Dynamical Behavior for the Solutions of the Navier-Stokes Equation

We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \begin{align} u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0(x). \label{NSa} \end{align} Leray and Giga obtained that for the weak and mild solutions $u$ of NS in $L^p(\mathbb{R}^d)$ which blow up at finite time $T>0$, respectively, one has that for $d<p \leq \infty$, $$ \|u(t)\|_p \gtrsim ( T-t )^{-(1-d/p)/2}, \ \ 0<t<T. $$ We will obtain the blowup profile and the concentration phenomena in $L^p(\mathbb{R}^d)$ with $d\leq p\leq \infty$ for the blowup mild solution. On the other hand, if the Fourier support has the form ${\rm supp} \ \widehat{u_0} \subset \{\xi\in \mathbb{R}^n: \xi_1\geq L \}$ and $\|u_0\|_{\infty} \ll L$ for some $L>0$, then \eqref{NSa} has a unique global solution $u\in C(\mathbb{R}_+, L^\infty)$. Finally, if the blowup rate is of type I: $$ \|u(t)\|_p \sim ( T-t )^{-(1-d/p)/2}, \ for \ 0<t<T<\infty, \ d<p<\infty $$ in 3 dimensional case, then we can obtain a minimal blowup solution $\Phi$ for which $$ \inf \{\limsup_{t \to T}(T-t)^{(1-3/p)/2}\|u(t)\|_{L^p_x}: \ u\in C([0,T); L^p) \mbox{\ solves \eqref{NSa}}\} $$ is attainable at some $\Phi \in L^\infty (0,T; \ \dot B^{-1+6/p}_{p/2,\infty})$.


Introduction
We study the Cauchy problem of the incompressible Navier-Stokes equations (NS) with initial data in L ∞ (R d ), d 2: u t − ∆u + u · ∇u + ∇p = 0, divu = 0, u(0, x) = u 0 (x), (1.1) where u = (u 1 , ..., u d ) denotes the flow velocity vector and p(t, x) describes the scalar pressure. ∇ = (∂ 1 , ..., ∂ d ), ∆ = ∂ 2 1 + ... + ∂ 2 d , u 0 (x) = (u 0 1 , ..., u 0 d ) is a given velocity with div u 0 = 0. It is easy to see that (1.1) can be rewritten as the following equivalent form: u t − ∆u + P div(u ⊗ u) = 0, u(0, x) = u 0 (x), (1.2) where P = I − ∇∆ −1 div is the projection operator onto the divergence free vector fields. The solution u of NS formally satisfies the conservation It is known that (1.1) is essentially equivalent to the following integral equation: u(t) = e t∆ u 0 + t 0 e (t−τ )∆ P div(u ⊗ u)(τ )dτ (1.4) and the solution of (1.4) is said to be a mild solution. Note that (1.1) is scaling invariant in the following sense: if u solves (1.1), so does u λ (t, x) = λu(λ 2 t, λx) and p λ (t, x) = λ 2 p(λ 2 t, λx) with initial data λu 0 (λx). A function space X defined in R d is said to be a critical space for (1.1) if the norms of u λ (0, x) in X are equivalent for all λ > 0 (i.e., u λ (0, ·) X ∼ u 0 X ). It is easy to see that L d andḢ d/2−1 are critical spaces of NS. For the sake of convenience, we will denote by N S(u 0 ) the solution of (1.1) (or simply denote it by u if there is no confusion), and by T (u 0 ) the supremum of all T > 0 so that the solution N S(u 0 ) exists in the time interval [0, T ].
Many years ago, Leray [33] showed that NS in 3D has at least one weak solution and he mentioned certain necessary blowup conditions for the weak solutions: (1.5) The existence of the mild solution in L p was established by Kato in [26] and the blowup rate (1.5) in all spatial dimensions was recovered by Giga [19] for the mild solution in C([0, T (u 0 )); L p ) with d < p < ∞. The blowup in the critical space L 3 (R 3 ) was first considered by Escauriaza, Seregin and Sverak [14] and they proved lim sup t→T (u 0 ) N S(u 0 )(t) L 3 = ∞ if T (u 0 ) < ∞, similar results in critical spacesḢ 1/2 and L 3 were obtained in [27,17] via the profile decomposition arguments developed by Kenig and Merle [28] together with the backward uniqueness in [14]. Seregin [38] proved that lim t→T (u 0 ) N S(u 0 )(t) L 3 = ∞ if T (u 0 ) < ∞. Blowup results in L d for higher spatial dimensions were obtained in Dong and Du [13] by following the approach in [14]. Recently, some generalizations for the blowup rates inḢ s (3/2 s 5/2) were obtained in [39,12], Some other kind of blowing up criteria can be found in Kozono, Ogawa and Taniuchi [32] and references therein.
