Journal of Function Spaces
Volume 2018 (2018), Article ID 3629179, 10 pages
https://doi.org/10.1155/2018/3629179

Research Article

Algebra Properties in Fourier-Besov Spaces and Their Applications

Xuhuan Zhou¹ and Weiliang Xiao²

¹Department of Information Technology, Nanjing Forest Police College, Nanjing 210023, China
²School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China

Correspondence should be addressed to Xuhuan Zhou; zhouxuhuan@163.com

Received 6 October 2018; Accepted 8 November 2018; Published 4 December 2018

Academic Editor: Maria Alessandra Ragusa

Copyright © 2018 Xuhuan Zhou and Weiliang Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [2–5]. 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 that

where . 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 j

Thus we can divide the norm by

The terms and are symmetrical. Using Young’s inequality and Hölder’s inequality

Using the inclusion , we have

In 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 have

When , 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 holds

Following 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 that

where . 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 gives

Multiplying and taking the -norm

where we denote . Multiplying and using Hölder’s inequality, we get

Using and taking -norm, we conclude

Similarly, 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 have

Particularly, the result also holds in the case

Proof. By Remark 7 and Hölder’s inequality, it is sufficient to prove

Note that all indices in the definition of are finite; we have

On the other hand, by Definition 8, we know

Thus, 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 satisfying

for 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 11

By Lemma 15 with and Lemma 16

By 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 that

Thus 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 that

then 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

Since converges to 0 as , and from (63) we know that

Thus we can conclude that the other two terms also converge to 0 as , so satisfies the Cauchy criterion at . Thus there exists an element in such that as . Now set , and consider the fractional Navier-Stokes equations starting by . Using Theorem 1 we obtain that a solution exists on a larger time interval than , which is a contradiction.

Proof of Theorem 4. Using the same method in proving Theorem 3, to prove Theorem 4, it is sufficient to re-estimate , and by Lemma 15 we only need to prove the following lemma.

Lemma 18. Let . Then there exists some constant depending on such that

Particularly, it is true for the case:

Proof. Similarly to Lemma 16, we only need to estimate . Using Bony’s decomposition

The terms and are symmetrical. Using Young’s inequality and Hölder’s inequality we have

Recalling in Proposition 11, there holds

In a similar way, we can prove

For the remaining term, we invoke Minkowski’s inequality. In fact when

In the last inequality we use for by (4) of Proposition 11. Collecting the above estimates we conclude our desired result.

4. Space Analyticity

The proof is similar with the proof of Theorem 3. Let

By the integral equation (34), we have

Since and are uniformly bounded on and , together with Lemmas 14 and 15 we only need to consider the estimate of . To apply Lemma 17, we need a bilinear estimate. To this end, we write as with and we prove the following lemma.

Lemma 19. Let . Then

Particularly, our result also holds in the case

Proof. Using the same discussion in the proof of Lemma 16, it is sufficient to consider the estimate of . By Bony’s decomposition, we have for fixed

The idea is that the -norm of can be bounded by that of . In fact,

Since is uniformly bounded on if , we can eliminate this term. Thus, we obtain

Similarly, we can also obtain that

Following the same step in the proof of Theorem 1 and Lemma 16, it is easy to conclude our desired result.

By this lemma and the above observations, we can obtain Theorem 5 by following the proof of Theorem 3 line by line. Similarly, to prove Theorem 6, it is sufficient to obtain the following lemma.

Lemma 20. Let . Then

Particularly, our result also holds in the case

Proof. Using the discussion in Lemma 19 and following the same steps in the proof of Lemma 18, it is easy to conclude our desired result.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the Pre research project under grant #LGY201604 and the NNSF of China under grant #11601223.

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