The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

for given 0 ∈ L 2 (R) with ∇ ⋅ 0 = 0 [1]. However, the regularity of Leray weak solutions is still an open problem in mathematical fluid mechanics even if much effort has been made [2–4]. It is an interesting problem to investigate the stability properties of the Navier-Stokes equations and related fluid models [5–11]. As regard to the above system (1), the asymptotic stability of weak solution of the generalized 3D Navier-Stokes equation is described as follows. If u is perturbed initially by 0 without any smallness assumption, then the perturbed system V is governed by the following equations:


Introduction and Main Result
In this study, we consider the Cauchy problem of the generalized 3D Navier-stokes equations: + (−Δ)   + ( ⋅ ∇)  + ∇ = , (, ) ∈ R 3 × (0, ∞) , ∇ ⋅  = 0,  (, 0) =  0 . ( Here, 0 <  < 1, and  and  denote unknown velocity and pressure, respectively. is the external force and  0 is a given initial velocity. It is well known that when  = 1, system (1) becomes the classic Navier-Stokes equations.For the Navier-Stokes equations, it is proved that it has a global weak solution  (, ) ∈  ∞ (0, ;  2 ) ∩  2 (0, ;  1 ) , ∀ > 0 (2) for given  0 ∈  2 (R 3 ) with ∇ ⋅  0 = 0 [1].However, the regularity of Leray weak solutions is still an open problem in mathematical fluid mechanics even if much effort has been made [2][3][4].It is an interesting problem to investigate the stability properties of the Navier-Stokes equations and related fluid models [5][6][7][8][9][10][11].As regard to the above system (1), the asymptotic stability of weak solution of the generalized 3D Navier-Stokes equation is described as follows.If  is perturbed initially by  0 without any smallness assumption, then the perturbed system V is governed by the following equations: where  0 is the initial perturbation.There is large literature on the stability issue of the classic Navier-Stokes equations and related fluid models [12][13][14][15][16][17].The aim of this paper is to show the stability of weak solution in the framework of the homogeneous Besov space.More precisely, with the use of the Littlewood-Paley decomposition and the classic Fourier splitting technique, we can show that when the initial perturbation  0 ∈  2 (R 3 ), then every weak solution V() of the perturbed system (2) converges asymptotically to () as ‖V() − ()‖  2 → 0,  → ∞.Now our result reads as follows.
Theorem 1.Let  ∈  2 (0, ;  − (R 3 )),  0 ∈  2 (R 3 ); Suppose that (, ) is a weak solution of (1) and that V(, ) is a weak solution of the perturbed problem (2), respectively.Moreover, if ∇ also lies in the following regular class: then The remainder of this paper is organized as follows.In the Section 2, we first recall the Littlewood-Paley decomposition and the Bony decomposition; then we give three key lemmas.And we prove asymptotic stability of the weak solution in the Section 3.

Some Auxiliary Lemmas
We recall some basic facts about the Littlewood-Paley decomposition (refer to [18]).Let S(R 3 ) be Schwartz class of rapidly decreasing functions; supposing  ∈ S(R 3 ), the Fourier transformation F is defined by Choose two nonnegative radial functions ,  ∈ S(R 3 ), sup- , respectively, such that Let ℎ = F −1  and h = F −1 , we define the dyadic blocks as follows: We can easily verify that Especially for any  ∈  2 (R 3 ), we have the Littlewood-Paley decomposition: Now we give the definition of the Besov space.Let  ∈ R and ,  ∈ [1, ∞]; the inhomogeneous Besov space   , (R 3 ) (see [18]) is defined by the full-dyadic decomposition, such as where and S  (R 3 ) is a dual space of S(R 3 ).The Bony decomposition (see [19]) will be frequently used; it is followed by where The following Bernstein inequality (see [18]) will be used in the next section.
and the constant  is independent of  and .
In the following, we will introduce two lemmas, which will be employed in the proof of our theorem.
Then the trilinear form is continuous and In particular, if  = , then Proof of Lemma 3. We borrow the idea of [20] to prove this lemma.By using of the Littlewood-Paley decomposition and the Bony decomposition, we obtain Then we estimate  1 ,  2 , and  3 one by one.Applying the Hölder inequality and the Bernstein inequality (40), we derive where Thanks to the Sobolev embedding ), we have the following estimate: Similarly, for  2 , we also have To estimate the last term  3 , by using the Hölder inequality and the Bernstein inequality we obtain Since | −   | ≤ 1, ,   ≥  − 3 and (2/) + (3/) = 2, 2 <  < ∞, we have  /  ) Applying the interpolation inequality, we have Then Especially if  = , by using the interpolation inequality, we get . (28) Hence, the proof of the lemma is complete.
Let (, ) = V(, ) − (, ) denote the difference of V(, ) and (, ), where (, ) is a weak solution of (1) and V(, ) is a weak solution of the perturbed problem (2).Thus (, ) satisfies the following equations: We can easily obtain Applying the operator ∇ div to the first equation of (38), we have and taking the Fourier transformation, we get which is the desired assertion of Lemma 4.

Proof of Theorem 1
The following argument is follows the classic Fourier splitting methods which is first used by Schonbek [21] (see also [22]).
Taking the inner product of the first equation in (38) with  together with the divergence-free condition of V,  we have which completes the proof of Theorem 1.