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

We prove that for initial data of the form \begin{equation}\nonumber u_0^\epsilon(x) = (v_0^h(x_\epsilon), \epsilon^{-1}v_0^n(x_\epsilon))^T,\quad x_\epsilon = (x_h, \epsilon x_n)^T, n \geq 4, \end{equation} the Cauchy problem of the incompressible Navier-Stokes equations on $\mathbb{R}^n$ is globally well-posed for all small $\epsilon>0$, provided that the initial velocity profile $v_0$ is analytic in $x_n$ and certain norm of $v_0$ is sufficiently small but independent of $\epsilon$.


Introduction
The Cauchy problem of the incompressible Navier-Stokes equations on R n is described by the following system    ∂ t u + u · ∇u − ∆u + ∇P = 0, x ∈ R n , t > 0, div u = 0, x ∈ R n , t > 0, 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 3 . In the seminal paper [23], Leray proved that the 3D incompressible Navier-Stokes equations are globally well-posed if the initial data u 0 is such that u 0 L 2 ∇u 0 L 2 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 tȯ where 3 < p < ∞. The space BMO −1 (R 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 [13,19,10,20] as references. We also mention that the work of Lei and Lin [21] was the first to quantify the smallness of the initial data to be 1 by introducing a new space X −1 .
We remark that the norm in the above scaling invariant spaces are always greater than the norm in the Besov spaceḂ −1 ∞,∞ defined by 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 largeḂ −1 ∞,∞ are usually called large data to the Navier-Stokes equations (for instance, see [7,25]).
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 ǫ 0 is of the form which allows slowly varying in the vertical variable x 3 when ǫ > 0 is a small parameter. This family of initial data are very interesting (as has been pointed out by V.Šverák, see the acknowledgement in [8]) and considered by Chemin, Gallagher and Paicu in [8]. They proved the global regularity of solutions to the Navier-Stokes equations when v 0 is analytic in x 3 and periodic in x h , and certain norm of v 0 is sufficiently small but independent of ǫ > 0. More precisely, they proved the following Theorem: Theorem 1.1 (Chemin-Gallagher-Paicu, Ann. Math. 2011). Let α be a positive number. There are two positive numbers ǫ 0 and η such that for any divergence free vector field v 0 satisfying e αD 3 v 0 the initial data. It is clear that theḂ −1 ∞,∞ norm of u ǫ 0 can tend to infinity as ǫ → 0. Let us first focus on the periodic constraint imposed on the initial data in Theorem 1. 1. As has been pointed out by Chemin, Gallagher and Paicu, the reason why the horizontal variable of the initial data in [8] is restricted to a torus is to be able to deal with very low horizontal frequencies. In the proof of Theorem 1.1 in [8], 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 v 0 on the horizontal variables. For instance, see [7,15,14,25] 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 n :    ∂ t u + u · ∇u + D s u + ∇P = 0, x ∈ R n , t > 0, div u = 0, x ∈ R n , t > 0, where D = √ −∆. The initial velocity field u ǫ 0 is of the form The horizontal variable x h = (x 1 , x 2 , · · · , x 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 1.2. Let α, ǫ 0 , p and s be four positive constants and (p, s) satisfy 1 ≤ p < n − 1, 1 ≤ s < min(n − 1, 2(n−1) p ). There exists a positive constant η such that for any 0 < ǫ < ǫ 0 and any divergence free vector field v 0 satisfying then the generalized Navier-Stokes equations (1.3) with initial data (1.4) generate a global smooth solution on R n . Remark 1.3. 1) When n ≥ 4, one can find that the classical Navier-Stokes equations on R n satisfy the assumption 1 ≤ s = 2 < min(n − 1, 2(n−1) p ) for 1 ≤ p < n − 1. Then according to Theorem 1.2, 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 ≤ s < 2. The main obstacle when s = 2 is that we can not get the product law inḂ (R 3 ). This anisotropic Besov space is induced by the a priori estimate of anisotropic pressure (∇ h q, ǫ 2 ∂ 3 q) (see equation (1.6) 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 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 p -type Besov space, in which the bilinear estimate of the solution can not be derived by the classical L 2 energy method. Particularly, one can obtain the L 1 estimate in time of the solution by introducing the new quantity in the a priori estimate. We also mention that in [8,25,26], authors did not get the L 1 -time estimate of the solution. Now we mention that many authors make efforts to remove the periodic restriction on horizontal variable. J. Chemin and I. Gallagher considered the well-prepared initial data in [7]. They proved the global well-posedness of (1.1) when u ǫ 0 is of the form . Later, G. Gui, J. Huang, and P. Zhang in [14] generalized this result to the density dependent Navier-Stokes equations with the same initial velocity. Recently, B. Han in [15] considered the global regularity of (1.1) if u ǫ 0 satisfies the form of ) for any 0 < δ < 1, then u ǫ 0 generates a global solution of (1.1) on R 3 . The case δ = 1 2 and δ ∈ (0, 1 2 ) were considered by M. Paicu and Z. Zhang in [25,26] if the initial data is allowed to be in Gevery class. All of the initial data is large inḂ −1 ∞,∞ , but still generates a global solution.
Main ideas of the Proof. We will prove our main result by constructing the bilinear estimate (independent of ǫ). Our strategy can be stated as follows.
Step 1. Rescaled system and simplification. As in [8], we define 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 ǫ): (1.6) where v h = (v 1 , · · · , v n−1 ). The rescaled pressure q can be recovered by the divergence free condition as The global regularity of solutions to system (1.4) for small initial data v 0 will be presented in Section 3 and 4 for any positive ǫ. But to best illustrate our ideas, let us here focus on the case of ǫ = 0. Formally, by taking ǫ = 0 in system (1.4), we have the following limiting system: Step 2. Set-up of the a priori estimate.
Observing that in the rescaled system (1.7), the viscosity is absent in the vertical direction. To make the full use of smoothing effect from operator ∂ t + D s 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: Note that pressure term doesn't explicitly appear in the equation of v n of the limiting system (1.7), which makes the estimate for v n easier. So here let us just focus on the equation of v h . Naturally, we define At this step, we assume that the initial data v h 0 belongs toḂ . This ensures that Ψ(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 According to the estimates of heat equation, one can formally has .

