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Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces
We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spaces . 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 . Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.
We study the mild solutions to the fractional Navier-Stokes equations in as follows: Here denotes the velocity vector, is the viscosity coefficient, and the scalar function denotes the pressure. The initial data is a divergence free vector field and the operator is the Fourier multiplier with symbol .
The fractional Navier-Stokes equations, which are also called generalized Navier-Stokes equations, enjoy an invariance under the scaling We say that a function space is -critical for (1) if its norm is invariant under the scaling . There are many examples of critical spaces, for instance, , , and the spaces we will discuss in this paper.
The classical incompressible Navier-Stokes equations (i.e., ) have been intensively studied. Leray first  introduced the concept of weak solutions and obtained the global existence of weak solutions. Fujita and Kato  gave a different approach to study the equations in their equivalent form of integral equations and proved the well-posedness in the space frame . A series study of mild solutions in different function spaces then arose, for instance, Kato  in Lebesgue space , Cannone  in Besov space , and the important well-posedness in by Koch and Tataru . These works naturally lead one to study the well-posedness in the largest critical space . In fact, all the above spaces are critical spaces and satisfy the following continuous embeddings in the 3 dimensions: However, in the space , the Navier-Stokes equations are ill-posedness (see Bourgain and Pavlović  and Cheskidov and Shvydkoy ).
As for the generalized case (1), Lions  proved the global existence of classical solutions in 3 dimensions when (see also Wu  in dimensions). For the important case , Wu [10, 11] studied the well-posedness in . Inspired by Xiao  in the classical case (), Li and Zhai [13, 14] studied (1) in some critical -type spaces for , and Zhai  showed the well-posedness in when . For the biggest critical space , Yu and Zhai  proved the well-posedness when , Cheskidov and Shvydkoy  showed the ill-posedness when . Very recently, Deng and Yao  studied (1) in Triebel-Lizorkin spaces and obtained the well-posedness in and ill-posedness in () in the case .
In this paper, we will study (1) in the Fourier-Besov spaces . We observe that although the Fourier-Besov spaces 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  worked in the space , which is in fact the space (see Section 2 for details). Biswas and Swanson  studied the Gevrey regularity of Navier-Stokes equations in . Konieczny and Yoneda  used to study the Navier-Stokes equations with Coriolis (see also Fang et al. ). Lei and Lin  proved global existence of mild solutions in , which is in fact equal to the space . Cannone and Wu  extended the result in  to the Fourier-Herz spaces . We may notice that . Also, some properties of solutions in the space have been studied recently; see Zhang and Yin  for the blow-up criterion and Benameur  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 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 . 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 . Moreover, the long time decay of the solutions in Fourier-Besov spaces is also proved, which fully extends the result of .
Throughout this paper, the notation means that there exist positive constants such that . We use to denote the classical homogenous Besov spaces and the homogenous Sobolev spaces. Also, denotes a positive constant which may differ in lines if not being specified; is the number satisfying for . 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.
2. 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 and We denote and the set of all polynomials. The space of tempered distributions is denoted by .
Definition 1. For , set One defines the homogeneous Fourier-Besov space 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  introduced the spaces as follows: We easily see that .
The norm of Fourier-Herz spaces in  is defined as Obviously, we have .
Proposition 2. Define the spaces as Then one has and the norms are equivalent:
We discuss some inclusion relationships in .
Proposition 3. Let , , . One has the follwoing.(1)If , then .(2)If , then .(3)If , then .(4)If , , and satisfy , then (5)If , , and for , then
Proof. (1) is a consequence of Plancherel’s identity, and Hausdorff-Young’s inequality gives (2). Equation (3) is just the inclusion for . To conclude (4), we use Hölder’s inequality to get Since and satisfy , 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 .
Definition 4. Let , , , and , . The space-time norm is defined on by
Our first result is on the well-posedness of (1).
