Stability Analysis of the Supercritical Surface Quasi-Geostrophic Equation

Abstract

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 𝜃(𝑥,𝑡) of the surface quasi-geostrophic equation with the force 𝑓∈𝐿²(0,𝑇;𝐻^−𝛼/2(ℝ²)) satisfies the growth condition in the critical BMO space ∇𝜃∈𝐿¹(0,∞;BMO), it is proved that every perturbed weak solution 𝜃(𝑡) converges asymptotically to solution 𝜃(𝑡) of the original surface quasi-geostrophic equation. The initial and external forcing perturbations are allowed to be large.

1. Introduction and Main Results

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 [1])—the dissipative surface quasi-geostrophic equation which is first introduced by Constantin et al. [2]: 𝜕𝜃𝜕𝑡+𝑢⋅∇𝜃+𝜅Λ^𝛼()𝜃=𝑓,𝑥,𝑡∈ℝ²×(),𝜃()0,∞𝑥,0=𝜃₀,𝑥∈ℝ². (1) Here 𝜅>0 is a dissipative coefficient and Λ is the Riesz potential operator defined by the fractional power of −Δ: Λ=(−Δ)^1/2,Λ^𝛼𝑔=(−Δ)^𝛼/2𝑔=𝜉^𝛼𝑔. (2)𝜃(𝑥,𝑡) is an unknown scalar function representing potential temperature. 𝑢(𝑥,𝑡) is the velocity field determined by 𝑢=−∇^⟂(−Δ)^−1/2𝜃=−ℛ^⟂𝜃=ℛ₂𝜃,−ℛ₁𝜃, (3) where ℛ_𝑗, 𝑗=1,2 is the 2D Riesz transform, 𝜃₀(𝑥) is the initial data, and 𝑓(𝑥,𝑡) is the nondecay external force.

The surface quasi-geostrophic equation shares many features with fundamental fluid motion equations. When 𝜅=0, it is comparable to the formation of sharp fronts between masses of hot and cold air (see Majda and Tabak [3]). When 𝛼=0 (1) can be looked to as a nonviscous wind driven circulation equation (see Pedlosky [1]). What is more, (1) with 𝛼=1 shares many similar features with three-dimensional incompressible Navier-Stokes equations [4]. Thus 𝛼=1 is therefore referred to as the critical case, while the cases 0<𝛼<1 and 1<𝛼≤2 are supercritical and subcritical, respectively.

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 1<𝛼≤2, Constantin and Wu [5] proved that this equation possesses a unique and global smooth solution for a sufficiently smooth initial function. When 𝛼=1, Kiselev et al. [6] showed the existence of global smooth solutions for smooth periodic initial data. At the same time, Caffarelli and Vasseur [7] also constructed a global smooth solution with 𝜃₀∈𝐿²(ℝ^𝑛). In the supercritical case 0<𝛼<1, however, the question of the global regularity is still generally not clear although many efforts have been made on the small regular solution or regularity criteria (see [8–16]). One may also refer to some interesting results of the generalized quasi-geostrophic equations (see [17–21]).

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: 𝜕𝜃+𝜕𝑡𝑢⋅∇𝜃+𝜅Λ^𝛼()𝜃=𝑓+𝑔,𝑥,𝑡∈ℝ²×(0,∞),𝜃(𝑥,0)=𝜃₀=𝜃₀+𝑤₀,𝑥∈ℝ², (4) where 𝑤₀(𝑥) and 𝑔(𝑥,𝑡) are the initial data and external forcing perturbations. The stability problem of (4) has been studied by many authors and a lot of good results have been obtained. To go directly to the main points of the present paper, in what follows we only review some former results which are closely related to our main results. First, when the surface quasi-geostrophic equations (1) has zero external force or time-decay external force, some interesting time decay results [22–25] have been obtained which essentially show the asymptotic stability for the trivial solution 𝜃(𝑥,𝑡)=0. For the stability issue of the nontrivial solutions, Chae and Lee [26] studied the asymptotic stability of critical quasi-geostrophic equation in Besov space under the small initial perturbation. The nonlinear stability and instability analysis of the subcritical quasi-geostrophic equation is recently investigated by Chen and Price [27]. Dong and Chen [28] recently considered the asymptotic stability problem in the Serrin-type critical Lebesgue space 𝐿^𝑝(ℝ²) (𝑝<∞) under the 𝐿²(ℝ²) external force. It is natural and interesting to further investigate the asymptotic stability for the nontrivial global solutions in the larger critical spaces, especially in the margin case 𝑝=∞.

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 (1).