It will be shown that
Step 3. Derivative loss: input the estimate of ∂ n v h .
For the quantity Ψ(t), we need to prove the bilinear estimate in the following form: Certainly, there is ∂ n -derivative loss! By the product law in anisotropic Besov spaces (Lemma 2.3), the strategy to bound the last term is dτ, and then we should add the new quantity is the hardest term to estimate.
Since we note that by Duhamel's principle, There is still ∂ n -derivative loss in the a priori estimates!
Here θ(t) is defined by which will be shown to be small to ensure that Φ(t, |ξ n |) satisfies the subadditivity. Denoting f Φ = e Φ(t,Dn) f , we then have Hence, we can recover the derivative loss by In this way, the losing derivative term can be absorbed by the left hand side of the above inequality by choosing λ sufficient large.
Step 5. The estimate of the pressure term ∇ h q. To estimate the pressure term, we write ) norm of the first term in ∇ h q can be estimated by 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 f g is hard to obtain since we can not control the low horizontal frequency part.
In this step, we want to prove that for any time t, θ(t) is a small quantity. This ensures that the phase function Φ satisfies the subadditivity property. We will go to derive a stronger estimate for However, when ǫ > 0, we can not get the closed estimate for Y (t). Our strategy is to add an extra term ǫv h under the same norm which is hidden in the pressure term ǫ 2 ∂ n q. See section 4 for details.
Step 7. Estimate of v n Φ . If this is done, we could get a closed a priori estimate (see Lemma 3.1) and finish the proof of Theorem 1.2. Observing that the nonlinear term v n ∂ n v n can be rewritten as −v n div h v h due to divergence free condition. Hence, in the limiting system (1.7), there is no loss of derivative in vertical direction on v n . Thus the estimate on v n is much easier than v 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, G. Raugel and G. Sell [27] were able to establish global solutions for a family of large initial data by using anisotropic Sobolev imbedding theorems (see also the paper [18] by D. Iftimie, G. Raugel and G. Sell). By choosing the initial data to transform the equation into a rotating fluid equations, A. Mahalov and B. Nicolaenko [24] obtained global solutions generated by a family of large initial data. A family of axi-symmetric large solutions were established in [16] by Thomas Hou, Z. Lei and C. M. Li. Recently, Z. Lei, F. Lin and Y. Zhou in [22] 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 [22] are not small inḂ −1 ∞,∞ even in the anisotropic sense. We also mention that for the general 3D incompressible Navier-Stokes equations which possess hyper-dissipation in horizontal direction, D. Fang and B. Han in [12] 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 θ(t) will be studied in Section 4. Finally, the proof of the main result will be given in Section 5.

