Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

Abstract

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spaces 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical case 𝛽=1/2. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.

1. Introduction

We study the mild solutions to the fractional Navier-Stokes equations in 𝑅⁺×𝑅³ as follows: 𝑢_𝑡+𝜇(−Δ)^𝛽𝑢+(𝑢⋅∇)𝑢+∇𝜋=0,(𝑡,𝑥)∈𝑅⁺×𝑅³;∇⋅𝑢=0,(𝑡,𝑥)∈𝑅⁺×𝑅³;𝑢(0,𝑥)=𝑢₀(𝑥),𝑥∈𝑅³. (1) Here 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),𝑢₃(𝑡,𝑥)) denotes the velocity vector, 𝜇>0 is the viscosity coefficient, and the scalar function 𝜋 denotes the pressure. The initial data 𝑢₀ is a divergence free vector field and the operator (−Δ)^𝛽 is the Fourier multiplier with symbol |𝜉|^2𝛽.

The fractional Navier-Stokes equations, which are also called generalized Navier-Stokes equations, enjoy an invariance under the scaling 𝑢_𝜆(𝑡,𝑥)=𝜆^2𝛽−1𝑢𝜆^2𝛽,𝑝𝑡,𝜆𝑥_𝜆(𝑡,𝑥)=𝜆^4𝛽−2𝑝𝜆^2𝛽,𝑢𝑡,𝜆𝑥_0,𝜆=𝜆^2𝛽−1𝑢₀(𝜆𝑥). (2) We say that a function space is 𝛽-critical for (1) if its norm is invariant under the scaling 𝑢₀(𝑥)→𝜆^2𝛽−1𝑢₀(𝜆𝑥). There are many examples of critical spaces, for instance, 𝐵𝑀𝑂^{−(2𝛽−1)}, 𝐵̇^{−(2𝛽−1)}_∞,∞, and the spaces we will discuss in this paper.

The classical incompressible Navier-Stokes equations (i.e., 𝛽=1) have been intensively studied. Leray first [1] introduced the concept of weak solutions and obtained the global existence of weak solutions. Fujita and Kato [2] gave a different approach to study the equations in their equivalent form of integral equations and proved the well-posedness in the space frame 𝐻̇^1/2. A series study of mild solutions in different function spaces then arose, for instance, Kato [3] in Lebesgue space 𝐿³(𝑅³), Cannone [4] in Besov space 𝐵̇^−1+3/𝑝_𝑝,∞, and the important well-posedness in 𝐵𝑀𝑂⁻¹ by Koch and Tataru [5]. These works naturally lead one to study the well-posedness in the largest critical space 𝐵̇⁻¹_∞,∞. In fact, all the above spaces are critical spaces and satisfy the following continuous embeddings in the 3 dimensions: 𝐻̇^1/2↪𝐿³↪𝐵̇^−1+3/𝑝_{𝑝,∞(𝑝<∞)}↪𝐵𝑀𝑂⁻¹↪𝐵̇⁻¹_∞,∞. (3) However, in the space 𝐵̇⁻¹_∞,∞, the Navier-Stokes equations are ill-posedness (see Bourgain and Pavlović [6] and Cheskidov and Shvydkoy [7]).

As for the generalized case (1), Lions [8] proved the global existence of classical solutions in 3 dimensions when 𝛽≥5/4 (see also Wu [9] in 𝑛 dimensions). For the important case 𝛽<5/4, Wu [10, 11] studied the well-posedness in 𝐵̇^{1−2𝛽+3/𝑝}_𝑝,𝑞. Inspired by Xiao [12] in the classical case (𝛽=1), Li and Zhai [13, 14] studied (1) in some critical 𝑄-type spaces for 𝛽∈(1/2,1), and Zhai [15] showed the well-posedness in 𝐵𝑀𝑂^{−(2𝛽−1)} when 𝛽∈(1/2,1). For the biggest critical space 𝐵̇^{−(2𝛽−1)}_∞,∞, Yu and Zhai [16] proved the well-posedness when 𝛽∈(1/2,1), Cheskidov and Shvydkoy [17] showed the ill-posedness when 𝛽∈[1,5/4). Very recently, Deng and Yao [18] studied (1) in Triebel-Lizorkin spaces 𝐹̇^−𝛽_𝛼,𝑟 and obtained the well-posedness in 𝐹̇^−𝛽_{3/(𝛽−1),2} and ill-posedness in 𝐹̇^−𝛽_{3/(𝛽−1),𝑟} (𝑟>2) in the case 𝛽∈(1,5/4).

In this paper, we will study (1) in the Fourier-Besov spaces 𝐹𝐵̇^𝑠_𝑝,𝑞. We observe that although the Fourier-Besov spaces 𝐹𝐵̇^𝑠_𝑝,𝑞 appear in the literature very recently, they have received a lot of attentions in studying Navier-Stokes equations, although sometime people gave these spaces several different names. An early paper by Cannone and Karch [19] worked in the space 𝒫ℳ^𝑎, which is in fact the space 𝐹𝐵̇^𝑎_∞,∞ (see Section 2 for details). Biswas and Swanson [20] studied the Gevrey regularity of Navier-Stokes equations in 𝐹𝐵̇^2−3/𝑝_𝑝,𝑝. Konieczny and Yoneda [21] used 𝐹𝐵̇^𝑠_𝑝,𝑞 to study the Navier-Stokes equations with Coriolis (see also Fang et al. [22]). Lei and Lin [23] proved global existence of mild solutions in 𝒳⁻¹, which is in fact equal to the space 𝐹𝐵̇⁻¹_1,1. Cannone and Wu [24] extended the result in [23] to the Fourier-Herz spaces ℬ̇^𝑠_𝑞. We may notice that ℬ̇^𝑠_𝑞= 𝐹𝐵̇⁻¹_1,𝑞. Also, some properties of solutions in the space 𝒳⁻¹ have been studied recently; see Zhang and Yin [25] for the blow-up criterion and Benameur [26] for the long time decay. All the above-mentioned works are involved in the classical Navier-Stokes equations. Those indicate that the Fourier-Besov spaces 𝐹𝐵̇^𝑠_𝑝,𝑞 might be good work frames in the study of Navier-Stokes equations. Inspired by these observations, in this paper, we will study generalized Navier-Stokes equations in 𝐹𝐵̇^𝑠_𝑝,𝑞. We obtain a global well-posedness result which is more general than those in [23, 24]. Particularly, our well-posedness is also valid in the critical case 𝛽=1/2. Moreover, the long time decay of the solutions in Fourier-Besov spaces is also proved, which fully extends the result of [26].

Throughout this paper, the notation 𝐴∼𝐵 means that there exist positive constants 𝐶₁≤𝐶₂ such that 𝐶₁𝐴≤𝐵≤𝐶₂𝐴. We use 𝐵̇^𝑠_𝑝,𝑞 to denote the classical homogenous Besov spaces and 𝐻̇^𝑠 the homogenous Sobolev spaces. Also, 𝐶 denotes a positive constant which may differ in lines if not being specified; 𝑝^′ is the number satisfying 1/𝑝+1/𝑝^′=1 for 1≤𝑝≤∞. The inverse Fourier transform is denoted by ℱ⁻¹.

