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Global Regularity to the Navier-Stokes Equations for a Class of Large Initial Data.

1 Introduction

The Cauchy problem of the incompressible Navier-Stokes equations on [R.sup.n] is described by the following system

[mathematical expression not reproducible] (1.1)

where u represents the velocity field and P is the scalar pressure.

First of all, let us recall some known results on the small-data global regularity of the Navier-Stokes equations on [R.sup.3]. In the seminal paper [18], Leray proved that the 3D incompressible Navier-Stokes equations are globally well-posed if the initial data [u.sub.0] is such that [mathematical expression not reproducible] is small enough. This quantity is invariant under the natural scaling of the Navier-Stokes equations. Later on, many authors studied different scaling invariant spaces in which Navier-Stokes equations are well-posed at least for small initial data, which include but are not limited to

[mathematical expression not reproducible],

where 3 < p < [infinity]. The space [BMO.sup.-1]([R.sup.3]) is known to be the largest scaling invariant space so that the Navier-Stokes equations (1.1) are globally well-posed under small initial data. The readers are referred to [3, 9, 14, 15] as references. We also mention that the work of Lei and Lin [16] was the first to quantify the smallness of the initial data to be 1 by introducing a new space [X.sup.-1].

We remark that the norm in the above scaling invariant spaces are always greater than the norm in the Besov space [mathematical expression not reproducible]defined by

[mathematical expression not reproducible]

Bourgain and Pavlovic in [2] showed that the Cauchy problem of the 3D Navier-Stokes equations is ill-posed in the sense of norm inflation. Partially because of the result of Bourgain and Pavlovic, data with a large [mathematical expression not reproducible] are usually called large data to the Navier-Stokes equations (for instance, see [4, 20]).

Towards this line of research, a well-oiled case is the family of initial data which is slowly varying in vertical variable. The initial velocity field [u.sup.[epsilon].sub.0] is of the form

[mathematical expression not reproducible], (1.2)

which allows slowly varying in the vertical variable [x.sub.3] when [epsilon] > 0 is a small parameter. This family of initial data is very interesting (as has been pointed out by V.Sverak, see the acknowledgement in [5]) and considered by Chemin, Gallagher and Paicu in [5]. They proved the global regularity of solutions to the Navier-Stokes equations when [v.sub.0] is analytic in [x.sub.3] and periodic in [x.sub.h], and certain norm of v0 is sufficiently small but independent of [epsilon] > 0. More precisely, they proved the following Theorem:

Theorem 1 [Chemin-Gallagher-Paicu, Ann. Math. 2011]. Let [alpha] be a positive number. There are two positive numbers [[epsilon].sub.0] and [eta] such that for any divergence free vector field [v.sub.0] satisfying

[mathematical expression not reproducible]

then, for any positive [epsilon] smaller than [[epsilon].sub.0], the initial data (1.2) generates a global smooth solution to (1.1) on [T.sup.2] x R.

The notation [B.sup.7/2.sub.2,1] in the above Theorem denotes the usual inhomogeneous Besov space. The significance of the result lies in that the global regularity of the 3D incompressible Navier-Stokes equations in [5] only requires very little smallness imposed on the initial data. It is clear that the [B.sup.-1.sub.[infinity][infinity]] norm of [u.sup.[epsilon].sub.0] can tend to infinity as [epsilon] [right arrow] 0. Let us first focus on the periodic constraint imposed on the initial data in Theorem 1.

As has been pointed out by Chemin, Gallagher and Paicu, the reason why the horizontal variable of the initial data in [5] is restricted to a torus is to be able to deal with very low horizontal frequencies. In the proof of Theorem 1 in [5], functions with zero horizontal average are treated differently to the others, and it is important that no small horizontal frequencies appear other than zero. Later on, many efforts are made towards removing the periodic constraint of v0 on the horizontal variables. For instance, see [4, 10, 20] and so on. We will review those results a little bit later.

In this paper, we consider the Cauchy problem of the following generalized Navier-Stokes equations on [R.sup.n]:

[mathematical expression not reproducible] (1.3)

where D = [square root of -[DELTA]]. The initial velocity field [u.sup.[epsilon].sub.0] is of the form

[mathematical expression not reproducible]. (1.4)

The horizontal variable [x.sub.h] = ([x.sub.1], [x.sub.2], ..., [x.sub.n-1]).

Our main result is the following theorem which generalizes the theorem of Chemin, Gallagher and Paicu to the whole space for the generalized Navier-Stokes equations with some appropriate number s. Definition of notations will be given in Section 2.

