Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

In this model, ψ is the geostrophic pressure, also called the geostrophic stream function, ξ = Δψ is the vertical component of the relative vorticity, ∇⊥ψ = (−∂ψ/∂y, ∂ψ/∂y) is a zeroth-order balance in the momentum equation, and F, β, and R e are the rotational Froude number, the Coriolis parameter, and the Reynolds number, respectively. Usually, ] = 1/R e is also called viscosity parameter. It has some features in common with the much studied two-dimensional surface quasigeostrophic equation (SQGE) (see [3–9] and references therein). However the quasi-geostrophic β-plane model has a number of novel and distinctive features. Recently, this equation has been intensively investigated because of both itsmathematical importance and its potential applications in meteorology and oceanography. The quasigeostrophic β-plane model is a simplified model for the shallow water β-plane model [2, 10, 11] when the Rossby number is small under several assumptions on the magnitude of the bottom topography variations, which is used to understand the atmospheric and oceanic circulation, the gulf stream, and the variability of this circulation on time scales from several months to several years. In this regime, quasi-geostrophic theory is an adequate approximation to describe the flow and is developed for the simulation of large-scale geophysical currents in the middle latitudes. Whenα = 1, this is the standard quasi-geostrophicmodel studied in [1], which was put forward as a simplifiedmodel of the shallowwatermodel (see also [2] for a review). In [12], the author studied a multilayer quasi-geostrophic model, which is a generalization of the single layer model in the case α = 1. The general fractional powerαwas considered by Pu andGuo [13]. The equation is


Introduction
This paper is concerned with the nonlocal quasigeostrophic -plane model with modified dissipativity [1,2] where (, ) ∈ Ω can be either the 2D torus T 2 or the whole space R 2 ,  ≥ 0,  = Δ −  + , and (1/  )(−Δ) 1+  with  ∈ (0, 1) being the modified dissipative term.Let (, ) =     −     denote the Jacobian operator; (1) can be notationally simplified as In this model,  is the geostrophic pressure, also called the geostrophic stream function,  = Δ is the vertical component of the relative vorticity, ∇ ⊥  = (−/, /) is a zeroth-order balance in the momentum equation, and , , and   are the rotational Froude number, the Coriolis parameter, and the Reynolds number, respectively.Usually, ] = 1/  is also called viscosity parameter.It has some features in common with the much studied two-dimensional surface quasigeostrophic equation (SQGE) (see [3][4][5][6][7][8][9] and references therein).However the quasi-geostrophic -plane model has a number of novel and distinctive features.Recently, this equation has been intensively investigated because of both its mathematical importance and its potential applications in meteorology and oceanography.The quasigeostrophic -plane model is a simplified model for the shallow water -plane model [2,10,11] when the Rossby number is small under several assumptions on the magnitude of the bottom topography variations, which is used to understand the atmospheric and oceanic circulation, the gulf stream, and the variability of this circulation on time scales from several months to several years.In this regime, quasi-geostrophic theory is an adequate approximation to describe the flow and is developed for the simulation of large-scale geophysical currents in the middle latitudes.
When  = 1, this is the standard quasi-geostrophic model studied in [1], which was put forward as a simplified model of the shallow water model (see also [2] for a review).In [12], the author studied a multilayer quasi-geostrophic model, which is a generalization of the single layer model in the case  = 1.The general fractional power  was considered by Pu and Guo [13].The equation is (, , 0) =  0 (, ) .
In [13], they proved the global existence of weak solutions by employing the Galerkin approximation method for initial data belonging to the (inhomogeneous) Sobolev space  2 (Ω).If the initial data is in the (homogeneous) Sobolev space Ḣ (Ω) ( > 2), it is natural for us to ask whether (3) has regular solutions.
In this paper, we only consider the 2D torus T 2 with periodic boundary conditions.And we will prove the wellposedness results of (3) under certain condition on initial data which belong to the (homogeneous) Sobolev space Ḣ (T 2 ) ( > 3 − 2).In Section 3, the local existence and uniqueness of the solutions of the problem are proved in Ḣ (T 2 ) when  > 3 − 2 for 1/2 <  < 1.That is, for any initial data  0 ∈ Ḣ (T 2 ) and  ∈  2 (0, ; Ḣ−−2 (T 2 )), there exists such that (3) has a uniqueness solution on [0, ], satisfying However, we may not obtain the global existence of solutions from energy (34), if the initial data has large Ḣ norm.
The main reason is that in the Ḣ energy estimate for (3), the integral (Λ 2(−1) , (, Δ)) ̸ = 0 for  > 2, where (, V) denotes the integral ∫ T 2 (, )V(, )  as usual.Thus, it is necessary to control it.To overcome this essential difficulty, we will make use of the properties of the product estimates (Proposition 2) as well as those of the Sobolev embedding inequality.
For the cases,  = 1/2 and  > 3, we also obtain the unique global solution in   proved by where  1 = 1 − 2/ and  2 = 1 − 3/.We conclude this introduction by mentioning the global existence result of weak solutions obtained [13].