Before stating our main result, we first give some notations. C 1, c 1 will denote constants which can be different at different places, we will use A B to denote A CB, A ∼ B means that A B and B A. We denote by L p = L p (R d ) the Lebesgue space on which the norm is written as · p . f Ḣs = (−∆) s/2 f 2 and H s = L 2 ∩Ḣ s for s 0. Let us write for any ρ > 0, R d +,ρ := {ξ = (ξ 1 , ..., ξ d ) ∈ R d : ξ 1 ρ}. (1.7) The standard iteration sequence for NS is defined in the following way: e (t−τ )∆ P div(u (n) ⊗ u (n) )(τ )dτ, u (0) = 0. (1.8) We will mainly consider the concentration behavior of the blowup solutions and obtain a global well-posedness result in L ∞ . The well-posedness of NS in L p with d < p < ∞ was established in Giga [19]: In Theorem 1.1 the left case is p = ∞. The Cauchy problem of the Navier-Stokes equations in L ∞ and in BU C spaces is studied by Cannone, Meyer [7,8], Giga et al. [20] and they proved a unique existence of a local-in-time solution in L ∞ and in BU C spaces. In [20], the authors also obtained the smoothness of the solutions. We will obtain a concentration phenomena of the blowup solutions with initial data only in L ∞ . The first main result of this paper is For any 1 p ∞, there exist x n ∈ R d and t n ր T (u 0 ) < ∞ such that Let {x j,n } ⊂ R d and {λ j,n } ⊂ (0, ∞) be two sequences. (λ j,n , x j,n ) ∞ n=1 (j ∈ N) are said to be orthogonal sequences of scales and cores, if for any j 1 = j 2 , j 1 , j 2 ∈ N, one has that lim n→∞ λ j,n λ k,n + λ k,n λ j,n + |x j,n − x k,n | λ j,n = ∞.
can be decomposed into the following profiles: where r J n is a reminder that satisfies lim J→∞ lim sup n→∞ r J n d = 0, moreover, we have α n = N S(u 0 )(t n ) Ḣd/2−1 , λ j (t n ) α −2/(d−2) n and in particular, for p 2, (1.15) Remark 1.4 Theorems 1.2 and 1.3 needs several remarks.
(i) Noticing that ω(t) c (T (u 0 ) − t) −1/2 , we have for any p d, 17) which implies that the solution has a concentration in a very small ball with radius less than or equals to C T (u 0 ) − t n in L d .
(ii) Taking p = d in (1.15) and noticing that lim which means that a very large potential norm is concentrated in a very small ball with radius less than or equals to C N S(u 0 )(t n ) H d/2−1 . However, up to now, it is not very clear for us how to unify the concentration phenomena of (1.17) and (1.18).
(iii) In the blowup profile decomposition (1.14), noticing that λ j (t n ) → 0 as t n ր T (u 0 ), we see that concentration blowup is the only way in all of the possible blowing up manners.
(iv) Taking n 0 = 0 in (1.12), we see that condition (1.12) can be substituted by the following condition: Noticing that u 0 is supported in R +,ρ , we see that condition (1.19) contains a class of large data in L ∞ if ρ ≫ 1 which are out of the control ofḂ −1+d/p p,q with p < ∞.