Definition 1. Letting 𝜃₀∈𝐿²(ℝ²), 𝑓∈𝐿²(0,𝑇;𝐻^−𝛼/2(ℝ²)). A measurable function 𝜃(𝑥,𝑡) is said to be a global weak solution of the surface quasi-geostrophic equation (1) if the following conditions hold true. (i)𝜃(𝑥,𝑡)∈𝐿^∞(0,𝑇;𝐿²(ℝ²))∩𝐿²(0,𝑇;𝐻^𝛼/2(ℝ²)) for all 𝑇>0.  (ii)𝜃(𝑥,𝑡) is continuous on [0,∞) in the weak topology of 𝐿²(ℝ²) and (𝜃(𝑡),𝜙)→(𝜃₀,𝜙) as 𝑡→0 for all 𝜙∈𝐿²(ℝ²).  (iii) For 0≤𝑠≤𝑡≤∞, and any test function 𝜑∈𝐶¹([𝑠,𝑡];𝐻^𝛼/2(ℝ²)), 𝜃(𝑡)(𝑡)+,𝜑_𝑡𝑠−𝜃,𝜕_𝜏𝜑+𝜅Λ^𝛼/2𝜃,Λ^𝛼/2𝜑−=𝜃(𝑠)(𝑠)+𝜃,𝑢⋅∇𝜑𝑑𝜏,𝜑^𝑡_𝑠𝑓,𝜑𝑑𝜏, (5) where (⋅,⋅) denotes the inner product of the space 𝐿²(ℝ²).  (iv) Energy-type inequality ‖𝜃(𝑡)‖²_𝐿2+2𝜅^𝑡_𝑠Λ^𝛼/2𝜃(𝜏)²_𝐿2≤‖𝜃(𝑠𝑑𝜏)‖²_𝐿2+2^𝑡_𝑠𝑓,𝜃𝑑𝜏0≤𝑠≤𝑡≤∞. (6)

Now our results read as the following.

Theorem 2. Letting 0<𝛼<1, 𝜃₀∈𝐿²(ℝ²). Assume 𝜃(𝑥,𝑡) is a global weak solution of the supercritical dissipative surface quasi-geostrophic equation (1) with the nondecay low-regular external force 𝑓∈𝐿²(0,𝑇;𝐻^−𝛼/2(ℝ²)) and satisfies the critical growth condition in BMO space: ∇𝜃∈𝐿¹(0,∞;𝐵𝑀𝑂). (7) Then, for any perturbations 𝑤₀∈𝐿²(ℝ²) and 𝑔∈𝐿²(0,∞;𝐻^−𝛼/2(ℝ²)), every weak solution 𝜃(𝑥,𝑡) of the perturbed quasi-geostrophic equation (4) converges asymptotically to solution 𝜃(𝑡) as 𝜃(𝑡)−𝜃(𝑡)_𝐿2⟶0(𝑡⟶∞). (8)

Remark 3. The critical space-time mixed space (7) in Theorem 2 is scaling invariant under the scaling transformation 𝜃_𝜆(𝑥,𝑡)=𝜆^𝛼−1𝜃(𝜆𝑥,𝜆^𝛼𝑡). Moreover, according to the result of Dong and Chen [29], the weak solution 𝜃(𝑥,𝑡) satisfying (7) is actually regular. Furthermore, our results also show that the weak solutions 𝜃(𝑥,𝑡) of the perturbed equation (4) do not need to satisfy the energy-type inequality (6) and then they do not have to be regular.

Remark 4. On one hand, compared with the stability result by Chae and Lee [26], there is no any smallness restriction on the perturbation functions 𝑤₀ and 𝑔 in Theorem 2. That is to say, the initial and external forcing perturbations from 𝜃(𝑡) are allowed to be large. On the other hand, our result here extends the previous result [28] to the margin case 𝑝=∞ due to the embedding relation 𝐿^∞⫋BMO. Moreover, in comparison with [28], we require less regularity for the external force.

Remark 5. Although Theorem 2 shows that the decay property of difference between the global solution and the perturbed solution ‖𝜃(𝑡)−𝜃(𝑡)‖_𝐿²→0, however, it is different from the time decay issue of the zero-forced surface quasi-geostrophic equations (see [22, 23] and references therein) ‖𝜃(𝑡)‖_𝐿²→0. The latter indeed shows the asymptotic stability of the trivial solution 𝜃=0 for the surface quasi-geostrophic equation with zero external force. However, when the surface quasi-geostrophic equation (1) has the nonzero low-regular external force 𝑓∈𝐿²(0,𝑇;𝐻^−𝛼/2(ℝ²)), 𝜃=0 is obviously not a solution of (1), therefore, our result here essentially shows the asymptotic stability of the nonzero global solution of (1).

Remark 6. The main idea in the proof of Theorem 2 is based on some new observations due to the special structure in nonlinear term 𝑢⋅∇𝜃 which is different from the classic Navier-Stokes equations. Additionally, in order to derive an auxiliary decay estimate in the framework of the critical BMO space, we also need to choose a suitable test function and derive some average decay properties. It is worth noting that although the study of the classic incompressible Navier-Stokes equations is beneficial to that of quasi-geostrophic equation, compared with some asymptotic stability results of three-dimensional Navier-Stokes equations (refer to [30–32]), we do not know, however, whether our result in the framework of the critical BMO space is still valid for the Navier-Stokes equations. We will focus on that problem in the future.