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 D. Iftimie in [17] for the study of incompressible Navier-Stokes equations in thin domains. Let us briefly explain how this may be built in R n . Let (χ, ϕ) be a couple of C ∞ functions satisfying and For u ∈ S ′ (R n )/P(R n ), we define the homogeneous dyadic decomposition on the horizontal variables by Similarly, on the vertical variable, we define the homogeneous dyadic decomposition by The anisotropic Littlewood-Paley decomposition satisfies the property of almost orthogonality: Similar properties hold for ∆ v j . In this paper, we shall use the following anisotropic version of Besov spaces [17]. In what follows, we denote for abbreviation The study of non-stationary equation requires spaces of the type L ρ 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 Chemin-Lerner type spaces: and define the space L ρ T (Ḃ σ,s p,r )(R n )to be the subset of distributions in u ∈ S ′ (0, T ) × S ′ (R n ) with finite L ρ T (Ḃ σ,s p,r ) 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 [3]. Let f, g ∈ S ′ (R n ), where the paraproducts T (f, g) and T (f, g) are defined by Similarly, we can define the decompositions for both horizontal variable x h and vertical variable x n . Indeed, we have the following split in x h .
The decomposition in vertical variable x n can be defined by the same line. Thus, we can write f g as Each term of (2.1) has an explicit definition. Here Similarly, 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 [14]. For completeness, here we prove a similar result in the L p framework.
then we have for any f ∈Ḃ (

2.2)
Proof. According to (2.1), we first give the bound of T h T v (f, g). Indeed, applying Hölder and Bernstein inequality, we get that Since σ 1 ≤ n−1 p , we obtain that where the sequence {c k,j } (k,j)∈Z 2 satisfies c k,j l 1 (Z 2 ) = 1. This gives the estimate of Again, σ 2 ≤ n−1 p implies that The estimate on the remainder operator which concerns the horizontal variable R h T v (f, g) may be more complicated. When 2 ≤ p, the strategy is following: In the case 1 ≤ p < 2, we have As 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, Φ denotes a locally bounded function on R + × R which satisfies the following subadditivity (see (3.5) for the explicit expression of Φ) For any function f in S ′ (0, T ) × S ′ (R n ), we define Let us keep the following fact in mind that the map f → f + preserves the norm of x h , x n ) represents the inverse Fourier transform of |f (t, x h , ξ n )| on vertical variable, defined as On the basis of these facts, we have the following weighted inequality as in Lemma 2.3.
Proof. We only prove theḂ Using the fact that f → f + preserves the norm of L p h (L 2 v ), we then get by the similar method as in Lemma 2.3 that The other terms in (2.1) can be estimated by the same method and finally, we have The following lemma is a direct consequence of Lemma 2.4.

Estimates for the re-scaled system
This section is devoted to obtaining the a priori estimate for the following system The pressure q can be computed by the formula Due to the divergence free condition, the pressure can be split into the following three parts It is worthwhile to note that there will lose one vertical derivative owing to the term v n ∂ n v h and pressure terms q 2 , q 3 which appear in the equation on v h . Thus, we assume that the initial data is analytic in the vertical variable. This method was introduced in [4] to compensate the losing derivative in x 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 θ(t) by and denote The phase function Φ(t, D n ) is defined by Φ(t, ξ n ) = (α − λθ(t))|ξ n |, (3.5) for some λ that will be chosen later on, α is a positive number. Obviously, we need to ensure that θ(t) < α λ which implies the subadditivity of Φ. The following lemma provides the a priori estimate of v Φ in the anisotropic Besov spaces, which is the key bilinear estimate. Lemma 3.1. There exist two constants λ 0 and C 1 such that for any λ > λ 0 and t satisfying θ(t) ≤ α 2λ , we have

Estimates on the horizontal component v h
According to the definition of v h Φ , we find that in each dyadic block, it verifies the following equation (3.7) We first estimate the linear term I 1 . In fact, we have where {c k,j } (k,j)∈Z 2 is a two dimensional sequence satisfying c k,j l 1 (Z 2 ) = 1. The term I 2 can be rewritten as By Young's inequality, we have .

Thus, we can get by Lemma 2.4 and 2.5 that
Now we are left with the study of the pressure term I 3 . The pressure can be split into q = q 1 + q 2 + q 3 with q 1 , q 2 , q 3 defined in (3.2). For convenience, we denote that Hence, using the fact that (−∆ ǫ ) −1 ∂ i ∂ j is a bounded operator applied for frequency localized functions in L p h (L 2 v ) when i, j = 1, 2, · · · , n − 1, we get . By the same method as in the estimate of I 2 , we have and as in the estimate of I 2 , it holds that we write I 33 as follows By Young's inequality and Lemma 2.5, as 1 ≤ s < min{n − 1, 2 n−1 p }, we have Now we are going to estimate the key quantity According to (3.6) , we find that in each dyadic block (3.12) Taking the L p h (L 2 v ) norm on both sides of (3.12), we have For fixed k, j, multiplying the (3.13) byθ(τ ) and integrating over (0, t), one can have The term I 4 containing initial data can be bounded by (3.14) By Fubini's theorem, the term I 5 can be rewritten as Thus, we can get by Lemma 2.4 and 2.5 that As for I 6 , for convenience, we denote that By the same method as in the estimate of I 5 , we have Finally, I 63 can be estimated as follows Together with the above estimates on