We organize the paper as follows. In Section 2 we give the definition of Fourier-Besov spaces and discuss some basic properties of these spaces. Our main results are also stated in this section. In Section 3 we prove the global well-posedness and in Section 4 we prove the long time decay property.

2. Preliminaries and Main Results

We first introduce the definition of Fourier-Besov spaces in 𝑛 dimensions. Let 𝜑∈𝐶^∞_𝑐(𝑅^𝑛) be a radial real-valued smooth function such that 0≤𝜑(𝜉)≤1 and supp𝜑⊂𝜉∈𝑅^𝑛∶34≤𝜉≤83_𝑗∈𝑍,𝜑2^−𝑗𝜉=1,forany𝜉≠0. (4) We denote 𝜑_𝑗(𝜉)=𝜑(2^−𝑗𝜉) and ℙ the set of all polynomials. The space of tempered distributions is denoted by 𝑆^′.

Definition 1. For 𝑠∈𝑅,1≤𝑝,𝑞≤∞, set 𝑓_{𝐹̇𝐵𝑠𝑝,𝑞}=_𝑗∈𝑍2^𝑗𝑠𝑞𝜑_𝑗𝑓^𝑞_𝐿𝑝^1/𝑞,𝑞<∞;sup_𝑗∈𝑍2^𝑗𝑠𝜑_𝑗𝑓_𝐿𝑝,𝑞=∞. (5) One defines the homogeneous Fourier-Besov space 𝐹𝐵̇^𝑠_𝑝,𝑞 as 𝐹𝐵̇^𝑠_𝑝,𝑞=𝑆𝑓∈^′ℙ∶𝑓_{𝐹̇𝐵𝑠𝑝,𝑞}<∞. (6)

We see that the Fourier-Besov spaces are defined in a similar way with the classical homogeneous Besov spaces, but there are lack of the inverse Fourier transform. This allows us to derive estimates by Hölder's inequality directly, instead of using Bernstein's inequality. Now we explain that Fourier-Besov spaces contain some known spaces applied in studying Navier-Stokes equations.

Cannone and Karch [19] introduced the spaces 𝒫ℳ^𝑎 as follows: 𝒫ℳ^𝑎=𝑣∈𝑆^′∶𝑣̂∈𝐿¹_loc,‖𝑣‖_𝒫ℳ^𝑎=esssup_{𝜉∈𝑅𝑛}𝜉^𝑎𝑣̂(𝜉)<∞. (7) We easily see that 𝒫ℳ^𝑎=𝐹𝐵̇^𝑎_∞,∞.

The norm of Fourier-Herz spaces ℬ̇^𝑠_𝑞 in [24] is defined as 𝑓_{̇ℬ𝑠𝑞}=_𝑗∈𝑍2^𝑗𝑠𝑞𝜑_𝑗𝑓^𝑞_𝐿1^1/𝑞,𝑞<∞;sup_𝑗∈𝑍2^𝑗𝑠𝜑_𝑗𝑓_𝐿1,𝑞=∞. (8) Obviously, we have ℬ̇^𝑠_𝑞=𝐹𝐵̇^𝑠_1,𝑞.

The space 𝒳⁻¹ introduced by Lei and Lin [23] is 𝒳⁻¹=𝑓∈𝑆^′𝑅^𝑛∶_𝑅𝑛𝜉⁻¹𝑓𝑑𝜉<∞. (9) We claim that 𝒳⁻¹=𝐹𝐵̇⁻¹_1,1. This fact can be seen by the following proposition (proof in [21]).

Proposition 2. Define the spaces 𝒳^𝑠 as 𝒳^𝑠=𝑆𝑓∈^′ℙ∶_𝑅𝑛𝜉^𝑠𝑝𝑓^𝑝𝑑𝜉^1/𝑝<∞. (10) Then one has 𝒳^𝑠=𝐹𝐵̇^𝑠_𝑝,𝑝 and the norms are equivalent: 𝑓_{𝐹̇𝐵𝑠𝑝,𝑝}∼_𝑅𝑛𝜉^𝑠𝑝𝑓^𝑝𝑑𝜉^1/𝑝. (11)

We discuss some inclusion relationships in 𝐹𝐵̇^𝑠_𝑝,𝑞.

Proposition 3. Let 𝑠∈𝑅, 1≤𝑝, 𝑞≤∞. One has the follwoing.(1)If 𝑝=2, then 𝐹𝐵̇^𝑠_2,𝑞=𝐵̇^𝑠_2,𝑞.(2)If 𝑝<2, then 𝐹𝐵̇^𝑠_𝑝,𝑞⊂𝐵̇^𝑠_𝑝′,𝑞.(3)If 𝑞₁≤𝑞₂, then 𝐹𝐵̇^𝑠_𝑝,𝑞1⊂𝐹𝐵̇^𝑠_𝑝,𝑞2.(4)If 1≤𝑞≤∞, 1≤𝑝₁≤𝑝₂≤∞, and 𝑠₁,𝑠₂∈𝑅 satisfy 𝑠₁+𝑛/𝑝₁=𝑠₂+𝑛/𝑝₂, then 𝐹𝐵̇^𝑠2_𝑝2,𝑞⊂𝐹𝐵̇^𝑠1_𝑝1,𝑞. (12)(5)If 𝑠=(1−𝜃)𝑠₁+𝜃𝑠₂, 1/𝑝=(1−𝜃)/𝑝₁+(1−𝜃)/𝑝₂, and 1/𝑞=(1−𝜃)/𝑞₁+(1−𝜃)/𝑞₂ for 0≤𝜃≤1, then 𝑓_{𝐹̇𝐵𝑠𝑝,𝑞}≤𝑓_𝐹̇𝐵^1−𝜃𝑠_1𝑝11,𝑞𝑓^𝜃_𝐹̇𝐵^𝑠_2𝑝22,𝑞. (13)

