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

Authors: Moez Benhamed; Sahar Mohammad Abusalim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we study the asymptotic behavior of the two-dimensional quasi-geostrophic equations with subcritical dissipation. More precisely, we establish thatθtX1−2α vanishes at infinity.


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## Body

## 1. Introduction and Statement of Main Results

In this paper, we consider the initial value problem for the 2D quasi-geostrophic equations with subcritical dissipationQGα:(1)QGα∂tθ+u·∇θ+kΛ2αθ=0,x∈ℝ2,t>0,u=R⊥θ=−ℜ2θ,ℜ1θ,θ0,x=θ0x,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:(2)Λ=−Δ1/2,Λg^=−Δ1/2g^=ξg^,and more generally(3)Λ2αg^=−Δαg^=ξ2αg^,where g^ξ denotes the Fourier transform of g. θx,t is an unknown scalar function representing potential temperature, and u=u1,u2 is the divergence free velocity which is determined by the Riesz transformation of θ in the following way:(4)u1=−ℜ2θ=−∂2−Δ−1/2θ,u2=−ℜ1θ=−∂1−Δ−1/2θ.Let us fixk=1 for the rest of the paper.The2D 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]). QGα with α=1/2 shares similar features with the three-dimensional Navier–Stokes equations. Thus, 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.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, α=1/2, Constantin et al. [9] proved that there exists of a unique global classical solution for any small initial data in L∞. The 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˙2,12−2α.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 X1−2αℝ2 defined as follows:(5)X1−2αℝ2=f∈S′ℝ2;ξ↦ξ1−2αf^ξ∈L1ℝ2,12<α≤1,which is equipped with the norm(6)fX1−2α=∫ℝ2ξ1−2αf^ξdξ.More precisely, their result is as follows..Theorem 1.
Letθ0∈X1−2αℝ2. There is a time T>0 and unique solution θ∈C0,T,X1−2αℝ2 of QGα; moreover, θ∈L10,T,X1ℝ2.
Ifθ0X1−2α<1/4, then the solution is global and(7)θtX1−2α+1−4θ0X1−2α∫0tθsX1ds≤θ0X1−2α,∀t≥0.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.
Let0<α≤1/2 and θ0∈L1ℝ2∩L2ℝ2. Then, there exists a weak solution θ of the QGα equation with initial data θ0 such that(8)θ.,tL2ℝ2≤C1+t−1/2α,Proof.
whereC is a constant depending on L1 and L2 norms of θ0.
The decay ofL2 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 PMa [13, 25, 26], Fourier–Besov ℱℬq,r2−3/q [7, 27, 28], Lei–Lin spaces Xσ [3, 29], and Fourier–Besov–Morrey ℱNp,μ,rs [11, 30]. In the case q=1, Fourier–Besov spaces ℱℬq,rsℝn were introduced by Iwabuchi [31] in the context of parabolic-elliptic Keller–Segel system. Later, Iwabuchi and Takada [7] used critical ℱℬ1,2−1.-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 Ω=0, they also obtained a global well-posedness result for the 3D Navier–Stokes equations with small initial data in ℱ˙ℬ1,2−1. 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 ℱℬp,∞2−3/q, with 1<p≤∞. Extensions of those results to critical Fourier–Besov–Morrey spaces ℱNp,μ,rs (larger than ℱ˙ℬp,rs) were obtained in [30, 32] for Navier–Stokes equations (and Navier–Stokes–Coriolis) and active scalar equations with fractional dissipation −Δα (including the 2D quasi-geostrophic equation QGα), respectively.
The main goal of this paper is to study the2D quasi-geostrophic equation in the framework of critical Lei–Lin spaces Xσ with σ=1−2α and 2/3<α≤1. We show that solution θ presents the asymptotic property θtX1−2α⟶0 as t⟶0 provided that θtX1−2α<1/4. For that, we use standard interpolation in the Fourier space, energy estimates in L2, and Young’s inequality of convolutions, among others. Our main result is the following.Theorem 3.
Let2/3<α<1,θ0X1−2α<1/4 and θ∈Cℝ+,X1−2αℝ2 be a global solution of QGα given by Theorem 1. Then,(9)limt⟶∞θtx1−2α=0.Remark 1.
