Regularity of global attractors and exponential attractors for 2D quasi-geostrophic equations with fractional dissipation

DISCRETE AND CONTINUOUS doi: DYNAMICAL SYSTEMS SERIES B REGULARITY OF GLOBAL ATTRACTORS AND EXPONENTIAL ATTRACTORS FOR 2D QUASI-GEOSTROPHIC EQUATIONS WITH FRACTIONAL DISSIPATION Lin Yang and Yejuan Wang∗ School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex Systems Lanzhou University, Lanzhou 730000, China Tom´ as Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla c/ Tarfia s/n, 41012-Sevilla, Spain Abstract. In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in H2α+s(T2) with α > 1 2and s > 1.We prove the existence of (H2α−+s(T2), H2α+s(T2))-global attractor A,that is, Ais compact in H2α+s(T2) and attracts all bounded subsets of H2α−+s(T2) with respect to the norm of H2α+s(T2).The asymptotic compactness of solutions in H2α+s(T2) is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in H2α+s(T2), whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of H2α−+s(T2) are all in the topology of H2α+s(T2). 1. Introduction. In this paper, we investigate the long-time behavior of solutions of the following 2D (surface) quasi-geostrophic equation with fractional dissipation defined on the 2D torus T2= [0,1]2:        ∂θ ∂t +u· ∇θ+κ(−∆)αθ=F(x, θ), θ(0) = θ0,ZT2 θ0(x)dx = 0, (1.1) where F(x, θ) is a given external forcing term in the form F(x, θ) = g1(x)f(θ) + g2(x),(1.2) 2010 Mathematics Subject Classification. Primary: 35B40, 35B41, 37L30, 35Q86. Key words and phrases. Global attractor, fractional dissipation, quasi-geostrophic equations, Asymptotic compactness, exponential attractor. This work was supported by the National Natural Science Foundation of China under grant 41875084, and by FEDER and Ministerio de Ciencia, Innovaci´on y Universidades of Spain (Grant PGC2018-096540-B-I00), and Junta de Andaluc´ıa, Spain (Grant US-1254251). ∗Corresponding author. 1 2 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO θrepresents the potential temperature, κ > 0 is a diffusivity coefficient, α∈1 2,1 is a fractional exponent, (−∆)αis the fractional Laplacian, and u= (u1, u2) is the velocity field determined by θthrough the relation: u=−∂ψ ∂x2 ,∂ψ ∂x1,where (−∆)1 2ψ=−θ, (1.3) or, in a more explicit way, u= (−R2θ, R1θ),(1.4) where Ri, i = 1,2 are the Riesz transforms (see [25, p.299]). In the following, we will restrict ourselves to flows which have zero average on the torus, i.e., ZT2 θ(t, x)dx = 0,∀t≥0. Equations (1.1)-(1.3) are important models in geophysical fluid dynamics, especially for atmospheric and oceanic fluid. Indeed, the system describes the evolution of the temperature on the 2D boundary of a rapidly rotating half-space with small Rossby and Ekman numbers (see [28]). Long time behavior of solutions to the 2D quasi-geostrophic equation with fractional dissipation has been studied in [3,11,22,26,27]. Using the framework of [31], the global weak attractor Ain the space of weak solutions W(f) was proved by Berselli for the sub-critical dissipative case α∈(1 2,1).Aattracts all bounded sets in the space of the generalized weak solutions GW(f),see [3] for more details. With an improvement of the positivity lemma of [10] and a generalized maximum principle, Ju [22] established the existence of the global attractor Ain Hs(T2) for any s > 2(1−α) and α∈(1 2,1].Aattracts all bounded subsets of Hs(T2) in the norm of Hr(T2) for any r≥s > 2(1−α),and for the case α∈(2 3,1],Aattracts all bounded subsets of L2(T2) in the norm of Hs(T2) for any s > 2(1−α).More detailed results on decay characterization of solutions to 2D dissipative quasi-geostrophic equations were given by Niche and Schonbek [26,27] for the homogeneous case (F=0).Wang and Tang [36] proved the existence of the global attractor in Lp(R2) for 2D quasigeostrophic equations with damping in the subcritical case α∈(1 2,1].Dlotko, Kania and Sun [12] studied the existence of the global attractor Ain Hs+2α−(R2) for any s > 1 and α∈(1 2,1].Very recently, the existence of the global attractor in Hs(R2) for any s > 2(1 −α) and α∈(1 2,1] was proved by Farwig and Qian [15] for 2D quasi-geostrophic equations with a nonlocal damping. For the critical case α=1 2, the existence of the global attractor in H3/2(T2) has been studied by Constantin, Coti Zelati, Kalita, Vicol and Tarfulea, see [7,9,11] for more details. However, to the best of our knowledge, there is no result available in the literature on the regularity of global attractors for 2D quasi-geostrophic equations. The first purpose of this work is to prove the regularity of global attractors of (1.1) in H2α+s(T2) for any s > 1 and α∈(1 2,1].More precisely, we will show that the global attractor Ain H2α−+s(T2) is actually an (H2α−+s(T2), H2α+s(T2))-global attractor in the sense that Ais compact in H2α+s(T2) and attracts all bounded subsets of H2α−+s(T2) with respect to the norm of H2α+s(T2) (see Theorem 5.3). The notion of exponential attractors, introduced by Eden, Foia¸s, Nicolaenko and Temam [13], has been shown to be one of the most important concepts of limit sets in the theory of dynamical systems in infinite-dimensional spaces (see [2,6, 30,34]).The exponential attractor, as an intermediate object between the global attractor and the inertial manifold, satisfies some nice properties like the inertial EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 3 manifold (e.g., finite fractal dimension, exponential attracting, stable with respect to some perturbations).However, contrarily to the global attractor, an exponential attractor is not necessarily unique, so that the concrete choice of an exponential attractor is in some sense artificial. The first technique to construct exponential attractors was developed in Hilbert spaces. This technique is based on the use of orthogonal projections (see [13]) and cannot be applied directly to dynamical systems defined in Banach spaces. A new method for constructing exponential attractors in Banach spaces was proposed by Efendiev, Miranville and Zelik in [14]. However, the existence of exponential attractors for 2D quasi-geostrophic equations is unsolved. The second purpose of this paper is to present some sufficient conditions for the construction of exponential attractors for autonomous dynamical systems on Banach spaces, and furthermore we apply our results to consider the existence of exponential attractors of (1.1) in H2α−+s(T2) and the regularity of exponential attractors of (1.1) with g2(x) instead of F(x, θ) in H2α+s(T2) for any s > 1 and α∈1 2,1. There are several results on estimation of the fractal and Hausdorff dimensions of the global attractor for quasi-geostrophic equations. Wang [35] proved the existence of a compact, connected global attractor to the 3D baroclinic quasi-geostrophic equations of large scale atmosphere, and derived an upper bound of the Hausdorff and fractal dimensions of the global attractor. A precise upper bound of the fractal dimension of the global attractor for 2D quasi-geostrophic equations with fractional dissipation in Hs(Ω) for any s≥2αand α∈1 2,1was obtained by Wang and Tang [37]. By using the fractional Lieb-Thirring inequality, estimates of the finite Hausdorff and fractal dimensions of the global attractor for 2D quasi-geostrophic equations with fractional dissipation were established by Farwig and Qian [15]. The paper is organized as follows. In the next section, we present some notation and recall the theory of global attractors for infinite dimensional dissipative dynamical systems and several preliminary results which will be used frequently. In Section 3, the global existence, uniqueness and regularity of solutions for problem (1.1) are established by using the theory of semilinear parabolic equations with sectorial operator. Section 4 is devote to a priori estimates which will yield the existence of bounded absorbing sets in H2α+s(T2).In Section 5, we first establish the existence of the global attractor Ain H2α−+s(T2) and then prove that Ais indeed the (H2α−+s(T2), H2α+s(T2))-global attractor. In Section 6, we study the existence of exponential attractors. 2. Preliminaries. We first recall some notations and basic results from harmonic analysis. The fractional Laplacian (−∆)s,with s∈Rmay be defined in this context as the Fourier multiplier with symbol |k|s,i.e., (−∆)sϕ(x) = X k∈Z2 ∗ |k|2sbϕ(k) exp(ik ·x), where Z2 ∗=Z2\ {0}, bϕ(k) = 1 (2π)2ZT2 ϕ(x)e−ik·xdx and ϕ(x) = X k∈Z2 ∗bϕ(k) exp (ik ·x). Notice that the eigenvalues of Λ = (−∆)1 2are given by |k|. Then we relabel them in increasing order as 0< λ1≤λ2≤ · · · ≤ λn≤ · · · , 4 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO and denote the eigenfunction associated to λjby ej.It is clear that {ej}j≥1is an orthonormal basis of L2(T2),and the sets {λj}j≥1and {|k|}k∈Z2 ∗are equal. As a consequence of the mean-free setting, for s∈Rwe define kϕkHs(T2)=kΛsϕkL2(T2) and Hs(T2) denotes the Sobolev space of all ffor which kfkHs(T2)is finite. Moreover, for s∈Rand p∈[1,+∞) we denote by Hs,p(T2) the space of mean-free Lp(T2) functions ϕ, which can be written as ϕ= Λ−sψ, with ψ∈Lp(T2).The Hs,p(T2) norm of ϕis defined to be the Lp(T2) norm of ψ, i.e., kϕkHs,p(T2)=kΛsϕkLp(T2). We recall the following important commutator and product estimates, cf. [23,24]. Lemma 1 (Commutator and Product Estimates).Suppose that γ > 0and p∈ (1,+∞).If f, g ∈ S(T2),the Schwartz class, then kΛγ(fg)−fΛγgkLp(T2)≤C(γ, p)k∇fkLp1(T2)kΛγ−1gkLp2(T2) +kΛγfkLp3(T2)kgkLp4(T2)(2.1) and kΛγ(fg)kLp(T2)≤C(γ, p)kfkLp1(T2)kΛγgkLp2(T2)+kΛγfkLp3(T2)kgkLp4(T2) (2.2) with p2, p3∈(1,+∞)such that 1 p1 +1 p2 =1 p3 +1 p4 =1 p. The following result can be obtained by the fact that the Riesz transforms commute with (−∆)land the boundedness of the Riesz transforms in Lp(T2),see [32, Chapter 3] for more details. Lemma 2. Let 1<p<∞and l≥0.Then there exists a constant C(l, p)such that k(−∆)lukLp(T2)≤C(l, p)k(−∆)lθkLp(T2).(2.3) If p= 2,the inequality (2.3)can be strengthened to k(−∆)lukL2(T2)=k(−∆)lθkL2(T2).