Proof. (1) is a consequence of Plancherel’s identity, and Hausdorff-Young’s inequality gives (2). Equation (3) is just the inclusion 𝑙^𝑞1⊂𝑙^𝑞2 for 1≤𝑞₁≤𝑞₂≤∞. To conclude (4), we use Hölder’s inequality to get 𝜑_𝑗𝑓_𝐿𝑝1≤𝐶2^{𝑗𝑛(1/𝑝1−1/𝑝2)}𝜑_𝑗𝑓_𝐿𝑝2. (14) Since 𝑠₁ and 𝑠₂ satisfy 𝑠₁+𝑛/𝑝₁=𝑠₂+𝑛/𝑝₂, we immediately get 2^𝑗𝑠1𝜑_𝑗𝑓_𝐿𝑝1≤𝐶2^𝑗𝑠2𝜑_𝑗𝑓_𝐿𝑝2. (15) Taking the 𝑙^𝑞-norm on the above inequality we have 𝑓_𝐹̇𝐵^𝑠_1𝑝1,𝑞≤𝐶𝑓_𝐹̇𝐵^𝑠_2𝑝2,𝑞. (16) To prove (5), we have 𝑓_{𝐹̇𝐵𝑠𝑝,𝑞}=_𝑗2^𝑗𝑠𝑞𝜑_𝑗𝑓^1−𝜃𝜑_𝑗𝑓^𝜃𝑞_𝐿𝑝^1/𝑞≤_𝑗2^{𝑗(1−𝜃)𝑠1𝑞}𝜑_𝑗𝑓_𝐿^{(1−𝜃)𝑞}_𝑝12^{𝑗𝜃𝑠2𝑞}𝜑_𝑗𝑓_𝐿^𝜃𝑞_𝑝2^1/𝑞≤𝑓_𝐹̇𝐵^1−𝜃𝑠_1𝑝11,𝑞𝑓^𝜃_𝐹̇𝐵^𝑠_2𝑝21,𝑞. (17)

Now we are ready to state our main results. From now on in this paper we take the dimension 𝑛=3.

Definition 4. Let 𝑠∈𝑅, 1≤𝑝, 𝑞≤∞, and 𝐼=[0,𝑇), 𝑇∈(0,∞]. The space-time norm is defined on 𝑓(𝑡,𝑥) by 𝑓(𝑡,𝑥)_{ℒ𝑟̇𝐵(𝐼;𝐹𝑠𝑝,𝑞)}∶=_𝑗∈𝑍2^𝑗𝑠𝑞𝜑_𝑗𝑓^𝑞_{𝐿𝑟𝐼;𝐿𝑝}^1/𝑞. (18)

Our first result is on the well-posedness of (1).

Theorem 5. Let 1≤𝑝, 𝑞≤∞, and 1/2<𝛽<min{1+3/𝑝^′,5/2}. Then there exists a constant 𝐶₀=𝐶₀(𝛽,𝑝,𝑞) such that, for any 𝑢₀∈𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞 with ∇⋅𝑢₀=0 satisfying 𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}<𝐶₀𝜇, (19) the Cauchy problem (1) admits a unique global mild solution 𝑢 and 𝑢∈𝒞[0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞∩ℒ¹[0,∞);𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞, (20) and it satisfies ‖𝑢‖_{ℒ∞̇𝐵([0,∞);𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}+𝜇‖𝑢‖_{ℒ1̇𝐵([0,∞);𝐹′1+3/𝑝𝑝,𝑞)}≤21+169^𝛽𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (21) Particularly, our result also holds in the critical case 𝑞=1 and 𝛽=1/2.

Remark 6. We emphasize that the case 𝛽=1/2 is important, since it is also the critical case for the fractional Navier-Stokes equations. Note that when 𝛽=1/2, the function spaces we work on are 𝐹𝐵̇^3/𝑝′_𝑝,1. All these spaces are embedded into 𝐹𝐵̇⁰_1,1, which is the space that consists of all functions whose Fourier transforms are in 𝐿¹ (see Proposition 2).

Remark 7. Note that 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞⊂𝐵̇^−2𝛽−1_∞,∞ by Proposition 3 and the space 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞 are also critical spaces. In fact, for 𝑢_0,𝜆=𝜆^2𝛽−1𝑢₀(𝜆𝑥), we have 𝑢_0,𝜆=𝜆^2𝛽−4𝑢₀(𝜆⁻¹𝜉). Set 𝑔_𝑗(𝜉)∶=𝜑2^{−𝑗+[log2𝜆]−log2𝜆}𝜉𝑢_0,𝜆(𝜉). (22) Then we have 2^{𝑗(1−2𝛽+3/𝑝′)}𝑔_𝑗𝐿𝑝=2^{𝑗(1−2𝛽+3/𝑝′)}𝜑2^{−𝑗+[log2𝜆]−log2𝜆}𝜉𝜆^2𝛽−4𝑢₀𝜆⁻¹𝜉_𝐿𝑝=2^{([log2𝜆]−log2𝜆)(1−2𝛽+3/𝑝′)}×2^{(𝑗−[log2𝜆])(1−2𝛽+3/𝑝′)}𝜑2^{−𝑗+[log2𝜆]}𝜂𝑢₀𝜂_𝐿𝑝∼2^{(𝑗−[log2𝜆])(1−2𝛽+3/𝑝′)}𝜑(2^{−𝑗+[log2𝜆]}𝜂)𝑢₀(𝜂)_𝐿𝑝. (23) This implies that _𝑗∈𝑍2^{𝑗𝑞(1−2𝛽+3/𝑝′)}𝑔_𝑗^𝑞_𝐿𝑝^1/𝑞∼𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (24) On the other hand, by 𝜑_𝑗(𝜉)𝑢_0,𝜆(𝜉)=_{𝑘−𝑗≤2}𝜑_𝑗(𝜉)𝑔_𝑘(𝜉), (25) we can easily deduce that 𝑢_{0,𝜆𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}∼_𝑗∈𝑍2^{𝑗𝑞(1−2𝛽+3/𝑝′)}𝑔_𝑗^𝑞_𝐿𝑝^1/𝑞. (26)

Unfortunately, Theorem 5 is not suitable for the case 𝛽=1,𝑝=1, in which similar existence has been proved by Cannone and Wu [24]. To address this case, we also get the following theorem.

Theorem 8. Let 1≤𝑝≤𝑞≤2 and 𝛽∈(1/2,1+3/2𝑝^′]. Then there exists a constant 𝐶₀=𝐶₀(𝛽,𝑝,𝑞) such that, for any 𝑢₀∈𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞 with ∇⋅𝑢₀=0 satisfying 𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}<𝐶₀𝜇, (27) the Cauchy problem (1) admits a unique global mild solution 𝑢 and 𝑢∈𝒞[0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞∩ℒ¹[0,∞);𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞, (28) and it satisfies ‖𝑢‖_{ℒ∞̇𝐵([0,∞);𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}+𝜇‖𝑢‖_{ℒ1̇𝐵([0,∞);𝐹′1+3/𝑝𝑝,𝑞)}≤21+169^𝛽𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (29) Particularly, our result also holds in the critical case 𝑝=𝑞=1 and 𝛽=1/2.

Remark 9. In comparison with Theorem 5, although we have a limitation 1≤𝑝≤𝑞≤2, the regularity index 𝛽 in Theorem 10 lies in a larger interval when 𝑝=1.

Our third result is on the decay property of the global solutions

Theorem 10. Let 1≤𝑝≤𝑞≤2 and 𝛽∈(5/6,1]. Assume that 𝑢∈𝐶([0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞) is a global solution of (1). One has limsup_𝑡→∞‖𝑢(𝑡)‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}=0. (30)

Remark 11. Recently, Benameur [26] obtained the same property in the space 𝒳⁻¹=𝐹𝐵̇⁻¹_1,1 for the classical Navier-Stokes equations (𝛽=1). Our result improves and extends his result.

