Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory
Abstract
The objective of this paper is to consider the asymptotical behavior of solutions for the two-dimensional partial dissipative Boussinesq system with memory and additive noise. We first establish the existence of a random absorbing set in the phase space. However, due to the presence of the memory term, we cannot obtain some kind of compactness of the corresponding cocycle through Sobolev compactness embedding theorem or by verifying the pullback flattening property. To overcome this difficulty, we first prove the asymptotical compactness of the velocity component of weak solutions, and then we prove the asymptotical compactness of other components based on some energy estimates and the Aubin–Lions compactness lemma, which implies the asymptotical compactness of the corresponding cocycle. Thus, the existence of a random attractor is obtained. Finally, we establish an abstract result about some kind of upper semi-continuity of the random attractor, which is applied to the two-dimensional partial dissipative Boussinesq system.
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Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory Haoran Dai⇤ ,BoYou † School of Mathematics and Statistics, Xi’an Jiaotong University Xi’an, 710049, P. R. 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 January 22, 2025 Abstract The objective of this paper is to consider the asymptotical behavior of solutions for the two-dimensional partial dissipative Boussinesq system with memory and additive noise. We first establish the existence of a random absorbing set in the phase space. However, due to the presence of the memory term, we cannot obtain some kind of compactness of the corresponding cocycle through Sobolev compactness embedding theorem or by verifying the pullback flattening property. To overcome this difficulty, we first prove the asymptotical compactness of the velocity component of weak solutions, and then we prove the asymptotical compactness of other components based on some energy estimates and the Aubin-Lions compactness lemma, which implies the asymptotical compactness of the corresponding cocycle. Thus, the existence of a random attractor is obtained. Finally, we establish an abstract result about some kind of upper semi-continuity of the random attractor, which is applied to the two-dimensional partial dissipative Boussinesq system. Keywords: Random attractor; Partial dissipative Boussinesq system; Memory term; Additive noise; Upper semi-continuity. Mathematics Subject Classification (2020) : 35B40, 35B41, 35Q35, 37L55, 60H15. ⇤Email address: [email protected] †Email address: y[email protected] ‡Email address: [email protected] 1
1 Introduction It is well-known that the Boussinesq system plays an important role in modelling geophysical flows, such as atmospheric fronts and oceanic circulation (see, e.g., [20, 28, 31]). However, in certain physical regimes, the dynamical system governing geophysical flows is described to be only partial dissipative or semi-dissipative. For example, the authors in [1] considered a 2D Boussinesq system with dissipation only in the velocity variable, while a semi-dissipative Boussinesq system without dissipation in temperature variable was considered in [6]. Actually, the Boussinesq equations with partial dissipation has been widely used to model the dynamics of geophysical flows in which the dissipation in some particular direction dominates (see, e.g., [12, 13, 29]). In this paper, we consider the following 2D partial dissipative Boussinesq system with memory and additive noise: 8 > > > > > > < > > > > > > : @tu1+u·ru1+@xp⌫@2 yu1=0,(x, t)2Q, @tu2+u·ru2+@yp⌫@2 xu2=',(x, t)2Q, r·u=0,(x, t)2Q, d''dt+u·r'dt+Rt 1 (tr)'(r)drdt =fdt+dWt,(x, t)2Q, u(x, t0)=ut0,'(x, t0+t)='0(x, t),(x, t)2D⇥(1,0] (1.1) equipped with periodic boundary conditions, where D=T2=[0,L]2,Q:= D⇥[t0,+1), t02R.Thevelocityu=(u1,u 