Global existence of the two-dimensional QGE with sub-critical dissipation

In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_\alpha)$. We prove that there exists a unique local-in-time solution for any large initial data $\theta_0$ in the space ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ defined by (\ref{ec}). Moreover we show that $({\bf S}_\alpha)$ has a global solution in time if the norms of the initial data in ${\bf{\mathcal X}}^{1-2\alpha}(\mathbb R^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}_\alpha)$.


Introduction
In this article, we study the initial value-problem for the two-dimensional quasi-geostrophic equation with sub-critical dissipation (S α ) , 1/2 < α ≤ 1, (S α ) ∂ t θ + (−∆) α θ + u.∇θ = 0 θ(0) = θ 0 where 1/2 < α ≤ 1 is a real number. The variable θ represents potential temperature, u = (∂ 2 (−∆) −1/2 θ, −∂ 1 (−∆) −1/2 θ) is the fluid velocity. In the following, we are interested in the case when 1/2 < α ≤ 1 as the case α = 1/2 is very much similar to the 2D Navier-Stokes which is known to be a wellposed problem, and therefore the case α > 1/2 is even easier to deal with. The mathematical study of the non-dissipative case has first been proposed by Constantin, Majda and Tabak in [13] where it is shown to be an analogue to the 3D Euler equations. The dissipative case has then been studied by Constantin and Wu in [11] when α > 1/2 and global existence in Sobolev spaces is studied by Constantin. In this paper, we study (S α ) in scaling invariant spaces. Solvability of evolution equations in scaling invariant spaces is well-developed in the context of the Navier-Stokes equations. For example, if we restrict the function spaces to the energy spaces, the optimal result is due to Fujita and Kato in [7]. Later, Chemin [10] proved similar results in the framework of Besov spacesḂ q p −1 p,q . Let us find the scaling invariant critical spaces for (S α ). The equation is invariant under the following scaling: θ λ (t, x) = λ 2α−1 θ(λ 2α t, λx), with initial data θ λ (0, x) = θ 0 λ (x) = λ 2α−1 θ 0 (λx).
In this paper, we will solve the system (S α ) in the critical space by means of a contraction argument. Thus, we can obtain a unique local solution to the system (S α ) for any initial data in the critical space and prove the corresponding solution will be global if the initial data is sufficient small. Before giving our main result let us first precise the notion of critical space.
For σ ∈ R, we define the functional space which is equipped with the norm Now we give our first result overcoming the above mentioned smallness assumption. Our global existence result reads as follows.
. There is a time T > 0 and unique solution θ ∈ C([0, T ], X 1−2α (R 2 )) of (S α ), moreover θ ∈ L 1 ([0, T ], X 1 (R 2 )). If θ 0 X 1−2α < 1/4, there are global existence and The proof of the above theorem is based on Fixed Point Theorem. We will establish a local existence result and will be able to get global existence that is essentially based on energy estimate of the solution in the space X 1−2α (R 2 ). We state now our second main result.
) be the maximal solution of (S α ) given by Theorem 1.1. Then The rest of this paper is divided into three sections. Section 2 is further divided into tow subsections. Section 2.1 recalls the notations, Section 2.2 preliminaries results. Section 3 is further divided into three subsections. Section 3.1 deals with the existence of solution with any large initial data in the spaces in X 1−2α (R 2 ) , section 3.2 deals with the uniqueness of solution, and section 3.3 s devoted to global existence. The section 4 proves a blow-up criterion of the non global solution given by Theorem 1.1.

Notations and Preliminaries results
2.1. Notations. In this short paragraph, we give some notations • For f , we denote u f the following • The inverse Fourier formula is • For any Banach space (B, . ), any real number 1 ≤ p ≤ ∞ and any time T > 0, we will denote by L p • The fractional Laplacian operator (−∆) α for a real number α is defined through the Fourier transform, namely (−∆) α f (ξ) = |ξ| 2α f (ξ).

Preliminaries results.
The main result of this section is the following lemmas that will play a crucial role in the proof of our main theorem.
We obtain, Hence, The following lemma, which is a direct consequence of the preceding one, will be useful in the proof of Theorem 1.1. Where, On the one hand, if |η| < |ξ − η|, we have Then Therefore By Young inequality, we get Then Hence
• Proof of (2.4). By including the inequalities (2.1) and (2.2) in (2.3), we obtain The proof of Theorem 1.1 requires the following lemma.

Using Lemma 2.3 and the fact
Also the following lemma will be useful in the sequel.

3.1.
Existence. The idea of the proof is to write the initial condition as sum higher and lower frequencies. For small frequencies, we give a regular solution of the associated linear system to (S α ) and for the higher frequencies we consider a partial differential equation very similar to (S α ) with small initial data in X 1−2α for which we can solve it by the fixed point theorem.
• Let r be a real number such that 0 < r < 1/20.
Put a 0 and b 0 defined by And put a = e −t|D| 2α a 0 , a is the unique solution of the heat equation We have Using Dominated Convergence Theorem, we get Let ε > 0 such that By equation (3.2), there is a time T = T (ε) > 0 such that Put b = θ − a, clearly b is a solution of the following system To prove the existence of b, put the following operator • Now, we introduce the space X T as follows with the norm . Using Lemmas 2.3 and 2.4, we can prove Ψ(X T ) ⊂ X T .
• Also, we note B r the subset of X T defined by Using equation (3.1), Lemma 2.3 and the fact b ∈ B r , we get Then Similarly, By using Lemma 2.4, we have Then Combining equation (3.6) and (3.7), we get Ψ(b) ∈ B r and we can deduce • Prove that Put B 1 and B 2 defined by We have Using Lemmas 2.2 and 2.3, we can deduce Then Using the fact r < 1/20, we obtain Then, combining equations (3.8)-(3.9) and the fixed point theorem, there is a unique b ∈ B r such that θ = a + b is solution of (S α ) with θ ∈ X T (R 2 ).