3. The Well-Posedness

First, we study the linear estimates of (1). For this purpose we consider the dissipative equation: 𝑢_𝑡+𝜇(−Δ)^𝛽𝑢=𝐹(𝑡,𝑥),(𝑡,𝑥)∈𝑅⁺×𝑅³;𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈𝑅³. (31) It is easy to see that the equivalent integral equation of (31) is 𝑢(𝑡,𝑥)=𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀−^𝑡₀𝑒^{−𝜇(𝑡−𝜏)(−Δ)𝛽}𝐹(𝜏,𝑥)𝑑𝜏. (32) By taking 𝐹(𝑡,𝑥)=0 or 𝑢₀(𝑥)=0, we obtain the linear term or the nonlinear term of the equation, respectively. This indicates that the following lemma is very useful in our later proof.

Lemma 12 (linear estimate). Let 𝐼=[0,𝑇), 0<𝑇≤∞, 𝑠∈𝑅, 1≤𝑝, and 𝑞≤∞. Assume that 𝑢₀∈𝐹𝐵̇^𝑠_𝑝,𝑞 and 𝐹∈ℒ¹(𝐼;𝐹𝐵̇^𝑠_𝑝,𝑞). Then the solution 𝑢(𝑡,𝑥) to the Cauchy problem (31) satisfies ‖𝑢‖_ℒ^∞_{̇𝐵(𝐼;𝐹}^𝑠_𝑝,𝑞)+𝜇‖𝑢‖_{ℒ1̇𝐵(𝐼;𝐹}^𝑠+2𝛽_𝑝,𝑞)≤1+169^𝛽𝑢_{0𝐹̇𝐵𝑠𝑝,𝑞}+‖𝐹‖_ℒ¹_{̇𝐵(𝐼;𝐹}^𝑠_𝑝,𝑞). (33)

Proof. By taking the Fourier transform we have 𝜕_𝑡𝑢+𝜇𝜉^2𝛽𝑢=𝐹. (34) Multiplying 𝜑_𝑗 and taking the 𝐿^𝑝-norm on both sides, 𝑑𝑢𝑑𝑡^𝑗𝐿𝑝+916^𝛽𝜇2^2𝛽𝑗𝑢_𝑗𝐿𝑝≤𝐹_𝑗𝐿𝑝, (35) where we denote 𝑢_𝑗=𝜑_𝑗𝑢. Integrating with respect to time on [0,𝑡), we get 𝑢_𝑗𝐿𝑝+916^𝛽𝜇2^2𝛽𝑗𝑢_{𝑗𝐿1(0,𝑡;𝐿𝑝)}≤𝑢₀𝑗_𝐿𝑝+𝐹_{𝑗𝐿1(𝐼;𝐿𝑝)}. (36) By the definition of 𝐹𝐵̇^𝑠_𝑝,𝑞 and the triangle inequality for 𝑙^𝑞, it is easy to obtain our desired inequality.

Next we consider the bilinear estimate, which is the key estimate in solving the Navier-Stokes equations.

Lemma 13 (bilinear estimate). Let 1≤𝑝, 𝑞≤∞, and 1/2<𝛽<min{1+3/𝑝^′,5/2} and set 𝑋=ℒ^∞𝐼;𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞∩ℒ¹𝐼;𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞, (37) with the norm ‖𝑢‖_𝑋=‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}+𝜇‖𝑢‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (38) Then there exists some constant 𝐶=𝐶(𝛽,𝑝,𝑞)>0 depending on 𝛽, 𝑝, and 𝑞 such that ‖∇⋅(𝑢⊗𝑣)‖_{ℒ1̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}≤𝐶𝜇⁻¹‖𝑢‖_𝑋‖𝑣‖_𝑋. (39) Particularly, it is true for the cases 𝑞=1 and 𝛽=1/2.