2), the pressure pand the temperature 'are the unknown functions. ⌫and are the viscosity and the thermal di↵usivity coefficients, respectively. In the following, we will always assume that ⌫== 1, which has no bearing on the mathematical analysis. (·)2C2(R+) is the so-called memory kernel, the memory term Rt 1 (tr)'(r)dr represents the integrated past history of temperature variable. f= f(x) is an external forcing term. W(x, t) is a Wiener process in L2(D)definedonacomplete probability space (⌦,F,P)withexpectationdenotedbyE,whichcanbewrittenas W(x, t)= 1 X j=1 jBj(t)ej, where {Bj}j2Z+are independent standard Brownian motions, {ej}j2Z+is an orthonormal basis of L2(D) consisting of the eigenfunctions of with periodic boundary conditions imposed the zero-average condition and the coefficients {j}j2Z+satisfies the following condition 1 X j=1 2 j 1 220 j <1(1.2) for some 0>1 2.Theexistenceofsuch0can be obtained by assumption (A2)below. The two-dimensional Boussinesq equations with partial dissipation has recently attracted considerable attention and made some progress, especially in terms of long-time behavior 2
of partial dissipative Boussinesq system (see, e.g., [2, 3, 8, 11, 23, 24, 26]). For example, the authors in [6] studied the long-time behavior of a semi-dissipative Boussinesq system which is dissipative only in the velocity variable, but not in the temperature. They proved such system has a global attractor. In [22], the authors considered a 2D Boussinesq system which is partially dissipative only in the velocity variable, they proved that such system is global well-posed under some weaker assumptions on the initial data. In addition, they also proved the existence of a weak sigma-attractor and established its upper semi-continuity under small perturbations. It is widely accepted that if some terms taking into account the past history of the system are incorporated into the equations, many physical phenomena can be better described, such as heat conduction in special materials (see, e.g., [21, 34, 35]), viscoelasticity of vibration in several materials (see, e.g., [7, 19]). In the past several decades, there are many works about the well-posedness and long-time behavior of solutions for partial di↵erential equations with memory term. For example, the authors in [30] established the existence and uniqueness of weak solutions of velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory, they also proved the existence of a uniform attractor of this system. In [36], the authors studied the dynamics of the three-dimensional globally modified NavierStokes equations with double delay in the forcing and convective terms, they established the existence of pullback attractors for the associated dynamical systems. As we know, the models of certain phenomena from the real world are more realistic if some kind of uncertainty is also considered in the formulation, such as some randomness or environmental noise. Thus, it is meaningful and necessary to consider the well-posedness and long-time behavior of solutions for stochastic partial di↵erential equations. For example, the authors have proved the existence of random attractor for the 2D stochastic Cahn-Hilliard-Navier-Stokes system with small additive noise and three dimensional damped Navier-Stokes equations with additive noise in [27, 38], respectively. Also in [33] a stochastic nonlocal reaction-di↵usion equation perturbed with additive and multiplicative noise is analyzed. The authors in [37] investigated mean dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay driven by multiplicative noise in unbounded domains, they proved such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. In [10], the authors studied the asymptotic behavior of a non-autonomous stochastic reaction-di↵usion equation with memory and proved the existence of a random pullback attractor. However, to the best of our knowledge, there is no known results