Journal of Applied Mathematics

Volume 2013 (2013), Article ID 321427, 6 pages

http://dx.doi.org/10.1155/2013/321427

Research Article

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

Wen-Juan Wang and Yan Jia

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Received 8 July 2013; Revised 30 September 2013; Accepted 21 October 2013

Academic Editor: Nicola Guglielmi

Copyright © 2013 Wen-Juan Wang and Yan Jia. 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 study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solution of the Navier-Stokes equations lies in the regular class , , , , then every weak solution of the perturbed system converges asymptotically to as , .

1. Introduction and Main Result

In this study, we consider the Cauchy problem of the generalized 3D Navier-stokes equations:

Here, , and and denote unknown velocity and pressure, respectively. is the external force and is a given initial velocity.

It is well known that when , system (1) becomes the classic Navier-Stokes equations. For the Navier-Stokes equations, it is proved that it has a global weak solution

for given with [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 is perturbed initially by without any smallness assumption, then the perturbed system is governed by the following equations:

where is the initial perturbation. There is large literature on the stability issue of the classic Navier-Stokes equations and related fluid models [12–17]. 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 , then every weak solution of the perturbed system (2) converges asymptotically to as .

Now our result reads as follows.

Theorem 1. Let , ; Suppose that is a weak solution of (1) and that 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.

2. Some Auxiliary Lemmas

We recall some basic facts about the Littlewood-Paley decomposition (refer to [18]). Let be Schwartz class of rapidly decreasing functions; supposing , the Fourier transformation is defined by

Choose two nonnegative radial functions , supported in and , respectively, such that

Let and , we define the dyadic blocks as follows:

We can easily verify that

Especially for any , we have the Littlewood-Paley decomposition:

Now we give the definition of the Besov space. Let and ; the inhomogeneous Besov space (see [18]) is defined by the full-dyadic decomposition, such as

where

and is a dual space of .

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.

Lemma 2. Assume that and , for , one has

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.

Lemma 3. Suppose that , for all , , , .

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 , , and one by one. Applying the Hölder inequality and the Bernstein inequality (40), we derive

where .

Since and with , then

Thanks to the Sobolev embedding , we have the following estimate:

Similarly, for , we also have

To estimate the last term , by using the Hölder inequality and the Bernstein inequality we obtain

Since and , we have

So, we can derive

Applying the interpolation inequality, we have

Then

Especially if , by using the interpolation inequality, we get

Hence, the proof of the lemma is complete.

Let denote the difference of and , where is a weak solution of (1) and is a weak solution of the perturbed problem (2). Thus satisfies the following equations:

Lemma 4. Let be the solution of the above problem. Then

Proof of Lemma 4. Taking the Fourier transformation of the first equation of (38), we get

We can easily obtain

Applying the operator to the first equation of (38), we have

and taking the Fourier transformation, we get

thus

Then we have

Thus solving the ordinary differential equation (31) and using (36) gives

which is the desired assertion of Lemma 4.

3. 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 we have

Applying Plancherel’s theorem to (38) yields

Let be a continuous function of with , and , we can derive the following:

By integrating in time from to for (40), we have

Noting that is a scalar function and applying Lemma 3, we get

Then,

Let , we have

Then,

In addition,

Choose , then

By using the Gronwall inequality, it follows that

Since

we derive

which completes the proof of Theorem 1.

Acknowledgments

The authors want to express their sincere thanks to the editor and the referees for their invaluable comments and suggestions. This work is partially supported by the NNSF of China (11271019), NSF of Anhui Province (11040606M02) and is also financed by the 211 Project of Anhui University (KJTD002B, KJJQ005).

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