LOCAL WELL-POSEDNESS FOR NAVIER-STOKES EQUATIONS WITH A CLASS OF ILL-PREPARED INITIAL DATA

. In this paper, we prove that for the ill-prepared initial data of the form the Cauchy problem of the incompressible Navier-Stokes equations on R 3 is locally well-posed for all (cid:15) > 0, provided that the initial velocity proﬁle v 0 is analytic in x 3 but independent of (cid:15) .

1. Introduction. In this paper, we consider the Cauchy problem of the 3D incompressible Navier-Stokes equations, which is described by the following system      ∂ t u + u · ∇u − ∆u + ∇P = 0, x ∈ R 3 , t > 0, div u = 0, x ∈ R 3 , t > 0, where u represents the velocity field and P is the scalar pressure. The initial velocity field u 0 is of the form which allows slowly varying in the vertical variable x 3 when > 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]) considered by Chemin, Gallagher and Paicu in [8] and Y. Chen, B. Han and Z. Lei in [10]. In [8], the authors 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 > 0. More precisely, they proved the following Theorem: ≤ η, then, for any positive smaller than 0 , the initial data (2) generates a global smooth solution to (1) on T 2 × R.
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 [7,15,14,25,10] and so on. Particularly, authors in [10] proved that 3D generalized Navier-Stokes equations with initial data (2) admits a global solution on R 3 by using the anisotropic Besov spaces. Futher more, authors in [10] proved that for there exists a global solution to the 4D Navier-Stokes equations with initial data (2).
In this paper, we want to show that for the initial data of the form (2), the Cauchy problem of 3D incompressible Navier-Stokes equations on R 3 is locally well-posed for all > 0, provided that v 0 is analytic in x 3 but independent of . Our main result states as follows.
the Navier-Stokes equations (1) with initial data (2) generates a local smooth solution on R 3 .
Before going any further, let us recall some known results on the small-data global regularity of the Navier-Stokes equations. 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 used by Leray 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 toḢ is known to be the largest scaling invariant space so that the Navier-Stokes equations (1) are globally well-posed under small initial data. The readers are referred to [13,19,4,20] 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 [3] showed that the cauchy problem of the 3D Navier-Stokes equation 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 well-prepared case was considered by J. Chemin and I. Gallagher in [7]. They proved the global well-posedness of (1) when u 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] consider global regularity of (1) if u 0 satisfies the form of for any δ > 0, then u 0 generates a global solution of (1) on R 3 . The case δ = 1 2 was considered by M. Paicu and Z. Zhang in [25]. The case δ = 0 was proved in [10] for the 3D generalized Navier-Stokes equations on R 3 and 4D Navier-Stokes equations on on R 4 . One can note that, all of the initial data is large inḂ −1 ∞,∞ , but still generates a global solution. 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.
In the following, we will introduce the sketch of the proof of Theorem 1.2.
2. Main ideas of the Proof. We will prove the result by construct the bilinear estimate (independent of ) of the rescaled system of (1). The following is the outline of our method. In preparation, we should rescale the system. As in [8], we define . Using the Navier-Stokes equations (1), it is easy to derive the equations governing the rescaled variables v and q (they are still depending on ): The rescaled pressure q can be recovered by the divergence free condition as The local existence result of the solutions to system (4) for small initial data v 0 will be presented in Section 3 for any positive . But to best illustrate our ideas, let us here focus on the case of = 0. Formally, by taking = 0 in system (4), we 2990 KEYAN WANG AND YAO XIAO have the following limiting system: The pressure q in (5) is given by Based on the new system, we try to find the best a priori estimate. An initial estimate. Observe that in the rescaled system (5), the viscosity is absent in the vertical direction. To make the full use of smoothing effect from horizontal heat kernel ∂ t − ∆ h , particularly in low frequency part, we will apply the theories in anisotropic homogeneous Besov spaces.
Naturally, we define The goal is to derive the certain a priori estimate of the form: According to the estimates of heat equation, one can formally has .

It is easy to see that
For the pressure term, we split it into three parts .
We have to deal with f g Ḃ δ−1, 1 2 2,1 type estimate as follows. Hence, we should choose that δ is a small positive number.
The ∂ 3 -derivative loss. Now we have the bilinear estimate in the following form:

LOCAL WELL-POSEDNESS FOR NAVIER-STOKES EQUATIONS 2991
However, it is not easy to estimate ∂ 3 v h . By the continuity property of the product in anisotropic Besov spaces (Lemma 3.5), the strategy to bound the last term is , and then we should add the new norm in the definition of Ψ(t). We is the hardest term to estimate. Since we note that by Duhumel's principle, There are still ∂ 3 -derivative loss in the a priori estimates.
Modification. Aiming at the ∂ 3 -derivative loss, motivated by Chemin-Gallagher-Paicu ([8]), we add an exponential weight e Φ(t,|D3|) with Here θ(t) is defined by which will be proved to be small to ensure that Φ(t, |ξ 3 |) satisfies the subadditivity. Denoting So that we recover the derivative loss by where the sequence {c k,j } (k,j)∈Z 2 satisfies c k,j l 1 (Z 2 ) = 1. Now we define the new Estimation for θ(t). For the introduced term θ(t), we should prove that for any time t, θ(t) is a small quantity. This can ensure that the phase function Φ satisfies the subadditivity property. Using the interpolation inequality (Lemma 3.4), we have This means that we should get the a priori estimate when the initial data v 3 0 in a low regularity spaceḂ δ, 1 2 2,1 . Then we are going to estimate .