(2.4) Denote Aγ,κ =κ(−∆)γwhere γ > 0 and κ > 0,and let Xγ,κ =u∈L2(T2) : Aγ,κu∈L2(T2) with norm kukXγ,κ =kAγ,κukL2(T2).We recall the following well-known results for the semigroup generated by the positive operator Aγ,κ (see [18,38] for the similar results). Proposition 3. Let 0< β1≤β2and u∈Xβ1,κ.Then there exists a constant C1=C1(β1, β2, γ, κ)such that ke−Aγ,κtukXβ2,κ ≤C1e−κλ2γ 1 2tt−β2−β1 γkukXβ1,κ , t > 0. We recall the following improved positivity lemma, cf. [22, Lemma 3.3], which we use in the proof of Lemma 11. EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 5 Lemma 4 (Improve Positivity Lemma).Suppose s∈[0,2] and θ, Λsθ∈Lp(T2), where p≥2.Then ZT2 |θ|p−2θΛsθdx ≥2 pZT2Λs 2|θ|p 22dx. The following variant of the classical Gronwall Lemma is due to Foia¸s and Prodi [16]. See also [5,8,34]. Lemma 5 (Uniform Gronwall Lemma).Let g, h and ybe non-negative locally integrable functions on [t0,+∞)such that dy dt ≤gy +h, ∀t≥t0, and Zt+r t g(s)ds ≤a1,Zt+r t h(s)ds ≤a2,Zt+r t y(s)ds ≤a3,∀t≥t0, where r > 0and a1, a2, a3are non-negative constants. Then y(t+r)≤a3 r+a2ea1,∀t≥t0. In the sequel, Cdenotes an arbitrary positive constant, which may be different from line to line and even in the same line. For r∈R,let r−denotes the number strictly less than rbut close to it. 2.1. Semigroup and attractor. In this subsection, we recapitulate basic concepts and results on the bi-spaces global attractor and the exponential attractor. Let (X, k·kX) and (Y, k·k) be two Banach spaces such that the injection Y ,→Xis continuous. The Hausdorff semidistance for nonempty subsets Aand Bof Xis written as distX(A, B) which is defined by distX(A, B) = sup{d(a, B) : a∈A}, where d(a, B) = inf{ka−bkX:b∈B},and the similar notation will be used for subsets of Y. Definition 6. Let Xbe a Banach space and {S(t)}t≥0be a family of operators on X. We say that {S(t)}t≥0is a continuous semigroup on X, if for all t, s ∈R+,the following conditions are satisfied: (i) S(0) = I(the identity); (ii) S(t)S(s) = S(t+s); (iii) S(t)xis continuous in xand t. Definition 7. Let {S(t)}t≥0be a semigroup on X. Then {S(t)}t≥0is said to be asymptotically compact in Xif for any bounded sequence {xn}∞ n=1 ⊂Xand any sequence tn→+∞,the sequence {S(tn)xn}∞ n=1 has a convergent subsequence in X. If, in addition, S(t) maps Xto Yfor every t∈R+,and the sequence {S(tn)xn}∞ n=1 has a convergent subsequence in Y for any bounded sequence {xn}∞ n=1 ⊂Xand any sequence tn→+∞,then {S(t)}t≥0 is said to be (X, Y )-asymptotically compact. Definition 8. Let {S(t)}t≥0be a semigroup on X. Then a set A ⊂ Xis called a global attractor of {S(t)}t≥0in Xif the following conditions (i)-(iii) are satisfied: 6 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO (i) Ais compact in X; (ii) Ais invariant, i.e., S(t)A=Afor all t≥0; (iii) Aattracts every bounded subset of X, that is, for any bounded set B⊂X, lim t→+∞distX(S(t)B, A)=0. If, in addition, the following conditions are satisfied: (iv) S(t) maps Xto Yfor every t∈R+; (v) Ais compact in Y; (vi) Aattracts every bounded subset of Xin Y, that is, for any bounded set B⊂X, lim t→+∞distY(S(t)B, A)=0, then Ais called an (X, Y )-global attractor of {S(t)}t≥0. The following existence result for a global attractor for a semigroup can be found in [2,5,20,29,30,33,34,40] (see also [19] for random case). Proposition 9. Suppose Xand Yare Banach spaces with continuous injection Y ,→Xsuch that all closed balls in Yare closed subsets of X. Let {S(t)}t≥0be a semigroup on X. If {S(t)}t≥0is asymptotically compact in Xand has a bounded absorbing set in X, then {S(t)}t≥0has a global attractor Ain X. If, in addition, S(t)maps Xto Yfor every t∈R+,and {S(t)}t≥0is (X, Y )- asymptotically compact, then Ais also an (X, Y )-global attractor of {S(t)}t≥0. 3. Global existence and uniqueness of solutions. In this section, we consider the global existence and uniqueness of solutions for problem (1.1) with initial data θ0∈H2α−+s(T2),and then define a continuous semigroup. Let θ1, θ2∈H2α−+s(T2) and u1= (−R2θ1, R1θ1), u2= (−R2θ2, R1θ2).Note that Hs(T2) is a Banach Algebra provided that s > 1 (see, e.g., [1, p.115]).Since Sobolev embeddings H2α−+s(T2)⊂H1+s(T2)⊂Hs(T2) are valid for the subcritical case, by (2.3) we have ku1· ∇θ1−u2· ∇θ2kHs(T2)≤R2(θ1−θ2)∂θ1 ∂x1 +R2θ2 ∂(θ1−θ2) ∂x1Hs(T2) +R1(θ1−θ2)∂θ1 ∂x2 +R1θ2 ∂(θ1−θ2) ∂x2Hs(T2) ≤Ckθ1−θ2kHs(T2)kθ1kH1+s(T2)+Ckθ2kHs(T2)kθ1−θ2kH1+s(T2) ≤Ckθ1kH2α−+s(T2)+kθ2kH2α−+s(T2)kθ1−θ2kH2α−+s(T2), (3.1) which is the required local Lipschitz condition. Following a standard approach in [6, p.55] and [21, Theorem 3.3.3] to semilinear parabolic equations, we obtain the local existence and uniqueness of solutions for problem (1.1) with initial data θ0∈H2α−+s(T2). Theorem 10. Let α∈(1 2,1], κ > 0and θ0∈H2α−+s(T2)with s > 1.Suppose further that g1, g2∈Hs(T2),and f:H2α−+s(T2)→Hs(T2)is Lipschitz continuous on bounded subsets of H2α−+s(T2).Then there exists a unique solution θto problem (1.1)such that θ∈C((0, τ); H2α+s(T2)) ∩C([0, τ); H2α−+s(T2)), θt∈C((0, τ); H2γ+s(T2)), EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 7 where γ < α is arbitrary and τ > 0is the maximal time of existence. Moreover, θ is given by the formula θ(t) = e−Aα,κtθ0+Zt 0 e−Aα,κ(t−s)(F(x, θ(s)) −u(s)· ∇θ(s))ds, ∀t∈[0, τ),(3.2) where e−Aα,κtdenotes the linear semigroup corresponding to the operator Aα,κ := κ(−∆)αin Hs(T2),and F(x, θ)is given by (1.2). Let θ∈H2α−+s(T2) and u= (−R2θ, R1θ).Similar to (3.1), we obtain ku· ∇θkHs(T2)≤CkθkHs(T2)kθkH2α−+s(T2).(3.3) Further, we assume that f:H2α−+s(T2)→Hs(T2) satisfies the sublinear growth restriction: kf(θ)kHs(T2)≤k11 + kθkH2α−+s(T2), θ ∈H2α−+s(T2) (3.4) for some k1>0.A priori estimate (4.47) below together with (3.3) and (3.4) will be used to guarantee that the local solution to problem (1.1) can be globally extended (see [6, p.71] for more details).On the other hand, following [21, Theorem 3.4.1], one can show that θ(t, θ0) is continuous with respect to θ0in H2α−+s(T2).Hence we now define a semigroup S:R+×H2α−+s(T2)→H2α−+s(T2) by S(t)θ0=θ(t, θ0) for all (t, θ0)∈R+×H2α−+s(T2).