Proof. We will use the technique of the paraproduct. Set Δ̇_𝑗𝑢=ℱ⁻¹𝜑_𝑗𝑆̇∗𝑢,_𝑗=_{𝑘≤𝑗−1}Δ̇_𝑘Δ̇𝑢,_𝑗𝑢=_{𝑘−𝑗≤1}Δ̇_𝑘𝑢,for∀𝑗∈𝑍. (40) By Bony’s decomposition, we have for fixed 𝑗Δ̇_𝑗(𝑢𝑣)=_{𝑘−𝑗≤4}Δ̇_𝑗𝑆̇_𝑘−1𝑢Δ̇_𝑘𝑣+_{𝑘−𝑗≤4}Δ̇_𝑗𝑆̇_𝑘−1𝑣Δ̇_𝑘𝑢+_{𝑘≥𝑗−3}Δ̇_𝑗Δ̇_𝑘𝑢Δ̇_𝑘𝑣∶=𝐼_𝑗+𝐼𝐼_𝑗+𝐼𝐼𝐼_𝑗. (41) For simplicity,we can view ∇⋅(𝑢⊗𝑣), as the first derivative of two scale functions 𝑢,𝑣. Consider ‖𝜕(𝑢𝑣)‖_{ℒ1̇𝐵(𝐼∶𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}≤83_𝑗2^{(1−2𝛽+3/𝑝′)𝑗𝑞}2^𝑗𝑞Δ̇_𝑗(𝑢𝑣)^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞≤83_𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞+_𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞+_𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞. (42) The terms 𝐼_𝑗 and 𝐼𝐼_𝑗 are symmetrical. Using Young’s inequality and Hölder’s inequality we have 𝐼_{𝑗𝐿1(𝐼;𝐿𝑝)}≤_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑢Δ̇_𝑘𝑣_{𝐿1(𝐼;𝐿𝑝)}≤_{𝑘−𝑗≤4}𝑣_{𝑘𝐿1(𝐼;𝐿𝑝)𝑙≤𝑘−2}𝑢_{𝑙𝐿∞(𝐼;𝐿1)}≤_{𝑘−𝑗≤4}𝑣_{𝑘𝐿1(𝐼;𝐿𝑝)}×_{𝑙≤𝑘−2}2^{(1−2𝛽)𝑙𝑞}𝑢_𝑙^𝑞_{𝐿∞𝐼;𝐿1}^1/𝑞×_{𝑙≤𝑘−2}2^{(2𝛽−1)𝑙𝑞′1/𝑞′}≤𝐶_{𝑘−𝑗≤4}2^{(2𝛽−1)𝑘}𝑣_{𝑘𝐿1(𝐼;𝐿𝑝)}‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹}^1−2𝛽_1,𝑞). (43) Using the conclusion 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞⊂𝐹𝐵̇^1−2𝛽_1,𝑞, we have _𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞≤𝐶‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}‖𝑣‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (44) In a similar way we can prove that _𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞≤𝐶‖𝑣‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}‖𝑢‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (45) For the remaining term, we first consider the case 𝑝≤2 in which 𝛽<1+3/𝑝^′. By Hölder's inequality with 1/𝑝=1/𝑝^′+1/𝑝−1/𝑝^′ and by Young's inequality with 1+1/𝑝−1/𝑝^′=1/𝑝+1/𝑝, we have 2^{(2−2𝛽+3/𝑝′)𝑗}𝐼𝐼𝐼_{𝑗𝐿1(𝐼;𝐿𝑝)}≤𝐶_{𝑘≥𝑗−3}2^{(2−2𝛽+3/𝑝′)𝑗}2^{(3/𝑝′)𝑗}𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿1(𝐼;𝐿′′𝑝𝑝/(𝑝−𝑝))}≤𝐶_{𝑘≥𝑗−3}2^{(2−2𝛽+6/𝑝′)𝑗}𝑢_{𝑘𝐿1(𝐼;𝐿𝑝)||𝑙−𝑘≤1}𝑣_{𝑙𝐿∞(𝐼;𝐿𝑝)}≤𝐶_{𝑘≥𝑗−3}2^{(2−2𝛽+6/𝑝′)(𝑗−𝑘)}2^{(1−2𝛽+3/𝑝′)𝑘}𝑢_{𝑘𝐿∞(𝐼;𝐿𝑝)}×_{||𝑙−𝑘≤1}2^{(1+3/𝑝′)𝑙}𝑣_{𝑙𝐿1(𝐼;𝐿𝑝)}. (46) When 𝑞>2, we take 𝑙^𝑞-norm of both sides of (46) and use Young’s inequality with 1+1/𝑞=1/𝑞^′+2/𝑞 to get _𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞≤𝐶2^{(1−2𝛽+3/𝑝′)𝑘}𝑢_{𝑘𝐿∞(𝐼;𝐿𝑝)}×_{|𝑙−𝑘|≤1}2^{(1+3/𝑝′)𝑙}𝑣_{𝑙𝐿1(𝐼;𝐿𝑝)𝑙𝑞/2(𝑘)}≤𝐶‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}‖𝑣‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (47) When 𝑞≤2, since 𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞⊂𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞′, we take 𝑙^𝑞-norm of both sides of (46) and use Young’s inequality with 1+1/𝑞=1+1/𝑞 to get _𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼𝐼_𝑗^𝑞_{𝐿1𝐼∶𝐿𝑝}^1/𝑞≤𝐶2^{(1−2𝛽+3/𝑝′)𝑘}𝑢_{𝑘𝐿∞(𝐼;𝐿𝑝)}×_{||𝑙−𝑘≤1}2^{(1+3/𝑝′)𝑙}𝑣_{𝑙𝐿1(𝐼;𝐿𝑝)𝑙1(𝑘)}≤𝐶‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}‖𝑣‖_{ℒ1̇𝐵(𝐼;𝐹′′1+3/𝑝𝑝,𝑞)}≤𝐶‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}‖𝑣‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (48) Next we consider the case 𝑝>2 and hence 𝛽≤5/2. By Hölder's inequality we have 2^{(2−2𝛽+3/𝑝′)𝑗}𝐼𝐼𝐼_{𝑗𝐿1(𝐼;𝐿𝑝)}≤𝐶_{𝑘≥𝑗−3}2^{(2−2𝛽+3/𝑝′)𝑗}2^(3/𝑝)𝑗𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿1(𝐼;𝐿∞)}≤𝐶_{𝑘≥𝑗−3}2^{(5−2𝛽)𝑗}𝑢_{𝑘𝐿∞(𝐼;𝐿𝑝′)||𝑙−𝑘≤1}𝑣_{𝑙𝐿1(𝐼;𝐿𝑝)}≤𝐶_{𝑘≥𝑗−3}2^{(5−2𝛽)(𝑗−𝑘)}2^{(1−2𝛽+3/𝑝′)𝑘}𝑢_{𝑘𝐿∞(𝐼;𝐿𝑝)}×_{|𝑙−𝑘|≤1}2^{(1+3/𝑝′)𝑙}𝑣_{𝑙𝐿1(𝐼∶𝐿𝑝)}. (49) Following the same steps as in the case 𝑝≤2, we obtain the same estimate for 𝑝>2. Collecting the above estimates we finish our proof.

Next we introduce an abstract lemma on the existence of fixed point solutions [16, 24].

Lemma 14. Let 𝑋 be a Banach space with norm ‖⋅‖_𝑋 and let 𝐵∶𝑋×𝑋↦𝑋 be a bounded bilinear operator satisfying ‖𝐵(𝑢,𝑣)‖_𝑋≤𝜂‖𝑢‖_𝑋‖𝑣‖_𝑋, (50) for all 𝑢,𝑣∈𝑋 and a constant 𝜂>0. Then for any fixed 𝑦∈𝑋 satisfying ‖𝑦‖_𝑋<𝜖<1/4𝜂, the equation 𝑥∶=𝑦+𝐵(𝑥,𝑥) has a solution 𝑥 in 𝑋 such that ‖𝑥‖_𝑋≤2‖𝑦‖_𝑋. Also, the solution is unique in 𝐵(0,2𝜖). Moreover, the solution depends continuously on 𝑦 in the sense that if ‖𝑦^′‖_𝑋<𝜖, 𝑥^′=𝑦^′+𝐵(𝑥^′,𝑥^′), and ‖𝑥^′‖_𝑋<2𝜖, then 𝑥−𝑥^′_𝑋≤11−4𝜖𝜂𝑦−𝑦^′_𝑋. (51)

This lemma allows us to solve the Cauchy problem (1) with bounded bilinear form and small data. The mild solution of (1) is the solution to the equivalent integral form: 𝑢(𝑡,𝑥)=𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀−^𝑡₀𝑒^{−𝜇(𝑡−𝜏)(−Δ)𝛽}⋅()𝒫∇𝑢⊗𝑢)(𝜏,𝑥𝑑𝜏=𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀+𝐵(𝑢,𝑢), (52) where 𝒫=𝐼+∇(−Δ)⁻¹div is the Leray-Hopf projector. To make 𝐵(𝑢,𝑣) become a bilinear form, we simply take (1/2)𝑢⊗𝑣+(1/2)𝑣⊗𝑢 instead of 𝑢⊗𝑣 in the integral.