Theorem 2. Let [alpha], [[epsilon].sub.0], p and s be four positive constants and (p, s) satisfy

1 [less than or equal to] p < 2(n-1), 1 [less than or equal to] s < min(n - 1, 2(n - 1)/p).

There exists a positive constant n such that for any 0 < [epsilon] < [[epsilon].sub.0] and any divergence free vector field [v.sub.0] satisfying

[mathematical expression not reproducible]

then the generalized Navier-Stokes equations (1.3) with initial data (1.4) generate a global smooth solution on [R.sup.n].

Remark 1. 1) When n [greater than or equal to] 4, one can find that the classical Navier-Stokes equations on [R.sup.n] satisfy the assumption 1 [less than or equal to] s = 2 < min(n-1, 2(np-1)/p) for 1 [less than or equal to] p [less than or equal to] 2(n-1). Then according to Theorem 2, the n-dimensional incompressible Navier-Stokes equations with initial data (1.4) have a global smooth solution in the whole space case.

2) In the case n = 3, we require that 1 [less than or equal to] s < 2. The main obstacle when s = 2 is that we can not get the product law in [mathematical expression not reproducible]). This anisotropic Besov space is induced by the a priori estimate of anisotropic pressure ([nabla].sub.h]q, [[epsilon].sup.2] [paragraph].sub.3]q) (see equation (1.5) and Step 5 for details). From this point of view, the global well-posedness of 3D incompressible Navier-Stokes equations with initial data (1.4) on [R.sup.3] is still unclear, even though the higher dimensional cases are settled down.

3) In the present paper, we establish the global solution in the [L.sup.p]-type Besov space, in which the bilinear estimate of the solution can not be derived by the classical [L.sup.2] energy method. Particularly, one can obtain the [L.sup.1] estimate in time of the solution by introducing the new quantity

[mathematical expression not reproducible]

in the a priori estimate. We also mention that in [5, 20, 21], authors did not get the [L.sup.1]-time estimate of the solution.

Now we mention that many authors make efforts to remove the periodic restriction on horizontal variable. Chemin and Gallagher considered the well-prepared initial data in [4]. They proved the global well-posedness of (1.1) when [u.sup.[epsilon].sub.0] is of the form [u.sup.[epsilon]].sub.0] = ([v.sup.h.sub.0] + [epsilon][w.sup.h.sub.0], [w.sup.3.sub.0])([x.sub.h], [epsilon][x.sub.3]). Later, Gui, Huang, and Zhang in [10] generalized this result to the density dependent Navier-Stokes equations with the same initial velocity. Recently, Paicu and Zhang in [20,21] considered the global regularity of (1.1) if [u.sup.[epsilon].sub.0] satisfies the form of

[mathematical expression not reproducible]

for [delta] = 1/2 and [delta] [member of] (0, 1/2), then [u.sup.[epsilon].sub.0] generates a global solution of (1.1) on [R.sup.3].

Main ideas of the Proof. We will prove our main result by constructing the bilinear estimate (independent of [epsilon]). Our strategy can be stated as follows.

Step 1. Rescaled system and simplification. As in [5], we define

[mathematical expression not reproducible].

Denote

[mathematical expression not reproducible]

Using the Navier-Stokes equations (1.1), it is easy to derive the equations governing the rescaled variables v and q (they are still depending on [epsilon]):

[mathematical expression not reproducible] (1.5)

where [v.sup.h] = ([v.sup.1], ..., [v.sup.n-1]). The rescaled pressure q can be recovered by the divergence free condition as

-[[DELTA].sub.[epsilon]]q = [summation over (i,j)] [[partial derivative].sub.i] [[partial derivative].sub.j] ([v.sup.i], [v.sup.j]).

The global regularity of solutions to system (1.4) for small initial data [v.sub.0] will be presented in Section 3 and 4 for any positive [epsilon]. But to best illustrate our ideas, let us here focus on the case of [epsilon] = 0. Formally, by taking [epsilon] = 0 in system (1.4), we have the following limiting system:

[mathematical expression not reproducible], (1.6)

where [D.sub.h] = [square root of -[[DELTA].sub.h]]. The pressure q in (1.6) is given by

-[[DELTA].sub.h]q = [summation over (i,j)] [[partial derivative].sub.i] [[partial derivative].sub.j] ([v.sup.i], [v.sup.j]).