(3.5) 4. Uniform estimates of solutions. In this section, we derive uniform estimates on the solution of (1.1). Such estimates imply the existence of a bounded absorbing set, and also will be used to prove the asymptotic compactness of the semiflow associated with the equation. We begin with uniform estimates of the solutions in Lq(T2). Lemma 11. Let the conditions in Theorem 10 hold. Assume further that (3.4) holds and f(v)v≤k2,∀v∈R(4.1) for some k2>0.Let q∈[2,∞).Then every solution θ(t, θ0)of (1.1)satisfies kθ(t, θ0)kq Lq(T2)≤ kθ0kq Lq(T2)e−κλ2α 1t+Ckg1k q 2 Lq 2(T2)+Ckg2kq Lq(T2).(4.2) Proof. Multiplying (1.1) with |θ|q−2θand then taking the inner product in L2(T2), we have 1 q d dt ZT2 |θ|qdx +κZT2 (−∆)αθ|θ|q−2θdx +ZT2 u· ∇θ|θ|q−2θdx =ZT2 F(x, θ)|θ|q−2θdx. (4.3) For the second term, by Lemma 4we obtain κZT2 (−∆)αθ|θ|q−2θdx ≥2κ qZT2Λα|θ|q 22dx ≥2κ qλ2α 1ZT2 |θ|qdx. (4.4) Due to ∇ · u= 0,by using integration by parts we have ZT2 u· ∇θ|θ|q−2θdx = 0.(4.5) 8 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO Since f(θ)θ≤k2,we deduce from Young’s inequality that for the case q > 2, ZT2 F(x, θ)|θ|q−2θdx =ZT2 (g1(x)f(θ) + g2(x))|θ|q−2θdx ≤k2ZT2 g1(x)|θ|q−2dx +ZT2 g2(x)|θ|q−2θdx ≤κ qλ2α 1ZT2 |θ|qdx +CZT2 |g1(x)|q 2dx +CZT2 |g2(x)|qdx. (4.6) For the case q= 2, ZT2 F(x, θ)θdx ≤k2ZT2 g1(x)dx +ZT2 g2(x)θdx ≤κ qλ2α 1ZT2 |θ|2dx +k2ZT2 |g1(x)|dx +CZT2 |g2(x)|2dx. (4.7) Inserting the above estimates into (4.3) gives d dt ZT2 |θ|qdx +κλ2α 1ZT2 |θ|qdx ≤CZT2 |g1(x)|q 2dx +CZT2 |g2(x)|qdx. (4.8) Then the desired result (4.2) follows from the Gronwall inequality. We next derive uniform estimates for θin Hα+s(T2) and for ∂tθin Hs(T2). Lemma 12. Let the conditions in Lemma 11 hold. Then, every solution θ(·)of problem (1.1)satisfies for all t≥0, kθ(t)k2 Hα+s(T2)≤e−κλ2α 1tkθ0k2 Hα+s(T2)+Ckg2k2 Hs(T2)+Ckg1k2 Hs(T2) +Cte−κλ2α 1tkθ0k2 L2(T2)kg1k 2 1−η0 Hs(T2)+CG2kg1k 2 1−η0 Hs(T2) +Ce−κλ2α 1tkθ0k2 L2(T2)kθ0k 2 1−η Lp2(T2)+CG1te−κλ2α 1tkθ0k2 L2(T2) +CG2e−κλ2α 1tkθ0k 2 1−η Lp2(T2)+CG1G2, (4.9) κZt+1 t kθ(r)k2 H2α+s(T2)dr ≤e−κλ2α 1tkθ0k2 Hα+s(T2)+Ckg2k2 Hs(T2)+Ckg1k2 Hs(T2) +Cte−κλ2α 1tkθ0k2 L2(T2)kg1k 2 1−η0 Hs(T2)+CG2kg1k 2 1−η0 Hs(T2) +Ce−κλ2α 1tkθ0k2 L2(T2)kθ0k 2 1−η Lp2(T2)+CG1(t+ 1)e−κλ2α 1tkθ0k2 L2(T2) +CG2e−κλ2α 1tkθ0k 2 1−η Lp2(T2)+CG1G2, (4.10) where p2=2 2α−−1, η0=2α−+s 2α+s, η ∈hs+1 2α+s,1is a constant, G1=kg1k 1 1−η L p2 2(T2)+kg2k 2 1−η Lp2(T2)and G2=kg1kL1(T2)+kg2k2 L2(T2). Proof. Multiplying (1.1) with (−∆)α+sθand then integrating over T2, we find that 1 2 d dt ZT2 [(−∆)α+s 2θ]2dx +κZT2 [(−∆)2α+s 2θ]2dx =ZT2 F(x, θ)(−∆)α+sθdx −ZT2 (u· ∇θ)(−∆)α+sθdx. (4.11) EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 9 We proceed to estimate the last two terms in (4.11). First, H¨older’s and Young’s inequalities imply that ZT2 g2(x)(−∆)α+sθdx≤ k(−∆)s 2g2kL2(T2)k(−∆)2α+s 2θkL2(T2) ≤κ 8kθk2 H2α+s(T2)+Ckg2k2 Hs(T2). (4.12) By Lemma 1, H¨older’s and Young’s inequalities, g1∈Hs(T2) and (3.4), we have ZT2 g1(x)f(θ)(−∆)α+sθdx ≤ k(−∆)s 2(g1f(θ))kL2(T2)k(−∆)2α+s 2θkL2(T2) ≤Ckg1kL∞(T2)k(−∆)s 2f(θ)kL2(T2)+k(−∆)s 2g1kL2(T2)kf(θ)kL∞(T2)kθkH2α+s(T2) ≤Ckg1kHs(T2)1 + kθkH2α−+s(T2)kθkH2α+s(T2) ≤Ckg1kHs(T2)1 + kθkη0 H2α+s(T2)kθk1−η0 L2(T2)kθkH2α+s(T2) ≤κ 8kθk2 H2α+s(T2)+Ckg1k2 Hs(T2)+Ckg1k 2 1−η0 Hs(T2)kθk2 L2(T2), (4.13) where we have used the Sobolev embedding Hs(T2)⊂L∞(T2) for s > 1 and the following Gagliardo-Nirenberg inequality: kθkH2α−+s(T2)≤Ckθkη0 H2α+s(T2)kθk1−η0 L2(T2), where η0:= 2α−+s 2α+s.Next, we analyze the last term in (4.11). By H¨older’s inequality, we have ZT2 (u· ∇θ)(−∆)α+sθdx≤ k(−∆)2α+s 2θkL2(T2)k(−∆)s 2(u· ∇θ)kL2(T2).(4.14) Note that ∇ · u= 0.Then by making use of Lemmas 1and 2, we obtain k(−∆)s 2(u· ∇θ)kL2(T2)=k(−∆)s+1 2(uθ)kL2(T2) ≤Ck(−∆)s+1 2ukLp1(T2)kθkLp2(T2)+CkukLp2(T2)k(−∆)s+1 2θkLp1(T2) ≤Ck(−∆)s+1 2θkLp1(T2)kθkLp2(T2), (4.15) where p1:= 1 1−α−, p2:= 2 2α−−1. To deal with the term k(−∆)s+1 2θkLp1(T2),we use the following Gagliardo-Nirenberg inequality: k(−∆)s+1 2θkLp1(T2)≤Ck(−∆)2α+s 2θkη L2(T2)kθk1−η L2(T2),(4.16) where η∈hs+1 2α+s,1.Then it follows from (4.14)-(4.16) and Young’s inequality that −ZT2 (u· ∇θ)(−∆)α+sθdx ≤Ck(−∆)2α+s 2θkη+1 L2(T2)kθk1−η L2(T2)kθkLp2(T2) ≤κ 4k(−∆)2α+s 2θk2 L2(T2)+Ckθk2 L2(T2)kθk 2 1−η Lp2(T2). (4.17) 16 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO Proof. Lemma 15 implies that {S(t)}t≥0has a bounded absorbing set in H2α+s(T2), and is asymptotically compact in H2α−+s(T2).Hence the existence of an (H2α−+s (T2), H2α−+s(T2))-global attractor for {S(t)}t≥0follows from Proposition 9. In order to obtain the existence of the (H2α−+s(T2), H2α+s(T2))-global attractor, we need the following auxiliary lemma. Lemma 17. Assume that the conditions of Lemma 14 hold. Then for any bounded set B⊂H2α−+s(T2),there exist T3=T3(B)>0and N0≥1such that any solution θ(t, θ0)of problem (1.1)with θ0∈Bsatisfies k(I−Pm)∂tθ(t)k2 Hs(T2)≤ε, ∀t≥T3, m ≥N0,(5.1) where Pm:L2(T2)→Hmis the projection operator and Hmis the space spanned by {ej}m j=1. Proof. Let w2= (I−Pm)w. Multiplying (4.30) by (∆)sw2and then integrating over T2,we have 1 2 d dt ZT2 [(−∆)s 2w2]2dx +κZT2 [(−∆)s+α 2w2]2dx =ZT2 g1(x)f0(θ)w(−∆)sw2dx −ZT2 (u· ∇w)(−∆)sw2dx −ZT2 (ut· ∇θ)(−∆)sw2dx. (5.2) For the first term on the right-hand side of (5.2), by similar arguments as in (4.32)- (4.34) we obtain ZT2 g1(x)f0(θ)w(−∆)sw2dx ≤ k(−∆)s−α 