The remains of this paper are organized as follows. In Section 2, we will investigate the global 𝐿² estimates of the difference between the original surface quasi-geostrophic equation and the perturbed equation. In Section 3, we will further derive an average decay estimates of the difference. Finally, we will prove the asymptotic stability of large solution to the surface quasi-geostrophic equation (1) under the large initial perturbation in Section 4.

Let us end this section by some notations. In what follows, 𝐶 stands for the abstractly positive constant. 𝐿^𝑝(ℝ²) with 1≤𝑝≤∞ denotes the usual Lebesgue space. 𝐻^𝑠(ℝ²) with 𝑠∈ℝ is the fractional Sobolev space with the norm: 𝑓_𝐻𝑠=_ℝ2𝜉^2𝑠𝑓²𝑑𝜉^1/2. (9)

BMO is the space of the bounded mean oscillation defined by BMO=𝑓∈𝐿¹_locℝ²;sup_𝑥,𝑟1𝐵_𝑟(𝑥×_{𝐵𝑟(𝑥)})𝑓𝑦−𝑓_{𝐵𝑟(𝑥)}𝑑𝑦<∞, (10) where 𝑓_{𝐵𝑟(𝑥)}=1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑓𝑦𝑑𝑦 (11) is the average of 𝑓 over 𝐵_𝑟(𝑥). In particular, BMO is the dual space of the Hardy space ℋ¹.

2. Global Estimate of 𝜃−𝜃

In this section, we will investigate the global 𝐿² estimates of the difference 𝑤=𝜃−𝜃 between the global solution 𝜃 of the original equation (1) and the weak solution 𝜃 of the perturbed equation (4). It is mentioned that some necessary smooth properties of the difference 𝑤 is required in the derivation of the global 𝐿² estimates below. As stated in Remark 3, the global weak solution 𝜃 of the original equation (1) satisfying the regular condition (6) is actually regular. However, the perturbed weak solution 𝜃 is not smooth, in order to derive the global estimates of the difference 𝑤=𝜃−𝜃. We first seek a smooth function to approximate the weak solution 𝜃 of the perturbed quasi-geostrophic equation (4). To do so, we apply the standard Galerkin method to construct the smooth approximate solutions: 𝜃_𝑚(𝑥,𝑡)=^𝑚_𝑘=1𝑔_𝑘𝑚(𝑡)𝜙_𝑘(𝑥),𝑚∈𝑁 (12) Here {𝜙_𝑘}^∞_𝑘=1∈𝐶^∞(ℝ²) is an orthonormal basis of 𝐿²(ℝ²).

Instead of the perturbed quasi-geostrophic equation (4), for any test function 𝜙∈𝑋_𝑚 = the space spanned by 𝜙₁,𝜙₂,…,𝜙_𝑚,𝜃_𝑚(𝑥,𝑡) satisfy 𝜕𝜃_𝑚+𝜕𝑡𝑢_𝑚⋅∇𝜃_𝑚+𝜅Λ^𝛼𝜃_𝑚,𝜙=𝑓_𝑚+𝑔_𝑚,𝜙,𝜃_𝑚(𝑥,0)=^𝑚_𝑘=1𝜃₀,𝜙_𝑘𝜙_𝑘, (13) where 𝑓_𝑚=^𝑚_𝑘=1𝑓,𝜙_𝑘𝜙_𝑘,𝑔_𝑚=^𝑚_𝑘=1𝑔,𝜙_𝑘𝜙_𝑘. (14)

It is worth noting that, for fixed integer 𝑚, equations (13) are indeed ordinary differential equations. The theory of existence and uniqueness for such ordinary differential equations is standard. Therefore, we can prove that the approximated solutions 𝜃_𝑚 exist globally and smoothly and are uniformly bounded in the following space: 𝜃_𝑚∈𝐿^∞0,𝑇;𝐿²ℝ²∩𝐿²0,𝑇;𝐻^𝛼/2ℝ²,∀𝑇>0. (15)

Furthermore, it allows us to prove that the approximates smooth solutions 𝜃_𝑚 converge to the weak solution of the perturbed quasi-geostrophic equation (4) when 𝑚 tends to infinity. Indeed, we only need to obtain the strong convergence of 𝜃_𝑚 in space 𝐿²_loc(ℝ²×(0,∞)). To do so, for example, with the aid of Hölder and Sobolev embedding estimates, it follows that 𝑢_𝑚⋅∇𝜃_𝑚,𝜙≤𝐶𝑢_𝑚𝐿2𝜃_{𝑚𝐿4/(2−𝛼)}∇𝜙_𝐿4/𝛼≤𝐶𝑢_𝑚𝐿2𝜃_{𝑚𝐻𝛼/2}𝜙_{𝐻(4−𝛼)/2} (16) for 𝜙∈𝐻^(4−𝛼)/2. This implies the uniform bounds of 𝜕𝜃_𝑚𝜕𝑡=−𝑢_𝑚⋅∇𝜃_𝑚−𝑘Λ^𝛼𝜃_𝑚+𝑓_𝑚+𝑔_𝑚inthespace𝐿²0,𝑇;𝐻^{−(4−𝛼)/2}ℝ². (17) By the compactness theorem [4, Chapter 3, Theorem 2.1], the sequence 𝜃_𝑚 admits a subsequence denoted also by 𝜃_𝑚 converging to the weak solution of the perturbed equation (4): 𝜃_𝑚⟶𝜃(𝑚⟶∞)stronglyin𝐿²_locℝ²×(0,∞), (18) from which and uniformly bounds of ‖𝜃_𝑚(𝑡)‖_𝐿2 due to (15), we have 𝜃_𝑚(𝑡)⟶𝜃(𝑡)(𝑚⟶∞)weaklyin𝐿²ℝ² (19) for almost all 𝑡∈(0,∞).

