Delta-b\nabla$ with Dirichlet boundary conditions was proved, under the conditions: $n=5$, $|\nabla^{2}b|\in L^{4/3}$, $b=0$ on the boundary and $divb=0$ ( c.f. Lemma 1.48 and Lemma1.11 [FR1]). The upshot of the result is that the bounds on the Green’s function is independent of the norm $|\nabla^{2}b|\in L^{4/3}$. The proof uses essentially the fact that the Green’s function vanishes on the boundary. So the drift term is integrated out. In contrast we do not have the benefit of a zero boundary.

In the three dimension case, we derive a further regularity result:

Corollary 1**.**

Assume $|b|\in L^{\infty}([0,T],L^{2}({\bf R}^{3}))\cap L^{q}({\bf R}^{3}\times[0,T])$ with $q>3$ and $divb=0$. Suppose $u$ is a weak solution of (1.1) in ${\bf R}^{3}\times[0,T]$ with $\int_{{\bf R}^{3}}u^{2}(x,0)dx<\infty$. Then $u$ is locally bounded and for almost every $t$, $u(\cdot,t)$ are Hölder continuous.

Proof.

According to Corollary 1 in [Z2], $b$ satisfies the conditions of Theorem 1.1. For completeness we provide the proof here.

Let us take $m=4/3$ and $p=2/m=3/2$. Then, by Hölder’s inequality,

[TABLE]

The last step is by Sobolev imbedding. This shows that condition (1.3) holds.

Hence Theorem 1.1 implies that $u$ is locally bounded. Notice the fact that

[TABLE]

is non-increasing in time since $b$ is divergence free. By this and Theorem 1.1, we know that $u\in L^{\infty}({\bf R}^{3}\times[t_{0},T])$ for any $t_{0}>0$.

Denote by $G_{0}$ the Gaussian heat kernel of the heat equation. Then, for $t>t_{0}$,

[TABLE]

Since $b$ is divergence free, we have

[TABLE]

Therefore, in the weak sense,

[TABLE]

It is well known that

[TABLE]

is a parabolic Calderon-Zygmond kernel (see [Lie] e.g.). Hence by our assumption on $b$ and the fact that $u$ is bounded in ${\bf R}^{3}\times[t_{0},T]$, the second term $I_{2}$ in the last integral is in $L^{q}({\bf R}^{3}\times[t_{0},T])$, $q>3$. It follows that

[TABLE]

for a.e. $t$. Sobolev imbedding theorem then shows that $u(\cdot,t)$ is Hölder continuous for a.e. $t$. ∎

Next we turn to equation (1.2), which is the first step in tackling the full Navier-Stokes equations. When $b=0$, (1.2) is just the Stokes equations which has been studied for long time. Our focus is on how to allow $b$ as singular as possible while retaining the boundedness of weak solutions. As far as equation (1.2) is concerned, our result does not improve the standard theory as dramatically as for equation (1.1). We have to restrict the data $b$ in a suitable Kato class for (1.2). Nevertheless, Theorem 1.2 still generalizes the key part of the important work [AS], [CrZ] on the elliptic equations to the case of linearized Navier-Stokes system. Moreover, we even obtain gradient estimates for solutions of (1.2) while only continuity was expected.

As pointed out in the papers [AS] and [Si], Kato class functions are quite natural objects in studying elliptic and parabolic equations with singular lower order terms. Roughly speaking, a function is in a Kato class with respect to an equation if its convolutions with certain kernel functions are small in some sense. The kernel function usually is related to the fundamental solution of the principal term of the equation. For instance, for the equation $\Delta u(x)+V(x)u(x)=0$ in ${\bf R}^{n}$, $n\geq 3$, the function $V$ is in Kato class if

[TABLE]

In [AS], it is proven that weak solutions to $\Delta u+Vu=0$ are continuous and satisfy a Harnack inequality when $V$ is in the above Kato class. Numerous papers have been written on this subject in the last thirty years, mainly in the context of elliptic and heat equations.

In the context of Navier-Stokes equations, the corresponding time dependent Kato class was defined recently in [Z3], which mirrors those for the heat equation [Z]. Normally, with data in the Kato class, weak solutions of elliptic equations are just continuous as proven in [AS], [CrZ]. It was proved in [Z3] that weak solutions to (1.2) are bounded when $b$ is in the Kato class. Here we prove that the spatial gradient of solutions to (1.2) are bounded provided that $b$ is in the Kato class locally. Let us mention that one can use the idea of Kato class to recover some (but not all) the decay estimate in the interesting papers [Scho1-2] and to prove some pointwise decay estimate (see [Z3]).

In order to make our statement precise, we introduce a number of notations. Throughout the paper, we write

[TABLE]

We write $Q_{r}(x,t)=B(x,r)\times[t-r^{2},t]$.

Definition 1.2*. A vector valued function $b=b(x,t)\in L^{1}_{loc}({\bf R}^{n+1})$ is in class $K_{1}$ if it satisfies the following condition:*

[TABLE]

For clarity of presentation, given $t>l$, we introduce the quantity

[TABLE]

By the example in Remark 1.2 in [Z3], we see that the function class $K_{1}$ permits solutions which are very singular. In case the spatial dimension is $3$, a function in this class can have an apparent singularity of certain type that is not $L^{p}_{loc}$ for any $p>1$ and of dimension $1$. One can also construct time dependent functions in $K_{1}$ with quite singular behaviors. The class $K_{1}$ also contains the space $L^{p,q}$ with $n/p+2/q<1$, which sometimes is referred to as the Prodi-Serrin class. For the nonlinear Navier-Stokes equation, if a weak solution is known to be in this class, then it is actually smooth. As for the linearized equation (1.2), following the argument in [Ser], it is clear that weak solutions are bounded if $b$ is in the above $L^{p,q}$ class. Now we are able to prove that the spatial gradient of weak solutions are bounded automatically, without resorting to the nonlinear structure. Using Hölder’s inequality, one can see that the class $K_{1}$ also contains the Morrey type space introduced in [O] by O’ Leary, where boundedness of weak solutions in Morrey space are proven.