Following [14,27,17], we see that, for the initial data u 0 in the critical spaces X = We can further ask what happens if X is a sub-critical spaces, say X =Ḣ s (R 3 ), s > 1/2, or X = L p (R 3 ), p > 3. Noticing the blowup rate as in (1.6), we have the following question in H s : [37] first considered such a kind of question and she introduced If M σs c < ∞, E. Poulon proved that M σs c can be attainable for some u 0 ∈Ḣ s and the corresponding solution is uniformly bounded in critical Besov spaceḂ 1/2 2,∞ (R 3 ). Following Theorem 1.1, we can define similar critical "minimal" quantity adapted to the L p (R 3 ) scale for 3 < p < ∞ (σ p = 2/(1 − 3/p)): If M σp c < ∞, Giga's Theorem 1.1 implies that there exists u 0 ∈ L p such that the solution of NS blows up at finite time T (u 0 ) with the blowing up rate Such a kind of solution is said to be type-I blowing up solution (cf. [31]). We have the following result: such that Φ := N S(Φ 0 ) blows up at time 1, and satisfies ( △ j := F −1 ϕ j F , j ∈ Z are said to be the dyadic decomposition operators. One easily sees that suppϕ j ⊂ B(0, 2 j+1 ) \ B(0, 2 j−1 ). For convenience, we denote The norms in homogeneous Besov spaces are defined as follows: Using the heat kernel, we have (see [6,40]) (1.26) The rest of this paper is organized as follows. In Section 2, we consider the wellposedness and blowup concentration of NS in L ∞ and prove Theorem 1.2. Using the "profile decomposition" techniques, in Section 3 we consider the blowup profile for the blowing up solution in H d/2−1 and show Theorem 1.3. In Section 5 we will prove our Theorem 1.5, whose proof consists of two steps, constructing a critical solution in L p anḋ B −1+6/p p/2,∞ respectively. The proof of Theorem 1.5 relies upon a profile structure theorem, whose proof will be given in Sections 6 and 7. Finally, in the Appendix, we state some basic estimates on NS and prove a perturbation result which is useful in obtaining the estimate of the remainder term in the profile structure theorem.
2 Initial data in L ∞

Local well-posedness and blowup analysis
We will frequently use the following Bernstein's multiplier estimate (see [4,42]): Recall that (see [10]) Similarly, we have Proof. The idea follows from [10] (see also [42]). By Young's inequality, we have By scaling argument and Lemma 2.1, we have In view of (2.4) and (2.5), we immediately have (2.3).
For convenience, we denote Proof. By the dyadic decomposition, Using Lemma 2.2, one sees that The result follows.
Lemma 2.4 (Short time estimates of lower frequency) Let j 0 ∈ Z. We have for any (2.10) Proof. Since e −t|ξ| 2 (ξ j ξ k ξ l |ξ| −2 ) ∈ L 1 (R d ) for any t > 0, whose Fourier transform has an integral form, it follows that which is the result, as desired. Proof. The proof of the local existence can be found in [20]. However, the blowup rate (1.10) is very important for our later purpose and we sketch the proof here. Put Let us consider the mapping (2.14) Using e t∆ : L ∞ → L ∞ , and Lemmas (2.3) and (2.4), we have Taking j 0 such that 2 −2j 0 ∼ t 0 , one has that Similarly, Further, one can choose t 0 verifying We easily see that T is a contraction mapping from D into itself. So, T has a unique fixed point in T , which is a solution of (1.4). It is easy to see that u ∈ C((0, T ]; L ∞ ) (see [20], for instance). The solution can be extended exactly in the same way as above. Indeed, considering the mapping: one has that Similarly, So, we can extend the solution from [0, Repeating the procedure as above, we can extend the solution step by step to [t 1 , Now let us assume that t i ր T . If T < ∞, we easily see that lim sup t→T N S(u 0 )(t) ∞ = ∞. Moreover, we can show (1.10) holds true. Assume for a contrary that there exist two sequences s n ր T and c n ց 0 satisfying Observing the integral equation similarly as in (2.20), we have for any s n s < T , Taking c n ≪ 1 and s → T , we see that (2.28) contradicts with (2.24) or with the fact that lim sup t→T N S(u 0 )(t) ∞ = ∞.
Lemma 2.6 (Concentration) Let T < ∞ and N S(u 0 ) ∈ L ∞ ([0, T ) × R d ) be the solution of (1.4) obtained in Lemma 2.5. Then for any 1 ≤ p < ∞, there exist two sequences {x n } and {τ n } of x n ∈ R d and τ n ր T satisfying Proof. Put τ 0 = 0. Since lim t→T ω(t) = ∞, we can find a t 1 > τ 0 such that We can further find a τ 1 ∈ (τ 0 , t 1 ] verifying ω(t). (2.31) Then we have Repeating this procedure, one can choose a monotone sequence {τ n } verifying Claim. For simplicity, we write u := N S(u 0 ). We have In fact, if (2.34) does not hold, then one has that Let us consider the integral equation It follows from Lemma 2.3 and (2.35) that A contradiction! So, we have (2.34). For convenience, we write Now we prove (2.29). There exists x n ∈ R d such that )u(τ n , y)dy By Hölder's inequality, Taking a suitable M ≫ 1, one easily sees that So, it follows from (2.41) and Hölder's inequality that So, From (2.34) and (2.42) we immediately have the result, as desired.