concerning the long-time behavior of solutions for the 2D partial dissipative Boussinesq system with memory and additive noise. Let us now comment on the main mathematical difficulties and novelties of this work. Due to the presence of the memory term, we cannot establish the existence of an absorbing set in more regular phase space, such that some kind of compactness of the corresponding cocycle cannot be obtained by verifying the pullback flattening property or using the Sobolev compactness embedding theorem. Meanwhile, the assumption on the memory kernel gives rise to another crucial difficulty that we cannot prove the asymptotic compactness of the corresponding cocycle by the methods of energy equation or semigroup decomposition. For3
tunately, the velocity component of the weak solutions possesses smoothing property, such that we can easily obtain the asymptotic compactness of the velocity component of weak solutions, and then we prove the corresponding one of other components based on some energy estimates and the Aubin-Lions compactness Lemma, which implies the asymptotic compactness of the corresponding cocycle. It is worth mentioning that another type of auxiliary Ornstein-Uhlenbeck process is introduced such that the stochastic term dW(t)canbe located in L2(D),which is di↵erent from the existing work about additive noise. This paper is organized as follows. In Section 2, we first introduce some notation and function spaces, then we recall some abstract results about random dynamical systems and some useful lemmas that will be used in this paper. In Section 3, we first provide a wellposedness result for the 2D partial dissipative Boussinesq system with memory and additive noise, then we establish the existence of random absorbing sets in an appropriate phase space. With the help of the Aubin-Lions compactness lemma and the energy estimates method, we prove the asymptotic compactness of the corresponding cocycle, which implies the existence of a random attractor. In Section 4, we establish an abstract result about the upper semi-continuity of random attractors, which is applied to the 2D partial dissipative Boussinesq system. Throughout this paper, we use Lr(D)(1 r<+1)andHm(D)(m2N)todenotethe usual Lebesgue and Sobolev spaces over Dwith zero-average condition. For convenience, we always use kfkrto denote the Lr(D) norm of function f, C denotes a generic constant which may change from line to line. A.Bmeans that there exists a generic constant C, which may be di↵erent on di↵erent lines, such as ACB. 2 Preliminaries In this section, we will first state some assumptions on the memory term and the stochastic term, then we introduce some function spaces on Dand recall some abstract results on random attractors. (A1)Thememorykernel(·)2C2(R+)satisfies lim s!1 (s) = 0 and the function µ(s)= 0(s)possessesthefollowingproperties µ(s)0,µ 0(s)+µ(s)0,8s2R+, where is a positive constant. A typical example is (s)=0ed0swith d0>0and0>0. Question: can have a singularity at zero, i.e. is the case (s)=0ed0s s1↵,↵2(0,1) included in this framework? (A2){W(t); t2R}is a two-sided L2(D)-valued Wiener process with covariance operator K=K⇤0,such that tr KA2↵⇤1<1 for some ↵⇤1, where A=Pis the Stokes operator and Pis the Leray-Helmholtz projection from (L2(D))2onto the free divergence subspace Hof (L2(D))2.Inourcaseof periodic boundary conditions, it is well known that A=(see, e.g., [15]). 4