Proof of Theorem 5. We begin with the bilinear operator 𝐵(𝑢,𝑣). Observing that 𝐵(𝑢,𝑣) can be viewed as the solution to the dissipative equation (31) with 𝑢₀=0,𝐹=−𝒫∇⋅(𝑢⊗𝑣). Thus we can use Lemma 12 with 𝑠=1−2𝛽+3/𝑝^′ and Lemma 13 to obtain ‖𝐵(𝑢,𝑣)‖_𝑋≤1+169^𝛽‖−𝒫∇⋅(𝑢⊗𝑣)‖_{ℒ1̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}≤1+169^𝛽𝐶𝜇⁻¹‖𝑢‖_𝑋‖𝑣‖_𝑋. (53) By Lemma 14 we know that if ‖𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀‖_𝑋<𝑅 with 𝑅=𝜇/4(1+(16/9)^𝛽)𝐶, then (52) has a unique solution in 𝐵(0,2𝑅), where 𝐵(0,2𝑅)∶=𝑥∈𝑋∶‖𝑥‖_𝑋≤2𝑅. (54)
Now we need to derive ‖𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀‖_𝑋<𝑅. Similarly, 𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢₀ is the solution to the dissipative equation (31) with 𝑢₀=𝑢₀ and 𝐹=0. By Lemma 12 we obtain 𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢_0𝑋≤1+169^𝛽𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (55) Thus we conclude that if ‖𝑢₀‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}<𝐶₀𝜇 with 𝐶₀=(4(1+(16/9)^𝛽)²𝐶)⁻¹, then (52) has a unique global solution 𝑢∈𝑋 satisfying ‖𝑢‖_𝑋≤2𝑒^{−𝜇𝑡(−Δ)𝛽}𝑢_0𝑋≤21+169^𝛽𝑢_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (56) The continuity with respect to time is standard and thus we finish our proof.

Proof of Theorem 8. The method is the same with the proof of Theorem 5. But we substitute Lemma 13 by the following lemma.

Lemma 15. Let 1≤𝑝≤𝑞≤2 and 𝛽∈(1/2,1+3/2𝑝^′] and 𝑋 is the same as in Lemma 13. Then there exists some constant 𝐶=𝐶(𝛽,𝑝,𝑞)>0 depending on 𝛽, 𝑝, and 𝑞 such that ‖∇⋅(𝑢⊗𝑣)‖_{ℒ1̇𝐵(𝐼;𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}≤𝐶𝜇⁻¹‖𝑢‖_𝑋‖𝑣‖_𝑋. (57)Particularly, it is true for the cases 𝑝=𝑞=1 and 𝛽=1/2.
Proof. The proof is also same with Lemma 13. In fact by Bony's decomposition, we divide Δ̇_𝑗(𝑢𝑣) into three parts 𝐼_𝑗, 𝐼𝐼_𝑗, and 𝐼𝐼𝐼_𝑗. The parts 𝐼_𝑗 and 𝐼𝐼_𝐽 satisfy the same estimate. Hence it is sufficient to deal with the part 𝐼𝐼𝐼_𝐽. In fact when 𝑞≥𝑝, we have _𝑗2^{(2−2𝛽+3/𝑝′)𝑗𝑞}𝐼𝐼𝐼_𝑗^𝑞_{𝐿1𝐼;𝐿𝑝}^1/𝑞≤_{𝑘≥𝑗−3𝐼}2^{(2−2𝛽+3/𝑝′)𝑗}𝜑_𝑗(𝜉)×𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿𝑝𝜉}𝑑𝑡_{𝑙𝑞(𝑗)}≤_{𝑘𝐼𝑅3}2^{(2−2𝛽+3/𝑝′)𝑝𝑗}×𝜑_𝑗(𝜉)^𝑝𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_𝑙^𝑝𝑑𝜉_𝑙^1/𝑝_{𝑞/𝑝𝑗≤𝑘+3}𝑑𝑡≤sup_𝜉𝑗𝜑_𝑗(𝜉)^𝑞1/𝑞_𝑘2^{(2−2𝛽+3/𝑝′)(𝑘+3)}×_𝐼𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿𝑝𝜉}𝑑𝑡≤2^{3(2−2𝛽+3/𝑝′)}_𝑘2^{(1−2𝛽)𝑘}𝑢_{𝑘𝐿∞(𝐼∶𝐿1)}×_{||𝑙−𝑘≤1}2^{(1+3/𝑝′)(𝑘−𝑙)}2^{(1+3/𝑝′)𝑙}𝑣_{𝑙𝐿1(𝐼;𝐿𝑝)}≤𝐶‖𝑢‖_{ℒ∞̇𝐵(𝐼;𝐹}^1−2𝛽_1,𝑞)‖𝑣‖_{ℒ1̇𝐵(𝐼;𝐹′1+3/𝑝𝑝,𝑞)}. (58) In the last inequality we use a similar conclusion with (3) in Proposition 3; that is, ℒ¹(𝐼;𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞)⊂ℒ¹(𝐼;𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞′), when 𝑞∈[1,2].

4. The Decay Property

We introduce some lemmas which have interest in themselves.

Lemma 16. Let 𝛽<5/4, 𝑠>5/2−2𝛽, and 1≤𝑝, 𝑞≤2. Then we have 𝑓_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝐶𝑓_𝐿^{1−(5/2−2𝛽)/𝑠}₂𝑓_̇𝐻^{(5/2−2𝛽)/𝑠}_𝑠. (59)

Proof. By definition and Hölder’s inequality we have 𝑓_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}=_𝑗∈𝑍2^{(1−2𝛽+3/𝑝′)𝑗𝑞}𝜑_𝑗𝑓^𝑞_𝐿𝑝^1/𝑞≤_𝑗∈𝑍2^{(1−2𝛽+3/𝑝′)𝑗𝑞}𝜑_𝑗𝑓^𝑞_𝐿22^{(3/𝑝−3/2)𝑗𝑞1/𝑞}≤_𝑗≤𝑀2^{(5/2−2𝛽)𝑗𝑞}𝜑_𝑗𝑓^𝑞_𝐿2^1/𝑞+_𝑗≥𝑀2^{(5/2−2𝛽−𝑠)𝑗𝑞}2^𝑠𝑗𝑞𝜑_𝑗𝑓^𝑞_𝐿2^1/𝑞≤𝐶₁2^{(5/2−2𝛽−𝑠)𝑀}_𝑗∈𝑍𝜑_𝑗𝑓²_𝐿2^1/2+𝐶₂2^{(5/2−2𝛽−𝑠)𝑀}_𝑗∈𝑍2^𝑠𝑗𝑞𝜑_𝑗𝑓²_𝐿2^1/2. (60) Since 𝐵̇^𝑠_2,2=𝐻̇^𝑠 and 𝐵̇⁰_2,2=𝐿² and by Proposition 3, we know that 𝐹𝐵̇^𝑠_2,2=𝐵̇^𝑠_2,2; we finish our proof by taking 𝑀 such that 2^𝑀=(‖𝑓‖_̇𝐻𝑠/‖𝑓‖_𝐿2)^1/𝑠.