2(g1f0(θ)w)kL2(T2)k(−∆)s+α 2w2kL2(T2) ≤Ckg1f0(θ)kLp3(T2)kwk Hs−α, 2p3 p3−2(T2) kw2kHs+α(T2) +Ckg1f0(θ)kHs−α, 2 1−α+p4(T2)kwkL 2 α−p4(T2)kw2kHs+α(T2) ≤C1 + kθkHs+α(T2)kg1kHs(T2)kwkHs(T2)kw2kHs+α(T2) ≤κ 4kw2k2 Hs+α(T2)+C1 + kθk2 Hs+α(T2)kg1k2 Hs(T2)kwk2 Hs(T2), (5.3) where p3and p4are given in (4.32), and we have used Young’s inequality and the Sobolev embeddings Hs(T2)⊂Hs−α, 2p3 p3−2(T2) and Hs(T2)⊂L2 α−p4(T2) for s > 1. For the last term in (5.2), by Lemmas 1-2, H¨older’s and Young’s inequalities, we EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 17 find that ZT2 (ut· ∇θ)(−∆)sw2dx ≤ k(−∆)s+α 2w2kL2(T2)k(−∆)s−α 2(ut· ∇θ)kL2(T2) ≤Ck(−∆)s+α 2w2kL2(T2)k(−∆)s−α 2utkLp5(T2)k∇θkLp6(T2) +Ck(−∆)s+α 2w2kL2(T2)kutkLp2(T2)k(−∆)s−α+1 2θkLp1(T2) ≤Ck(−∆)s+α 2w2kL2(T2)k(−∆)s−α 2wkLp5(T2)k∇θkLp6(T2) +Ck(−∆)s+α 2w2kL2(T2)kwkLp2(T2)k(−∆)s−α+1 2θkLp1(T2) ≤Ckw2kHs+α(T2)kwkHs(T2)kθkHs+α(T2) ≤κ 8kw2k2 Hα+s(T2)+Ckwk2 Hs(T2)kθk2 Hs+α(T2), (5.4) where p1=1 1−α−, p2=2 2α−−1, p5=2 1−α−, p6=2 α−, and we have used the Sobolev embeddings Hs(T2)⊂Hs−α,p5(T2), Hs(T2)⊂ Lp2(T2), Hs+α(T2)⊂H1,p6(T2) and Hs+α(T2)⊂Hs−α+1,p1(T2) in the last inequality. For the second term on the right-hand side of (5.2), noticing that (−∆)s 2 and ∇are commutable, and hu· ∇((−∆)s 2w2),(−∆)s 2w2i= 0, hence we have −ZT2 (u· ∇w)(−∆)sw2dx =−h(−∆)s 2(u· ∇w2)−u· ∇((−∆)s 2w2),(−∆)s 2w2i − h(−∆)s 2(u· ∇w1),(−∆)s 2w2i =−h(−∆)s 2(u· ∇w2)−u·((−∆)s 2∇w2),(−∆)s 2w2i − h(−∆)s 2(u· ∇w1),(−∆)s 2w2i := I6+I7, (5.5) where h·,·i denotes the inner product in L2(T2) and w1=Pmw. Then using Lemmas 1-2and H¨older’s inequality, we obtain |I6| ≤ Ckw2kHs(T2)k∇ukLp6(T2)k(−∆)s−1 2∇w2kLp5(T2) +k(−∆)s 2ukLp6(T2)k∇w2kLp5(T2) ≤Ckw2kHs(T2)k∇θkLp6(T2)kw2kHs,p5(T2)+kθkHs,p6(T2)k∇w2kLp5(T2) ≤CkwkHs(T2)kθkHs+α(T2)kw2kHs+α(T2), (5.6) where p5=2 1−α−, p6=2 α−,and we have used the Sobolev embeddings Hs+α(T2)⊂ Hs,p6(T2) and Hs+α(T2)⊂Hs,p5(T2).By similar arguments as in (4.36), using the 18 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO equivalence property of norm in the finite dimensional case, I7is bounded by |I7|≤k(−∆)s 2w2kL2(T2)k(−∆)s 2(u· ∇w1)kL2(T2) ≤Ck(−∆)s 2w2kL2(T2)k(−∆)s+1 2(uw1)kL2(T2) ≤Ck(−∆)s 2w2kL2(T2)k(−∆)s+1 2ukLp1(T2)kw1kLp2(T2) +Ck(−∆)s 2w2kL2(T2)kukLp2(T2)k(−∆)s+1 2w1kLp1(T2) ≤Ck(−∆)s 2w2kL2(T2)k(−∆)s+1 2θkLp1(T2)kw1kLp2(T2) +Ck(−∆)s 2w2kL2(T2)kθkLp2(T2)k(−∆)s+1 2w1kLp1(T2) ≤Ckw2kHs+α(T2)kθkH2α+s(T2)kwkHs(T2), (5.7) where p1=1 1−α−and p2=2 2α−−1are given in (4.15), and we have used the Sobolev embeddings Hα+s(T2)⊂Hs(T2), H2α+s(T2)⊂Hs+1,p1(T2) and H2α+s(T2)⊂ Lp2(T2).Inserting (5.6) and (5.7) into (5.5) gives ZT2 (u· ∇w)(−∆)sw2dx≤Ckw2kHs+α(T2)kθkH2α+s(T2)kwkHs(T2) ≤κ 8kw2k2 Hs+α(T2)+Ckwk2 Hs(T2)kθk2 H2α+s(T2), (5.8) due to Young’s inequality and the Sobolev embedding H2α+s(T2)⊂Hs+α(T2). Combining (5.2)-(5.4) and (5.8), we have d dtkw2k2 Hs(T2)+κkw2k2 Hs+α(T2) ≤C1 + kθk2 Hs+α(T2)kg1k2 Hs(T2)kwk2 Hs(T2)+Ckwk2 Hs(T2)kθk2 H2α+s(T2). (5.9) From (4.9), (4.29) and (4.47) we see that for any given bounded set B⊂Hs+2α−(T2), there exists a T0 3=T0 3(B)>0 such that any solution θ(t, θ0) of problem (1.1) with θ0∈Bsatisfies kθ(t)k2 Hs+α(T2)≤C, ∀t≥T0 3,(5.10) kθ(t)k2 H2α+s(T2)≤C, ∀t≥T0 3,(5.11) k∂tθ(t)k2 Hs(T2)≤C, ∀t≥T0 3.(5.12) Then (5.9)-(5.12) and g1∈Hs(T2) imply that d dtkw2k2 Hs(T2)+κλ2α m+1kw2k2 Hs(T2)≤C, ∀t≥T0 3. Finally, by Gronwall’s inequality we obtain (5.1), and thus the proof of this lemma is completed. We are now ready to prove the main result of this section. Theorem 18. Assume that the conditions of Lemma 14 hold. Then the semigroup {S(t)}t≥0associated with problem (1.1)has an (H2α−+s(T2), H2α+s(T2))-global attractor A. Proof. Thanks to Theorem 16 and Proposition 9, now it only remains to show that the semigroup {S(t)}t≥0is (H2α−+s(T2), H2α+s(T2))-asymptotically compact. Let a bounded set B⊂H2α−+s(T2),sequences {tn}∞ n=1 with tn→+∞and {θn 0}∞ n=1 ⊂Bare given arbitrarily. We will show that the sequence {S(tn)θn 0}∞ n=1 has a convergent subsequence in H2α+s(T2).Notice that Lemma 15 implies that EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 19 {S(tn)θn 0}∞ n=1 is bounded in H2α+s(T2).By the compactness of embedding H2α+s(T2)⊂ H2α−+s(T2),it follows that there is ξ∈H2α−+s(T2) such that, up to a subsequence, S(tn)θn 0→ξstrongly in H2α−+s(T2).(5.13) This together with (3.1) implies that {un(tn)· ∇θn(tn)}∞ n=1 is a Cauchy sequence in Hs(T2).(5.14) Since f:H2α−+s(T2)→Hs(T2) is Lipschitz continuous on bounded subset of H2α−+s(T2),in view of (1.2) and (5.13), we have that {F(x, θn(tn))}∞ n=1 is a Cauchy sequence in Hs(T2).(5.15) Observe that ∂θn(tn) ∂tn −∂θn0(tn0) ∂tn0 +un(tn)· ∇θn(tn)−un0(tn0)· ∇θn0(tn0) +κ(−∆)α(θn(tn)−θn0(tn0)) = F(x, θn(tn)) −F(x, θn0(tn0)). (5.16) Taking the inner product of (5.16) with (−∆)α+s(θn(tn)−θn0(tn0)),we obtain κkθn(tn)−θn0(tn0)k2 H2α+s(T2) =−*(−∆)s 2 ∂θn(tn) ∂tn −∂θn0(tn0) ∂tn0!,(−∆)2α+s 2θn(tn)−θn0(tn0)+ −D(−∆)s 2un(tn)· ∇θn(tn)−un0(tn0)· ∇θn0(tn0), (−∆)2α+s 2θn(tn)−θn0(tn0)E +D(−∆)s 2F(x, θn(tn)) −F(x, θn0(tn0)),(−∆)2α+s 2θn(tn)−θn0(tn0)E ≤κ 2kθn(tn)−θn0(tn0)k2 H2α+s(T2)+C ∂θn(tn) ∂tn −∂θn0(tn0) ∂tn0 2 Hs(T2) +Ckun(tn)· ∇θn(tn)−un0(tn0)· ∇θn0(tn0)k2 Hs(T2) +kF(x, θn(tn)) −F(x, θn0(tn0))k2 Hs(T2), (5.17) thanks to Young’s and H¨older’s inequalities. Lemma 17 implies that for every ε > 0, there exist N0 0>0 and m0≥1 such that (I−Pm0)∂θn(tn) ∂tnHs(T2) +(I−Pm0)∂θn0(tn0) ∂tn0Hs(T2) ≤ε, ∀n, n0≥N0 0. (5.18) On the other hand, by (4.29) we find that the sequence nPm0 ∂θn(tn) ∂tno∞ n=1 is bounded in Hs(T2),and thus nPm0 ∂θn(tn) ∂tno∞ n=1 is precompact in Hs(T2).This together with (5.18) shows that n∂θn(tn) ∂tno∞ n=1 has a finite open covering of balls with radii less than εin Hs(T2).This implies that, up to a subsequence, ∂θn(tn) ∂tn∞ n=1 is a Cauchy sequence in Hs(T2).