Step 2. Set-up of the a priori estimate. Observing that in the rescaled system (1.6), the viscosity is absent in the vertical direction. To make the full use of smoothing effect from operator [[partial derivative].sub.t] + [D.sup.s.sub.h], particularly in low frequency parts, we will apply the tool of anisotropic homogeneous Besov spaces. The goal is to derive certain a priori estimate of the form:

[mathematical expression not reproducible].

Note that pressure term doesn't explicitly appear in the equation of [v.sup.n] of the limiting system (1.6), which makes the estimate for [v.sup.n] easier. So here let us just focus on the equation of [v.sup.h]. Naturally, we define

[mathematical expression not reproducible]

At this step, we assume that the initial data [v.sup.h.sub.0] belongs to [mathematical expression not reproducible]. This ensures that [PSI] (t) is a critical quantity with respect to the natural scaling of the generalized Navier-Stokes equations. By Duhamel's principle, we can write the integral equation of [v.sup.h] by

[mathematical expression not reproducible]

According to the estimates of heat equation, one can formally has

[mathematical expression not reproducible]

It will be shown that

[mathematical expression not reproducible]

Step 3. Derivative loss: input the estimate of [[partial derivative].sub.n] [v.sup.h]. For the quantity [PSI](t), we need to prove the bilinear estimate in the following form:

[mathematical expression not reproducible]

Certainly, there is [[partial derivative].sub.n]-derivative loss! By the product law in anisotropic Besov spaces (Lemma 1), the strategy to bound the last term is

[mathematical expression not reproducible]

and then we should add the new quantity

[mathematical expression not reproducible]

in the definition of [[PSI](t). We find that [mathematical expression not reproducible] is the hardest term to estimate. Since we note that by Duhamel's principle,

[mathematical expression not reproducible]

There is still [[partial derivative].sub.n]-derivative loss in the a priori estimates!

Step 4. Recover the derivative loss: analyticity in [x.sub.n]. Motivated by Chemin-Gallagher-Paicu [5], we add an exponential weight [mathematical expression not reproducible] with

[mathematical expression not reproducible]

Here [theta](t) is defined by

[mathematical expression not reproducible]

which will be shown to be small to ensure that [PHI] (t, [absolute value of [[xi].sub.n]]) satisfies the subadditivity. Denoting [mathematical expression not reproducible], we then have

[mathematical expression not reproducible]

Hence, we can recover the derivative loss by

[mathematical expression not reproducible]

In this way, the losing derivative term can be absorbed by the left hand side of the above inequality by choosing [lambda] sufficient large.

Step 5. The estimate of the pressure term [[nabla].sub.h]q. To estimate the pressure term, we write

[mathematical expression not reproducible]

The [mathematical expression not reproducible] norm of the first term in [[nabla].sub.h]q can be estimated by

[mathematical expression not reproducible]

Here we should point out that when s = 2, n = 3, system (1.3) is nothing but the 3D incompressible Navier-Stokes equations. In this case we have to deal with [mathematical expression not reproducible] type estimate. Unfortunately, the product law in [mathematical expression not reproducible] is hard to obtain since we can not control the low horizontal frequency part.

Step 6. Estimate of [theta](t). In this step, we want to prove that for any time t, [theta](t) is a small quantity. This ensures that the phase function [PHI] satisfies the subadditivity property. We will go to derive a stronger estimate for

[mathematical expression not reproducible]

However, when [epsilon] > 0, we can not get the closed estimate for Y(t). Our strategy is to add an extra term [epsilon][v.sup.h] under the same norm which is hidden in the pressure term [[epsilon].sup.2] [[partial derivative].sub.n]q. See Section 4 for details.

Step 7. Estimate of [v.sup.n.sub.[PHI]]. If this is done, we could get a closed a priori estimate (see Lemma 4) and finish the proof of Theorem 2. Observing that the nonlinear term [v.sup.n] [[partial derivative].sub.n] [v.sup.n] can be rewritten as -[v.sup.n] [div.sub.h] [v.sup.h] due to divergence free condition. Hence, in the limiting system (1.6), there is no loss of derivative in vertical direction on [v.sup.n]. Thus the estimate on [v.sup.n] is much easier than [v.sup.h].