Global well-posedness
For convenience, we denote Let us recall the iteration sequence and induction, it follows from (2.47) that (2.46) holds true.
Proof. By (2.47), we have for any n > n 0 , By Lemmas 2.7 and 2.8, it follows from (2.50) that Using a similar way, one can estimate the second term in (2.50). So, in view of (2.50) and (2.51) we have Repeating the procedure as in (2.52), one can obtain that for any n > n 0 , 2M 0 , n > n 0 + 1. We prove the Claim. Taking n = n 0 + 1 in (2.52), we have from (2.49) that 2M 0 . Now let us assume the following induction assumption holds true: (2.55) We show that (2.55) also holds for m = n + 1. Applying (2.53), one has that for any ℓ = n 0 + 1, ..., n − 1, We have Finally, we show that {u (n) } is a Cauchy sequence. Again, in view of (2.56), we have for any n > m ≫ n 0
However, Theorem 1.2 implies the well-posedness for a class of large data inḂ −1+d/p p,q . Let ϕ be as in (1.23), ϕ(ξ) = ϕ(ξ)χ {1/2 · 2} (ξ) 1 and where c ≪ 1 is a small constant. It is easy to see that divu 0 = 0. We have, It follows from Theorem 1.2 that for the initial data as in (2.59), NS is globally well-posed in L ∞ . Also, we see that

Blowup profile
If we consider the profile decomposition inḢ s , the orthogonal condition (1.13) can be replaced by In this section we always assume that u 0 ∈ H d/2−1 and div u 0 = 0. Let the solution of NS blow up at T > 0. According to the results in [13,14,27,38], we see that For convenience, we denote for any λ j,n > 0, x j,n ∈ R d , Let us recall the profile decomposition for a bounded sequence inḢ d/2−1 (see [23]).
Theorem 3.1 Let u 0,n be a bounded sequence of divergence-free vector fields inḢ d/2−1 .
Then there exists {x j,n } ⊂ R d and {λ j,n } ⊂ (0, ∞) which are orthogonal in the sense of (3.1), and a sequence of divergence-free vector fields where ω J n is a reminder in the sense that Moreover, we have Using the conservation law (1.3) in L 2 , we can further show that Proof. In view of (1.3) and where ω J n satisfies (3.5) and (3.6). Let j 0 ∈ N. For any J j 0 , one has (3.7). As So we can take a R 0 > 0 such that In particular, we have λ j 0 ,n → 0 as n → ∞.
Proof. We can assume that the reminder term ω J n satisfying (3.10). Otherwise we can choose another profile decomposition with J J 0 . Let us write Taking the inner product of (3.7) and In view of (3.12), we have By Hölder's inequality, (3.5) and (3.8), We have First, we estimate I. By Hölder's inequality, we have Next, we estimate II and divide the proof into the following three cases. Case 1. λ j 0 ,n = λ j,n and lim n→∞ |x j 0 ,n − x j,n |/λ j,n = ∞. Since ϕ j ∈ C ∞ 0 (R d ), and one can also approximate ϕ j 0 by a C ∞ 0 (R d ) function, in view of (3.12) we see that II → 0 if n is sufficiently large.
can be decomposed in the following way: where ω J n satisfies lim J→∞ lim sup n→∞ ω J n d = 0 and moreover, α n ∼ N S(u 0 )(τ n ) Ḣd/2−1 , Proof. Let us observe that we have from (3.26) that Again, in view of (3.8), one has that for any 2 p ∞, (3.33) Combining (3.32) and (3.33), Taking x j,n = x j (τ n ), λ j (τ n ) = λ j,n , we see that the result follows from the profile decomposition in Theorem 3.1.