One can also define a slightly bigger Kato class by requiring the limit in (1.6) to be a small positive number rather than [math]. We will not seek such generality this time. The appearance of two kernel functions is due to the asymmetry of the equation in time direction.

Let $D$ for a domain in ${\bf R}^{3}$ and $T>0$. Following standard practice, we will use this definition for solutions of (1.2) throughout the paper.

Definition 1.3.* A divergence free vector field $u\in L^{\infty}(0,T;L^{2}(D))\cap L^{2}(0,T;W^{1,2}(D))$ is called a (weak) solution of (1.2) if:*

for any vector valued $\phi\in C^{\infty}(D\times[0,T])$ with $div\phi=0$ and $\phi=0$ on $\partial D\times[0,T]$, $u$ satisfies

[TABLE]

Next we state the theorem on equation (1.2), the linearized Navier-Stokes equations.

Theorem 1.2**.**

Let $u$ be a solution of (1.2) in a domain $\Omega\subset{\bf R}^{3}\times{\bf R}$. Suppose $Q_{4r}(x,t)\subset\Omega$, $divb=0$ and that $b|_{Q_{2r}(x,t)}$ is in class $K_{1}$ and $b\in L^{2}_{loc}$. Then both $u$ and $|\nabla u|$ are bounded functions in $Q_{2r}(x,t)$.

Moreover, for some positive constants $C=C(b)$ and $r_{0}$, depending on the size of the Kato norm of $b$, there hold, when $0<r<r_{0}$,

[TABLE]

Here $\overline{u}_{Q_{2r}}$ is the average of $u$ in $Q_{2r}(x,t)$.

If in addition that $u(\cdot,t)\in W^{1,2}_{0}(B(x,2r))$ for a.e. $t$, then

[TABLE]

Remark 1.2. One may think that the last term on the right hand side of the gradient estimate (1.7) is too singular to be natural, especially when $r\to 0$. However, both (1.7) and equation (1.2) are actually scaling invariant under the scaling: $u_{\lambda}=u(\lambda x)$, $b_{\lambda}=\lambda b(\lambda x)$. So (1.7) is in the right setting. Moreover, the gradient estimate in (1.7) immediately simplifies to (1.8) when $u$ vanishes on the lateral boundary or when $u$ enjoys a suitable extension property in $W^{1,2}$ space.

Remark 1.3. The reader may wonder whether any extra regularity in the time direction is possible. The answer is no as indicated in the example in [Ser], restated in [O]. Let $\phi$ be a harmonic function in ${\bf R}^{3}$ and $a=a(t)$ be an integrable function. Then $u=a\nabla\phi$ is a weak solution of the Navier-Stokes equation. Obviously in the time direction $u$ is no more regular than $a$.

Theorem 1.1 and 1.2 will be proven in section 3 and 4 respectively. The idea for the proof of Theorem 1.2 is to combine a recent localization argument in [O] with a refined iteration. Using the idea in the proof of Theorem 1.1, in Section 4 we introduce a sufficient condition on the velocity that implies boundedness of weak solutions of 3 dimensional Navier-Stokes equations. The main improvement is that no absolute value of the velocity is involved.

2. Proof of Theorem 1.1

Since the drift term $b$ in (1.1) can be much more singular than those allowed by the standard theory, the existence and uniqueness of weak solutions of (1.1) can not be taken for granted. In order to proceed first we need some approximation results whose proof can be found in [Z2], with only small modifications.

Proposition 2.1**.**

Suppose $u$ is a weak solution of equation (1.1) in the cube $Q=D\times[0,T]$, where $b$ satisfies the condition in Theorem 1.1, part (a). Here $D$ is a domain in ${\bf R}^{n}$. Then $u$ is the $L^{1}_{loc}$ limit of functions $\{u_{k}\}$. Here $\{u_{k}\}$ is a weak solution of (1.1) in which $b$ is replaced by smooth $b_{k}$ such that $divb_{k}\leq 0$ and $b_{k}\to b$ strongly in $L^{2}(Q)$, $k\to\infty$.

Proof.

The proof is almost identical to that of Proposition 2.4 in [Z2]. The only difference is that we are assuming $divb\leq 0$ instead of $divb=0$. Let $G=G(x,t;y,s)$ be the fundamental solution of (1.1) with $divb\leq 0$ and $b$ smooth. Then it is easy to show by differentiation that

[TABLE]

The rest of the proof is identical to that of Proposition 2.4 [Z2]. ∎

Now we are ready to give a

Proof of Theorem 1.1.

Step 1. $L^{2}$ gradient estimate.

By the above approximation result, we can and do assume that the vector $b$ is smooth and we will differentiate freely as we wish.

In this section, we actually will prove the following local ”mean value” property.