Theorem 5. Let , , and . Then there exists a constant such that, for any with satisfying the Cauchy problem (1) admits a unique global mild solution and and it satisfies Particularly, our result also holds in the critical case and .
Remark 6. We emphasize that the case is important, since it is also the critical case for the fractional Navier-Stokes equations. Note that when , the function spaces we work on are . All these spaces are embedded into , which is the space that consists of all functions whose Fourier transforms are in (see Proposition 2).
Remark 7. Note that by Proposition 3 and the space are also critical spaces. In fact, for , we have . Set Then we have This implies that On the other hand, by we can easily deduce that
Theorem 8. Let and . Then there exists a constant such that, for any with satisfying the Cauchy problem (1) admits a unique global mild solution and and it satisfies Particularly, our result also holds in the critical case and .
Our third result is on the decay property of the global solutions
Theorem 10. Let and . Assume that is a global solution of (1). One has
Remark 11. Recently, Benameur  obtained the same property in the space for the classical Navier-Stokes equations (). 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: It is easy to see that the equivalent integral equation of (31) is By taking or , 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 , , , , and . Assume that and . Then the solution to the Cauchy problem (31) satisfies
Proof. By taking the Fourier transform we have Multiplying and taking the -norm on both sides, where we denote . Integrating with respect to time on , we get By the definition of 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.
Lemma 13 (bilinear estimate). Let , , and and set with the norm Then there exists some constant depending on , , and such that Particularly, it is true for the cases and .
Proof. We will use the technique of the paraproduct. Set By Bony’s decomposition, we have for fixed For simplicity,we can view , as the first derivative of two scale functions . Consider The terms and are symmetrical. Using Young’s inequality and Hölder’s inequality we have Using the conclusion , we have In a similar way we can prove that 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 (46) and use Young’s inequality with to get When , since , we take -norm of both sides of (46) and use Young’s inequality with to get Next we consider the case and hence . By Hölder's inequality we have Following the same steps as in the case , we obtain the same estimate for . Collecting the above estimates we finish our proof.
Lemma 14. Let be a Banach space with norm and let 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 that if , , and , 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 is the Leray-Hopf projector. To make become a bilinear form, we simply take instead of in the integral.
Proof of Theorem 5. We begin with the bilinear operator . Observing that can be viewed as the solution to the dissipative equation (31) with . Thus we can use Lemma 12 with and Lemma 13 to obtain
By Lemma 14 we know that if with , then (52) has a unique solution in , where
Now we need to derive . Similarly, is the solution to the dissipative equation (31) with and . By Lemma 12 we obtain Thus we conclude that if with , then (52) has a unique global solution satisfying The continuity with respect to time is standard and thus we finish our proof.
Lemma 15. Let and and is the same as in Lemma 13. Then there exists some constant depending on , , and such that
Particularly, it is true for the cases and .
Proof. The proof is also same with Lemma 13. In fact by Bony's decomposition, we divide 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, , when .
4. The Decay Property
We introduce some lemmas which have interest in themselves.
Lemma 16. Let , , and , . Then we have
Proof. By definition and Hölder’s inequality we have Since and and by Proposition 3, we know that ; we finish our proof by taking such that .
Lemma 17. Let and , . Consider
Proof. We use the equivalence . To conclude the result we only need to show that since we have the conclusions and by Proposition 3. The method is similar with the proof of Lemma 13. Consider where , , and are the same with the proof of Lemma 13. Consider Thus we get To estimate the term , we make a minor modification to get By (4) in Proposition 3, we know that . Thus Finally we derive the estimate of the last part as
Proof of Theorem 10. Let be any constant small enough such that , where is the constant in Theorem 5 and is the viscosity coefficient in (1). For , define Obviously converges to in . So there exists some such that Set Thus and we have shown that . Now we consider the following equations: Since , by Theorem 5, there exists a unique global solution of (72) such that Moreover, An easy computation gives and for all , we have