There are also some other type of large initial data so that the Navier-Stokes equations are globally well-posed. For instance, when the domain is thin in the vertical direction, Raugel and Sell [22] were able to establish global solutions for a family of large initial data by using anisotropic Sobolev imbedding theorems (see also the paper [13] by Iftimie, Raugel and Sell). By choosing the initial data to transform the equation into a rotating fluid equations, Mahalov and Nicolaenko [19] obtained global solutions generated by a family of large initial data. A family of axi-symmetric large solutions were established in [11] by Hou, Lei and Li. Recently, Lei, Lin and Zhou in [17] proved the global well-posedness of 3D Navier-Stokes equations for a family of large initial data by making use of the structure of Helicity. The data in [17] are not small in [[??]-1.sub.[infinity],[infinity]] even in the anisotropic sense. We also mention that for the general 3D incompressible Navier-Stokes equations which possess hyper-dissipation in horizontal direction, Fang and Han in [8] obtain the global existence result when the initial data belongs to the anisotropic Besov spaces.

The remaining part of the paper is organized as follows. In Section 2, we present the basic theories of anisotropic Littlewood-Paley decomposition and anisotropic Besov spaces. Section 3 is devoted to obtaining the a priori estimates of solution. The [theta](t) will be studied in Section 4. Finally, the proof of the main result will be given in Section 5.

2 Anisotropic Littlewood-Paley theories and preliminary lemmas

In this section, we first recall the definition of the anisotropic Littlewood-Paley decomposition and some properties about anisotropic Besov spaces. It was introduced by Iftimie in [12] for the study of incompressible Navier-Stokes equations in thin domains. Let us briefly explain how this may be built in [R.sub.n]. Let (x, [phi]) be a couple of [C.sup.[infinity]] functions satisfying

[mathematical expression not reproducible]

For u [member of] S'([R.sup.n])/P([R.sup.n]), we define the homogeneous dyadic decomposition on the horizontal variables by

[mathematical expression not reproducible].

Similarly, on the vertical variable, we define the homogeneous dyadic decomposition by

[mathematical expression not reproducible].

The anisotropic Littlewood-Paley decomposition satisfies the property of almost orthogonality:

[mathematical expression not reproducible],

where [S.sup.h.sub.l] is defined by [mathematical expression not reproducible]. Similar properties hold for [[DELTA].sup.v.sub.j]. In this paper, we shall use the following anisotropic version of Besov spaces [12]. In what follows, we denote for abbreviation [mathematical expression not reproducible].

Definition 1 [Anisotropic Besov space]. Let (p, r) [member of] [[1, [infinity]].sup.2], [sigma], s [member of] R and u [member of] S'([R.sup.n])/P([R.sup.n]), we set

[mathematical expression not reproducible]

(1) For [sigma] < n-1/p, s < 1/2 ([sigma] = n-1/p or s = 1/2 if r =1), we define

[mathematical expression not reproducible].

(2) If [mathematical expression not reproducible], is defined as the subset of u [member of] S'([R.sup.n]) such that [mathematical expression not reproducible].

The study of non-stationary equation requires spaces of the type [L.sup.p.sub.T] (X) for appropriate Banach spaces X. In our case, we expect X to be an anisotropic Besov space. So it is natural to localize the equations through anisotropic Littlewood-Paley decomposition. We then get estimates for each dyadic block and perform integration in time. As in [6], we define the so called CheminLerner type spaces:

Definition 2. Let (p, r) [member of] [[1, [infinity]].sup.2], [sigma], s [member of] R and T [member of] (0, [infinity]], we set

[mathematical expression not reproducible]

and define the space [mathematical expression not reproducible] to be the subset of distributions in u [member of] S'(0, T) x S'([R.sup.n]) with finite [mathematical expression not reproducible] norm.

In order to investigate the continuity properties of the products of two temperate distributions f and g in anisotropic Besov spaces, we then recall the isotropic product decomposition which is a simple splitting device going back to the pioneering work by J.-M. Bony [1]. Let f, g [member of] S'([R.sup.n]),

fg = T (f, g) + [??](f, g) + R(f, g),

where the paraproducts T(f, g) and [??](f, g) are defined by

[mathematical expression not reproducible]

and the remainder

[mathematical expression not reproducible]

Similarly, we can define the decompositions for both horizontal variable [x.sub.h] and vertical variable [x.sub.n]. Indeed, we have the following split in [x.sub.h].

[mathematical expression not reproducible],

with

[mathematical expression not reproducible]

The decomposition in vertical variable [x.sub.n] can be defined by the same line. Thus, we can write fg as

[mathematical expression not reproducible]. (2.1)

Each term of (2.1) has an explicit definition. Here

[mathematical expression not reproducible]

Similarly,

[mathematical expression not reproducible]

and so on.

At this moment, we can state an important product law in anisotropic Besov spaces. The case p = 2, n = 3 was proved in [10]. For completeness, here we prove a similar result in the [L.sup.p] framework.