Let $u$ be a nonnegative solution of (1.1) in the parabolic cube $Q_{\sigma_{0}r}=B(x,\sigma_{0}r)\times[t-(\sigma_{0}r)^{2},t]$. Here $x\in{\bf R}^{n}$, $r>0$, $t>0$ and $\sigma_{0}$ is a suitable number greater than $1$. Suppose $b$ satisfies Condition A in $Q_{\sigma_{0}r}$. Then there exist $C=C(r,b)>0$ such that

[TABLE]

Pick a solution $u$ of (1.1) in the parabolic cube $Q_{\sigma r}=B(x,\sigma r)\times[t-(\sigma r)^{2},t]$, where $x\in{\bf R}^{n}$, $\sigma>1$, $r>0$ and $t>0$. By direct computation, for any rational number $p\geq 1$, which can be written as quotient of two integers with the denominator being odd, one has

[TABLE]

Here the condition on $p$ is to ensure that $u^{p}$ makes sense when $u$ changes sign. Actually $u^{2}$ is a sub-solution to (1.1). Hence one can also assume that $u$ is a non-negative sub-solution to (1.1) by working with $u^{2}$.

Choose $\psi=\phi(y)\eta(s)$ to be a refined cut-off function satisfying

[TABLE]

[TABLE]

[TABLE]

By modifying the following function

[TABLE]

and scaling, it is easy to show that such a function $\phi$ exists. Here $k$ is a sufficiently large number.

Denoting $w=u^{p}$ and using $w\psi^{2}$ as a test function on (2.1), one obtains

[TABLE]

Using integration by parts, one deduces

[TABLE]

By direct calculation,

[TABLE]

Substituting this to (2.2), we obtain

[TABLE]

Next, notice that

[TABLE]

Combining this with (2.3), we see that

[TABLE]

The first term on the righthand side of (2.4) is already in good shape. So let us estimate the second term as follows.

[TABLE]

Here we just used the assumption that $divb\leq 0$.

The next paragraph contains a key argument of the paper.

Let $D>0$ be a number to be chosen later

[TABLE]

Using the property that $\psi=\phi\eta$ and

[TABLE]

we have

[TABLE]

By our assumptions on $b$,

[TABLE]

and the fact that $\phi|\ln\phi|^{3/2}$ is a bounded function, we deduce

[TABLE]

Now we choose $D$ so that $\frac{A}{(\sigma-1)r(\ln D)^{1/2}}=\frac{1}{2}$, i.e. $D=e^{(2A/[(\sigma-1)r])^{2}}$, then

[TABLE]

Here $c_{0}$ and $c_{1}$ are positive constants independent of $r$ and $\sigma$.

Combining (2.4) with (2.6), we reach

[TABLE]

step 2. $L^{2}-L^{\infty}$ bounds.

By modifying Moser’s iteration moderately, we deduce from (2.7) the following $L^{2}-L^{\infty}$ estimate.

[TABLE]

Indeed, by Hölder’s inequality,

[TABLE]

Using Sobolev inequality, one obtains

[TABLE]

The last inequality, together with (2.7) implies, for some $C_{1}>0$,

[TABLE]

where $\theta=1+(2/n)$ and $\sigma^{\prime}<\sigma$.

Take a number $\rho>1$ so that $\rho^{2}<\theta$. We set $\tau_{i}=\rho^{-i}$, $\sigma_{0}=1/(1-\rho^{-1})$, $\sigma_{i}=\sigma_{i-1}-\tau_{i}=\sigma_{0}-\Sigma^{i}_{1}\tau_{j}$, $p=\theta^{i}$, $i=1,2,...$. The above then yields, for some $c_{2},c_{3}>0$,

[TABLE]

After iterations the above implies, for some $c_{4}>0$,

[TABLE]

Observe that $\rho^{2}/\theta<1$. Letting $i\rightarrow\infty$ and observing that $\sigma_{i}\to 1$ as $i\to\infty$, we obtain

[TABLE]

This completes proof the theorem. ∎

3. Proof of Theorem 1.2

First we will need a short lemma concerning kernel function $K_{1}$ defined in (1.5). It was proved, among others things, in [Z3]. We give a proof for completeness.

Lemma 3.1**.**

The following inequality holds for all $x,y,z\in{\bf R}^{n}$ and $t>\tau>0$.

[TABLE]

Here, recalling from (1.6’),

[TABLE]

Proof.

Since

[TABLE]

we have, either

[TABLE]

or

[TABLE]

Suppose (3.3) holds then

[TABLE]

That is

[TABLE]

Suppose (3.4) holds but (3.3) fails, then

[TABLE]

This shows

[TABLE]

Substituting this to (3.1), we obtain

[TABLE]

That is

[TABLE]

Clearly the only remaining case to consider is when both (3.3) and (3.4) holds. However this case is already covered by (3.5). Thus (3.1) is proven. ∎

Next we state and prove a representation formula for solutions of (1.2) and their spatial gradient, following and extending the idea in [O]. The formula for solutions is contained in [O]. However, we will outline the proof since it is useful in the proof of the formula for the gradient, which is a new contribution of this paper.

Remark 3.1. Let us note that at this moment, the representation formula (3.6’) below for the gradient is understood as a comparison of two $L^{1}_{loc}$ functions in space-time. This is legal for two reasons. First we assumed that $\nabla u$ is a $L^{2}$ function a priori. Second, it is easy to check $K_{1}(\cdot,\cdot;y,s)$ is $L^{1}_{loc}$ and $b\nabla u$ is $L^{1}_{loc}$ by the assumption that $b$ is $L^{2}$. Therefore the last function on the righthand side of (3.6’) is a $L^{1}_{loc}$ function.