Profile decomposition in L p
First, let us recall the following theorem concerning the profile decomposition of bounded sequence in L p (R d ). Using the diagonal method, we can assume that, up to an extraction, lim n→∞ λ j,n exists for all j ∈ N (whose limit can be +∞). We further denote For j ∈ J 1 , we will simply call λ j,n constant scale. Now f n can be rewritten as . Next, we show that, if the scales λ j,n are very small or very large, then the corresponding profiles Λ d/p j,n φ j are very small in appropriate function spaces.   Proof. First, we prove (1). Let p > p 1 , J, η fixed, then The proof of (2) is similar to (1), and the details of the proof are omitted. Finaly, we prove (3). By the assumptions on p, q, r and Proposition A.1, we can easily see L p ֒→Ḃ sr,p r,q , thus By the Lebesgue dominated convergence theorem and the definition of φ j,η c , we see that So, we have the result, as desired. Now, let us denote lim n→∞ λ j,n = λ j , Λ j φ j = 1 Proposition 4.2 indicates that the scales with indices in J 0 ∪ J ∞ may not take the major roles to the Navier-Stokes evolution, as they can be considered to be the error term in some function spaces. In fact, we have the following Theorem 4.3 (NS evolution of the bounded L p initial data) Fix 3 < p < ∞, let (u 0,n ) n 1 be a bounded sequence of divergence-free vector fields in L p (R 3 ), whose profile decomposition is given in the following form as in T heorem 4.1: We have the following conclusions. and T (u 0,n ) = ∞ if J 1 is empty.
(ii) There exists a J 0 ∈ N and N (J) ∈ N, up to an extraction on n, such that is well-defined for J > J 0 , n > N (J) and t < T , moreover, for any T < T , The solution satisfies the orthogonality in the sense Remark 4.4 T is attainable, provided J 1 is nonempty, i.e. there exists some j 0 ∈ J 1 such that T = T (Λ j 0 φ j 0 ). In fact, this is a simple consequence of (1.9) and (4.3). The proof of Theorem 4.3 will be given in Sections 6 and Section 7.

Proof of Theorem 1.5
Denote In this Section, we will prove Theorem 1.5 by using Theorem 4.3, whose proof are separated into two steps. First, we show that there is a minimal solution N S(u 0 ) ∈ D c . Secondly, using the smoothing effect of the bilinear term of the Navier-Stokes equation, we finally construct a new solution which belongs to both L p andḂ −1+6/p p/2,∞ . The proof follows the same ideas as in Poulon [37]. Then the following conclusions hold (passing to a subsequence if necessary)

Existence of minimal solution in subcritical
• In the profile decomposition of v 0,n , there exists a unique scale profile Λ k 0 φ k 0 such that N S(Λ k 0 φ k 0 ) blows up at time 1.
Proof. Assume the profile decomposition of (v 0,n ) has the following form (i) We show that T = T (Λ k 0 φ k 0 ) = 1. In view of (4.6) in Theorem 4.3, one has that T 1. Assume T < 1, for any τ < T , by (4.9), p σp on both sides, and using the assumption (5.4), one has that (5.7) Taking limit with regard to n and J on both sides, we have The left hand side of (5.8) can be arbitrarily small if we take τ sufficently close to T , however, the right hand side of (5.8) has a lower bound as in Theorem 1.1, which leads to a contradiction. Hence T = T (Λ k 0 φ k 0 ) = 1.
(iii) Profile with constant scale blowing up at time 1 is unique. We also prove it by contradiction. Assume there is another constant scale profile Λ j 0 φ j 0 which blows up at time 1. Using (4.9) again, we find Repeating the the argument in (i), we can deduce that Due to N S(Λ k 0 φ k 0 ) ∈ D c , we must have N S(Λ j 0 φ j 0 ) = 0, so φ j 0 = 0. The proof of Lemma 5.2 is finished.

Existence of solutions in
In general, we could not expect that the linear term e t∆ u 0 belongs toḂ −1+6/p p/2,∞ , provided we only assume the initial data u 0 ∈ L p . To get the desired result, we need to use the regularization effect of the biliner term B(u, u). In fact, based on our assumption, the following proposition holds.
Proof. First, we prove (5.9). It follows from (2.2) that Since u(t) is a blowup solution of type-I, Let us observe that (5.9) is obtained.
Next we prove a lemma which allows us to exclude profiles without constant scales in the profile decomposition of a bounded sequence in L p (R 3 ).