(5.19) 20 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO Combining (5.14)-(5.15), (5.17) and (5.19), we have that, up to a subsequence, {θn(tn)}∞ n=1 is a Cauchy sequence in H2α+s(T2), and thus {S(t)}t≥0is (H2α−+s(T2), H2α+s(T2))-asymptotically compact. The proof of this theorem is completed. 6. Exponential attractors. In this section, we are concerned with the construction of exponential attractors. First, we recall the definition of exponential attractors and the fractal dimension for a general set. Definition 19. Let Abe a compact subset of a Banach space X. Then the fractal dimension dimf X(A) of Ais defined by dimf X(A) = lim sup ε→0 log b N(A, ε) log 1 ε , where ε > 0 and b N(A, ε) is the minimal number of closed balls in Xhaving a radius εwhich cover the set A. In particular, when dimf X(A)<∞, A is said to have a finite fractal dimension. Definition 20. Let {S(t)}t≥0be a semigroup on a Banach space X, and let A ⊂ X be a global attractor of {S(t)}t≥0in X. Then a set M ⊂ Xis called an exponential attractor for {S(t)}t≥0in Xif the following properties hold: (i) Mis a compact subset of Xsuch that A ⊂ M ⊂ X; (ii) Mis positively invariant, i.e., S(t)M⊂Mfor all t≥0; (iii) Mhas finite fractal dimension in X; (iv) Mattracts exponentially every bounded subset of X, i.e., there exists an exponent σ > 0 such that for any bounded set B⊂X, distX(S(t)B, M)≤CBe−σt, t ≥btB with two positive constants CBand btBdepending on B. For the convenience of applications, we reformulate the abstract result on the construction of exponential attractors [4,14,17,39] for {S(t)}t≥0under slightly modified conditions. Theorem 21. Let {S(t)}t≥0be a semigroup on a Banach space X, and let A ⊂ X be a global attractor of {S(t)}t≥0in X. Assume that there exists a closed bounded set X⊂Xsatisfying: (i) it is positively invariant, i.e., S(t)X⊂Xfor all t≥0; (ii) it is an absorbing set of {S(t)}t≥0,i.e., for any bounded set B⊂X, there exists a time tB>0such that S(t)B⊂Xfor all t≥tB; (iii) there exist 0< γ1≤1,0< γ2≤1, t∗>0and L=L(t∗)>0such that for any t1, t2∈[t∗,2t∗]and u, v ∈X, kS(t1)u−S(t2)vkX≤L(|t1−t2|γ1+ku−vkγ2 X) ; (iv) there exist a positive constant δ∈0,1 2and a N-dimensional subspace XN of Xsuch that the bounded projection PN:X→XNsatisfies that for any u, v ∈X, k(I−PN)(S(t∗)u−S(t∗)v)kX≤δku−vkγ2 X. Then the semigroup {S(t)}t≥0has an exponential attractor Esatisfying the following properties: EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 21 (1) Econtains a global attractor Aof {S(t)}t≥0; (2) Eis a compact subset of Xwith finite fractal dimension dimf X(E)≤1 γ0 −log b Nµ log aµ + 1!; (3) Eis positively invariant, i.e., S(t)E ⊂ E for all t≥0; (4) distX(S(t)X,E)≤L(Rγ2)γ2a−2γ2 µe−(γ2 t∗log a−1 µ)tfor every t≥t∗. Here γ0= min{γ1, γ2}, R is the diameter of X,0< aµ<1is the exponent given by aµ= 2(δ+µL)with 0<µ<1−2δ 2L,and b Nµis the minimal number of closed balls of Xwith radius µwhich cover the closed unit ball of XN,BN(0; 1) = {x∈XN: kxkX≤1}centered at 0. Proof. Following a similar procedure to the method given in [17,39], we can construct the exponential attractor Efor {S(t)}t≥0. (1) Let Sn=S(nt∗) for n∈N.We will first construct an exponential attractor E∗for the discrete dynamical system {Sn}n≥0. Let µbe any exponent such that 0 < µ < 1−2δ 2L,and let aµ= 2(δ+µL).Then it is clear that 0 < aµ<1.The closed unit ball, BN(0; 1) = {x∈XN:kxkX≤1} centered at 0,is a finite N-dimensional compact ball in X. Therefore, BN(0; 1) can be covered by finite closed balls of Xwith radius µ. Denote by b Nµthe minimal number of balls of Xwith radius µwhich cover BN(0; 1).Following the arguments in [39, Theorem 6.12] step by step, we obtain that for n= 0,1,2,· · · ,there exists a finite covering of SnXsuch that SnX⊂ b Nn µ [ i=1 B(Wn,i;Rγ2an µ), with centers Wn,i ∈SnX,1≤i≤b Nn µ,where Ris the diameter of X. Let P={Wn,i : 0 ≤n < ∞,1≤i≤b Nn µ}and E∗=S∞ n=0 SnP.Then E∗is an exponential attractor for the discrete dynamical system {Sn}n≥0.More precisely, E∗contains a global attractor A∗of {Sn}n≥0,the fractal dimension of E∗is dimf X(E∗)≤ −log b Nµ log aµ ,(6.1) SnE∗⊂ E∗for each nand distX(SnX,E∗)≤Rγ2an µfor all n≥0.(6.2) (2) Now we consider the continuous case. To this aim, we define E=[ t∈[t∗,2t∗] S(t)E∗. Thanks to the condition (iii),it is obvious that Eis compact. Then it follows from (6.1) and the condition (iii) that dimf X(E)≤1 γ01 + dimf X(E∗)≤1 γ0 1−log b Nµ log aµ!, where γ0= min{γ1, γ2}.Moreover, by the positive invariance for the discrete exponential attractor, we have S(t)E ⊂ E for any t≥0. 22 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO For t≥t∗,writing t=nt∗+τwith n≥1 and τ∈[0, t∗],by the condition (iii) we obtain that distX(S(t)X,E)≤distX(S(t)X, S(t∗)E∗)≤L(distX(S(t−t∗)X,E∗))γ2.(6.3) Furthermore, using the condition (i) and (6.2), we have distX(S(t−t∗)X,E∗)≤distX(S((n−1)t∗)S(τ)X,E∗) ≤distX(Sn−1X,E∗) ≤Rγ2an−1 µ=Rγ2a−1 µa t−τ t∗ µ ≤Rγ2a−2 µe−(1 t∗log a−1 µ)t. This and (6.3) imply that for all t≥t∗, distX(S(t)X,E)≤L(Rγ2)γ2a−2γ2 µe−(γ2 t∗log a−1 µ)t.