Lemma 3.2**.**

(mean value inequality)

(a). Let $u$ be a solution of (1.2) in the region $\Omega$. Suppose $Q_{2r}(x,t)\subset\Omega$. Then there exists a constant $\lambda$ such that

[TABLE]

(b). Under the same assumption as (a), there exists a constant $\lambda$ such that

[TABLE]

Proof.

of (a).

Let $E=E(x,t;y,s)$ be the fundamental solution (matrix) of the Stokes system in ${\bf R}^{3}\times(0,\infty)$ and $E_{k}$ be the $k$-th column of $E$. This function has been studied for a long time. All of its basic properties we are using below can be found in [So] and [FJR]. Fixing $(x,t)$, we construct a standard cut-off function $\eta$ such that $\eta(y,s)=1$ in $Q_{r/2}(x,t)$, $\eta(y,s)=0$ outside of $Q_{r}(x,t)$, $0\leq\eta\leq 1$ and $|\nabla\eta|^{2}+|\Delta\eta|+|\partial_{s}\eta|\leq c/r^{2}$.

Define a vector valued function

[TABLE]

It is clear that when $t>s$, $\Phi_{k}$ is a valid test function for equation (1.2) since $\Phi_{k}$ is smooth, compactly supported and divergence free. Using $\Phi_{k}$ as a test function on (1.2), by Definition (1.2), we obtain

[TABLE]

Here and later,we will suppress the superscript $(x,t)$ on $\Phi$, unless there is a confusion.

Since $E_{k}$ is divergence free $curl\ curlE_{k}=-\Delta E_{k}$. Thus

[TABLE]

Using the property of the fundamental matrix $E$ and the fact that $\overrightarrow{Z}$ is a lower order term, it is easy to see that

[TABLE]

where $u_{k}$ is the k-th component of $u$. Hence

[TABLE]

Here and later $ub\nabla\Phi_{k}=u\cdot\sum^{3}_{i=1}b_{i}\partial_{i}\Phi_{k}$.

Note that $\Delta E_{k}+\partial_{s}E_{k}=0$ when $s<t$. The above implies

[TABLE]

We are going to estimate $J_{1},J_{2},J_{3}$ separately. By well known estimates on $E$ (see [So] or [FJR] for example), for $(y,s)\in Q_{r}(x,t)-Q_{r/2}(x,t)$,

[TABLE]

[TABLE]

Using these and the property of $\eta$, we see that

[TABLE]

Next we give an estimate $J_{2}$, which follows verbatim from $[O]$ p626. Recall that

[TABLE]

Here $G$ is the fundamental solution of the heat equation and $\{e_{1},e_{2},e_{3}\}$ is the standard orthonormal basis of ${\bf R}^{3}$. Since $curl\nabla=0$, from (3.8), $\overrightarrow{Z}$ defined in (3.8) takes a very simple form

[TABLE]

Hence

[TABLE]

From here direct computation, using the estimates ([So], Chapter 2, Section 5),

[TABLE]

and the fact that $|x-y|\geq r/2$ here, it is easy to show that

[TABLE]

Finally direct computation using (3.12) shows that

[TABLE]

Substituting (3.10), (3.13) and (3.14) to (3.9), we finish the proof of part (a) of the lemma.

Proof of part (b).

Our next task is to prove the representation formula for $\nabla u$. In (3.9) the cut-off function apparently depends on $(x,t)$. In order to prove the gradient estimate, we need modify it a little. For $(w,l)\in Q_{r/4}(x,t)$, we take

[TABLE]

as a text function for (1.2). Since $\eta(w,l)=1$ in this case, following the computation before (3.9), we obtain

[TABLE]

Here $\overrightarrow{Z}=\overrightarrow{Z}(w,l;y,s)$ is defined as $\overrightarrow{Z}$ in (3.8) except that $E_{k}(x,t;y,s)$ is replaced by $E_{k}(w,l;y,s)$. i.e.

[TABLE]

We would like to differentiate (3.9) in the spatial variables. However, since $\nabla u$ is only known as a $L^{2}$ function, we have to consider the weak derivatives.

Let $\rho=\rho(w)$ be a smooth cut-off function supported in $B(x,r/4)$. Then (3.9’) implies, for $i=1,2,3$, and a.e. $l$,

[TABLE]

Here we just used integration by parts which is legitimate since we will show that $M_{1},M_{2}$ are bounded functions and $M_{3}$ is $L^{1}_{loc}$. Note also we should have integrated in the time direction as well since $\nabla u$ is only known to $L^{2}$ in space time. However, since all the estimates below are uniform for $l\in[t-(r/4)^{2},t]$, our estimates are valid.

Let us estimate $M_{1}$ first. Noting that $\partial_{w_{i}}E_{k}(w,l;y,s)=-\partial_{y_{i}}E_{k}(w,l;y,s)$, we deduce

[TABLE]

Using integration by parts, we obtain

[TABLE]

By standard properties of $E$ and the bounds on $\eta$ and its derivatives, we have

[TABLE]

Here we also utilize the fact that the arguments $(w,l)$ and $(y,s)$ have a parabolic distance of at least $r/4$

Next we need to find an upper bound for $M_{2}$. Recall from the formula before (3.11) that

[TABLE]

Using the vector identity $F\times(G\times H)=(F\cdot H)G-H(F\cdot G)$, we have

[TABLE]

Since $\partial_{w_{i}}G(w,l;y+z,s)=-\partial_{y_{i}}G(w,l;y+z,s)$, the above shows

[TABLE]

Hence

[TABLE]

Substituting the above into the defining formula for $M_{2}$ ((3.15)) and use integration by parts, we obtain

[TABLE]

Just like the estimate of $J_{2}$ in part (a), we have

[TABLE]

Here, as before, we used the fact that the parabolic distance between $(w,l)$ and $(y,s)$ is a least $r/4$.