Thanks to the regular solution 𝜃∈𝐿^∞(0,𝑇;𝐿²(ℝ²)), we then derive from (19) that 𝜃(𝑡)−𝜃_𝑚(𝑡)⟶𝜃(𝑡)−𝜃(𝑡)(𝑚⟶∞)weaklyin𝐿²ℝ² (20) for almost all 𝑡∈(0,∞), which implies by Fatou Lemma that 𝜃(𝑡)−𝜃(𝑡)_𝐿2≤liminf_𝑚→∞𝜃(𝑡)−𝜃_𝑚(𝑡)_𝐿2 (21) for almost all 𝑡∈(0,∞).

Denoting by 𝑤_𝑚=𝜃−𝜃_𝑚 the difference between solution 𝜃 of original equation (1) and approximated smooth solution 𝜃_𝑚 of the (13), with the direct computation due to the smoothness of 𝑤_𝑚, it is easy to examine 𝑤_𝑚 satisfying 𝑤_𝑚(𝑡)²_𝐿2+2𝜅^𝑡₀Λ^𝛼/2𝑤_𝑚²_𝐿2≤𝜃𝑑𝜏₀−𝜃_𝑚(0)²_𝐿2+2^𝑡₀𝑢⋅∇𝜃,𝜃_𝑚−𝑢_𝑚⋅∇𝜃_𝑚,𝜃𝑑𝜏+2^𝑡₀𝑓_𝑚+𝑔_𝑚−𝑓,𝑤_𝑚≤𝜃𝑑𝜏₀−𝜃_𝑚(0)²_𝐿2−2^𝑡₀𝑢−𝑢_𝑚⋅∇𝜃,𝑤_𝑚𝑑𝜏+2^𝑡₀𝑓_𝑚+𝑔_𝑚−𝑓,𝑤_𝑚𝑑𝜏, (22) where the divergence free property of the velocity is used. In order to deal with the nonlinear term of the right hand side of (22), we require the following important lemma.

Lemma 7 (Marchand [33]). Assume that ℎ∈𝐿²(ℝ²), ℛ_𝑖 (𝑖∈{1,2}) is the Riesz transform, then there exists a constant 𝐶 such that ℎℛ_𝑖ℎ_ℋ1≤𝐶‖ℎ‖²_𝐿2 (23) is valid. Here ℋ¹ is the Hardy space.

Thus we have the following crucial estimates for the nonlinear term of (23): −2^𝑡₀𝑢−𝑢_𝑚⋅∇𝜃,𝑤_𝑚𝑑𝜏≤𝐶^𝑡_0ℝ2ℛ^⟂𝑤_𝑚⋅∇𝜃𝑤_𝑚𝑑𝑥𝑑𝑠≤𝐶^𝑡₀𝑤_𝑚ℛ₁𝑤_𝑚ℋ1+𝑤_𝑚ℛ₂𝑤_𝑚ℋ1‖∇𝜃‖_BMO𝑑𝑠≤𝐶^𝑡₀𝑤_𝑚²_𝐿2‖∇𝜃‖_BMO𝑑𝑠, (24) where we have also used the fact that BMO is the dual space of the Hardy space ℋ¹.

With the aid of the Hölder inequality and the Young inequality, one shows that for the external force term 2^𝑡₀𝑓_𝑚+𝑔_𝑚−𝑓,𝑤_𝑚𝑑𝜏≤𝐶^𝑡₀𝑔_{𝑚𝐻−𝛼/2}𝑤_{𝑚𝐻𝛼/2}𝑑𝜏+𝐶^𝑡₀𝑓_𝑚−𝑓_{𝐻−𝛼/2}𝑤_{𝑚𝐻𝛼/2}𝑑𝜏≤𝐶^𝑡₀𝑔_𝑚²_{𝐻−𝛼/2}+𝑓_𝑚−𝑓²_{𝐻−𝛼/2}𝑑𝜏+𝜅^𝑡₀𝑤_𝑚²_𝐻𝛼/2𝑑𝜏. (25)