Thus, we can obtain that
(3.17)

Estimates on the vertical component v n
We begin this part by studying the equation of v n , which is stated as follows 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 n ∂ n v n can be rewritten as −v n div h v h . Thus the estimate on v n is different from v h . Applying the anisotropic dyadic decomposition operator ∆ k,j to the equation of v n , then in each dyadic block, v n satisfies We write the solution of (3.18) as follows Taking the L p h (L 2 v ) norm, we infer that (3.20) By the Young's inequality, we deduce that (3.21) Multiplying both sides of (3.21) by 2 k( n−1 p +1−s) 2 j 1 2 and taking the sum over k, j, we have . (3.22) According to Lemma 2.5, we can obtain the estimates of nonlinear term by the following: .

This implies that
.
While for the pressure term, we use the decomposition q = q 1 + q 2 + q 3 in (3.2). For . Therefore, we get by using Lemma 2.5 that Similarly, the fact that ǫ 2 (−∆ ǫ ) −1 ∂ 2 n is a bounded operator applied for frequency localized functions in L p h (L 2 v ) implies .

Thus, we have
.
Then we obtain that Combining the above estimates, we can get the bound of v n Φ as follows: (3.23) Together (3.23) with (3.17), we finally get that This completes the proof of Lemma 3.1 by choosing λ large enough.

Estimates for θ(t)
In the above section, we have used the fact that Φ(t) is a subadditivity function. This means we should ensure that θ(t) < α λ . Thus, it is sufficient to prove that for any time t, θ(t) is a small quantity. By the definition of θ(t), naturally, we assume that e αDn v n ). However, we can not enclose the estimate for ). Our observation is to add an extra term ǫv h under the same norm which is hidden in the pressure term ǫ 2 ∂ n q 1 . Hence, we first denote that .
In order to get the desired estimates, it suffices to prove the following lemma.
Lemma 4.1. There exists a constant C 2 such that for any λ > 0 and t satisfying θ(t) ≤ α 2λ , we have Proof. We apply the same method as in the above section to prove Y (t). Indeed, multiplying both sides of (3.21) by 2 ( n−1 p −s)k 2 1 2 j and taking the sum over k, j, we can get that According to Lemma 2.5, we can obtain the estimates of nonlinear terms that While for the pressure term, we use the decomposition q = q 1 + q 2 + q 3 in (3.2). For q 1 , since ǫ(−∆ ǫ ) −1 ∂ i ∂ n is a bounded operator applied for frequency localized functions in L p h (L 2 v ) if i = 1, 2, · · · , n − 1, we have Similarly, the fact that ǫ 2 (−∆ ǫ ) −1 ∂ 2 n is a bounded operator applied for frequency localized functions in L p h (L 2 v ) implies Thus, applying Lemma 2.5, we have .
Then we obtain that Combining all the above estimates, we can get the bound This completes the proof of Y (t) in Lemma 4.1.
The following is devoted to getting the estimate of X(t). The horizontal component v h in each dyadic block satisfies Here and in what follows, if s = 1, the quantity ∆ k,j . Therefore, taking L ∞ norm and L 1 norm on [0, t], we deduce that (4.5) Multiplying both sides of (4.5) by 2 ( n−1 p −s)k 2 1 2 j and taking the sum over k, j, we finally get For the pressure term q = q 1 + q 2 + q 3 , we find that where we have used that (−∆ ǫ ) −1 ∂ i ∂ j is a bounded operator for frequency localized functions in L p h (L 2 v ). Similarly, According to Lemma 2.5, the right hand side of (4.6) can be bounded by following: .

Proof of the main result
In this section, we will prove the Theorem 1.2. It relies on a continuation argument. For any λ > λ 0 and η 1 , we define τ by In what follows, we shall prove that τ = ∞ under the assumption (1.3) for some small number η 1 . Assume that this is not true. We choose η 1 small enough such that For such fixed η 1 , we select the following norms of initial data sufficiently small enough such that C 1 Ψ(0) + C 2 (X(0) + Y (0)) ≤ Cη ≤ η 1 4 .

(5.2)
This implies that However, this contradicts (5.1) and hence completes the proof.