Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces

( ⋅∇ ) = ∂ + ∂ + ∂ u u u u u u u u 1 1 2 2 3 3 , and = ( ( ) ( ) ( )) u u x u x u x , , 0 1 2 3 is an initial given velocity. If u0 is quite regular, then the divergence-free condition determines the pressure p. Several authors have studied the local existence of solutions to the (NSE), for example, Leray [1,2] and Kato [3]. The global existence of weak solutions goes back to Leray [2] and Hopf [4]. The global wellposedness of strong solutions for small initial data in the critical Sobolev space Ḣ 1 2 is due to Fujita and Kato [5]. In [6], Chemin considered initial data that belong to the space Ḣ s for > s 1 2 . In [7], Kato has proved the case of the Lebesgue space L3. In [8], Koch and Tataru considered the space − BMO 1 (see also [9–11]). In all theseworks, the norms in the corresponding spaces of the initial data are assumed tobe very small.More precisely, the norm was supposed to be bounded by the viscosity ν multiplied by some positive constants. More results and details in this direction can be found in the book by Cannone [12]. In [13], the authors consider a new critical space that is contained in − BMO 1, where they show it is sufficient to assume that the norms of initial data are less than exactly the viscosity coefficient ν. Then, the space used in [13] is the following


Introduction
The 3D incompressible Navier-Stokes equations (NSEs) are , and = ( ) p p t x , denote, respectively, the unknown velocity and the unknown pressure of the fluid at the point ( ) ∈ × + t x , 3 , ( ⋅∇ ) = ∂ + ∂ + ∂ u u u u u u u u is an initial given velocity. If u 0 is quite regular, then the divergence-free condition determines the pressure p.
Several authors have studied the local existence of solutions to the (NSE), for example, Leray [1,2] and Kato [3]. The global existence of weak solutions goes back to Leray [2] and Hopf [4]. The global wellposedness of strong solutions for small initial data in the critical Sobolev space Ḣ 1 2 is due to Fujita and Kato [5]. In [6], Chemin considered initial data that belong to the space Ḣ s for > s 1 2 . In [7], Kato has proved the case of the Lebesgue space L 3 . In [8], Koch and Tataru considered the space − BMO 1 (see also [9][10][11]). In all these works, the norms in the corresponding spaces of the initial data are assumed to be very small. More precisely, the norm was supposed to be bounded by the viscosity ν multiplied by some positive constants. More results and details in this direction can be found in the book by Cannone [12].
In [13], the authors consider a new critical space that is contained in − BMO 1 , where they show it is sufficient to assume that the norms of initial data are less than exactly the viscosity coefficient ν. Then, the space used in [13] is the following which is equipped with the norm 1 3 3 We will also use the notation, for = i 0, 1, For the small initial data, the global existence is proved in [13]: . Then, there is a unique ∈ ( ( )) On the other hand, in [14] the authors proved the local existence for the large initial data and blow-up criteria if the maximal time is finite, precisely: There exists time T such that the system (NSE) has a unique solution In [15], to improve the result [13,14], we introduced the Fourier Lei-Lin space which is defined as follows: In the same study, as ( ) σ 3 is not a Banach space for > σ 0, we introduced the following non-homogeneous spaces: Precisely, we proved the following theorem: Our problem is to show that the norm of the global solution to (NSE) in − σ 1, tends to zero when the time grows to infinity. The behavior of the norm of the solution to infinity, in the different Banach spaces, was studied by several researchers. Wiegner proved in [16] that the L 2 norm of the solutions vanishes for any square integrable initial data, as time goes to infinity and gives a decay rate that seems to be optimal for a class of initial data.
Now we are ready to state the main result.
3 be the global solution to (NSE). Then In the following, we give a natural application of Theorem 1.5, which is the stability of global solutions of (NSE) system.
Then, Navier-Stokes system starting by v 0 has a global solution. Moreover, if v is the corresponding global solution, then The paper is organized in the following way: in Section 2, we give some notations and important preliminary results. Section 3 is devoted to prove the principal result. In Section 4, we prove the stability result for the global solutions.

Notations
In this section, we collect some notations and definitions that will be used later.
First, the Fourier transformation is normalized as and We denote by the Leray projection operator defined by the formula: such that 0, . p

Preliminary results
In this section, we recall some classical results and we give a few technical lemmas.
it suffices to prove that On the other hand, we have Taking the Fourier transform with respect to the space variable, we obtain  Using the fundamental property ⋅∇ = ( ⊗ ) Now, for ≥ t t 0 , we have