(6.4) By the condition (ii),we conclude that for the global attractor Aof {S(t)}t≥0, there exists a time tA>0 such that S(t)A ⊂ Xfor all t≥tA.This and the invariance of Aensure that distX(A,E) = distX(S(t)A,E)≤distX(S((t−tA)X,E)). Thanks to (6.4), we see that A⊂E. The proof is complete. In order to construct an exponential attractor Efor the semigroup {S(t)}t≥0 associated with (1.1) in H2α+s(T2) with α > 1 2and s > 1,now it suffices to show that there exists a closed bounded set X⊂H2α+s(T2) having the properties (i)-(iv) in Theorem 21. Lemma 15 implies that {S(t)}t≥0has a closed bounded absorbing set Bin H2α+s(T2),and there exists a time tB>0 such that S(t)B⊂Bfor every t≥tB. We define X=[ t≥tB S(t)B.(6.5) Then it follows that Xis a closed bounded set in H2α+s(T2),and the conditions (i) and (ii) in Theorem 21 are fulfilled. In order to verify the H¨older continuity property, we need the following lemma on estimates of ∂tθ. Lemma 22. Assume that the conditions of Lemma 14 hold. Then for any fixed eη with 0<eη≤2α−2α−,there exists T∗ 0=T∗ 0(X)>0such that any solution θ(t, θ0) of problem (1.1)with θ0∈Xsatisfies for any t1, t2≥T∗ 0, Zt2 t1 k∂tθk2 H2α+s−eη(T2)dτ ≤C|t2−t1|+C. (6.6) Proof. Multiplying (4.30) by (−∆)α+s−eηwand then integrating over T2,we obtain 1 2 d dt ZT2 [(−∆)α+s−eη 2w]2dx +κZT2 [(−∆)s−eη+2α 2w]2dx =ZT2 g1(x)f0(θ)w(−∆)α+s−eηwdx −ZT2 (u· ∇w)(−∆)α+s−eηwdx −ZT2 (ut· ∇θ)(−∆)α+s−eηwdx. (6.7) EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 23 For the first term on the right-hand side of (6.7), by similar arguments as in (4.32)- (4.34) we have ZT2 g1(x)f0(θ)w(−∆)α+s−eηwdx ≤ k(−∆)s−eη 2(g1f0(θ)w)kL2(T2)k(−∆)s−eη+2α 2wkL2(T2) ≤Ckg1f0(θ)kLp0 3(T2)kwk H s−eη, 2p0 3 p0 3−2(T2) kwkHs−eη+2α(T2) +Ckg1f0(θ)k H s−eη, 2 1−eη+p0 4(T2) kwk L 2 eη−p0 4(T2) kwkHs−eη+2α(T2) ≤C1 + kθkHs+α(T2)kg1kHs(T2)kwkHs(T2)kwkHs−eη+2α(T2) +Ckf0(θ)kLp0 3(T2)kg1k H s−eη, 2 1−eη+p0 4−2/p0 3(T2) +kg1kLp0 3(T2)kf0(θ)k H s−eη, 2 1−eη+p0 4−2/p0 3(T2)kwkHs(T2)kwkHs−eη+2α(T2) ≤C1 + kθkHs+α(T2)kg1kHs(T2)kwkHs(T2)kwkHs−eη+2α(T2) ≤κ 4kwk2 Hs−eη+2α(T2)+C1 + kθk2 Hs+α(T2)kg1k2 Hs(T2)kwk2 Hs(T2) (6.8) for some 0 <2 p0 3< p0 4<eη, and we have used Lemma 1, Young’s inequality and the Sobolev embeddings Hs(T2)⊂Hs−eη, 2p0 3 p0 3−2(T2), Hs(T2)⊂L∞(T2), Hs(T2)⊂ L 2 eη−p0 4(T2), Hs(T2)⊂Lp0 3(T2) and Hs(T2)⊂ Hs−eη, 2 1−eη+p0 4−2/p0 3(T2) for s > 1.For the last term in (6.7), using Lemmas 1-2, H¨older’s and Young’s inequalities, we deduce that ZT2 (ut· ∇θ)(−∆)α+s−eηwdx ≤ k(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη 2(ut· ∇θ)kL2(T2) ≤Ck(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη 2utkLp0 5(T2)k∇θkLp0 6(T2) +Ck(−∆)s−eη+2α 2wkL2(T2)kutkLp0 2(T2)k(−∆)s−eη+1 2θkLp0 1(T2) ≤Ck(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη 2wkLp0 5(T2)k∇θkLp0 6(T2) +Ck(−∆)s−eη+2α 2wkL2(T2)kwkLp0 2(T2)k(−∆)s−eη+1 2θkLp0 1(T2) ≤CkwkHs−eη+2α(T2)kwkHs(T2)kθkHs+2α(T2) ≤κ 8kwk2 Hs−eη+2α(T2)+Ckwk2 Hs(T2)kθk2 Hs+2α(T2), (6.9) where p0 1=1 1−α, p0 2=2 2α−1, p0 5=2 1−eη, p0 6=2 eη, and we have used the Sobolev embeddings Hs(T2)⊂Hs−eη,p0 5(T2), Hs+2α(T2)⊂ H1,p0 6(T2), Hs(T2)⊂Lp0 2(T2), Hs+2α(T2)⊂Hs−eη+1,p0 1(T2) in the last inequality. Arguing as in (4.36)-(4.39), the second term on the right-hand side of (6.7) is 24 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO bounded by ZT2 (u· ∇w)(−∆)α+s−eηwdx ≤ k(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη 2(u· ∇w)kL2(T2) ≤ k(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη+1 2(uw)kL2(T2) ≤Ck(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη+1 2ukLp1(T2)kwkLp2(T2) +Ck(−∆)s−eη+2α 2wkL2(T2)kukLp2(T2)k(−∆)s−eη+1 2wkLp1(T2) ≤Ck(−∆)s−eη+2α 2wkL2(T2)k(−∆)s−eη+1 2θkLp1(T2)kwkLp2(T2) +Ck(−∆)s−eη+2α 2wkL2(T2)kθkLp2(T2)k(−∆)s−eη+1 2wkLp1(T2) ≤Ck(−∆)s−eη+2α 2wk1+η0 4 L2(T2)kwk1−η0 4 L2(T2)k(−∆)s−eη+1 2θkLp1(T2) +Ck(−∆)s−eη+2α 2wk1+η0 3 L2(T2)kwk1−η0 3 L2(T2)kθkLp2(T2) ≤κ 8kwk2 Hs−eη+2α(T2)+Ckwk2 L2(T2)kθk 2 1−η0 4 Hs+2α(T2)+Ckwk2 L2(T2)kθk 2 1−η0 3 Hs+2α(T2). (6.10) Here p1=1 1−α−, p2=2 2α−−1,and we have used the Sobolev embeddings Hs+2α(T2)⊂ Hs−eη+1,p1(T2), Hs+2α(T2)⊂Lp2(T2),and the following Gagliardo-Nirenberg inequalities: k(−∆)s−eη+1 2wkLp1(T2)≤Ck(−∆)s−eη+2α 2wkη0 3 L2(T2)kwk1−η0 3 L2(T2), kwkLp2(T2)≤Ck(−∆)s−eη+2α 2wkη0 4 L2(T2)kwk1−η0 4 L2(T2), where η0 3∈hs−eη+1 s−eη+2α,1and η0 4∈(0,1). Inserting (6.8)-(6.10) into (6.7) gives d dtkwk2 Hα+s−eη(T2)+κkwk2 Hs−eη+2α(T2) ≤C1 + kθk2 Hs+α(T2)kg1k2 Hs(T2)kwk2 Hs(T2)+Ckwk2 Hs(T2)kθk2 Hs+2α(T2) +Ckwk2 L2(T2)kθk 2 1−η0 4 Hs+2α(T2)+Ckwk2 L2(T2)kθk 2 1−η0 3 Hs+2α(T2). (6.11) Recall that θ0∈Xand Xis a closed bounded set in H2α+s(T2) having the positively invariant property. In view of Lemma 14, we deduce that there exists T∗∗ 0=T∗∗ 0(X)>0 such that any solution θ(t, θ0) of problem (1.1) with θ0∈X satisfies k∂tθ(t)k2 Hs(T2)≤C, ∀t≥T∗∗ 0, k∂tθ(t)k2 L2(T2)≤C, ∀t≥T∗∗ 0, and by (4.46), we have that for any t1, t2≥T∗∗ 0, Zt2 t1 k∂tθk2 Hs+α(T2)dτ ≤C|t2−t1|+C. EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 25 Using g1∈Hs(T2) and the Sobolev embedding Hs+2α(T2)⊂Hs+α(T2),it follows from (6.11) that d dtkwk2 Hα+s−eη(T2)+κkwk2 Hs−eη+2α(T2)≤C, ∀t≥T∗∗ 0.(6.12) Applying the uniform Gronwall lemma to (6.12) results in k∂tθ(t)k2 Hα+s−eη(T2)≤C, ∀t≥T∗∗ 0+ 1.