Lemma 1. Let 1 [less than or equal to] p < [infinity] and ([[sigma].sub.1], [[sigma].sub.2]) be in [R.sup.2]. If ([[sigma].sub.1], [[sigma].sub.2]) [less than or equal to] n-1/p and [[sigma].sub.1] + [[sigma].sub.2] > (n-1) max (0,2/p-1),

then we have for any [mathematical expression not reproducible],

[mathematical expression not reproducible]

Proof. According to (2.1), we first give the bound of [T.sup.h][T.sup.v](f, g). Indeed, applying Holder and Bernstein inequality, we get that

[mathematical expression not reproducible]

Since [[sigma] [less than or equal to] n-1/p, we obtain that

[mathematical expression not reproducible]

where the sequence [mathematical expression not reproducible]. This gives the estimate of [T.sup.h] [T.sup.v] (f, g).

Similarly, for [mathematical expression not reproducible], we have

[mathematical expression not reproducible]

Again, [[sigma].sub.2] [less than or equal to] n-1/p implies that

[mathematical expression not reproducible]

The estimate on the remainder operator which concerns the horizontal variable [R.sup.h][T.sup.v] (f, g) may be more complicated. When 2 [less than or equal to] p, the strategy is following:

[mathematical expression not reproducible]

As [[sigma].sub.1] + [[sigma].sub.2] > 0 if 2 [less than or equal to] p, we have

[mathematical expression not reproducible]

In the case 1 [less than or equal to] p < 2, we have

[mathematical expression not reproducible]

As [[sigma].sub.1] + [[sigma].sub.2] > (n - 1) (2/p - 1) if 1 [less than or equal to] p < 2, we have

[mathematical expression not reproducible]

The other terms can be followed exactly in the same way, here we omit the details. These complete the proof of this lemma.

Throughout this paper, [PHI] denotes a locally bounded function on [R.sup.+] x R which satisfies the following subadditivity (see (3.2) for the explicit expression of [PHI])

[mathematical expression not reproducible].

For any function f in S'(0, T) x S'([R.sup.n]), we define

[mathematical expression not reproducible].

Let us keep the following fact in mind that the map f [??] [f.sup.+] preserves the norm of [L.sup.p.sub.h] ([L.sup.2.sub.v]), where [f.sup.+](t, [x.sub.h], [x.sub.n]) represents the inverse Fourier transform of [mathematical expression not reproducible] on vertical variable, defined as

[mathematical expression not reproducible]

On the basis of these facts, we have the following weighted inequality as in Lemma 1.

Lemma 2. Let 1 [less than or equal to] p < [infinity] and [[sigma].sub.1], [[sigma].sub.2]) be in [R.sup.2]. If [sigma].sub.1], [[sigma].sub.2] [less than or equal to] (n--1)/p and

[sigma].sub.1] + [[sigma].sub.2] > (n-1) max (0,2/p-1),

then we have for any [mathematical expression not reproducible],

[mathematical expression not reproducible]

Proof. We only prove the [mathematical expression not reproducible]. For fixed k, j, we have

[mathematical expression not reproducible]

Using the fact that f [??] [f.sup.+] preserves the norm of [L.sup.p.sub.h] ([L.sup.2.sub.v]), we then get by the similar method as in Lemma 1 that

[mathematical expression not reproducible]

The other terms in (2.1) can be estimated by the same method and finally, we have

[mathematical expression not reproducible]

The following lemma is a direct consequence of Lemma 2.

Lemma 3. Let [mathematical expression not reproducible] Assume that

[mathematical expression not reproducible]

If [mathematical expression not reproducible]

3 Estimates for the re-scaled system

This section is devoted to obtaining the a priori estimate for the following system

[mathematical expression not reproducible]

The pressure q can be computed by the formula

[mathematical expression not reproducible]

Due to the divergence free condition, the pressure can be split into the following three parts

[mathematical expression not reproducible] (3.1)

It is worthwhile to note that there will lose one vertical derivative owing to the term [v.sup.n] [[partial derivative].sub.n] [v.sup.h] and pressure terms [q.sup.2], [q.sup.3] which appear in the equation on [v.sup.h]. Thus, we assume that the initial data is analytic in the vertical variable. This method was introduced in [7] to compensate the losing derivative in [x.sub.n]. Therefore, we introduce two key quantities which we want to control in order to obtain the global bound of v in a certain space. We define the function [theta](t)

[mathematical expression not reproducible]

and denote

[mathematical expression not reproducible]

The phase function [PHI](t, [D.sub.n]) is defined by

[PHI](t, [[xi].sub.n]) = ([alpha]- [lambda][theta](t)) [absolute value of [[xi].sub.n]], (3.2)

for some [lambda] that will be chosen later on, a is a positive number. Obviously, we need to ensure that [theta](t) < [sigma]/[lambda] which implies the subadditivity of [PHI].

