Long Time Behavior of the Solution of the Two-Dimensional Dissipative QGE in Lei–Lin Spaces

<jats:p>In this paper, we study the asymptotic behavior of the two-dimensional quasi-geostrophic equations with subcritical dissipation. More precisely, we establish that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mfenced open="‖" close="‖" separators="|"><mml:mrow><mml:mi>θ</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="script">X</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math> vanishes at infinity.</jats:p>


Introduction and Statement of Main Results
In this paper, we consider the initial value problem for the 2D quasi-geostrophic equations with subcritical dissipation (QG) α : where 1/2 < α ≤ 1 is a real number and k > 0 is a dissipative coefficient. Λ is the operator defined by the fractional power of − Δ: Λ � (− Δ) 1/2 , Λg � (− Δ) 1/2 g � |ξ|g, (2) and more generally where g(ξ) denotes the Fourier transform of g. θ(x, t) is an unknown scalar function representing 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: Let us fix k � 1 for the rest of the paper. e 2D 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 k � 0, this equation is comparable to the vorticity formulation of the Euler equations (see [2]). (QG) α with α � 1/2 shares similar features with the three-dimensional Navier-Stokes equations. us, α � 1/2 is therefore referred as the critical case, while the cases α � 1/2 and α � 1/2 are subcritical and supercritical, respectively. e existence of a global weak solution was established by several researchers. e reader is referred to [3][4][5][6][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, α � 1/2, Constantin et al. [9] proved that there exists of a unique global classical solution for any small initial data in L ∞ . e hypothesis requiring smallness in L ∞ was removed by Caffarelli and Vasseur [10] and Dong and Du [5]. In [7], the authors proved persistence of a global solution in C ∞ for to any C ∞ periodic initial data. Chae and Lee [6] established the global existence and uniqueness of solution for any small initial data in the Besov space _ B 2− 2α 2,1 . e global existence for the quasi-geostrophic equation has been studied in the previous work of Benameur and Benhamed [3]. e authors have introduced new spaces X 1− 2α (R 2 ) defined as follows: which is equipped with the norm More precisely, their result is as follows..

, 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.
Proof. where C is a constant depending on L 1 and L 2 norms of θ 0 . e decay of L 2 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][12][13][14][15][16][17][18][19][20][21][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 PM a [13,25,26], [7,27,28], Lei-Lin spaces X σ [3,29], and Fourier-Besov-Morrey FN s p,μ,r [11,30]. In the case q � 1, Fourier-Besov spaces FB s q,r (R n ) were introduced by Iwabuchi [31] in the context of parabolic-elliptic Keller-Segel system. Later, Iwabuchi and Takada [7] -spaces in order to obtain a global wellposedness class (uniformly with respect to the angular velocity) for the Navier-Stokes-Coriolis system. Taking in particular Ω � 0, they also obtained a global well-posedness result for the 3D Navier-Stokes equations with small initial data in _ FB − 1 1,2 . Konieczny and Yoneda [28] also showed global well-posedness and asymptotic stability of small solutions for 3D Navier-Stokes equations (and Navier-Stokes-Coriolis) in critical Fourier-Besov spaces FB 2− (3/q) p,∞ , with 1 < p ≤ ∞. Extensions of those results to critical Fourier-Besov-Morrey spaces FN s p,μ,r (larger than _ FB s p,r ) were obtained in [30,32] for Navier-Stokes equations (and Navier-Stokes-Coriolis) and active scalar equations with fractional dissipation (− Δ) α (including the 2D quasi-geostrophic equation (QG) α ), respectively. e main goal of this paper is to study the 2D quasigeostrophic equation in the framework of critical Lei-Lin spaces X σ with σ � 1 − 2α and 2/3 < α ≤ 1. We show that solution θ presents the asymptotic property ‖θ(t)‖ X 1− 2α ⟶ 0 as t ⟶ 0 provided that ‖θ(t)‖ X 1− 2α < 1/4. For that, we use standard interpolation in the Fourier space, energy estimates in L 2 , and Young's inequality of convolutions, among others. Our main result is the following. □ Theorem 3. Let 2/3 < α < 1, ‖θ 0 ‖ X 1− 2α < 1/4 and θ ∈ C(R + , X 1− 2α (R 2 )) be a global solution of (QG) α given by eorem 1. en, lim t⟶∞ ‖θ(t)‖x 1− 2α � 0.
Remark 1. Remark that our main result is not implied by eorem 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 L 2 while that the result of eorem 2 concerns the study in the space L 1 ∩L 2 . e proof techniques (for eorem 3 and Lemma 1) appeal to fairly standard interpolation in the Fourier space (which creates the need for α > 2/3 instead of the more natural α > 1/2), energytype L 2 estimates for θ n that exploit the natural appearance of the X − 1 -norm from Young's inequality of convolutions, and two uses of ( eorem 3) (proved in the earlier work). e remaining part of the paper is organized as follows. e main results are given in Section 1. We explain the framework in Section 2. e long time behaviour ( eorem 3) is established in Section 3.

Preliminary
Let us recall that in [29], Lei and Lin introduced a new space, named Lei-Lin space X − 1 , which belongs to a class whose definition of norm is based on the Fourier transform but is not contained in L 2 . 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 σ ∈ R, we define equipped with the norm To prove eorem 3, we need the following lemma.

Proof of the Main Theorem
e main aim of this section is to study the asymptotic behavior of the global solutions given by eorem 1. e proof is inspired from [33].