Plugging the above inequalities into (22) and taking the Gronwall inequality into consideration, we derive the global 𝐿² estimate of 𝑤_𝑚: 𝑤_𝑚(𝑡)²_𝐿2+𝜅^𝑡₀Λ^𝛼/2𝑤_𝑚(𝜏)²_𝐿2≤𝜃𝑑𝜏₀−𝜃_𝑚(0)²_𝐿2+𝐶^𝑡₀𝑔_𝑚²_{𝐻−𝛼/2}+𝑓_𝑚−𝑓²_{𝐻−𝛼/2}𝐶𝑑𝜏×exp^𝑡₀‖∇𝜃‖_BMO𝑑𝜏. (26) Passing the limit 𝑚→∞ in the above inequality and applying (21), we have ‖𝑤(𝑡)‖²_𝐿2+𝜅^𝑡₀Λ^𝛼/2𝑤(𝜏)²_𝐿2𝑑𝜏≤liminf_𝑚→∞𝑤_𝑚(𝑡)²_𝐿2+𝜅^𝑡₀Λ^𝛼/2𝑤_𝑚(𝜏)²_𝐿2≤𝑤𝑑𝜏_02𝐿2+𝐶^𝑡₀𝑔²_{𝐻−𝛼/2}𝑑𝜏exp𝐶^𝑡₀‖∇𝜃‖_BMO≤𝑤𝑑𝜏_02𝐿2+𝐶^∞₀𝑔²_{𝐻−𝛼/2}𝐶𝑑𝜏×exp^∞₀‖∇𝜃‖_BMO𝑑𝜏=∶𝐸₀<∞ (27) for the constant 𝐸₀ independent of 𝑡>0.

Moreover, by the slight modification of the derivation of (22) and (27), it is not difficult to derive the following global estimates of 𝑤: ‖𝑤(𝑡)‖²_𝐿2+𝜅^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2≤‖𝑤(𝑠𝑑𝜏)‖²_𝐿2+𝐶^𝑡_𝑠𝑔²_{𝐻−𝛼/2}𝑑𝜏+𝐶^𝑡_𝑠‖∇𝜃‖_BMO𝑑𝜏 (28) for 0≤𝑠≤𝑡≤∞.

3. Average Decay of 𝜃−𝜃

In this section, we are devoted to investigating the average decay of the difference 𝑤=𝜃−𝜃 which will play an important role in the proof of Theorem 2.

It is worth noting that Kozono [30] and Zhou [32] have investigated the average decay properties of the classic Navier-Stokes equation. For the surface quasi-geostrophic equation considered here, however, the fractional dissipative operators Λ^𝛼=(−Δ)^𝛼/2 are nonlocal singular integral operators. the new difficulty on the uniformly estimates of 𝑤 will rise if we directly follow the similar argument of the classic Navier-Stokes equations [30, 32].

Fortunately, we can avoid those additional difficulties with the aid of the theory of the analytic semigroup (see Pazy [34]). Since the Laplacian operator −Δ in whole space ℝ² generates a bounded analytic semigroup in each 𝐿^𝑝(ℝ²) (1<𝑝<∞), thus the fractional power Λ^𝛼=(−Δ)^𝛼/2 can be redefined with the aid of the spectral decomposition: Λ^𝛼=^∞₀𝜆^𝛼/2𝑑𝐸(𝜆), (29) where 𝐸(𝜆) is a family of projection operators and 𝜆(0<𝜆<∞) is the spectral of the Laplacian operator −Δ. Namely, the analytic semigroup 𝑒^{−𝑡Λ𝛼} generated by the nonlocal singular integral operators Λ^𝛼 is also defined in a satisfactory form: 𝑒^{−𝑡Λ𝛼}=^∞₀𝑒^{−𝑡𝜆𝛼/2}𝑑𝐸(𝜆). (30) Furthermore, we may derive the embedding relation in a natural form ‖𝜃‖_𝐿^𝑝≤𝐶‖𝜃‖_𝐻^𝛼=𝐶𝜉^𝛼𝜃_𝐿2=𝐶Λ^𝛼𝜃_𝐿2,1with𝑝=12−𝛼2, (31) where Parseval equality is used in the last two equalities.

Now we carry out to study the average decay of the difference 𝑤=𝜃−𝜃. Firstly, according to the definitions of the weak solutions in Definition 1, we choose a special test function 𝜑_𝜀 as 𝜑_𝜀(𝜏)=^𝑡_𝑠𝜂_𝜀(𝜏−𝜎)1+Λ^{2−(3−𝛼)/2}𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}×𝑤(𝜎)𝑑𝜎,𝜀>0, (32) where 𝜂_𝜀(𝑡)=(1/𝜀)𝜂(|𝑡/𝜀|)>0 is the standard mollifier function with 𝜂(𝑡)∈𝐶^∞₀(−1,1) and ∫¹₋₁𝜂(𝑡)𝑑𝑡=1. Thus 𝑤(𝑥,𝑡) satisfies the following weak formation ^𝑡_𝑠𝜕_𝜏𝑤+𝜅Λ^𝛼𝑤,𝜑_𝜀𝑑𝜏+^𝑡_𝑠𝑢⋅∇𝜃−𝑢⋅∇𝜃,𝜑_𝜀+𝑑𝜏_𝑡𝑠𝑔,𝜑_𝜀𝑑𝜏=0 (33) for 0≤𝑠≤𝑡<∞.

