#### Abstract

We estimate the norm of the product of two scale functions in Fourier-Besov spaces. As applications of these algebra properties, we establish the global well-posedness for small initial data and local well-posedness for large initial data of the generalized Navier-Stokes equations. Particularly, we give a blow-up criterion of the solutions in Fourier-Besov spaces as well as a space analyticity of Gevrey regularity.

#### 1. Introduction

In this paper, we study the mild solutions to the generalized Navier-Stokes equations (NSE) in Here , is a constant, and the operator is the Fourier multiplier with symbol . The bilinear operator denotes the map of the form where and are real numbers.

The incompressible NSE is a particular case of (1), by taking with the Leray projector defined as where if and if ; that is, If we set the initial data to be a divergence free vector field, the above system is exactly the (fractional) incompressible NSE.

For the classical incompressible NSE (), the study of mild solution started by Fujita and Kato [1] in the space frame and was extended by many mathematicians in different spaces [2â€“5]. As for the generalized case , Lions [6] proved the global existence of classical solutions in 3-dimension when (see also Wu [7] in n-dimension). For the important case , Wu [8, 9] studied the well-posedness in . Inspired by Xiao [10] in the classical case (), Li and Zhai [11, 12] studied (1) in some critical -type spaces for , and Zhai [13] showed the well-posedness in when . For the biggest critical space , Yu and Zhai [14] proved the well-posedness when , and Cheskidov and Shvydkoy [15] showed the ill-posedness when . Deng and Yao [16] studied (1) in Triebel-Lizorkin spaces in 3-dimension and obtained the well-posedness in and ill-posedness in ( in the case .

We focus on the results in Fourier-Besov spaces. Early results came from Cannone and Karch [17] in pseudomeasure type spaces , in which they discuss the singular and regular solution of NSE. Iwabuchi [18] introduced a Besov type space (named Fourier-Herz space in [19]) to study the well-posedness and ill-posedness of Keller-Segel system. Scapellato and Ragusa in [20] introduced a Morrey-type sapce (named mixed Morrey space) to study qualitative properties of partial differential equations with discontinuous coefficients. Lei and Lin [21] study the global well-posedness for NSE in spaces . All those spaces are special cases of Fourier-Besov spaces, which were first introduced by Konieczny and Yoneda [22], to study the dispersive effect of the Coriolis for NSE.

Our first result on the estimates of the product in Fourier-Besov spaces has the similar form with that in Sobolev spaces [23], which can be seen as follows.

Theorem 1. *Let and such thatwhere . Then for , one has *

*Remark 2. *An important case is , since by Plancherelâ€™s identity we know . This theorem gives that if , then . This is exactly the case in homogeneous Sobolev spaces [24]. A similar result in Besov space can be seen in p. 61 of [25].

As an application of this theorem, we study the Cauchy problem of (1) in Fourier-Besov spaces .

Theorem 3. *Let . Then for any with , the Cauchy problem (1) admits a unique mild solution and Moreover, let denote the maximal time of existence of such a solution, then**(i) there is a constant such that if , then (ii) if , then Particularly, our result also holds in the case *

This well-posedness result corresponds to the classical case for initial data [2] if we take in Theorem 3. Unfortunately, our result is not suitable for the case , which has been proved in [26]. To address this special case, we also prove the following theorem.

Theorem 4. *Let . Then for any with , the Cauchy problem (1) admits a unique mild solution . Moreover, let denote the maximal time of existence of such a solution, then**(i) there is a constant such that if , then (ii) if , then Particularly, our result also holds in the case *

Now we focus on the space analyticity. Our main method is Gevrey estimate, which was introduced by Foias and Temam [27], since that Gevrey class technique has become an effective approach in the study of space analyticity of solutions. Ferrari and Titi [28] established Gevrey regularity for a very large class of parabolic equation with analytic nonlinearity. GrujiÄ‡ and Kukavica [29] prove the Gevrey regularity for NSE in . More results on the analyticity of solution for NSE can be seen in Lemarie-Rieusset [5] and references therein. Biswas [30] established Gevrey class regularity of solutions to a large class of dissipative equations in Besov type spaces defined via caloric extension. Bae [31] proved the Gevrey estimate of solution for NSE in the spaces . Inspired by this, we establish the Gevrey class regularity for the generalized NSE in the Fourier-Besov spaces. We indicate that any order derivative of the solution enjoys the same behavior with in some sense. In fact, denote by the Fourier multiplier with symbol , then we have the following result.

Theorem 5. *Let . Then for any , the Cauchy problem (1) admits a unique mild solution such that . Particularly, our result also holds in the case *

Theorem 6. *Let . Then for any , the Cauchy problem (1) admits a unique mild solution such that . Particularly, our result also holds in the case *

*Remark 7. *In our later proof, we can also obtain that conclusions (i), (ii) in Theorems 3 and 4 are also valid for in Theorems 5 and 6, respectively.

Throughout this paper, the notation means that there exist positive constants such that . We use to denote the classical homogeneous Besov spaces and the homogeneous Sobolev spaces. Also, denotes a positive constant which may differ in lines if not being specified, and is the number satisfying for . The inverse Fourier transform is denoted by .