Lemma 17. Let 𝛽∈(1/2,1] and 1≤𝑝, 𝑞≤2. Consider ‖𝑢𝑣‖_̇𝐻^1−𝛽≤𝐶‖𝑢‖_𝐿²‖𝑣‖_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}+𝐶‖𝑢‖_̇𝐻^𝛽‖𝑣‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (61)

Proof. We use the equivalence 𝐻̇^𝑠=𝐵̇^𝑠_2,2=𝐹𝐵̇^𝑠_2,2. To conclude the result we only need to show that ‖𝑢𝑣‖_̇𝐻^1−𝛽≤𝐶‖𝑢‖_𝐿²‖𝑣‖_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,2}+𝐶‖𝑢‖_̇𝐻^𝛽‖𝑣‖_𝐹̇𝐵^1−2𝛽_1,2, (62) since we have the conclusions 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,2⊂𝐹𝐵̇^1−2𝛽_1,2 and 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞⊂𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,2 by Proposition 3. The method is similar with the proof of Lemma 13. Consider ‖𝑢𝑣‖_𝐹̇𝐵^1−𝛽_2,2≤_𝑗2^{(1−𝛽)𝑗2}𝐼_𝑗²_𝐿2^1/2+_𝑗2^{(1−𝛽)𝑗2}𝐼𝐼_𝑗²_𝐿2^1/2+_𝑗2^{(1−𝛽)𝑗2}𝐼𝐼𝐼_𝑗²_𝐿2^1/2, (63) where 𝐼_𝑗, 𝐼𝐼_𝑗, and 𝐼𝐼𝐼_𝑗 are the same with the proof of Lemma 13. Consider 𝐼_𝑗𝐿2≤_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑢Δ̇_𝑘𝑣_𝐿2≤_{𝑘−𝑗≤4}𝑣_{𝑘𝐿2𝑙≤𝑘−2}𝑢_𝑙𝐿1≤𝐶_{𝑘−𝑗≤4}𝑣_𝑘𝐿2×_{𝑙≤𝑘−2}2^{(1−2𝛽)𝑙2}𝑢_𝑙²_𝐿1^1/2×_{𝑙≤𝑘−2}2^{(2𝛽−1)𝑙21/2}≤𝐶_{𝑘−𝑗≤4}2^{(2𝛽−1)𝑘}𝑣_𝑘𝐿2‖𝑢‖_𝐹̇𝐵^1−2𝛽_1,2. (64) Thus we get _𝑗2^{(1−𝛽)𝑗2}𝐼_𝑗²_𝐿2^1/2≤𝐶‖𝑣‖_̇𝐻^𝛽‖𝑢‖_𝐹̇𝐵^1−2𝛽_1,2. (65) To estimate the term 𝐼𝐼_𝑗, we make a minor modification to get 𝐼𝐼_𝑗𝐿2≤_{𝑘−𝑗≤4}𝑆̇_𝑘−1𝑣Δ̇_𝑘𝑢_𝐿2≤_{𝑘−𝑗≤4}𝑢_{𝑘𝐿𝑝𝑙≤𝑘−2}𝑣_{𝑙𝐿2𝑝/(3𝑝−2)}≤𝐶_{𝑘−𝑗≤4}𝑢_𝑘𝐿𝑝×_{𝑙≤𝑘−2}2^{−2𝑙(3/𝑝′)}𝑣_𝑙²_{𝐿2𝑝/(3𝑝−2)}^1/2×_{𝑙≤𝑘−2}2^{2𝑙(3/𝑝′)1/2}≤𝐶_{𝑘−𝑗≤4}2^{(3/𝑝′)𝑘}𝑢_𝑘𝐿𝑝‖𝑣‖_{𝐹̇𝐵′−3/𝑝2𝑝/(3𝑝−2),2}. (66) By (4) in Proposition 3, we know that 𝐿²=𝐹𝐵̇⁰_2,2⊂𝐹𝐵̇^−3/𝑝′_{2𝑝/(3𝑝−2),2}. Thus _𝑗2^{(1−𝛽)𝑗2}𝐼𝐼_𝑗²_𝐿2^1/2≤𝐶‖𝑢‖_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,2}‖𝑣‖_𝐿². (67) Finally we derive the estimate of the last part as _𝑗2^{(1−𝛽)𝑗2}𝐼𝐼𝐼_𝑗²_𝐿2^1/2≤_{𝑘≥𝑗−3}2^{(1−𝛽)𝑗}𝜑_𝑗(𝜉)×𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿2𝜉𝑙2(𝑗)}≤_{𝑘𝑅3𝑗≤𝑘+3}2^{(1−𝛽)2𝑗}𝜑_𝑗(𝜉)²×𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_𝑙²𝑑𝜉^1/2≤sup_𝜉𝑗𝜑_𝑗(𝜉)^21/2×_𝑘2^{(1−𝛽)(𝑘+3)}𝑢_𝑘∗_{||𝑙−𝑘≤1}𝑣_{𝑙𝐿2𝜉}≤𝐶_𝑘2^𝛽𝑘𝑢_{𝑘𝐿2||𝑙−𝑘≤1}2^{(1−2𝛽)(𝑘−𝑙)}2^{(1−2𝛽)𝑙}𝑣_𝑙𝐿1≤𝐶‖𝑢‖_̇𝐻^𝛽‖𝑣‖_𝐹̇𝐵^1−2𝛽_1,2. (68)

Now we can begin our proof of Theorem 10. The method is based on the work from Gallagher-Iftimie-Planchon [27].