Notations and Preliminaries
We now review the notations used throughout the paper.Let us denote Λ = (−Δ) 1/2 .The Fourier transform f of a tempered distribution () on T 2 is defined as Generally, Λ   for  ∈ R can be identified with the Fourier series (T 2 ) denotes the space of the th-power integrable functions normed by For any tempered distribution  on Ω and  ∈ R, we define Ḣ denotes the homogeneous Sobolev space of all  for which ‖‖ Ḣ is finite.The homogeneous counterparts of Ḣ are denoted by   .
Next, this section contains a few auxiliary results used in the paper.In particular, we recall, by now, the classical, product, and commutator estimates, as well as the Sobolev embedding inequalities.Proofs of these results can be found for instance, in [14][15][16].
Proposition 2 (product estimate).If  > 0, then, for all ,  ∈   ∩  ∞ , one has the estimates where In the case of a commutator we have the following estimate.
We will use as well the following Sobolev inequality.

Local Existence and Large Data
In [7], the authors studied and established the existence and uniqueness of local and global solutions to the twodimensional SQGE.It is natural that (3) is more complex than SQGE.However, we also establish an analogue.In this section we will prove that (3) is locally well-posed in Ḣ (T 2 ) when  > 3 − 2 for 1/2 <  < 1. Regarding arbitrarily large initial data, we obtain the following result.
Proof.First of all, multiplying (3) by Λ 2(−1) , we get the following energy inequality: We estimate the first term on the right side by To handle the second term, we proceed as follows.
In order to estimate ‖Λ where  1 = (2 − )/.From Sobolev embedding, we have Then, using the same method as in Case 1, we can complete Theorem 6.

Global Existence and Small Data
The main result of this section concerns global wellposedness in case of small initial data.
Proof.We proceed as in the proof of Theorem 6.The product term in ( 27) is now estimated by where  = 1/(1 − ), so that Similarly, in order to estimate ‖Λ − ‖  2 ‖Δ‖  2 in (40), we split it into two cases.
Case 3 (3 − 2 < 2 < ).From Sobolev imbedding, we have Case 4 (3 − 2 <  < 2).Using Sobolev imbedding, we have So, we can always obtain the following estimate With this choice of  and the above embedding, the product estimate gives us Combining ( 24) with (45) and proceeding as in (34) we obtain where  is sufficiently small, thereby concluding the proof of the theorem.
Note also that the proof of Theorem 7 fails for the value  = 1/2.Thus,  = 1/2 indeed is the limit of the local wellposedness theory.Nonetheless, we still can prove that the considered system is globally well-posed for small data.
Proof.We proceed as in the proof of Theorem 7 and obtain the energy estimate (62) We obtain the desired result as in the proof of Theorem 7.