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 [25]. 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 Plancherel’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. Grujić 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 Minkowski’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 Bony’s decomposition, which can be found in [8, 17].

Proof of Theorem 1. We will use the technique of the paraproduct. Set By Bony’s decomposition, we have for fixed jThus we can divide the norm byThe terms and are symmetrical. Using Young’s inequality and Hölder’s inequalityUsing the inclusion , we haveIn a similar way, we can prove For the remaining term, we first consider the case , in which . By Hölder’s inequality with and by Young’s inequality with , we haveWhen , we take -norm of both sides of (28) and use Young’s inequality with to get When , then ; by definition, taking -norm of both sides of (28) and using Young’s inequality with , we get For the case , we have . By Hölder’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 Hölder’s inequality, we getUsing and taking -norm, we concludeSimilarly, multiplying Taking -norm with respect to time on and using Young’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 Hölder’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</