For the rest of the terms, using (3.12) and the same argument as in the estimate of $K_{1}$, we deduce

[TABLE]

Finally we bound $M_{3}$. Using integration by parts on the $y$ variable and using the assumption that $div\ b=0$, we can write

[TABLE]

The last step is by (3.8), with $(w,l)$ replacing $(x,t)$ there. From the well known property of the Stokes system

[TABLE]

This together with (3.16’) and (3.12) yield

[TABLE]

Therefore

[TABLE]

Recall (see Remark 3.1 just before Lemma 3.2), the integral on the righthand side of (3.18) is a $L^{1}_{loc}$ function of space time.

Combining (3.15)-(3.18), we have, for $(w,l)\in Q_{r/4}(x,t)$, a.e.

[TABLE]

Since $\rho$ is arbitrary and $\nabla u$ and $M_{3}$ is $L^{1}_{loc}$, we deduce, for $(w,l)\in Q_{r/4}(x,t)$,

[TABLE]

This proves the lemma. ∎

Now we are ready to give the

Proof of Theorem 1.2.

(a) We will prove the first formula, i.e. the bound on $u(x,t)$.

The idea of the proof is to iterate (3.6) in a special manner since simple iteration will double the domain of integration in each step and will not yield a local formula. The key is to cut in half the size of the cube in (3.6) after each iteration. Here are the details.

In order to simplify the presentation, we will use capital letters to denote points in space-time. For example $X=(x,t),Y=(y,s),Z=(z,\tau)$, etc. From (3.6), we have

[TABLE]

For $Y\in Q_{r}(X)$, we apply the representation formula for cubes of half of the previous size to get

[TABLE]

Combining the above two inequalities, we obtain

[TABLE]

Notice that $Q_{r/2}(Y)\subset Q_{3r/2}(X)$ when $Y\in Q_{r}(X)$. The above thus shows

[TABLE]

Applying Lemma 3.1, we deduce

[TABLE]

here and throughout this section $B(b)=B(b,t-(4r)^{2},t)$ defined in (1.6’).

For $u(Z)$, we use the representation formula in the cube $Q_{r/4}(Z)$:

[TABLE]

Note that for $Z\in Q_{3r/2}(X)$, we have $Q_{r/4}(Z)\subset Q_{7r/4}(X)\subset Q_{2r}(X)$. Hence

[TABLE]

Exchanging the integrals and using Lemma 3.1 again, we deduce

[TABLE]

We iterate the above process, halving the size of each cube. By induction, it is clear that for some $C,c_{1}>0$,

[TABLE]

When $2^{5}c_{1}B(b)<1$ the above series converges to yield the mean value inequality for $u$. Since $b$ is in the Kato class defined in Definition 1.2, we know that $2^{5}c_{1}B(b)=2^{5}c_{1}B(b,t-(4r)^{2},t)<1$ when $r$ is sufficiently small. This proves the bound on $u$.

(b) We prove the gradient bound.

Since $u+c$ is also a solution of (1.2) for any constant $c$, we will assume that $\overline{u}_{Q_{2r}}$, the average of $u$ in $Q_{2r}(x,t)$ is [math].

The idea is to iterate (3.6’) in the above manner. From (3.6’), using the same notation as in part (a), we have

[TABLE]

where

[TABLE]

Here, we remind the reader that both sides of (3.20) are $L^{1}_{loc}$ functions and hence finite almost everywhere (see Remark 3.1 just before Lemma 3.2).

Applying (3.20) to $\nabla u(Y)$ and $Q_{r/2}(Y)$, we obtain

[TABLE]

For $Y\in Q_{r}(X)$, it is clear that there exists a $\mu>0$, independent of $r$ such that

[TABLE]

Hence

[TABLE]

Substituting (3.21) to (3.20) we have

[TABLE]

Therefore

[TABLE]

Lemma 3.1 then implies

[TABLE]

Now, using (3.20) on $|\nabla u(Z)|$ and the cube $Q_{r/4}(Z)$ and repeat the above argument, halving the size of the cube in each step, we finally reach

[TABLE]

As before this implies the desired gradient bound when $B(b)$ is small.

The last statement of Theorem 1.2 is a simple consequence of the gradient estimate and Poincarè inequality. ∎

We end the section by showing that if the sum of the entries of the drift term $b$ is zero, then (1.2) in torus has bounded solutions for many initial values, regardless of the singularity of $b$. To state the result rigourously, we will assume that $b$ is bounded. However all coefficients are independent of the bounds of $b$.

Proposition 3.1**.**

Given bounded vector fields $b=(b_{1}(x,t),...,b_{n}(x,t))$, consider the linearized Navier-Stokes equation with periodic boundary condition i.e. in a torus.

[TABLE]

Here $D=[0,2\pi]^{n}$. The functions $b(\cdot,t)$, $u_{0}(\cdot)$ and $u(\cdot,t)$ have period $2\pi$.

Suppose $\sum^{n}_{j=1}b_{j}(x,t)=\lambda$, a constant and $u_{0}$ is any finite linear combination of

[TABLE]

where $k$ is a positive integer and $f$ is a bounded, divergence free vector field with period $2\pi$. Then there exists a constant $c$ independent of $b$ and $\lambda$ such that

[TABLE]

Proof.