Lemma 5.4 Let 3 < p < ∞, 0 < s < 1 − 3/p, then the following statements hold: then there are no scales which tend to infinity in the L p profile decomposition of f n . x − x j,n λ j,n + ψ J n .
It follows from lim sup n→∞ f n Ḃ0 p,∞ > 0 that f n ≡ 0, As we know, each φ j is a weak limit of some rigid tranformation of f n , more precisely, we have in the sense of tempered distribution. Without loss of generality, we assume φ j = 0 for each j. Observing due to the hypothesis on f n and λ k,n n→∞ −−−→ ∞. By lower semi-continuity, we get so φ k = 0, this contradicts our assumption. The proof of (2) proceeds in a similar way as that of (1) and the details are omitted.
Proof of Theorem 1.5. We only outline the main procedures in the proof of existence of a critical element inḂ −1+6/p p/2,∞ , as the argument is quite similar to [37]. Indeed, we already have a minimal solution Ψ = N S(Ψ 0 ) ∈ D c from Proposition 5.1, so there exists a time sequence t n → T (Ψ 0 ) and ǫ n → 0 such that In this circumstance, ψ J n + τ 1 2 n e tn∆ Ψ 0 (τ 1 2 n x) can be considered as the remainders, and according to Lemma 5.4, in the profile decomposition of ρ n , only profiles with constant scales are left, and each φ j (x) is the weak limit of λ 3/p j,n ρ n (λ j,n x + x j,n ), so all these profiles lies in L p ∩Ḃ −1+6/p p/2,∞ ∩Ḃ s p,∞ . Now it can be easily checked that (v 0,n ) n 1 satisfy the assumptions of Lemma 5.2, hence there exists a unique scale profile Λ l 0 φ l 0 ∈ L p ∩Ḃ −1+6/p p/2,∞ such that N S(Λ l 0 φ l 0 ) ∈ D c and blows up at time 1. Moreover for any 0 < τ < 1, we have where we have used Proposition 5.3. Therefore we finish the proof of Theorem 1.5.

Proof of Theorem 4.3
This section is devoted to the proof of (i) and (ii) in Theorem 4.3. Let (u 0,n ) n 1 be a bounded sequence in L p and the corresponding profile decomposition is given below where Q(a, b) := a · ∇b + b · ∇a = 2∇ · (a ⊗ b) for any divergence free vector fields a, b and Now (i) and (ii) of Theorem 4.3 will be consequences of the following two lemmas and Proposition A.6 in the appendix. We remark that by the lower semi-continuity in Corollary A.7, for arbitrary ε > 0, there exists a N := N (ε), such that n > N , Let us turn to the proof of Lemma 6.1 and Lemma 6.2. First, we give a lemma, which will be used in the sequel. Lemma 6.3 Let J ∈ N, 3 < p < ∞, r = 5p/3. Let φ j ∈ L p and (x j,n , λ j,n ) n 1 be orthogonal scales and cores in the sense of (1.13), then for any T inf j∈J 1 (J) T j , T j < T (Λ j φ j ) is arbitrary. Then there exists N (T, J) such that for any n > N (T, J), where lim n→∞ ε(J, n) = 0. Furthermore, for any 0 t T , up to an extraction to n ∈ N, From (6.5) it follows that Taking n → ∞, up to an extraction to n, one has that lim sup In view of Remark 4.5, we see that there exists a J c ∈ N such that for j J c , Therefore, the left hand side of (6.6) can be bounded by Using the same way as above, in view of (6.4)and l p ֒→ l  Combining the estimate on I and II, we finally complete the proof of Lemma 6.1.
Next, we prove Lemma 6.3 by following the ideas in Lemma 2.6 of [17]. However, in the subcritical cases Λ Proof of Lemma 6.3. For r = 5p/3, we have where ε(J, n) : = Here we have used an elementary inequality (cf.,e.g., [23]) In order to show the result, we divide the proof into the following three steps.
Step 2. For any j ∈ J 1 and φ j ∈ L p , we show that In fact, in view of the translation invariance of NS, one can regard x j,n = 0 in (6.16). Now, let v n := N S(Λ 3/p j,n φ j ) − N S(Λ j φ j )(x), obviously it satisfies the following perturbation equation Applying Proposition A.6 and (6.13), we see that for sufficiently large n > N (T ), v n Hence, we have (6.16) and So, by (6.17) and (6.9), Step 3. We show that By (6.16), it suffices to show that Noticing that |x j,n − x k,n | → ∞, we see that the left hand side of (6.20) is zero when n is sufficiently large. Hence, we have (6.19). Now (6.4) follows from the estimates of Steps 1-3. Next, we turn to prove (6.5).