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However, when > 0, we can not get the closed estimate for Y (T ). Our observation is to add an extra term v h under the same norm which is hidden in the pressure term 2 ∂ 3 q. Then we define a new quantity: We will get the closed estimate for X(t) + Y (t).
Estimation for v 3 . To complete the prove, we should estimate v 3 . In fact, the estimate for v 3 is much easier than v h . From the limiting system (5), there is no loss of derivative in vertical direction. Due to divergence free condition, the nonlinear 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.
3. 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 3 . Let (χ, ϕ) be a couple of C ∞ functions satisfying For u ∈ S (R 3 )/P(R 3 ), 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) = L ρ (0, 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 denote by L ρ Before giving some properties of the anisotropic Besov spaces, we recall the Hölder and Young's inequalities in the framework of anisotropic Lebesgue spaces.
where 1 + 1 r = 1 r + 1 r and 1 + 1 p = 1 p + 1 p . 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 [2].
Let f, g ∈ S (R 3 ), 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 3 . Indeed, we have the following split in x h .
The decomposition in vertical variable x 3 can be defined by the same line. Thus, we can write f g as Each term of (8) has an explicit definition. Here Similarly, and so on. At this moment, we can state some properties concerning the continuity of the product in anisotropic Besov spaces. The first lemma is about the spaces in the framework of L 2 . And similar result was proved in [14], for completeness, we also give the details of the proof here.
Proof. According to (8), we first give the bound of T h T v (f, g). Indeed, applying Lemma 3.3 and Bernstein inequality, we get that Since σ 1 , σ 2 ≤ 1, we obtain by applying Young's inequality that The remainder operator which concerns with the horizontal variable R h T v (f, g) can be bounded as follows: As

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For the remainder operator on both horizontal and vertical variables R h R v (f, g), we get by applying Lemma 3.3 and Bernstein inequality that This implies that The other terms can be followed exactly in the same way, here we omit the details. This completes the proof of this lemma.
In the proof of our main result, we require the similar continuity results in the Chemin-Lerner type spaces L ρ T (Ḃ σ,s 2,1 ) as in Lemma 3.4.
If σ 1 + σ 2 is positive and σ 1 , σ 2 ≤ 1, then for any f ∈ L ρ1 T (Ḃ σ1, 1 2 2,1 ) and g ∈ L ρ2 T (Ḃ σ2, 1 2 2,1 ), we have . Throughout this paper, Φ denotes a locally bounded function on R + × R which satisfies the following subadditivity For any function f in Let us keep the following fact in mind that the map f → f + preserves the norm of Base on these facts, we have the following weighted inequalities as in Lemma 3.4. Proof. For fixed k, j, we have Using the fact that f → f + preserves the norm of L 2 , we then get by the similar method as in Lemma 3.4 that The other terms in (8) can be estimated by the same method and finally, we have The following lemma is a direct consequence of Lemma 3.6.

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 2998 KEYAN WANG AND YAO XIAO It is worthwhile to note that there will lose one vertical derivative owing to the term v 3 ∂ 3 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 [5] to compensate the losing derivative in x 3 . 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 For any δ ∈ (0, 1), denote where v Φ is defined as in (10). The phase function Φ(t, D 3 ) is defined by 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 estimate of v Φ in the anisotropic Besov spaces, which is the key bilinear estimate.
Lemma 4.1. There exists two constants C 1 and λ 0 such that for any λ > λ 0 and t satisfying θ(t) ≤ α 2λ , we have For the horizontal component v h , it is unavoidable that we will lose one derivative in vertical direction. Formally, the terms v 3 ∂ 3 v h , ∇ h q 2 and ∇ h q 3 are the main bad elements.

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 Then, taking the L 2 norm, we deduce that We first estimate the linear term I 1 . In fact, we have and 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 For I 21 , by Young's inequality, we have .
Thus, we can get by Lemma 3.7 that .

Using Hölder inequality and interpolation, we have
where the exponents satisfy the conditions 1 p + 1 p = 1 and 1 p = δ 2 .
Then one can deduce that

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Similarly, we can obtain that Now let us estimate I 22 . By Lemma 3.6, we have Hence, by Young's inequality, we have 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 (12). For convenience, we denote that Hence, using the fact that (−∆ ) −1 ∂ i ∂ j is a bounded operator applied for frequency localized functions in L 2 when i, j = 1, 2, we get By the same method as in the estimate of I 21 , we have Noting that and as the estimate of (23), we have Similarly, by Hölder and interpolation inequality, we have where the exponents satisfy the conditions 1 q + 1 q = 1 and 1 q = 1+δ 2 . Using we write I 33 = I 1 33 + I 2 33 and first estimate that Thus, we have For the term I 2 33 , Then one can get that Now we are going to estimate the L ∞ t (B δ, 1 2 2,1 ) norm of ∂ 3 v h Φ . According to (16) , we find that in each dyadic block Then, taking the L 2 norm on both side of (27), we have For fixed k, j, the L ∞ t norm of I 4 can be bounded by The term I 5 can be rewritten as By Young's inequality, we can deduce that Thus, we can get by Lemma 3.7 that .