Under the assumption that $b$ is bounded, the existence of solutions to (3.22) follows from the standard theory. Let $E=E(x,t;y,s)$ be the fundamental solution of (3.22). The existence of $E$ is also standard.

First we just assume that $u_{0}$ has one term, i.e.

[TABLE]

Fixing $(x,t)$, consider

[TABLE]

By the fact that the rows of $E$ satisfies the conjugate equation of (3.22), we have

[TABLE]

Here

[TABLE]

with $P_{1},...,P_{n}$ being scalar functions. Using integration by parts and the divergence free property of $b$, we deduce

[TABLE]

Noticing that $\Delta u_{0}=-nk^{2}u_{0}$ and $\partial_{y_{j}}u_{0}(y)=iku_{0}$, we have

[TABLE]

By our assumption that $\sum^{n}_{j=1}b_{j}=\lambda$, the above shows

[TABLE]

Hence

[TABLE]

From (3.23) and the fact that $E$ is the fundamental solution to (3.22), we have $I(0)=u(x,t)$ and $I(t)=u_{0}(x)$. This shows

[TABLE]

Now let $S$ be a set of finite positive integers and

[TABLE]

Here $k_{l}$ is a positive integer and $f_{l}$ is a bounded, divergence free vector field with period $2\pi$. By (3.24) one has

[TABLE]

∎

4. A regularity condition for Navier-Stokes equations not involving

absolute values

In this section we introduce another sufficient condition on the velocity for boundedness of weak solutions of 3 dimensional Navier-Stokes equations. The novelty is that no absolute value of $u$ is involved. This is useful since it allows more cancellation effect to be taken into account. Throughout the years, various conditions on $u$ that imply regularity have been proposed. One of them is the Prodi-Serrin condition which requires that $u\in L^{p,q}$ with $\frac{3}{p}+\frac{2}{q}\leq 1$ for some $3<p\leq\infty$ and $q\geq 2$. See ([Ser], [Str] e.g.) Recently the authors in [ESS] showed that the condition $p=3$ and $q=\infty$ also implies regularity. In another development the author of [Mo] improved the Prodi-Serrin condition by a log factor, i.e. by requiring

[TABLE]

where $3/p+2/q=1$ and $3<p<\infty$, $2<q<\infty$

Most recently in [Z3], a form boundedness condition on velocity was introduced, which will imply the boundedness of weak solutions. The form boundedness condition, with its root in the perturbation theory of elliptic operators and mathematical physics, seems to be different from all the previous conditions. It seems to be one of the most general condition under the available tools. This fact has been well documented in the theory of linear elliptic equations. See [Si] e.g. Moreover, as indicated in [Z3], it contains the Prodi-Serrin condition except when $p$ or $q$ are infinite. It also includes suitable Morrey-Compamato type spaces. However we are not sure this condition contains the one in [Mo].

More precisely we proved

Theorem 5.1 ([Z3])* Let $u$ be a Leray-Hopf solution to the 3 dimensional Navier-Stokes equation in ${\bf R}^{3}\times(0,\infty)$.*

[TABLE]

Suppose for every $(x_{0},t_{0})\in{\bf R}^{3}\times(0,\infty)$, there exists a cube $Q_{r}=B(x_{0},r)\times[t_{0}-r^{2},t_{0}]$ such that $u$ satisfies the form bounded condition

[TABLE]

Here $\phi$ is any smooth function vanishing on the parabolic side of $Q_{r}$ and $B=B(t)$ is any given locally bounded function of $t\in{\bf R}^{1}$. Then $u$ is a classical solution when $t>0$.

In this section we are able to extend the form boundedness condition further. The main improvement is that our new condition in $u$ ((4.3)) below does not involve the absolute value of $u$. This differs significantly from known conditions on $u$ so far where $|u|$ is always present. By a simple integration by parts argument, it is clear that Condition (4.3) is more general than Condition (4.2).

Theorem 4.1**.**

Let $u$ be a Leray-Hopf solution to the 3 dimensional Navier-Stokes equation in ${\bf R}^{3}\times(0,\infty)$.

Suppose for every $(x_{0},t_{0})\in{\bf R}^{3}\times(0,\infty)$, there exists a cube $Q_{r}=B(x_{0},r)\times[t_{0}-r^{2},t_{0}]$ such that $u$ satisfies the form bounded condition: for a given $\delta>0$,

[TABLE]

Here $\phi$ is any smooth vector field vanishing on parabolic the side of $Q_{r}$ and $B=B(t)$ is any given locally bounded function of $t\in{\bf R}^{1}$. Then $u$ is a classical solution when $t>0$.

Remark 4.1. Condition 4.3 is actually a condition on the strain tensor $\nabla u+(\nabla u)^{T}$. Theorem 4.1 immediately implies that weak solutions to the 3 dimensional Navier-Stokes equations are locally bounded in any open subset of the region where the strain tensor is negative definite.

Proof of Theorem 4.1. Let $t_{0}$ be the first moment of singularity formation. We will reach a contradiction. It is clear that we only need to prove that $u$ is bounded in $Q_{r/8}=Q_{r/8}(x_{0},t_{0})$ for some $r>0$. In fact the number $8$ is not essential. Any number greater than $1$ would work.