We organize the paper as follows. In Section 2, we give some basic properties of Fourier-Besov spaces. Then we prove Theorem 1 as well as a corollary. In Section 3, we give the proof of Theorems 3 and 4. And in Section 4, we prove the space analyticity of Theorems 5 and 6.

#### 2. Algebra Properties in Fourier-Besov Spaces

Let be a radial real-valued smooth function such that and We denote and the set of all polynomials. The space of tempered distributions is denoted by .

*Definition 8. *For , set We define the homogeneous Fourier-Besov space as

The Fourier-Besov spaces look similar to the classical Besov spaces, but without the inverse Fourier transform. In fact, there are close relationships between them [32]. These spaces are, also, similar to central Morrey spaces studied in [33]. In order to apply in PDE, we also need to derive the properties of Fourier-Besov spaces with space-time norm.

*Definition 9. *Let and . The space-time norm is defined on by Here denotes the classical homogeneous Besov space. Besides, we also define the norm for some Banach space with norm by

*Remark 10. *By the definition and Minkowskiâ€™s inequality, we easily deduce that

Proposition 11. *Let . The following inclusions hold.** If , then .** If , then .** If , then .** If , then .** If satisfy , then If satisfy and for , then *

The special case has an interesting equivalent norm, which can be seen by the following proposition (see [22] for the proof).

Proposition 12. *Define the spaces as Then we have and the norms are equivalent *

Now we give the proof of Theorem 1. The tools we use are the paraproduct and Bonyâ€™s decomposition, which can be found in [8, 17].

*Proof of Theorem 1. *We will use the technique of the paraproduct. Set By Bonyâ€™s decomposition, we have for fixed jThus we can divide the norm byThe terms and are symmetrical. Using Youngâ€™s inequality and HÃ¶lderâ€™s inequalityUsing the inclusion , we haveIn a similar way, we can prove For the remaining term, we first consider the case , in which . By HÃ¶lderâ€™s inequality with and by Youngâ€™s inequality with , we haveWhen , we take -norm of both sides of (28) and use Youngâ€™s inequality with to get When , then ; by definition, taking -norm of both sides of (28) and using Youngâ€™s inequality with , we get For the case , we have . By HÃ¶lderâ€™s inequality, there holdsFollowing the same steps as in the case , we obtain the same estimate. Collecting the above estimates we finish our proof.

By a slight modification of the proof, we can also obtain the following.

Corollary 13. *Let and such thatwhere . Then for , one has*

#### 3. The Well-Posedness

To prove the well-posedness, we invoke the fix point principle. We consider the mild solution which means the equivalent integral equation

Lemma 14 (linear estimate). *Let . Set ; we have*

*Proof. *By the Fourier transform, we have Multiplying and taking the -norm imply where we denote . Multiplying and taking -norm Similarly, multiplying and taking -norm with respect to time on , Since , taking -norm we get

Lemma 15 (nonlinear estimate). *Let . Set ; we have *

*Proof. *Fourier transform givesMultiplying and taking the -normwhere we denote . Multiplying and using HÃ¶lderâ€™s inequality, we getUsing and taking -norm, we concludeSimilarly, multiplying Taking -norm with respect to time on and using Youngâ€™s inequality, Now taking -norm, we obtain our desired inequality

Lemma 16 (bilinear estimate). *Let . We haveParticularly, the result also holds in the case *

*Proof. *By Remark 7 and HÃ¶lderâ€™s inequality, it is sufficient to proveNote that all indices in the definition of are finite; we have On the other hand, by Definition 8, we knowThus, we just need to consider the estimate of the product . However, set , then by Theorem 1 and Corollary 13, we obtain our desired result.

Next we introduce an abstract lemma on the existence of fixed point solutions [14, 19].

Lemma 17. *Let be a Banach space with norm and be a bounded bilinear operator satisfyingfor all and a constant . Then for any fixed satisfying , the equation has a solution in such that . Also, the solution is unique in . Moreover, the solution depends continuously on in the sense: if , then *

This lemma allows us to solve the Cauchy problem (1) with bounded bilinear form and small data. Now we begin our proof.

*Proof of Theorem 3. *We first seek the solution in the spaces . By (6) of Proposition 11By Lemma 15 with and Lemma 16By Lemma 17, we know that if with , then (34) has a unique solution in , where Now we need to derive . First, we consider small initial data. Lemma 14 and (55) imply thatThus we can take such that with . Next, for the large initial data , we divide by , where is a large real number determined later. Since converges to 0 in as , by (58) there exists some large enough such that Now for , there holds Thus we can choose small enough such that We now conclude that (34) has a unique solution . By (34), Lemmas 14 and 15, and Theorem 1, we conclude The continuity with respect to time is standard and thus we prove Theorem 3 up to the blow-up criterion. Next we prove the blow-up criterion. Suppose is the maximal time of existence of mild solution associated with . If we have a solution of (1) on such thatthen the integral equation (34), Lemmas 14 and 15, and Theorem 1 imply that for all Using the integral equation (34), and by similar method with the proof of Lemmas 14, 15, and 16