LOCAL WELL-POSEDNESS FOR NAVIER-STOKES EQUATIONS 3003
By Hölder and interpolation inequality, we have where the exponents satisfy the conditions 1 r + 1 r = 1 2 and 1 r = δ 2 .
Then one can deduce that Now let us estimate I 52 . By Lemma 3.6, we have .
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 estimate as follows dτ.

KEYAN WANG AND YAO XIAO
Thus, we can get that Together with the above estimates (18)-(32), we obtain that

4.2.
Estimates on the vertical component v 3 . We begin this part by studying the equation of v 3 , 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 Applying the anisotropic dyadic decomposition operator ∆ k,j to the equation of v 3 , then in each dyadic block, v 3 satisfies Writing the solution of (36) in terms of the Fourier transform in vertical variable, we get that Taking the L 2 (R 3 ) norm on both sides of (35), we can obtain that By Young's inequality, one can infer that Multiplying both sides of (37) by 2 δk 2 1 2 j and taking the sum over k, j, we finally get Then Lemma 3.7 implies that For the pressure term, due to the fact that (−∆ ) −1 ∂ i ∂ 3 and 2 (−∆ ) −1 ∂ 2 3 are bounded operators in L 2 for i = 1, 2, we get by applying Lemma 3.7 that .
By the result of interpolation inequalities above, we infer that Together with (33), we finally get that This completes the proof of Lemma 4.1 by choosing λ large enough.

5.
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 α|D3| v 3 0 belongs toḂ ). Our observation is to add an extra term v h under the same norm which is hidden in the pressure term 2 ∂ 3 q 1 . Hence, we first denote that In order to get the desired estimates, it suffices to prove the following lemma.
Lemma 5.1. There exists a constant C 2 such that for any λ and t satisfying θ(t) ≤ α 2λ , for any M 1 > 0 ,there exists N 1 ∈ Z + , we have Proof. We apply the same method as in the above section to prove Y (t). For (36), we use Young's inequality to obtain Multiplying both sides of (42) by 2 kδ 2 1 2 j and taking the sum over k, j, we can get that Similarly, we can get that Concerning the first term in (43)  and a large part which could be controlled by Y 0 . For some fixed M 1 > 0, there exits some positive real number N 1 , such that Thus, we have and k,j According to Lemma 3.7, we can obtain the estimates of nonlinear terms that .

Using interpolation inequality, this implies that
While for the pressure term, we use the decomposition q = q 1 + q 2 + q 3 in (12). For q 1 , since (−∆ ) −1 ∂ i ∂ 3 is a bounded operator applied for frequency localized functions in L 2 if i = 1, 2, we have . Therefore, we get by using Lemma 3.7 that Similarly, the fact that 2 (−∆ ) −1 ∂ 2 3 is a bounded operator applied for frequency localized functions in L 2 implies . Thus, applying Lemma 3.7 again, we have Then we obtain that Combining all the above estimates, we can get the bound of Y (t) by the following: This completes the proof of Y (t) in Lemma 5.1. The following is devoted to getting the estimate of X(t). The horizontal component v h in each dyadic block satisfies We note that Then we get by Young's inequality that and t 0 e −c(2 2k + 2 2 2j )(t−τ ) ∆ k,j ∂ 3 (v 3 v h ) Φ L 2 dτ L 1

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Therefore, by taking L 2 norm on [0, t] for (50), we deduce that Multiplying both sides of (51) by 2 kδ 2 1 2 j and taking the sum over k, j, we finally get For the first term about v h 0 , by (46), for some fixed M 1 > 0 ,we have k,j 2 kδ 2 1 2 j e −c(2 2k + 2 2 2j )t For the pressure term q = q 1 + q 2 + q 3 , we find that where we have used that (−∆ ) −1 ∂ i ∂ j is a bounded operator for frequency localized functions in L 2 . Similarly, we have . According to Lemma 3.7, the right hand side of (52) can be bounded by following: and .
These imply that Then, taking L 1 norm on [0, t] for (50) , by the same method, we deduce that Multiplying both sides of (55) by 2 (δ+1)k 2 1 2 j and taking the sum over k, j, we finally get For the first term about v h 0 , by (47), for some fixed M 1 > 0 ,we have k,j 2 k(δ+1) 2 1 2 j e −c(2 2k + 2 2 2j )t By the same methods as the above estimates, we can also get the bound of v h Φ L 1 t (Ḃ δ+1, 1 2 2,1 ) by the following: Combining (48) with (58) and (54), we finally obtain that there exists a constant C 2 such that This completes the proof of Lemma 5.1.
Proof of Theorem 1.2. In this section, we will prove the Theorem 1.2. It relies on a continuation argument. For any λ > λ 0 and η, we assume that: We can choose M large enough such that