Long-Time Decay to the Global Solution of the 2 D Dissipative Quasigeostrophic Equation

and Applied Analysis 3 iii For s ∈ R, H R2 denotes the usual nonhomogeneous Sobolev space on R2 and 〈·, ·〉Hs R2 its scalar product. iv For s ∈ R, Ḣ R2 denotes the usual homogeneous Sobolev space on R2 and 〈·, ·〉Ḣs R2 its scalar product. v For s, s′ ∈ R and t ∈ 0, 1 , ∥ ∥f ∥ ∥ Hts 1−t s′ ≤ ∥ ∥f ∥ ∥ t Hs ∥ ∥f ∥ ∥ 1−t Hs′ , 2.3 ∥ ∥f ∥ ∥ Ḣts 1−t s′ ≤ ∥ ∥f ∥ ∥ t Ḣs ∥ ∥f ∥ ∥ 1−t Ḣs′ . 2.4 These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces. i For any Banach space B, ‖ · ‖ , any real number 1 ≤ p ≤ ∞, and any time T > 0, we denote by LpT B the space of measurable functions t ∈ 0, T → f t ∈ B such that t → ‖f t ‖ ∈ L 0, T . ii If f f1, f2 and g g1, g2 are two vector fields, we set f ⊗ g : (g1f, g2f ) , div ( f ⊗ g) : (div(g1f ) ,div ( g2f )) . 2.5 We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces. Lemma 2.1 see 6 . Let s1, s2 be two real numbers such that s1 < 1, s1 s2 > 0. 2.6 There exists a constant C : C s1, s2 , such that for all f, g ∈ Ḣ1 R2 ∩ Ḣs2R2 , ∥ ∥fg ∥ ∥ Ḣs1 s2−1 R2 ≤ C (∥ ∥f ∥ ∥ Ḣs1 R2 ∥ ∥g ∥ ∥ Ḣs2 ∥ ∥f ∥ ∥ Ḣs2 ∥ ∥g ∥ ∥ Ḣs1 ) . 2.7 If s1, s2 < 1 and s1 s2 > 0, there exists a constant c c s1, s2 such that for all f ∈ Ḣ1 R2 and g ∈ Ḣs2R2 , ∥ ∥fg ∥ ∥ Ḣs1 s2−1 R2 ≤ c ∥ ∥f ∥ ∥ Ḣs1 ∥ ∥g ∥ ∥ Ḣs2 . 2.8 For the proof of the main result, we need the following lemma. Lemma 2.2. With the same conditions of Theorem 1.1, for all σ ≥ 0,


Introduction
We consider the 2D dissipative quasi-geostrophic equation with subcritical exponent 1/2 < α ≤ 1, where x ∈ R 2 , t > 0, θ θ x, t is the unknown potential temperature, and u u 1 , u 2 is the divergence free velocity which is determined by the Riesz transformation of θ in the following way:

1.1
The local well-posedness of S α with Ḣ2−2α R 2 data is established by 2 and 3 separately if α ∈ 0, 1/2 .In 4 , Dong and Du study the critical case α 1/2 in the critical space Ḣ1 R 2 .They prove the global existence if the initial condition is in the critical space The global existence when α ∈ 1/2, 1 is an open problem.We have only the local existence.In this case 5 , Niche and Schonbek prove that if the initial data θ 0 is in L 2 R 2 , then the L 2 norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay.If θ 0 ∈ L p R 2 , with 1 ≤ p ≤ 2, they obtain a uniform decay rate in L 2 .They consider also the solution in other L q spaces.For the proof of their results, they use the kernel P α t, x associated to the operator ∂ t −Δ α , and they use the Littlewood-Paley decomposition.Our main result is the following.

Notations and Technical Lemmas
In this short section, we collect some notations and definitions that will be used later, and we give some technical lemmas.
i The Fourier transformation in R 2 is normalized as ii The inverse Fourier formula is iii For s ∈ R, H s R 2 denotes the usual nonhomogeneous Sobolev space on R 2 and •, • H s R 2 its scalar product.
iv For s ∈ R, Ḣs R 2 denotes the usual homogeneous Sobolev space on R 2 and •, • Ḣs R 2 its scalar product.
v For s, s ∈ R and t ∈ 0, 1 , These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces.
i For any Banach space B, • , any real number 1 ≤ p ≤ ∞, and any time T > 0, we denote by ii If f f 1 , f 2 and g g 1 , g 2 are two vector fields, we set

2.5
We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.
Lemma 2.1 see 6 .Let s 1 , s 2 be two real numbers such that There exists a constant C : C s 1 , s 2 , such that for all f, g 2.8 For the proof of the main result, we need the following lemma.

Lemma 2.2. With the same conditions of Theorem 1.1, for all
2.9 Remark 2.3.i In the case where σ 0, the formula 2.9 gives

2.10
In the case where σ 2 − 2α, the formula 2.9 gives

Existence Theorem
In 7 , Wu proves an existence and uniqueness theorem of S α in the well-known Besov spaces Ḃr p,q .We recall this theorem in the special case, where p q 2. Theorem 2.4.Assume that α ∈ 0, 1 and θ 0 ∈ Ḣ2−2α R 2 , then there exists a constant c α > 0 such that if where C b R , Ḣ2−2α R 2 is the space of continuous and bounded functions from In use of the fact that Ḣ2−2α R 2 is a Hilbert space, one deduces the following.

2.19
Then the quadratic term can be absorbed, Taking the integral on the interval 0, t , we obtain

Proof of the Main Theorem
The proof of the first part will be in two steps.

First Step (Small Initial Data)
In this case, we suppose that with c α a sufficient small number.Then from Corollary 2.5, For a strictly positive real number δ and a given distribution f, we define the operators A δ D and B δ D , respectively, by the following:

3.4
We define w δ A δ D θ and v δ B δ D θ; F θ F w δ F v δ .Then,

3.7
The function v δ satisfies

3.9
Abstract and Applied Analysis 7 Using Remark 2.3 and 3.3 , we get

3.10
We set

3.17
Thus, lim t → ∞ θ t Ḣ2−2α 0, and this finishes the proof in this case.

Second Step (Large Initial Data)
To prove the result for any initial data, it suffices to prove the existence of some t 0 ≥ 0 such that θ t 0 Ḣ2−2α < c α .

3.19
Now, consider the following system:

3.20
By Corollary 2.5, there is a unique solution r then a is a solution of the following system: Taking a scalar product in L 2 R 2 , we obtain

3.26
Now define the set as a measurable with respect to the Lebesgue measure.We have

10 Abstract and Applied Analysis
So , then there is Then, a t 0 Ḣ2−2α < ε, 3.30 and then

3.31
Applying the conclusion of Theorem 1.1 for S α system starting at θ t 0 , we can deduce the desired result.

3.33
Thus, it suffices to prove that lim t → ∞ θ t L 2 0.

3.34
Let δ > 0, then we recall the operators

3.35
We define w δ A δ D θ and v δ B δ D θ .Then,

3.38
Then from the dominate convergence theorem and the following L 2 energy estimate