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

and Applied Analysis 3 We claim that X = F?̇?−1 1,1 . This fact can be seen by the following proposition (proof in [21]). Proposition 2. Define the spacesX as

( 2) We say that a function space is -critical for (1) if its norm is invariant under the scaling  0 () →  2−1  0 ().There are many examples of critical spaces, for instance,  −(2− 1) , Ḃ −(2−1) ∞,∞ , and the spaces we will discuss in this paper.The classical incompressible Navier-Stokes equations (i.e.,  = 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 Ḣ1/2 .A series study of mild solutions in different function spaces then arose, for instance, Kato [3] in Lebesgue space  3 ( 3 ), Cannone [4] in Besov space Ḃ −1+3/ ,∞ , and the important well-posedness in  −1 by Koch and Tataru [5].These works naturally lead one to study the wellposedness 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: However, in the space Ḃ −1 ∞,∞ , the Navier-Stokes equations are ill-posedness (see Bourgain and Pavlović [6] and Cheskidov and Shvydkoy [7]).
In this paper, we will study (1) in the Fourier-Besov spaces  Ḃ  , .We observe that although the Fourier-Besov spaces  Ḃ  , 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  , which is in fact the space  Ḃ  ∞,∞ (see Section 2 for details).Biswas and Swanson [20] studied the Gevrey regularity of Navier-Stokes equations in  Ḃ 2−3/ , .Konieczny and Yoneda [21] used  Ḃ  , 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  Ḃ −1 1,1 .Cannone and Wu [24] extended the result in [23] to the Fourier-Herz spaces Ḃ  .We may notice that Ḃ  =  Ḃ −1 1, .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  Ḃ  , 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  Ḃ  , .We obtain a global wellposedness result which is more general than those in [23,24].Particularly, our well-posedness is also valid in the critical case  = 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  ∼  means that there exist positive constants  1 ≤  2 such that  1  ≤  ≤  2 .We use Ḃ  , to denote the classical homogenous Besov spaces and Ḣ the homogenous Sobolev spaces.Also,  denotes a positive constant which may differ in lines if not being specified;   is the number satisfying 1/ + 1/  = 1 for 1 ≤  ≤ ∞.The inverse Fourier transform is denoted by 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.

Preliminaries and Main Results
We first introduce the definition of Fourier-Besov spaces in  dimensions.Let  ∈  ∞  (  ) be a radial real-valued smooth function such that 0 ≤ () ≤ 1 and We denote   () = (2 − ) and P the set of all polynomials.The space of tempered distributions is denoted by   .
One defines the homogeneous Fourier-Besov space  Ḃ  , as 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  as follows: We easily see that PM  =  Ḃ  ∞,∞ .The norm of Fourier-Herz spaces Ḃ  in [24] is defined as Obviously, we have Ḃ  =  Ḃ  1, .The space X −1 introduced by Lei and Lin [23] is We claim that X −1 =  Ḃ −1 1,1 .This fact can be seen by the following proposition (proof in [21] We discuss some inclusion relationships in  Ḃ  , .
Proposition 3. Let  ∈ , 1 ≤ ,  ≤ ∞.One has the follwoing. ( (5 Since  1 and  2 satisfy  1 + / 1 =  2 + / 2 , we immediately get Taking the   -norm on the above inequality we have To prove ( 5), we have Now we are ready to state our main results.From now on in this paper we take the dimension  = 3.
Our first result is on the well-posedness of (1).
Theorem 5. Let 1 ≤ ,  ≤ ∞, and 1/2 <  < min{1 + 3/  , 5/2}.Then there exists a constant  0 =  0 (, , ) such that, for any the Cauchy problem (1) admits a unique global mild solution  and and it satisfies Particularly, our result also holds in the critical case  = 1 and  = 1/2.Remark 6.We emphasize that the case  = 1/2 is important, since it is also the critical case for the fractional Navier-Stokes equations.Note that when  = 1/2, the function spaces we work on are  Ḃ 3/  ,1 .All these spaces are embedded into  Ḃ 0 1,1 , which is the space that consists of all functions whose Fourier transforms are in  1 (see Proposition 2).
are also critical spaces.In fact, for Then we have This implies that On the other hand, by we can easily deduce that Unfortunately, Theorem 5 is not suitable for the case  = 1,  = 1, in which similar existence has been proved by Cannone and Wu [24].To address this case, we also get the following theorem.
Then there exists a constant  0 =  0 (, , ) such that, for any the Cauchy problem (1) admits a unique global mild solution  and and it satisfies ) Particularly, our result also holds in the critical case  =  = 1 and  = 1/2.
Remark 9.In comparison with Theorem 5, although we have a limitation 1 ≤  ≤  ≤ 2, the regularity index  in Theorem 10 lies in a larger interval when  = 1.
Our third result is on the decay property of the global solutions ) is a global solution of (1).One has Remark 11.Recently, Benameur [26] obtained the same property in the space X −1 =  Ḃ −1 1,1 for the classical Navier-Stokes equations ( = 1).Our result improves and extends his result.

