This paper is devoted to the study of the stability issue of the supercritical dissipative surface quasi-geostrophic equation with nondecay low-regular external force. Supposing that the weak solution

Mathematical models in fluid dynamics play an important role in theoretical and computational studies in meteorological and oceanographic sciences and petroleum industries, and so forth. In this paper, we consider a simple mathematical model of large scalar ocean and atmosphere dynamics (see Pedlosky [

The surface quasi-geostrophic equation shares many features with fundamental fluid motion equations. When

Due to the importance in mathematics, there is much attention on the well-posedness and large time behaviors of the surface quasi-geostrophic equation. When

However, it is desirable to understand the asymptotic stability issue of the supercritical dissipative surface quasi-geostrophic equation. More precisely, we consider the perturbed quasi-geostrophic equation:

The main purpose of this study is to investigate the asymptotic stability for the global solution of the supercritical surface quasi-geostrophic equation in the critical BMO space with nondecay low-regular external force. To state our main results, we first give the definition of the weak solution of the dissipative surface quasi-geostrophic equation (

Letting

For

where

Energy-type inequality

Now our results read as the following.

Letting

The critical space-time mixed space (

On one hand, compared with the stability result by Chae and Lee [

Although Theorem

The main idea in the proof of Theorem

The remains of this paper are organized as follows. In Section

Let us end this section by some notations. In what follows,

BMO is the space of the bounded mean oscillation defined by

In this section, we will investigate the global

Instead of the perturbed quasi-geostrophic equation (

It is worth noting that, for fixed integer

Furthermore, it allows us to prove that the approximates smooth solutions

Thanks to the regular solution

Denoting by

Assume that

Thus we have the following crucial estimates for the nonlinear term of (

With the aid of the Hölder inequality and the Young inequality, one shows that for the external force term

Plugging the above inequalities into (

Moreover, by the slight modification of the derivation of (

In this section, we are devoted to investigating the average decay of the difference

It is worth noting that Kozono [

Fortunately, we can avoid those additional difficulties with the aid of the theory of the analytic semigroup (see Pazy [

Now we carry out to study the average decay of the difference

We now give the estimates of (

Under the same condition in Theorem

We will give the proof at the end of this section. Once the crucial uniform estimates (

then one shows that for (

and then letting

Furthermore, employing the Parseval equality and the Hölder inequality, we obtain that for

Now it remains to prove Lemma

For convenience, the three terms of the left hand side of (

Applying the Fubini theorem, it is not difficult to obtain

For

Hence, combining the above inequalities, one derives the estimates of

For

For

Similarly, for

Plugging the estimates of

In order to estimate

Inserting (

With the aid of the global estimates and average decay of the difference

The integration of (

The authors would like to express their sincere thanks to Professor D. Córdoba for the valuable comments and suggestions. This work is partially supported by the NNSF of China (11271019) and the NSF of Anhui Province (11040606M02), and is also financed by the 211 Project of Anhui University (KJTD002B, KJJQ005).