1. Introduction

We consider the dissipative quasi-geostrophic equation with subcritical exponent , where , , is the unknown potential temperature, and is the divergence free velocity which is determined by the Riesz transformation of in the following way: This equation is a two-dimensional model of the incompressible Euler equations, and if , the equation is the Navier-Stokes equation. We refer the reader to [1] where the authors explain the physical origin and the signification of the parameters of this equation.

The critical homogeneous Sobolev space of the system is , and we have

The local well-posedness of () with data is established by [2] and [3] separately if . In [4], Dong and Du study the critical case in the critical space . They prove the global existence if the initial condition is in the critical space .

The global existence when is an open problem. We have only the local existence. In this case [5], Niche and Schonbek prove that if the initial data is in , then the norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay. If , with , they obtain a uniform decay rate in . They consider also the solution in other spaces. For the proof of their results, they use the kernel associated to the operator , and they use the Littlewood-Paley decomposition. Our main result is the following.

Theorem 1.1. Assume that .(i)If is a global solution of (), then (ii)If is a global solution of (), then

2. Notations and Preliminary Results

2.1. 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 is normalized as (ii)The inverse Fourier formula is (iii)For , denotes the usual nonhomogeneous Sobolev space on and its scalar product.(iv)For , denotes the usual homogeneous Sobolev space on and its scalar product.(v)For and , These two inequalities are called the interpolation inequalities, respectively, in the homogeneous and nonhomogeneous Sobolev spaces.(i)For any Banach space , any real number , and any time , we denote by the space of measurable functions such that .(ii)If and are two vector fields, we set We recall a fundamental lemma concerning some product laws in homogeneous Sobolev spaces.

Lemma 2.1 (see [6]). Let , be two real numbers such that There exists a constant , such that for all , If and , there exists a constant such that for all and ,

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 ,

Remark 2.3. (i) In the case where , the formula (2.9) gives
In the case where , the formula (2.9) gives

Proof of Lemma 2.2. From the Cauchy-Schwarz inequality, we have Using the weak derivatives properties, the product laws (Lemma 2.1), with , , and , we can dominate the nonlinear part as follows:

2.2. Existence Theorem

In [7], Wu proves an existence and uniqueness theorem of () in the well-known Besov spaces . We recall this theorem in the special case, where .

Theorem 2.4. Assume that and , then there exists a constant such that if then the initial value problem () has a unique solution in . Moreover, where is the space of continuous and bounded functions from to .

In use of the fact that is a Hilbert space, one deduces the following.

Corollary 2.5. Assume that and , then there exists a constant such that if then the initial value problem () has a unique solution in . Moreover,

Proof . Taking the scalar product in , we get
Using Lemma 2.1 with and , we obtain Then the quadratic term can be absorbed, Taking the integral on the interval , we obtain

3. 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 a sufficient small number. Then from Corollary 2.5,
For a strictly positive real number and a given distribution , we define the operators and , respectively, by the following: We define and ; . Then, We deduce that Since , then from the dominate convergence theorem and (3.3), we have The function satisfies Multiplying this equation by , we deduce that Using Remark 2.3 and (3.3), we get We set Then, Let , from (3.7), there exists such that Let , then where is the Lebesgue measure of . If then . For , there exists such that , and it results that Equation (3.13) and (3.16) give that Thus, , 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 such that Let , with
Now, consider the following system: By Corollary 2.5, there is a unique solution such that Let , then is a solution of the following system: Taking a scalar product in , we obtain Using the product law in Lemma 2.1, with and , then, for all , then , with , Then, Now define the set as a measurable with respect to the Lebesgue measure. We have So and , then there is Then, and then

Applying the conclusion of Theorem 1.1 for () system starting at , we can deduce the desired result.

In the nonhomogeneous case, we suppose that , then

We can suppose that , and for all ,

Thus, it suffices to prove that

Let , then we recall the operators We define and . Then, and from Lemma 2.2, We deduce that Then from the dominate convergence theorem and the following energy estimate we deduce that

Multiplying this equation by , we have We set Then, Let , from (3.40), then there exists such that Let , then where is the Lebesgue measure of . If then . For , there exists such that , then The equations (3.45) and (3.48) give that Thus, , and this finishes the proof.