Consider the equation for vorticity $w=\nabla\times u$. It is well known that, in the interior of $Q_{r}$, $w$ is a classical solution to the parabolic system with singular coefficients

[TABLE]

Let $\psi=\psi(y,s)$ be the refined cut-off function defined right after (2.1) such that $\psi=1$ in $Q_{r/2}$, $\psi=0$ in $Q^{c}_{r}$ and such that $0\leq\psi\leq 1$, $|\nabla\psi|\leq C/r$ and $|\psi_{t}|\leq C/r^{2}$. We can use $w\psi^{2}$ as a test function on (4.4) to obtain

[TABLE]

The term $I_{1}$ is already in good shape. Next, using integration by parts and the divergence free condition on $u$, we have

[TABLE]

Since $\nabla u\in L^{2}_{loc}({\bf R}^{3}\times[0,\infty))$ and $\|u(\cdot,t)\|_{{\bf R}^{3}}$ is non-increasing in time, it is easy to prove by Sobolev imbedding and Hölder’s inequality that $u|_{Q_{r}}$ satisfies the form boundedness condition (1.3). In fact this has been proven in Corollary 2 in [Z2]. Hence we can bound $I_{2}$ in exactly the same way as $T_{2}$ in (2.4) with $b$ being chosen as $u$ here. Following the argument between (2.4) and (2.6), we obtain, for any given $\delta>0$,

[TABLE]

Note that in (2.6), $\delta$ was chosen as $2$. However, a closer look at the proof shows that (4.6) is true.

Next we estimate $I_{3}$. From Condition (4.3)

[TABLE]

Substituting (4.6)-(4.7) to (4.5) we obtain,

[TABLE]

Repeating the above process, but restrict the integrals to $Q_{r}\cap\{(y,s)\ |s<t\}$ with $t<t_{0}$, we obtain, for any $t\in[t_{0}-r^{2},t_{0}]$,

[TABLE]

Combining the last two estimates, we deduce,

[TABLE]

Here $B_{1}$ is a locally bounded, one variable function.

Using standard results, we know that (4.8) implies that $u$ is regular. Here is the proof.

From (4.8), it is clear that $\int_{Q_{r}}|curl\ (w\psi)|^{2}dyds\leq C.$ Hence, since $\psi=1$ in $Q_{r/2}$,

[TABLE]

Let $\eta=\eta(y)$ be a cut-off function such that $\eta=1$ in $B(x_{0},r/4)$ and $\eta=0$ in $B(x_{0},r/2)^{c}$. Then for each $s\in[t_{0}-(r/4)^{2},t_{0}]$, we have, in the weak sense

[TABLE]

in $Q_{r/2}$. By standard elliptic estimates, using the fact that $u\eta=0$ on the boundary,

[TABLE]

This shows that

[TABLE]

By Sobolev imbedding

[TABLE]

Next, from (1.22) on p316 [Te],

[TABLE]

Here all norms are over the ball $B(x_{0},r/2)$. Therefore

[TABLE]

It follows

[TABLE]

From Sobolev imbedding we know that

[TABLE]

We treat $u$ and $\nabla u$ as coefficients in the vortex equation (4.4). By (4.10) and (4.11), the standard parabolic theory (see [Lieb] e.g.), shows that $w$ is bounded and Hölder continuous in $Q_{r/8}$. Here the bound depends only on the $L^{2}$ norm of $w$ in $Q_{r}$ and $r$. This is so because of the relation $3/6+2/\infty<1$ for the norm of $u$ and $3/6+2/2<2$ for the norm of $\nabla u$. Now a standard bootstrapping argument shows that $u$ is smooth. ∎

Acknowledgement I thank Professors Z. Grujic, I. Kukavica and V. Liskevich for helpful discussions.

5. Addendum May 2019

The purpose of the addendum is to fill in a missing term in Theorem 1.7 and Lemma 3.3 in the current paper that is published in Pacific Journal of Mathenatics, Vol. 223, No.2, 2006, which will be cited as [Z4]. This is due to the use of a formula in a cited reference, which omitted a term. The main conclusion that local solutions of certain linearized Navier-Stokes equation have bounded spatial gradient is intact. This includes bounded local Leray-Hopf solutions to the Navier Stokes equation without condition on the pressure.

Let us recall the basic set up of [Z4]. The equation is: for $(x,t)\in\Omega\subset{\bf R}^{3}\times{\bf R}$,

[TABLE]

Here $\Delta$ is the standard Laplacian and $b=b(x,t)$ is a given $L^{2}_{loc}$ vector field to be specified later. $\Omega$ is a domain. The parabolic Kato norm is:

[TABLE]

We say $b$ is in class $K_{1}$ if $\lim_{l\to t}B(b,l,t)=0$. $K_{1}$ class in a bounded space time domain contains the usual $L^{p,q}$ functions with $(3/p)+(2/q)<1.$

Let $D$ for a domain in ${\bf R}^{3}$ and $T>0$, we will use this definition for solutions of (5.1).

Definition 5.1**.**

A divergence free vector field $u\in L^{\infty}(0,T;L^{2}(D))\cap L^{2}(0,T;W^{1,2}(D))$ is called a (weak) solution of (5.1) if: for any vector valued $\phi\in C^{\infty}(D\times[0,T])$ with $div\,\phi=0$ and $\phi=0$ on $\partial D\times[0,T]$, $u$ satisfies

[TABLE]

The corrected version of Theorem 1.7 in [Z4] is:

Theorem 5.1**.**

Let $u$ be a solution of (5.1) in a domain $\Omega\subset{\bf R}^{3}\times{\bf R}$. Suppose $Q_{4r}(x,t)\subset\Omega$, $divb=0$ and that $b|_{Q_{2r}(x,t)}$ is in class $K_{1}$ and $b\in L^{2}_{loc}$. Then both $u$ and $|\nabla u|$ are bounded functions in $Q_{2r}=Q_{2r}(x,t)$, the standard parabolic cube of size $2r$.