(6.13) Integrating the differential inequality (6.12), then the assertion of the lemma follows immediately from (6.13). The Lipschitz continuity of the solutions with respect to the initial data is presented in the following lemma. Lemma 23. Assume that the conditions of Lemma 14 hold. Let θand ξbe the solutions of problem (1.1)with the initial data θ0and ξ0∈X,respectively. Then, for any fixed eηwith 0<eη < 2α−2α−, kθ(t)−ξ(t)kH2α−+s(T2)≤Ckθ0−ξ0kH2α−+s(T2)eC νte−κλ2α 1 4t, kθ(t)−ξ(t)kH2α+s−eη(T2)≤Ce−κλ2α 1 2tt−α−α−− eη 2 αkθ0−ξ0kH2α−+s(T2) +Ce−κλ2α 1 4teC νtkθ0−ξ0kH2α−+s(T2), (6.14) where νis a constant with ν > α α−α−. Proof. Thanks to (3.2), we have θ(t)−ξ(t) = e−Aα,κt(θ0−ξ0) + Zt 0 e−Aα,κ(t−r)(F(x, θ(r)) −F(x, ξ(r))) −(u(r)· ∇θ(r)−v(r)· ∇ξ(r))dr, (6.15) where u= (−R2θ, R1θ) and v= (−R2ξ, R1ξ).Denote Aα−+s 2,κ =κ(−∆)α−+s 2, As 2,κ =κ(−∆)s 2. By Proposition 3and (6.15), we deduce that Aα−+s 2,κ(θ(t)−ξ(t))L2(T2) ≤Aα−+s 2,κe−Aα,κt(θ0−ξ0))L2(T2) +Zt 0Aα−+s 2,κe−Aα,κ(t−r)(g1(x)(f(θ(r)) −f(ξ(r))))L2(T2)dr +Zt 0Aα−+s 2,κe−Aα,κ(t−r)(u(r)· ∇θ(r)−v(r)· ∇ξ(r))L2(T2)dr ≤Ce−κλ2α 1 2tAα−+s 2,κ(θ0−ξ0)L2(T2) +CZt 0 e−κλ2α 1 2(t−r)(t−r)−α− αAs 2,κ (g1(x)(f(θ(r)) −f(ξ(r))))L2(T2)dr +CZt 0 e−κλ2α 1 2(t−r)(t−r)−α− αAs 2,κ(u(r)· ∇θ(r)−v(r)· ∇ξ(r))L2(T2)dr. (6.16) 32 LIN YANG, YEJUAN WANG AND TOM´ AS CARABALLO [7] Constantin, P., Coti Zelati, M., Vicol, V., Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity 29 (2016), 298-318. [8] Constantin, P., Foia¸s, C., Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. [9] Constantin, P., Tarfulea, A., Vicol, V., Long time dynamics of forced critical SQG, Comm. Math. Phys. 335 (2015), 93-141. [10] C´ordoba, A., C´ordoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), 511-528. [11] Coti Zelati, M., Kalita, P., Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1857-1873. [12] Dlotko, T., Kania, M.B., Sun, C., Quasi-geostrophic equation in R2,J. Differential Equations 259 (2015), 531-561. [13] Eden, A., Foias, C., Nicolaenko, B., Temam, R., Exponential Attractors for Dissipative Evolution Equation, Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [14] Efendiev, M., Miranville, A., Zelik, S., Exponential attractors for a nonlinear reactiondiffusion system in R3,C. R. Acad. Sci. Paris S´er. I Math. 330 (2000), 713-718. [15] Farwig, R., Qian, C., Asymptotic behavior for the quasi-geostrophic equations with fractional dissipation in R2,J. Differential Equations 266 (2019), 6525-6579. [16] Foia¸s, C., Prodi, G., Sur le comportement global des solutions non-stationnaires des ´equations de Navier-Stokes en dimension 2,Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34. [17] Gatti, S., Grasselli, M., Miranville, A., Pata, V., A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc. 134 (2006), 117-127. [18] Giga, Y., Miyakawa, T., Solutions in Lrof the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281. [19] Gu, A., Li, D., Wang, B., Yang, H., Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn,J. Differential Equations 264 (2018), 7094-7137. [20] Hale, J.K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. [21] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. [22] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasigeostrophic equations, Comm. Math. Phys. 255 (2005), 161-181. [23] Kato, T., Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. [24] Kenig, C.E., Ponce, G., Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4(1991), 323-347. [25] Mart´ınez Carracedo, C., Sanz Alix, M., The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001. [26] Niche, C.J., Schonbek, M.E., Decay of weak solutions to the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys. 276 (2007), 93-115. [27] Niche, C.J., Schonbek, M.E., Decay characterization of solutions to dissipative equations, J. Lond. Math. Soc. 91 (2015), 573-595. [28] Pedlosky, J., Geophysical Fluid Dynamics, Springer, Berlin, 1982. [29] Raugel, G., Global Attractors in Partial Differential Equations, Handbook of dynamical systems, vol. 2, 885-982, North-Holland, Amsterdam, 2002. [30] Robinson, J.C., Infinite-Dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. [31] Sell, G.R., Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations 8(1996), 1-33. [32] Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Princeton University Press, Princeton, N.J., 1970. [33] Sun, C., Zhong, C., Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal. 63 (2005), 49-65. [34] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. EXPONENTIAL ATTRACTORS FOR QUASI-GEOSTROPHIC EQUATIONS 33 [35] Wang, S., Attractors for the 3D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl. 165 (1992), 266-283. [36] Wang, M., Tang, Y., Long time dynamics of 2D quasi-geostrophic equations with damping in Lp,J. Math. Anal. Appl. 412 (2014), 866-877. [37] Wang, M., Tang, Y., On dimension of the global attractor for 2D quasi-geostrophic equations, Nonlinear Anal. Real World Appl. 14 (2013), 1887-1895. [38] Wu, J., Dissipative quasi-geostrophic equations with Lpdata, Electron. J. Differential Equations 2001 (2001), 1-13. [39] Yagi, A., Abstract Parabolic Evolution Equations and their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. [40] Zhong, C., Yang, M., Sun, C., The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations 223 (2006), 367-399. E-mail address:[email protected] E-mail address:[email protected] E-mail address:[email protected]