Proof of Theorem 10. Let 𝜖>0 be any constant small enough such that 𝜖≤𝐶₀𝜇, where 𝐶₀ is the constant in Theorem 5 and 𝜇 is the viscosity coefficient in (1). For 𝑘∈𝑁⁺, define 𝒜_𝑘∶=𝜉∈𝑅³∶𝜉≤𝑘and𝑢₀(𝜉)≤𝑘. (69) Obviously ℱ⁻¹(𝜒_𝒜𝑘𝑢₀) converges to 𝑢₀ in 𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞. So there exists some 𝑘∈𝑁⁺ such that 𝑢₀−ℱ⁻¹(𝜒_𝒜𝑘𝑢₀)_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}<𝜖2. (70) Set 𝑢_0,𝑘=ℱ⁻¹𝜒_𝒜𝑘𝑢₀,𝑤_0,𝑘=𝑢₀−ℱ⁻¹𝜒_𝒜𝑘𝑢₀. (71) Thus 𝑢_0,𝑘∈𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞∩𝐿² and we have shown that ‖𝑤_0,𝑘‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}<𝜖/2. Now we consider the following equations: 𝑤_𝑡+𝜇(−Δ)^𝛽()𝑤()𝑤+𝑤⋅∇+∇𝜋=0,𝑡,𝑥∈𝑅⁺×𝑅³;∇⋅𝑤=0,(𝑡,𝑥)∈𝑅⁺×𝑅³;𝑤(0,𝑥)=𝑤_0,𝑘(𝑥),𝑥∈𝑅³. (72) Since 𝜖/2≤𝐶₀𝜇/2≤𝐶₀𝜇, by Theorem 5, there exists a unique global solution 𝑤_𝑘 of (72) such that 𝑤_𝑘∈𝒞[0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞∩ℒ¹[0,∞);𝐹𝐵̇^1+3/𝑝′_𝑝,𝑞. (73) Moreover, 𝑤_{𝑘ℒ∞̇𝐵([0,∞);𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}+𝜇𝑤_{𝑘ℒ1̇𝐵([0,∞);𝐹′1+3/𝑝𝑝,𝑞)}≤𝐶𝑤_{0,𝑘𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (74) An easy computation gives 𝑤_𝑘∈𝐶([0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞) and for all 𝑡∈[0,∞), we have 𝑤_𝑘(𝑡)_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}+𝜇𝑤_{𝑘ℒ1̇𝐵([0,𝑡);𝐹′1+3/𝑝𝑝,𝑞)}≤𝐶𝑤_{0,𝑘𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (75) Now put 𝑢_𝑘=𝑢−𝑤_𝑘. Then 𝑢_𝑘∈𝐶([0,∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞) and it satisfies 𝜕_𝑡𝑢_𝑘+𝜇(−Δ)^𝛽𝑢_𝑘+𝑢_𝑘⋅∇𝑢_𝑘+𝑢_𝑘⋅∇𝑤_𝑘+𝑤_𝑘⋅∇𝑢_𝑘+∇𝜋−∇𝜋_𝑘=0,∇⋅𝑢_𝑘𝑢=0,_𝑘(0,𝑥)=𝑢_0,𝑘(𝑥), (76) where 𝜋 and 𝜋_𝑘 are the correspond pressures to the solutions 𝑢 and 𝑤_𝑘, respectively. Taking the 𝐿² inner product with 𝑢_𝑘, we have 12𝑑𝑢𝑑𝑡^𝑘2𝐿2+𝜇(−Δ)^𝛽/2𝑢_𝑘²_𝐿2≤𝑢_𝑘⋅∇𝑤_𝑘+𝑤_𝑘⋅∇𝑢_𝑘,𝑢_𝑘. (77) To estimate |⟨(𝑢_𝑘⋅∇)𝑤_𝑘,𝑢_𝑘⟩|, we have 𝑢_𝑘⋅∇𝑤_𝑘,𝑢_𝑘=∇⋅𝑢_𝑘⊗𝑤_𝑘,𝑢_𝑘≤(−Δ)^1/2−𝛽/2(𝑢_𝑘⊗𝑤_𝑘)_𝐿2×(−Δ)^𝛽/2𝑢_𝑘𝐿2. (78) By Lemma 17, we have 𝑢_𝑘⋅∇𝑤_𝑘,𝑢_𝑘≤𝐶𝑤_{𝑘𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}𝑢_𝑘𝐿2𝑢_{𝑘̇𝐻𝛽}+𝐶𝑤_{𝑘𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}𝑢_𝑘²_̇𝐻𝛽. (79) By (75), we have ‖𝑤_𝑘‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝐶𝜖/2. We further assume 𝜖 small enough such that 𝐶²𝜖≤𝜇/4. Using 𝑎𝑏≤𝑎²/2+𝑏²/2, we have 𝑢_𝑘⋅∇𝑤_𝑘,𝑢_𝑘≤2𝐶²𝜇𝑤_𝑘²_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}𝑢_𝑘²_𝐿2+𝜇4𝑢_𝑘²_̇𝐻𝛽. (80) Thus we conclude that 𝑑𝑢𝑑𝑡^𝑘2𝐿2+𝜇𝑢_𝑘²_̇𝐻𝛽≤8𝐶²𝜇𝑤_𝑘²_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}𝑢_𝑘²_𝐿2. (81) Gronwall lemma gives 𝑢_𝑘²_𝐿2+𝜇^𝑡₀𝑢_𝑘²_̇𝐻𝛽≤𝑢_0,𝑘²_𝐿2×exp8𝐶²𝜇^𝑡₀𝑤_𝑘²_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}. (82) Since 𝑞≤2, by Minkowski's inequality, we get ^𝑡₀𝑤_𝑘²_{𝐹̇𝐵′1−𝛽+3/𝑝𝑝,𝑞}≤_𝑗∈𝑍2^{𝑗(1−𝛽+3/𝑝′)𝑞}×^𝑡₀𝜑_𝑗𝑤_𝑘²_𝐿𝑝^𝑞/22/𝑞≤_𝑗∈𝑍2^{𝑗(1−2𝛽+3/𝑝′)𝑞/2}𝜑_𝑗𝑤_𝑘𝐿^𝑞/2_{∞[)0,𝑡;𝐿𝑝}×2^{𝑗(1+3/𝑝′)𝑞/2}𝜑_𝑗𝑤_𝑘𝐿^𝑞/2_{1)[0,𝑡;𝐿𝑝}^2/𝑞≤𝑤_{𝑘ℒ∞̇𝐵([0,𝑡);𝐹′1−2𝛽+3/𝑝𝑝,𝑞)}𝑤_{𝑘ℒ1̇𝐵([0,𝑡);𝐹′1+3/𝑝𝑝,𝑞)}. (83) Thus, combining with (74) we can obtain 𝑢_𝑘²_𝐿2+𝜇^𝑡₀𝑢_𝑘²_̇𝐻𝛽≤𝑢_0,𝑘²_𝐿2exp8𝐶⁴𝜇²𝑤_0,𝑘²_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}. (84) Using Lemma 16 with 𝑠=𝛽 and (75), we know that 𝑢_𝑘∈𝐿^{4𝛽/(5−4𝛽)}([0,+∞);𝐹𝐵̇^{1−2𝛽+3/𝑝′}_𝑝,𝑞) and ^∞₀𝑢_{𝑘𝐹̇𝐵}^{4𝛽/(5−4𝛽)}_{′1−2𝛽+3/𝑝𝑝,𝑞}≤𝐶^{4𝛽/(5−4𝛽)}𝜇⁻¹𝑢_0,𝑘𝐿^{4𝛽/(5−4𝛽)}₂×exp16𝐶⁴𝛽𝜇²𝑤5−4𝛽^{0,𝑘2𝐹̇𝐵′1−2𝛽+3/𝑝}_𝑝,𝑞. (85) So by continuity, there is a time 𝑡₀ such that ‖𝑢_𝑘(𝑡₀)‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝜖/2. Then we have 𝑢𝑡_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝑢_𝑘𝑡_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}+𝑤_𝑘𝑡_{0𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝜖2+𝜖2≤𝜖. (86) Finally, we consider the fractional Navier-Stokes equations starting at 𝑡=𝑡₀; by Theorem 5 and using a method as estimating (75) we conclude that ‖𝑢(𝑡)‖_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}+𝜇‖𝑢‖_{ℒ1([𝑡0̇𝐵,𝑡);𝐹′1+3/𝑝𝑝,𝑞)}≤𝐶𝑢(𝑡₀)_{𝐹̇𝐵′1−2𝛽+3/𝑝𝑝,𝑞}≤𝐶𝜖. (87) for all 𝑡>𝑡₀. Thus we finish our proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research was supported by the NNSF of China under Grants no. 11271330 and no. 11201103.