Moreover, for some positive constants $C=C(b)$ and $r_{0}$, depending on the size of the Kato norm of $b$, there hold, when $0<r<r_{0}$,

[TABLE]

The proof of the theorem is through a scaled iteration process starting with Lemma 3.3 in [Z4], which contains mean value inequalities for: (a) solutions of (5.1); (b) spatial gradient of solutions. Inequality (a) is first obtained in [O], which is also the basis for inequality (b). Recently Professor Hongjie Dong kindly informed us that inequality (a) and hence (b) misses a term. The corrected version of Lemma 3.3 in [Z4] is the following. In our opinion the main localization idea in [O] is still nice.

Lemma 5.1**.**

(mean value inequalities, replacing Lemma 3.3 in [Z4])

(a). Let $u$ be a solution of (5.1) in the region $\Omega$. Suppose $Q_{2r}(x,t)\subset\Omega$. Then there exists a constant $\lambda$ such that

[TABLE]

(b). Under the same assumption as (a), there exists a constant $\lambda$ such that

[TABLE]

Proof.

of (a). The under-braced terms were missing. We indicate the changes needed for the proof, dividing into 3 steps.

Step 1. Let $E=E(x,t;y,s)$ be the fundamental solution (matrix) of the Stokes system in ${\bf R}^{3}\times(0,\infty)$ and $E_{k}$ be the $k$-th column of $E$. Then

[TABLE]

where $G$ is the fundamental solution of the heat equation. Fixing $(x,t)$, we construct a standard cut-off function $\eta$ such that $\eta(y,s)=1$ in $Q_{r/2}(x,t)$, $\eta(y,s)=0$ outside of $Q_{r}(x,t)$, $0\leq\eta\leq 1$ and $|\nabla\eta|^{2}+|\Delta\eta|+|\partial_{s}\eta|\leq c/r^{2}$.

Define for $k=1,2,3$, after [O], a vector valued function

[TABLE]

It is clear that when $t>s$, $\Phi_{k}$ is a valid test function for equation (5.1) since $\Phi_{k}$ is smooth, compactly supported and divergence free. Using $\Phi_{k}$ as a test function on (5.1), by Definition 5.1, we obtain

[TABLE]

Here,we will suppress the superscript $(x,t)$ on $\Phi_{k}$, unless there is a confusion. Also $u$ is regarded as a row vector so that $u\Phi_{k}$ etc is a scalar.

Since $E_{k}$ is divergence free $curl\ curlE_{k}=-\Delta E_{k}$. Thus

[TABLE]

Step 2. Using formula (5.3) of the fundamental matrix $E$ and that of $\overrightarrow{Z}$ in (5.6), direct computation shows that

[TABLE]

where $u_{k}$ is the k-th component of $u$ and $\Gamma=\frac{1}{4\pi|x-y|}$ is the Green’s function on ${\bf R}^{3}$.

Alternatively, from (5.6),

[TABLE]

First we treat $I$. Consider the Helmholtz decomposition

[TABLE]

where $\Delta f=0$, $div\,X=0$. Since $\Delta f=\nabla\eta\cdot u$ and $\eta(\cdot,s)$ is compactly supported, we can take $f(y,s)=-\int\Gamma(y,z)(\nabla\eta\cdot u)(z,s)dz.$ Then

[TABLE]

Using the decay estimates $|\nabla_{y}E_{k}(x,t;y,s)|\leq\frac{c}{(|x-y|+\sqrt{t-s})^{4}},$ $|f(y,s)|\leq\frac{c}{|y|}$ and the fact that $E_{k}$ is a divergence vector field of variable $y$, we can integrate by parts to deduce that $\int\nabla fE_{k}(x,t;y,s)dy=0.$ Therefore

[TABLE]

In order to compute $II$, we see from (5.3) and (5.6) that

[TABLE]

This implies

[TABLE]

A combination of (5.10), (5.11) and (5.8) proves (5.7). By (5.7) and (5.5):

[TABLE]

Here and later $ub\nabla\Phi_{k}=u\cdot\sum^{3}_{i=1}b_{i}\partial_{i}\Phi_{k}$.

Note that $\Delta E_{k}+\partial_{s}E_{k}=0$ when $s<t$. The above infers, with $\Gamma=\Gamma(x,y)$, a local representation formula which may be of independent interest:

[TABLE]

Step 3. It is clear that $|J_{4}|\leq\frac{C}{r^{3}}\int_{B_{r}(x)-B_{r/2}(x)}|u(y,t)|dy.$ The estimates $J_{1},J_{2},J_{3}$ are done on p381 of [Z4] following [O]. This proves of part (a) of the lemma.

Proof of part (b). Given $(w,l)\in Q_{r/4}(x,t)$, (5.12) still holds when $(x,t)$ is replaced by $(w,l)$. It is clear that, for $(w,l)\in Q_{r/4}(x,t)$,

[TABLE]

The rest of the proof of the lemma is the same as that on p382-285 of [Z4] by taking the gradient of $J_{1},J_{2},J_{3}$. ∎

With the lemma in hand, the proof of Theorem 5.1 is then the same as the scaled iteration on p385-388 of [Z4] with the additional mild integral terms $J_{4}$ and $|\nabla J_{4}|$. Note these additional terms are spatial integrals only. These account for the difference between the result of Navier Stokes equation and related result for the heat equation.

Acknowledgement We wish to sincerely thank Professor Hongjie Dong for pointing out the issue of the theorem.