The following lemma provides the a priori estimate of [v.sub.[PHI]] in the anisotropic Besov spaces, which is the key bilinear estimate.

Lemma 4. There exist two constants [[lambda].sub.0] and [C.sub.1] such that for any [lambda] > [[lambda].sub.0] and t satisfying [theta](t) [less than or equal to] [alpha]/2[lambda] , we have

[PSI](t) [less than or equal to] [C.sub.1] [PSI] (0) + [C.sub.1] [PSI] [(t).sup.2].

3.1 Estimates on the horizontal component [v.sup.h]

According to the definition of [v.sup.h.sub.[PHI]], we find that in each dyadic block, it verifies the following equation

[mathematical expression not reproducible]. (3.3)

Taking the [L.sup.p.sub.h] ([L.sup.2.sub.v]) norm, we deduce that

[mathematical expression not reproducible]

We first estimate the linear term [I.sub.1]. In fact, we have

[mathematical expression not reproducible]

where [mathematical expression not reproducible] is a two dimensional sequence satisfying [mathematical expression not reproducible].

The term [I.sub.2] can be rewritten as

[mathematical expression not reproducible]

By Young's inequality, we have

[mathematical expression not reproducible]

Thus, we can get by Lemma 2 and 3 that

[mathematical expression not reproducible]

Now we are left with the study of the pressure term [I.sub.3]. The pressure can be split into q = [q.sup.1] + [q.sup.2] + [q.sup.3] with [q.sup.1], [q.sup.2], [q.sup.3] defined in (3.1). For convenience, we denote that

[mathematical expression not reproducible]

Hence, using the fact that [(-[[DELTA].sub.[epsilon]]).sup.-1] [[partial derivative].sub.i] [[partial derivative].sub.j] is a bounded operator applied for frequency localized functions in [L.sup.p.sub.h]([L.sup.2.sub.v]) when i, j = 1, 2,..., n--1, we get

[mathematical expression not reproducible]

By the same method as in the estimate of [I.sub.2], we have

[mathematical expression not reproducible]

Noting that

[mathematical expression not reproducible]

and as in the estimate of [I.sub.2], it holds that

[mathematical expression not reproducible]

Using

[mathematical expression not reproducible]

we write [I.sub.33] as follows

[mathematical expression not reproducible].

By Young's inequality and Lemma 3, as 1 [less than or equal to] s < min{n-1,2 n-1/p}, we have

[mathematical expression not reproducible]

[mathematical expression not reproducible]

Now we are going to estimate the key quantity

[mathematical expression not reproducible]

According to (3.3), we find that in each dyadic block [[partial derivative].sub.n] [v.sup.h.sub.[PHI]] verifies

[mathematical expression not reproducible] (3.4)

Taking the [L.sup.p.sub.h] ([L.sup.2.sub.v]) norm on both sides of (3.4), we have

[mathematical expression not reproducible] (3.5)

For fixed k, j, multiplying the (3.5) by [??]([tau]) and integrating over (0, t), one can have

[mathematical expression not reproducible]

The term [I.sub.4] containing initial data can be bounded by

[mathematical expression not reproducible]

By Fubini's theorem, the term [I.sub.5] can be rewritten as

[mathematical expression not reproducible]

Thus, we can get by Lemma 2 and 3 that

[mathematical expression not reproducible]

As for [I.sub.6], for convenience, we denote that

[mathematical expression not reproducible]

By the same method as in the estimate of [I.sub.5], we have

[mathematical expression not reproducible]

Finally, [I.sub.63] can be estimated as follows

[mathematical expression not reproducible]

Thus, we can obtain that

[mathematical expression not reproducible]

Together with the above estimates on [I.sub.1] - [I.sub.6], we get that

[mathematical expression not reproducible] (3.6)

3.2 Estimates on the vertical component [v.sup.n]

We begin this part by studying the equation of [v.sup.n], which is stated as follows

[[partial derivative].sub.t] + [D.sup.s.sub.[epsilon]] [v.sup.n] + v x [nabla] [v.sup.n] + [[epsilon].sup.2] [[partial derivative].sub.n]q = 0.