We now give the estimates of (33) by passing the limit 𝜀→0. More precisely, we will prove the following lemma.

Lemma 8. Under the same condition in Theorem 2, we have the following uniform estimates of the difference 𝑤: 1+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2−𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2≤𝐶^𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏+𝐶^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏+𝐶^𝑡_𝑠𝑔²_{𝐻−𝛼/2}𝑑𝜏. (34)

We will give the proof at the end of this section. Once the crucial uniform estimates (34) is obtained, we can obtain the average decay of the difference 𝑤. In fact, letting 𝑡→∞ in (34), since lim_𝑡→∞𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2=0, (35)

then one shows that for (34) limsup_𝑡→∞1+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2≤𝐶^∞_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏+𝐶^∞_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏+𝐶^∞_𝑠𝑔²_{𝐻−𝛼/2}𝑑𝜏, (36)

and then letting 𝑠→∞, we have limsup_𝑡→∞1+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2=0. (37)

Furthermore, employing the Parseval equality and the Hölder inequality, we obtain that for ‖𝑤‖_𝐿²‖𝑤‖²_𝐿2=𝐶_ℝ2|𝑤|²𝑑𝜉=𝐶_ℝ21+𝜉^{2(𝛼2−3𝛼)/6}|𝑤|^2𝛼/3×1+𝜉^{2(3𝛼−𝛼2)/6}|𝑤|^{(6−2𝛼)/3}𝑑𝜉≤𝐶_ℝ21+𝜉^{2−(3−𝛼)/2}|𝑤|²𝑑𝜉^𝛼/3×_ℝ21+𝜉^2𝛼/2|𝑤|²𝑑𝜉^(3−𝛼)/3≤𝐶_ℝ21+𝜉^{2−(3−𝛼)/2}|𝑤|²𝑑𝜉^𝛼/3×_ℝ2|𝑤|²+𝜉^𝛼|𝑤|²𝑑𝜉^(3−𝛼)/3≤𝐶1+Λ^{2−(3−𝛼)/4}𝑤_𝐿^2𝛼/3₂×‖𝑤‖²_𝐿2+Λ^𝛼/2𝑤²_𝐿2^1−(𝛼/3). (38) Integrating in time from 𝑡/2 to 𝑡 yields 1𝑡^𝑡_𝑡/2‖𝑤(𝜏)‖²_𝐿2≤𝐶𝑑𝜏𝑡^𝑡_𝑡/21+Λ^{2−(3−𝛼)/4}𝑤_𝐿^2𝛼/3₂×‖𝑤‖²_𝐿2+Λ^𝛼/2𝑤²_𝐿2^1−(𝛼/3)1𝑑𝜏≤𝐶𝑡^𝑡_𝑡/21+Λ^{2−(3−𝛼)/4}𝑤²_𝐿2𝑑𝜏^𝛼/3×1𝑡^𝑡_𝑡/2‖𝑤‖²_𝐿2+Λ^𝛼/2𝑤²_𝐿2𝑑𝜏^1−(𝛼/3)1≤𝐶𝑡^𝑡_𝑡/21+Λ^{2−(3−𝛼)/4}𝑤²_𝐿2𝑑𝜏^𝛼/311+𝑡^1−(𝛼/3)≤𝐶limsup_{𝑡/2<𝜏<𝑡}1+Λ^{2−(3−𝛼)/4}𝑤(𝜏)²_𝐿2. (39) Together with (37) we derive the desired average decay of the difference 𝑤=𝜃−𝜃 between the global solution 𝜃 of the original quasi-geostrophic equation (1) and the weak solution 𝜃 of the perturbed quasi-geostrophic equation (4): lim_𝑡→∞1𝑡^𝑡_𝑡/2‖𝑤(𝜏)‖²_𝐿2𝑑𝜏=0. (40)

Now it remains to prove Lemma 8.