Remark that our main result is not implied by Theorem2 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 L1∩L2. The proof techniques (for Theorem 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), energy-type L2 estimates for θn that exploit the natural appearance of the X−1-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 Section1. 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 X−1, which belongs to a class whose definition of norm is based on the Fourier transform but is not contained in L2. 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(10)Xσℝ2=f∈S′ℝ2;f^∈Lloc1ℝ2andξ↦ξσf^ξ∈L1ℝ2,equipped with the norm(11)fχσ=∫ℝ2ξσf^ξdξ.To prove Theorem3, we need the following lemma.Lemma 1.
For2/3<α<1, H˙αℝ2⟶X1−2αℝ2. More precisely, if f∈H˙αℝ2∩L2ℝ2. Then, we have(12)fX1−2α≤CfL23α−2/αfH˙α2−2α/α.Proof.
Forλ>0, put fX1−2α=Aλ+Bλ, where(13)Aλ=∫ξ<λξ1−2αf^ξdξ,Bλ=∫ξ<λξ1−2αf^ξdξ.
Let us start by controlling the first term; using Cauchy–Schwarz inequality, we get(14)Aλ=∫ξ<λξ1−2α×f^ξdξ≤∫ξ<λf^ξdξ1/2∫ξ<λξ2−4αdξ1/2≤4πfL2∫0λr3−4αdr1/2≤4π4−4αfL2λ4−4α1/2.
Therefore,(15)Aλ≤4π4−4αλ2−2αfL2.
A similar calculation to the foregoing yields(16)Bλ=∫ξ>λξαf^ξ×ξ1−3αdξ≤∫ξ>λξ2αf^ξdξ1/2∫ξ>λξ2−6αdξ1/2≤4πfH˙α∫λ∞r3−6αdr1/2≤4π6α−4λ2−3αfH˙α.
Then,(17)Bλ≤4π6α−4λ2−3αfH˙α.
Combining (15) and (17), we get(18)fX1−2α≤4π4−4αλ2−2αfL2+4π6α−4λ2−3αfH˙α.
OptimizingfX1−2α, it suffices to choose λ=fH˙α/fL21/α, 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 Theorem1. The proof is inspired from [33].First, we Takeθ0∈X1−2α.Letε>0, such that ε≤1/8. For n∈ℕ, put(19)An=ξ∈ℝ2;ξ≤nandθ^0ξ≤n.We haveℱ−111An.θ0^ converges in X1−2α to θ0. Then, there is n0∈ℕ such that(20)θ0−ℱ−111Anθ0^X1−2α≤ε/4,∀n≥n0.Letn≥n0 be a fixed integer. Put(21)θn0=ℱ−111An.θ0^,wn0=θ0−θn0.Then,(22)wn0X1−2α≤ε/4,θn0∈X1−2αℝ2∩L2ℝ2.Consider the system(23)QGαn∂twn+−Δαwn+uwn·∇wn=0,x∈ℝ2,t>0,wn0,x=wn0x.For alln≥n0, we have(24)wn0X1−2α≤ε4.Use Theorem1; then, there exists a unique global solution wn∈Cℝ+,X1−2αℝ2∩L1ℝ+,X1ℝ2.Furthermore,(25)wntX1−2α+1−4wn0X1−2α∫0twnsX1ds≤wn0X1−2α,∀t≥0.Moreover,(26)∂tθ+−Δαθ+u·∇θ=0,θ0=θ0∈X1−2α.Putθ=θ−wn︸θn+wn. Then, θn is a solution of the following system:(27)∂tθn+−Δαθn+uθn·∇θn+uθn·∇wn+uwn·∇θn=0,θn0=θn0∈X1−2α∩L2.By taking the inner product inL2ℝ2 with θn, we get(28)12ddtθnL22+θnH˙α2≤uθn·∇wn,θnL2.Then,(29)12ddtθnL22+θnH˙α2≤uθn·∇wn,θnL2≤uθn·∇wnL2θnL2≤Cℱuθn·∇wnL2θnL2≤Cℱuθn∗ℱ∇wnL2θnL2.By Young’s inequality, we obtain(30)12ddtθnL22+θnH˙α2≤CℱuθnL2ℱ∇wnL1θnL2≤Cnℱ∇wnL1θnL22.Therefore,(31)12ddtθnL22+θnH˙α2≤CnwnX1θnL22.Integrating with respect to time, we obtain(32)θnL22+2∫0tθnH˙α2≤θn0L22+Cn∫0twnX1θnL22.By Gronwall’s lemma, we get(33)θnL22≤θn0L22expCn∫0twnX1≤Cnθn0L22.Combining (32) and (33), we obtain(34)θnL22+2∫0tθnH˙α2≤θn0L22+C0θn0L22∫0∞wnX1≤Mn.Applying Lemma1 to θn, we infer(35)θnX1−2α≤CθnL23α−2/αθnH˙α2−2α/α.Then,(36)θnX1−2αα/1−α≤CθnL23α−2/1−αθnH˙α2.From (33) it follows that(37)θnX1−2αα/1−α≤CθnH˙α2.Therefore, by integrating in time between 0 and∞, we get(38)∫0∞θnX1−2αα/1−α≤C0∫0∞θnH˙α2.Indeed consider the following subset of0,∞:(39)Pε=t≥0;θntX1−2αℝ2≥ε4.We have(40)θntX1−2αℝ2≥1Pεtε4.Then, for all1/2<α≤1,(41)θntX1−2αα/1−αℝ2≥1Pεtε4α/1−α.Thus,(42)∫0∞θntX1−2αℝ2α/1−αdt≥λ1Pεε4α/1−α.Using the inequality (38), we get λ1Pε<∞ and 0,∞∖Pε≠∅.Therefore, there existst0∈0,∞∖Pε. Particularly,(43)θnt0X1−2αℝ2<ε4.Now, put(44)θt0=θt0−wnt0︸+wnt0=θnt0+wnt0.Thus, by (24), we obtain(45)θt0X1−2α≤θt0X1−2α+wnt0X1−2α≤ε4+wn0X1−2α≤ε4+ε4.Therefore,(46)θt0X1−2α≤ε2≤ε.Let us consider the following equation:(47)∂tγ+−Δαγ+uγ·∇γ=0,x∈ℝ2,t>0,γ0,x=γ0=θt0.Using inequality (46) and Theorem 1, we infer that there exists a unique solution γ∈Cℝ+,X1−2αℝ2∩L1ℝ+,X1ℝ2 of QGα such that(48)γX1−2γ+1−4γ0X1−2α∫0tγX1≤γ0X1−2α,∀t≥0.The existence and uniqueness of a solution to the quasi-geostrophic equation gives∀t≥0γt=θt0+t. Then,(49)θt0+tX1−2α=γtX1−2α≤γ0X1−2α≤ε.Thus, Theorem 3 is proved.



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