Observing that in the above equation, one can expect that there is no loss of derivative in vertical direction. More precisely, due to divergence free condition, the nonlinear term [v.sup.n] [[partial derivative].sub.n] [v.sup.n] can be rewritten as -[v.sup.n] [div.sub.h] [v.sup.h]. Thus the estimate on [v.sup.n] is different from [v.sup.h].

Applying the anisotropic dyadic decomposition operator [[DELTA].sub.k,j] to the equation of [v.sup.n], then in each dyadic block, [v.sup.n] satisfies

[mathematical expression not reproducible]. (3.7)

Let us define [mathematical expression not reproducible]. We write the solution of (3.7) as follows

[mathematical expression not reproducible]

Taking the [L.sup.p.sub.h]([L.sup.2.sub.v]) norm, we infer that

[mathematical expression not reproducible]

By the Young's inequality, we deduce that

[mathematical expression not reproducible] (3.8)

Multiplying both sides of (3.8) by [mathematical expression not reproducible] and taking the sum over k, j, we have

[mathematical expression not reproducible]

According to Lemma 3, we can obtain the estimates of nonlinear term by the following:

[mathematical expression not reproducible]

This implies that

[mathematical expression not reproducible]

While for the pressure term, we use the decomposition q = [q.sup.1] + [q.sup.2] + [q.sup.3] in (3.1). For [q.sup.1], since [epsilon][(-[[DELTA].sub.[epsilon]]).sup.-1] [[partial derivative].sub.i][[partial derivative].sub.n] is a bounded operator applied for frequency localized functions in [L.sup.p.sub.h] ([L.sup.2.sub.v]) if i =1, 2, ... , n-1, we have

[mathematical expression not reproducible]

Therefore, we get by using Lemma 3 that

[mathematical expression not reproducible]

Similarly, the fact that [[epsilon].sup.2][(-[[DELTA].sub.[epsilon]]).sup.-1] [[partial derivative].sup.2.sub.n] is a bounded operator applied for frequency localized functions in [L.sup.p.sub.h] ([L.sup.2.sub.v]) implies

[mathematical expression not reproducible]

Thus, we have

[mathematical expression not reproducible]

Then we obtain that

[mathematical expression not reproducible]

Combining the above estimates, we can get the bound of [v.sup.n.sub.[PHI]] as follows:

[mathematical expression not reproducible]. (3 9)

Together (3.9) with (3.6), we finally get that

[mathematical expression not reproducible].

This completes the proof of Lemma 4 by choosing A large enough.

4 Estimates for [theta](t)

In the above section, we have used the fact that [PHI](t) is a subadditivity function. This means we should ensure that [theta](t) < [alpha]/[lambda]. Thus, it is sufficient to prove that for any time t, [theta](t) is a small quantity. By the definition of [theta](t), naturally, we assume that [e.sup.[alpha]Dn] [v.sup.n.sub.0] belongs to [mathematical expression not reproducible]. According to the property of the operator [[partial derivative].sub.t] + [D.sup.s.sub.[epsilon]], then we can get the bound for [mathematical expression not reproducible]. However, we can not enclose the estimate for [mathematical expression not reproducible]. Our observation is to add an extra term [[epsilon].sup.n.sub.v] under the same norm which is hidden in the pressure term [[epsilon].sup.2] [[partial derivative].sub.n] [q.sup.1]. Hence, we first denote that

[mathematical expression not reproducible]

In order to get the desired estimates, it suffices to prove the following lemma.

Lemma 5. There exists a constant [C.sub.2] such that for any [lambda] > 0 and t satisfying [theta](t) [less than or equal to] [alpha]/2[lambda] 2A, we have

X(t) + Y(t) [less than or equal to] [C.sub.2]([X.sub.0] + [Y.sub.0]) + [C.sub.2](X(t) + Y(t)) [PSI](t).

Proof. We apply the same method as in the above section to prove Y (t). Indeed, multiplying both sides of (3.8) by [mathematical expression not reproducible] and taking the sum over k, j, we can get that

[mathematical expression not reproducible]

According to Lemma 3, we can obtain the estimates of nonlinear terms that

[mathematical expression not reproducible]

This implies that

[mathematical expression not reproducible]

While for the pressure term, we use the decomposition q = [q.sup.1] + [q.sup.2] + [q.sup.3] in (3.1). For [q.sub.1], since [epsilon][(-[[DELTA].sub.[epsilon]].sup.-1] [[partial derivative].sub.i] [[partial derivative].sub.n] is a bounded operator applied for frequency localized functions in [L.sup.p.sub.h] ([L.sup.2.sub.v]) if i =1, 2, ... , n-1, we have