Proof of Lemma 8. For convenience, the three terms of the left hand side of (33) are denoted by 𝐼, 𝐽, and 𝐾, respectively. For 𝐼, we have 𝐼=^𝑡_𝑠𝜕_𝜏𝑤+𝜅Λ^𝛼𝑤,𝜑_𝜀=𝑑𝜏_𝑡𝑠𝑑𝑑𝜏_ℝ²𝑤𝜑_𝜀+𝑑𝑥𝑑𝜏_𝑡𝑠ℝ2−𝑤𝜕_𝜏𝜑_𝜀+𝜅𝑤Λ^𝛼𝜑_𝜀=𝑑𝑥𝑑𝜏_ℝ2𝑤𝜑_𝜀(𝑡)𝑑𝑥−_ℝ2𝑤𝜑_𝜀𝑠)+(𝑑𝑥_𝑡𝑠ℝ2𝑤(𝜏)−𝜕_𝜏𝜑_𝜀+𝜅Λ^𝛼𝜑_𝜀(𝜏)=𝑑𝑥𝑑𝜏_ℝ2𝑤𝜑_𝜀(𝑡)𝑑𝑥−_ℝ2𝑤𝜑_𝜀(𝑠)−𝑑𝑥_{𝑡𝑠ℝ2𝑡𝑠}𝜕_𝜏𝜂_𝜀(𝜏−𝜎)1+Λ^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}𝑤(𝜎)𝑤(𝜏)𝑑𝜎𝑑𝑥𝑑𝜏=𝐼₁+𝐼₂. (41)
Applying the Fubini theorem, it is not difficult to obtain lim_𝜀→0ℝ2𝑤𝜑_𝜀(𝑡)𝑑𝑥=lim_𝜀→0ℝ2^𝑡_𝑠𝜂_𝜀(𝑡−𝜎)1+Λ^{2−(3−𝛼)/2}𝑒^{−𝜅(𝑡−𝜎)Λ𝛼}(𝜎)𝑤(𝑡)=1×𝑤𝑑𝜎𝑑𝑥2_ℝ²1+Λ^{2−(3−𝛼)/2}𝑤(𝑡)𝑤(𝑡)=1𝑑𝑥21+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2 (42) and similarly lim_𝜀→0ℝ2𝑤𝜑_𝜀(𝑠)1𝑑𝑥=2𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2, (43) thus we have for 𝐼₁lim_𝜀→0𝐼₁=121+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2−12𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2. (44)
For 𝐼₂ we have 𝐼₂=−^𝑡_𝑠ℝ2^𝑡_𝑠𝜕_𝜏𝜂_𝜀(𝜏−𝜎)1+Λ^{2−(3−𝛼)/2}𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}(𝜎)𝑤(𝜏)1×𝑤𝑑𝜎𝑑𝑥𝑑𝜏=−𝜀^𝑡_𝑠ℝ^2𝑡_𝑠𝜕_𝜏𝜂𝜏−𝜎𝜀1+Λ^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}𝑤(𝜎)𝑤(𝜏)=1𝑑𝜎𝑑𝑥𝑑𝜏𝜀^𝑡_𝑠ℝ^2𝑡_𝑠𝜕_𝜎𝜂𝜏−𝜎𝜀1+Λ^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}𝑤(𝜎)𝑤(𝜏)=1𝑑𝜎𝑑𝑥𝑑𝜏𝜀^𝑡_𝑠ℝ^2𝑡_𝑠𝜕_𝜎𝜂𝜎−𝜏𝜀1+Λ^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)Λ𝛼}𝑤(𝜎)𝑤(𝜏)𝑑𝜎𝑑𝑥𝑑𝜏=−𝐼₂ (45) which implies 𝐼₂=0.
Hence, combining the above inequalities, one derives the estimates of 𝐼: lim_𝜀→01𝐼=21+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2−12𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2. (46)
For 𝐽, employing integrating by parts gives 𝐽=^𝑡_𝑠𝑢⋅∇𝜃−𝑢⋅∇𝜃,𝜑_𝜀=𝑑𝜏_𝑡𝑠ℝ2(𝑢−𝑢)⋅∇𝜑_𝜀𝑤𝑑𝑥𝑑𝜏−^𝑡_𝑠ℝ2𝑢⋅∇𝜑_𝜀−𝑤𝑑𝑥𝑑𝜏_𝑡𝑠ℝ2(𝑢−𝑢)⋅∇𝜑_𝜀𝜃𝑑𝑥𝑑𝜏=∶𝐽₁+𝐽₂+𝐽₃. (47)
For 𝐽₁, applying the Hölder inequality and the Gagliardo-Nirenberg inequality and (27) gives 𝐽₁=^𝑡_𝑠ℝ2(𝑢−𝑢)⋅∇𝜑_𝜀≤𝑤𝑑𝑥𝑑𝜏_𝑡𝑠ℛ^⟂𝑤_{𝐿8/(4−𝛼)}∇𝜑_{𝜀𝐿4/𝛼}‖𝑤‖_𝐿^8/(4−𝛼)𝑑𝜏≤𝐶^𝑡_𝑠∇𝜑_{𝜀𝐿4/𝛼}‖𝑤‖_𝐿²Λ^𝛼/2𝑤_𝐿2𝑑𝜏≤𝐶sup_0<𝜏<∞‖𝑤‖_𝐿^2𝑡_𝑠∇𝜑_𝜀²_𝐿4/𝛼𝑑𝜏^1/2×^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏^1/2≤𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏, (48) where we have used the following estimates: ^𝑡_𝑠∇𝜑_𝜀(𝜏)²_𝐿4/𝛼𝑑𝜏≤𝐶^𝑡_𝑠∇𝜑_𝜀(𝜏)²_{𝐻1−(𝛼/2)}𝑑𝜏=𝐶^𝑡_𝑠𝜉^1−(𝛼/2)𝑖𝜉𝜑_𝜀(𝜏)²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠𝜉^2−(𝛼/2)𝜑_𝜀(𝜏)²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠𝜉^2−𝛼𝑡_𝑠𝜂_𝜀(𝜏−𝜎)1+𝜉^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)|𝜉|𝛼}𝜉^𝛼/2𝑤(𝜎)𝑑𝜎²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠^𝑡_𝑠𝜂_𝜀(𝜏−𝜎)𝜉^𝛼/2𝑤(𝜎)𝑑𝜎²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏. (49)
Similarly, for 𝐽₂,𝐽₃, it follows that 𝐽₂+𝐽₃=−^𝑡_𝑠ℝ2𝑢⋅∇𝜑_𝜀−𝑤𝑑𝑥𝑑𝜏_𝑡𝑠ℝ2(𝑢−𝑢)⋅∇𝜑_𝜀𝜃𝑑𝑥𝑑𝜏≤𝐶^𝑡_𝑠ℛ^⟂𝜃_{𝐿4/(2−𝛼)}∇𝜑_{𝜀𝐿4/𝛼}‖𝑤‖_𝐿²+𝑑𝜏_𝑡𝑠ℛ^⟂𝑤_𝐿2∇𝜑_{𝜀𝐿4/𝛼}‖𝜃‖_𝐿^4/(2−𝛼)𝑑𝜏≤𝐶^𝑡_𝑠Λ^𝛼/2𝜃_𝐿2∇𝜑_{𝜀𝐿4/𝛼}‖𝑤‖_𝐿²𝑑𝜏≤𝐶sup_0<𝜏<∞‖𝑤‖_𝐿^2𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏^1/2×^𝑡_𝑠∇𝜑_𝜀²_𝐿4/𝛼𝑑𝜏^1/2≤𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏^1/2𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏^1/2≤𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏+𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏. (50)
Plugging the estimates of 𝐽_𝑖 (𝑖=1,2,3) into (47) yields |𝐽|≤𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏+𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏. (51)
In order to estimate 𝐾, thanks to ^𝑡_𝑠𝜑_𝜀(𝜏)²_𝐻𝛼/2𝑑𝜏≤𝐶^𝑡_𝑠𝜉^𝛼/2𝜑_𝜀(𝜏)²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠^𝑡_𝑠𝜂_𝜀(𝜏−𝜎)1+𝜉^{2−(3−𝛼)/2}×𝑒^{−𝜅(2𝑡−𝜏−𝜎)|𝜉|𝛼}𝜉^𝛼/2𝑤(𝜎)𝑑𝜎²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠^𝑡_𝑠𝜂_𝜀(𝜏−𝜎)𝜉^𝛼/2𝑤(𝜎)𝑑𝜎²_𝐿2𝑑𝜏≤𝐶^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏, (52) together with the Hölder inequality and the Young inequality gives that 𝐾=−^𝑡_𝑠𝑔,𝜑_𝜀𝑑𝜏≤^𝑡_𝑠𝑔_{𝐻−𝛼/2}𝜑_{𝜀𝐻𝛼/2}𝑑𝜏≤𝐶^𝑡_𝑠𝑔²_{𝐻−𝛼/2}𝑑𝜏+𝐶^𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏. (53)
Inserting (46), (51), and (53) into (33), we obtain 1+Λ^{2−(3−𝛼)/4}𝑤(𝑡)²_𝐿2−𝑒^{−𝜅(𝑡−𝑠)Λ𝛼}1+Λ^{2−(3−𝛼)/4}𝑤(𝑠)²_𝐿2≤|𝐽|+|𝐾|≤𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝜃²_𝐿2𝑑𝜏+𝐶+𝐸₀^1/2𝑡_𝑠Λ^𝛼/2𝑤²_𝐿2𝑑𝜏+𝐶^𝑡_𝑠𝑔²_{𝐻−𝛼/2}𝑑𝜏 (54) which completes the proof of Lemma 8.