In view of (6.16), (6.6) and (6.7), we see that lim n→∞ sup 0≤t≤T |α 1 (t, J, n)| = 0. (6.22) Let us write It is easy to see that The desired conclusion follows if we obtain that (up to a subsequence for n ∈ N) Now let us turn our attention to the proof of Lemma 6.2, which shows the smallness of the forcing term. First we present the main inequalities that will be used, whose proof relies heavily on the space-time estimate of heat kernel. Proposition 6.4 Let 3 < p < ∞, 1/β(p) = 1/2 − 3/10p.
Proof of Lemma 6.2: we rewrite G J n so as to make use of the smallness of profiles without constant scales. We make a cut-off on φ j and denote It follows that G J n = G J,1 n,η + G J,2 n,η + G J,3 n,η + G J,4 n , where G J,1 n,η = 2 First, we consider the estimate of G J,1 n,η . Let r be as in (ii) of Proposition 6.4, we see that As a result of (6.9) and (6.10), one has that lim J→∞ lim sup In addition, r < p, by (1)  Next, using (i) of Proposition 6.4 and (2) of Proposition 4.2, the estimate of G J,2 n,η is quite similar to G J,1 n,η and we omit the details. Thirdly, we estimate of G J,3 n,η . Let σ(p), s σ(p),p be as in Proposition 6.4 (iii), so the following inequality holds, Finally, the smallness of G J,4 n has been given as in Lemma 6.3. Combining the estimate on G J,1 n,η , G J,2 n,η , G J,3 n,η and G J,4 n , we complete the proof of Lemma 6.2.

Orthogonality of the profiles of L p -solutions
This section is intended to prove the orthogonality property of the Navier-Stokes solutions, i.e. formula (4.9) in Theorem 4.3. Let us introduce some simplified notations first, define where U 0 n,η , U ∞ n,η , ψ J n,η are defined as (6.26). Recall that we have Proposition 7.1 LetĀ J n , B J n be as above, 0 < t < T be fixed, then it holds Proof: Let ǫ > 0 be arbitrary, as we know Concerning |Ā J n | p−1 |B J n |, it suffices to verify the smallness of |Ā J n | p−1 |e t∆ U 0 n,η |, |Ā J n | p−1 |e t∆ U ∞ n,η | and |Ā J n | p−1 |e t∆ ψ J n,η |. Let us estimate them separately.
First, we show that For each j ∈ J 1 , we approximate N S(Λ j φ j ) by a smooth function with compact support in L p spaces, denote this function as Θ j,t (x), so we see where we have used Hölder inequality in the last inequality and 1 < a < p, a ′ satisfies 1 a + |Ā J n | p−1 |e t∆ ψ J n,η |dx = 0.
Proposition A.3 Let 1 ≤ r ≤ p ≤ ∞ and 1 < γ, γ 1 < ∞ satisfy Then we have Then As a direct consequence, we have we now give several estimates adapted to our needs.
Proposition A.6 Let 3 < p < ∞, w ∈ X p T be vector fields in R 3 , f ∈ L β(p) T L 5p/6 x is a 3 × 3 matrix function. β(p) is the same as Lemma 6.1, 0 < T < ∞, v satisfies the following perturbative NS equation (PNS) Assume that there exist two constants ǫ 0 ≪ 1 and C ≫ 1 such that Then v belongs to X p T , moreover we have  .
As a direct consequence, we have the following corollary.
Corollary A.7 The mapping w 0 → T (w 0 ) is lower semi-continuous for (NS) with initial in L p , 3 < p < ∞, i.e. for arbitrary ε > 0, there exists a η > 0, such that if v 0 p < η, then T (w 0 + v 0 ) T (w 0 ) − ε. So, if u ∈ D T := {u ∈ X p T : u X p T 2C u 0 p }, we can choose T satisfying T (1−3/p)/2 u 0 p 1/4C 2 . It follows that T u ∈ D T . Using contraction mapping principle, we obtain that NS has a unique solution u ∈ X p T and the above proof implies the result of Remark 4.5.