[mathematical expression not reproducible]

Similarly, the fact that [[epsilon].sup.2][(-[[DELTA].sub.[epsilon]]).sup.-1] [[partial derivative].sup.2.sub.n] is a bounded operator applied for frequency localized functions in [L.sup.p.sub.h] ([L.sup.2.sub.v]) implies

[mathematical expression not reproducible]

Thus, applying Lemma 3, we have

[mathematical expression not reproducible]

Then we obtain that

[mathematical expression not reproducible]

Combining all the above estimates, we can get the bound of [v.sup.n.sub.[PHI]] in [mathematical expression not reproducible] by the following:

[mathematical expression not reproducible]. (4.1)

This completes the proof of Y(t) in Lemma 5.

The following is devoted to getting the estimate of X(t). The horizontal component [v.sup.h] in each dyadic block satisfies

[mathematical expression not reproducible]

Denote [mathematical expression not reproducible], then we infer that

[mathematical expression not reproducible]

By Young's inequality, it holds that

[mathematical expression not reproducible]

Here and in what follows, if s = 1, the quantity [mathematical expression not reproducible] should be regarded as [mathematical expression not reproducible]. Therefore, taking [L.sup.[infinity]] norm and [L.sup.1] norm on [0, t], we deduce that

[mathematical expression not reproducible]. (4.2)

Multiplying both sides of (4.2) by [mathematical expression not reproducible] and taking the sum over k, j, we finally get

[mathematical expression not reproducible]. (4.3)

For the pressure term q = [q.sup.1] + [q.sup.2] + [q.sup.3], we find that

[mathematical expression not reproducible]

where we have used that [(-[[DELTA].sub.[epsilon]]).sup.-1] [[partial derivative].sub.i] [[partial derivative].sub.i] [[partial derivative].sub.j] is a bounded operator for frequency localized functions in [L.sup.p.sub.h]([L.sup.2.sub.v]). Similarly,

[mathematical expression not reproducible]

According to Lemma 3, the right hand side of (4.3) can be bounded by following:

[mathematical expression not reproducible]

These imply that

[mathematical expression not reproducible] (4.4)

Combining (4.4) with (4.1), we finally obtain that there exists a constant [C.sub.2] such that

X(t) + Y(t) [less than or equal to] [C.sub.2] ([X.sub.0] + [Y.sub.0]) + [C.sub.2](X(t) + Y(t)) [PSI] (t).

This completes the proof of Lemma 5.

5 Proof of the main result

In this section, we will prove the Theorem 2. It relies on a continuation argument. For any [lambda] > [[lambda].sub.0] and [[eta].sub.1], we define [tau] by

[tau] = max{t [greater than or equal to] 0| X(t)+ Y(t) [less than or equal to] [[eta].sub.1], [psi](t) [less than or equal to] [[eta].sub.1]}. (5.1)

In what follows, we shall prove that [tau] = [infinity] under the assumption (1.3) for some small number [[eta].sub.1]. Assume that this is not true. We choose [[eta].sub.1] small enough such that

[mathematical expression not reproducible].

For such fixed [[eta].sub.1], we select the following norms of initial data sufficiently small enough such that

[C.sub.1] [PSI] (0) + [C.sub.2](X(0) + Y(0)) [less than or equal to] [C.sub.[eta]] [less than or equal to] [[eta].sub.1]/4.

Hence, we obtain from Lemma 4 and 5 that

[mathematical expression not reproducible].

This implies that

X([tau])+ Y([tau]) [less than or equal to] [[eta].sub.1]/2, [PSI]([tau]) [less than or equal to] [[eta].sub.1]/2.

However, this contradicts (5.1) and hence completes the proof.

https://doi.org/ 10.3846/mma.2018.017

Acknowledgement

The first author was in part supported by NSFC (No. 11626075 and No. 11701131), Zhejiang Province Science fund for Youths(No. LQ17A010007) and Scientific research fund of Hangzhou Dianzi Unversity (No. GK158800299004/005).

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Bin Han (a) and Yukang Chen (b)

(a) Department of Mathematics, Hangzhou Dianzi University Xiasha Higher Education Zone, 310018 Hangzhou, Zhejiang Province, China

(b) School of Mathematical Sciences, Fudan University 220 Handan Rd., Yangpu District, 200433 Shanghai, China

E-mail(corresp.): hanbinxy@163.com

E-mail: yukangchen@hotmail.com

Received August 29, 2017; revised February 23, 2018; accepted February 23, 2018
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