4. Proof of Theorem 2

With the aid of the global estimates and average decay of the difference 𝑤=𝜃−𝜃 between the original surface quasi-geostrophic equation (1) and the perturbed surface quasi-geostrophic equation, the proof of stability is straight forward.

The integration of (28) with respect to 𝑠 on the interval ((𝑡/2),𝑡) gives ‖𝑤(𝑡)‖²_𝐿2≤2𝑡^𝑡_𝑡/2‖𝑤(𝑠)‖²_𝐿2+𝐶𝑑𝑠𝑡^𝑡_𝑡/2^𝑡_𝑠‖∇𝜃‖_BMO𝑑𝜏𝑑𝑠. (55) From the average decay property of 𝑤 in Section 3, the first term of the right hand side of the above inequality tends to zero as 𝑡→∞. For the second term of the right hand side, we have 1𝑡^𝑡_𝑡/2^𝑡_𝑠‖∇𝜃‖_BMO≤𝐶𝑑𝜏𝑑𝑠𝑡^𝑡_𝑡/2𝑡𝜏−2‖∇𝜃‖_BMO𝑑𝜏≤𝐶^𝑡_𝑡/2‖∇𝜃‖_BMO𝑑𝜏, (56) according to the regular condition on 𝜃 in Theorem 2, the second term also tends to zero as 𝑡→∞. Hence, we derive the following stability property: ‖𝑤(𝑡)‖_𝐿²⟶0as𝑡⟶∞ (57) which completes the proof of Theorem 2.

Acknowledgments

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).