Abstract
In this paper, we study the asymptotic behavior of the two-dimensional quasi-geostrophic equations with subcritical dissipation. More precisely, we establish that vanishes at infinity.
1. Introduction and Statement of Main Results
In this paper, we consider the initial value problem for the 2D quasi-geostrophic equations with subcritical dissipation :where is a real number and is a dissipative coefficient. is the operator defined by the fractional power of :and more generallywhere denotes the Fourier transform of . is an unknown scalar function representing potential temperature, and is the divergence free velocity which is determined by the Riesz transformation of in the following way:
Let us fix for the rest of the paper.
The quasi-geostrophic fluid is an important model in geophysical fluid dynamics, which are special cases of the general quasi-geostrophic approximations for atmospheric and oceanic fluid flow with the small local Rossby number which ensures the validity of the geostrophic balance between the pressure gradient and the Coriolis force (see [1]). Furthermore, this quasi-geostrophic fluid motion equation shares many features with fundamental fluid motion equations. When , this equation is comparable to the vorticity formulation of the Euler equations (see [2]). with shares similar features with the three-dimensional Navier–Stokes equations. Thus, is therefore referred as the critical case, while the cases and are subcritical and supercritical, respectively.
The existence of a global weak solution was established by several researchers. The reader is referred to [3–7] and their references. Furthermore, in the subcritical case, Constantin and Wu [8] proved that every sufficiently smooth initial data give rise to a unique global smooth solution. For the critical case, , Constantin et al. [9] proved that there exists of a unique global classical solution for any small initial data in . The hypothesis requiring smallness in was removed by Caffarelli and Vasseur [10] and Dong and Du [5]. In [7], the authors proved persistence of a global solution in for to any periodic initial data. Chae and Lee [6] established the global existence and uniqueness of solution for any small initial data in the Besov space .
The global existence for the quasi-geostrophic equation has been studied in the previous work of Benameur and Benhamed [3]. The authors have introduced new spaces defined as follows:which is equipped with the norm
More precisely, their result is as follows..
Theorem 1. Let . There is a time and unique solution of ; moreover, .
If , then the solution is global and
We will recall further that the paper titled “Behavior of solutions of 2D quasi-geostrophic equations” by Constantin and Wu [8] (published in the SIAM Journal of Mathematical Analysis 30, 1999) has several results along the same lines. In particular, it includes the following result.
Theorem 2. Let and . Then, there exists a weak solution of the equation with initial data such that
Proof. where is a constant depending on and norms of .
The decay of and Sobolev norms and asymptotic behaviour of solutions to the quasi-geostrophic equations have also been addressed in many articles (see, for example, [5, 11–22]
Global-in-time well-posedness, time-decay, and asymptotic behavior of solutions are core properties in understanding how fluid mechanics models work. In fact, there is a rich literature about those properties for fluid dynamics PDEs via several approaches and different frameworks. In this direction, there are studies in frameworks containing singular data and invariant under the scaling (critical spaces), where the smallness conditions are taken in the weak norms of the critical spaces (e.g., see the review book [23, 24]). Of particular interest is the analysis of PDEs in critical frameworks whose structure is based on the Fourier transform. Navier–Stokes and quasi-geostrophic equations have been studied in several spaces such as [13, 25, 26], Fourier–Besov [7, 27, 28], Lei–Lin spaces [3, 29], and Fourier–Besov–Morrey [11, 30]. In the case , Fourier–Besov spaces were introduced by Iwabuchi [31] in the context of parabolic-elliptic Keller–Segel system. Later, Iwabuchi and Takada [7] used critical -spaces in order to obtain a global well-posedness class (uniformly with respect to the angular velocity) for the Navier–Stokes–Coriolis system. Taking in particular , they also obtained a global well-posedness result for the Navier–Stokes equations with small initial data in . Konieczny and Yoneda [28] also showed global well-posedness and asymptotic stability of small solutions for Navier–Stokes equations (and Navier–Stokes–Coriolis) in critical Fourier–Besov spaces , with . Extensions of those results to critical Fourier–Besov–Morrey spaces (larger than ) were obtained in [30, 32] for Navier–Stokes equations (and Navier–Stokes–Coriolis) and active scalar equations with fractional dissipation (including the quasi-geostrophic equation ), respectively.
The main goal of this paper is to study the quasi-geostrophic equation in the framework of critical Lei–Lin spaces with and . We show that solution presents the asymptotic property as provided that . For that, we use standard interpolation in the Fourier space, energy estimates in , and Young’s inequality of convolutions, among others. Our main result is the following.
Theorem 3. Let and be a global solution of given by Theorem 1. Then,
Remark 1. Remark that our main result is not implied by Theorem 2 of Constantin- Wu, because our works concern the asymptotic behavior of solution in the Lei‐Lin space who belongs to a class whose definition of the norm is based on Fourier transform, but it is not contained in L2 while that the result of Theorem 2 concerns the study in the space . The proof techniques (for Theorem 3 and Lemma 1) appeal to fairly standard interpolation in the Fourier space (which creates the need for instead of the more natural ), energy-type estimates for that exploit the natural appearance of the -norm from Young’s inequality of convolutions, and two uses of (Theorem 3) (proved in the earlier work).
The remaining part of the paper is organized as follows. The main results are given in Section 1. We explain the framework in Section 2. The long time behaviour (Theorem 3) is established in Section 3.
2. Preliminary
Let us recall that in [29], Lei and Lin introduced a new space, named Lei–Lin space , which belongs to a class whose definition of norm is based on the Fourier transform but is not contained in . In [3], Benameur and Benhamed defined the spaces that are useful for the study of well-posedness of PDEs of parabolic, elliptic, and dispersive types. More precisely, for , we defineequipped with the norm
To prove Theorem 3, we need the following lemma.
Lemma 1. For , . More precisely, if . Then, we have
Proof. For , put , whereLet us start by controlling the first term; using Cauchy–Schwarz inequality, we getTherefore,A similar calculation to the foregoing yieldsThen,Combining (15) and (17), we getOptimizing , it suffices to choose , to obtain (12).
3. Proof of the Main Theorem
The main aim of this section is to study the asymptotic behavior of the global solutions given by Theorem 1. The proof is inspired from [33].
First, we Take .
Let , such that . For , put
We have converges in to . Then, there is such that
Let be a fixed integer. Put
Then,
Consider the system
For all , we have
Use Theorem 1; then, there exists a unique global solution .
Furthermore,
Moreover,
Put . Then, is a solution of the following system:
By taking the inner product in with , we get
Then,
By Young’s inequality, we obtain
Therefore,
Integrating with respect to time, we obtain
By Gronwall’s lemma, we get
Combining (32) and (33), we obtain
Applying Lemma 1 to , we infer
Then,
From (33) it follows that
Therefore, by integrating in time between 0 and , we get
Indeed consider the following subset of :
We have
Then, for all ,
Thus,
Using the inequality (38), we get and .
Therefore, there exists . Particularly,
Now, put
Thus, by (24), we obtain
Therefore,
Let us consider the following equation:
Using inequality (46) and Theorem 1, we infer that there exists a unique solution of such that
The existence and uniqueness of a solution to the quasi-geostrophic equation gives . Then,Thus, Theorem 3 is proved.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.