Algebra Properties in Fourier-Besov Spaces and Their Applications

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 NavierStokes equations. Particularly, we give a blow-up criterion of the solutions in Fourier-Besov spaces as well as a space analyticity of Gevrey regularity.

As an application of this theorem, we study the Cauchy problem of ( 1 ) . ( Moreover, let  * denote the maximal time of existence of such a solution, then (i) there is a constant  0 such that if ‖ 0 ‖  Ḃ Particularly, our result also holds in the case  > 1,  = 1,  = 1.
This well-posedness result corresponds to the classical case for initial data  0 ∈ Ḣ1/2 (R 3 ) [2] if we take  = 3,  =  = 2,  = 1,  = 1 in Theorem 3. Unfortunately, our result is not suitable for the case  =  = 1,  = 1, which has been proved in [26].To address this special case, we also prove the following theorem.with ∇ ⋅  0 = 0, 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  0 such that if Particularly, our result also holds in the case  =  = 1,  = 1.
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 Ẋ−1 .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.
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  1 ≤  2 such that  1  ≤  ≤  2 .We use Ḃ  , to denote the classical homogeneous Besov spaces and Ḣ the homogeneous Sobolev spaces.Also,  denotes a positive constant which may differ in lines if not being specified, and   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 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.

Algebra Properties in Fourier-Besov Spaces
We denote   () = (2 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.
The following inclusions hold. ( 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 Ẋ𝑠,𝑝 as
Then we have Ẋ, =  Ḃ  , 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  ≤ 2, in which  1 +  2 > 0. By Hölder's inequality with 1/ = 1/  + 1/ − 1/  and by Young's inequality with 1 + 1/ − 1/  = 1/ + 1/, we have When  > 2, we take   -norm of both sides of ( 28) and use Young's inequality with 1 + 1/ = 1/  + 2/ to get When  ≤ 2, then  Ḃ  2 , ⊂  Ḃ  2 ,  ; by definition, taking  norm of both sides of (28) and using Young's inequality with 1 + 1/ = 1 + 1/, we get For the case  > 2, we have  1 +  2 > /  − /.By Hölder's inequality, there holds Following the same steps as in the case  ≤ 2, 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. where

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).
Proof.By Remark 7 and Hölder's inequality, it is sufficient to prove ‖ (, V)‖  Ḃ 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 , V ∈  and a constant  > 0. Then for any fixed  ∈  satisfying ‖‖  <  < 1/4, the equation  fl  + (, ) has a solution  in  such that ‖‖  ≤ 2‖‖  .Also, the solution is unique in (0, 2).Moreover, the solution depends continuously on  in the sense: if 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  = 1 − 2 + /  and Lemma 16 By Lemma 17, we know that if has a unique solution in (0, 2), where Now we need to derive .First, we consider small initial data.Lemma 14 and (55) imply that Thus we can take  0 such that ‖ 0 ‖ Next, for the large initial data  0 , we divide  0 by as  → +∞, by (58) there exists some  large enough such that Now for   ) . (62) 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  0 .If we have a solution of ( 1) on [0,  * ) such that then the integral equation ( 34  . 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  Ḃ  . (72)