The Well-Posedness
First, we study the linear estimates of (1).For this purpose we consider the dissipative equation: It is easy to see that the equivalent integral equation of ( 31) is By taking (, ) = 0 or  0 () = 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.
By the definition of  Ḃ  , and the triangle inequality for   , 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.
Proof.We will use the technique of the paraproduct.Set By Bony's decomposition, we have for fixed For simplicity,we can view ∇ ⋅ ( ⊗ V), as the first derivative of two scale functions , V. Consider Using the conclusion In a similar way we can prove that For the remaining term, we first consider the case  ≤ 2 in which  < 1 + 3/  .By Hölder's inequality with 1/ = 1/  + 1/ − 1/  and by Young's inequality with 1 + 1/ − 1/  = 1/ + 1/, we have  . ( Next we consider the case  > 2 and hence  ≤ 5/2.By Hölder's inequality we have Following the same steps as in the case  ≤ 2, we obtain the same estimate for  > 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  be a Banach space with norm ‖ ⋅ ‖  and let  :  ×   →  be a bounded bilinear operator satisfying for all , V ∈  and a constant  > 0. Then for any fixed  ∈  satisfying ‖‖  <  < 1/4, the equation  :=  + (, ) has a solution  in  such that ‖‖  ≤ 2‖‖  .Also, the solution is unique in (0, 2).Moreover, the solution depends continuously on  in the sense that if ‖  ‖  < ,   =   + (  ,   ), and ‖  ‖  < 2, then 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: where P =  + ∇(−Δ) −1 div is the Leray-Hopf projector.To make (, V) become a bilinear form, we simply take Proof of Theorem 5. We begin with the bilinear operator (, V).Observing that (, V) can be viewed as the solution to the dissipative equation ( 31) with  0 = 0,  = −P∇ ⋅ ( ⊗ V).
Proof of Theorem 8.The method is the same with the proof of Theorem 5.But we substitute Lemma 13 by the following lemma.
Proof.The proof is also same with Lemma 13.In fact by Bony's decomposition, we divide Δ (V) into three parts   ,   , and   .The parts   and   satisfy the same estimate.Hence it is sufficient to deal with the part   .In fact when  ≥ , we have In the last inequality we use a similar conclusion with (3) in Proposition 3; that is,

The Decay Property
We introduce some lemmas which have interest in themselves.
Proof of Theorem 10.Let  > 0 be any constant small enough such that  ≤  0 , where  0 is the constant in Theorem 5 and  is the viscosity coefficient in (1).For  ∈  + , define Obviously . So there exists some  ∈  + such that Set Since /2 ≤  0 /2 ≤  0 , by Theorem 5, there exists a unique global solution   of (72) such that ) .
( ) and it satisfies where  and   are the correspond pressures to the solutions  and   , respectively.Taking the  2 inner product with   , we have for all  >  0 .Thus we finish our proof.