Remark 2.1. The condition ↵⇤1in (A2) implies the existence of 0in (1.2). In fact, we have tr KA2↵⇤1:= 1 X j=1 ⌦KA2↵⇤1ej,e j↵= 1 X j=1 ⌦K2↵⇤1 jej,e j↵ = 1 X j=1 2↵⇤1 jhKej,e ji= 1 X j=1 2↵⇤1 j⌦2 jej,e j↵ = 1 X j=1 2 j2↵⇤1 j<1. Thus, in order to ensure condition (1.2) holds, i.e. 1 P j=1 2 j201 2 j<1, it is sufficient to require that 0satisfies the following condition: 201 22↵⇤1, i.e. 0↵⇤1 4.Therefore, there always exists 02(1 2,↵ ⇤1 4],such that condition (1.2) holds. Let Vbe the set of all vector-valued L-periodic trigonometric polynomials from R2to R2that are divergence-free and have zero average. Denote by H, V and Zthe closures of V in L2(D),H 1(D)andH2(D), respectively. Define the bilinear operator B:V⇥V!V0by hB(u, v),wi=ZD [(u·r)v]·wdx, 8u, v, w 2V. In order to carry out the analysis of the memory term, let M:= L2 µ(R+;H1(D)) be the Hilbert space with inner product and norm given by (⇠,⇣)M:= Z1 0 µ(s)(⇠(s),⇣(s))H1(D)ds and k⇠k2 M:= Z1 0 µ(s)k⇠(s)k2 H1(D)ds, for any ⇠,⇣2H1(D). In our case of periodic boundary conditions, the norm on spaces H1(D)andH2(D) with zero-average condition are defined by k⇠kH1(D):= kr⇠k2,k⇣kH2(D):= k⇣k2 for any ⇠2H1(D), ⇣2H2(D), respectively. 5
Now, we define an auxiliary Ornstein-Uhlenbeck process z(t). For any ↵0andany !2⌦, let z(t;!)=Zt 1 e(A+↵)(ts)dW(s) be the solution of the stochastic equation dz +(A+↵)zdt=dW(t). Note that zis a stationary Gaussian process, its trajectories are P-a.s. continuous (see [18]). We can also involve a perfection procedure to define z(t;!)=¯z(✓t!)forall!2⌦ (see Proposition 3.1 in [14]). Moreover, the mapping t!¯z(✓t!) is continuous from Rinto D(A↵⇤)foreach!2⌦andsatisfiesthefollowingcondition: sup t2R{kA↵⇤¯z(✓t!)k2e|t|}<1for any >0andany!2⌦. By introducing a new variable #='z, problem (1.1) can be rewritten as 8 > > > > > > < > > > > > > : @tu1+u·ru1+@xp@2 yu1=0,(x, t)2Q, @tu2+u·ru2+@yp@2 xu2=#+z, (x, t)2Q, r·u=0,(x, t)2Q, @t##+u·r#+Rt 1 (tr)#(r)dr =fu·rz+z(t)+↵z, (x, t)2Q, u(x, t0)=ut0,#(x, t0+t)='(x, t0+t)z(t0+t),(x, t)2D⇥(1,0], (2.1) where z(t)isaprocessdefinedby z(t, !):=Zt 1 (tr)z(r, !)dr =Zt 1 (tr)Az(r, !)dr. Here, we provide an important result for z(t)whichplaysakeyroleintheproofofthe existence of a random attractor. Lemma 2.2. (see [10]) The process t!z(t, !)is continuous with valued in D(A↵⇤1). Moreover, z(t, !)=¯z(✓t!), where ¯z(!)=Z1 0 (s)A¯z(✓s!)ds is a tempered random variable with valued in D(A↵⇤1). In what follows, we recall a useful lemma used in the sequel. Lemma 2.3. (see [1]) For any u2V, we have kruk2 2=kr⇥uk2 22k(@yu1,@ xu2)k2 2. Moreover, for any u2V\(H2(D))2, we have kuk2 22ZD @2 yu1u1dxdy +2ZD @2 xu2u2dxdy. 6
Now, in order to carry out our analysis, we introduce the new variable which reflects the integrated past history of problem (2.1) given by ⌘t(x, s)=⌘(x, t, s)=Zs 0 #(x, t r)dr =Zt ts #(x, r)dr, s 0,tt0, then we have @t⌘t(x, s)=#(x, t)@s⌘t(x, s)and⌘t(x, 0) := lim s!0⌘t(x, s)=0. Thus, problem (2.1) can be reformulated into the following form: 8 > > > > > > > > < > > > > > > > > : @tu1+u·ru1+@xp@2 yu1=0,(x, t)2Q, @tu2+u·ru2+@yp@2 xu2=#+z, (x, t)2Q, r·u=0,(x, t)2Q, @t##+u·r#R+1 0µ(r)⌘t(r)dr =fu·rz+z(t)+↵z, (x, t)2Q, @t⌘t(s)=@s⌘t(s)+#(t),(x, t)2Q, s 0, u(x, t0)=ut0,#(x, t0)='0(t0)z(t0),⌘ t0(x, s)=⌘0(s),x2D, s 0. (2.2) Let (X, k·kX)beaseparableBanachspacewithBorel-algebra B(X), we recall some abstract results from random dynamical system. Definition 2.4. (see [4, 5]) (⌦,F,P,{✓t}t2R)is called a metric dynamical system, if ✓:R⇥⌦!⌦is (B(R)⇥F,F)-measurable, ✓0is the identity on ⌦,✓s+t=✓s✓tfor all s, t 2Rand ✓tP=Pfor all t2R. Definition 2.5. (see [4, 16, 25]) A mapping :R+⇥⌦⇥X!Xis called a continuous cocycle on Xover a metric dynamical system (⌦,F,P,{✓t}t2R), if it is (B(R+)⇥F⇥ B(X),B(X))-measurable and satisfies for P-a.e. !2⌦, (i) (0,!)is the identity on X, (ii) (t+s, !)=(t, ✓s!)(s, !)for all t, s 2R+, (iii) (t, !): X!Xis continuous for all t2R+. Definition 2.6. (see [16, 25]) A random subset {B(!):!2⌦}of Xis called tempered with respect to {✓t}t2R, if for P-a.e. !2⌦, lim t!+1etd(B(✓t!)) = 0 for all >0, where d(B)=sup x2BkxkX. 7
Definition 2.7. (see [9, 17]) Assume that is a continuous cocycle on a Banach space Xover a metric dynamical system (⌦,F,P,{✓t}t2R),let Dbe a collection of random subsets of X. ˆ K={K(!):!2⌦}is called a D-random absorbing set for , if for every ˆ B= {B(!):!2⌦}2Dand P-a.e. !2⌦,there exists some T=T(!,ˆ B)>0,such that (t, ✓t!)B(✓t!)⇢K(!) for any tT. Definition 2.8. (see [25]) Let Dbe a collection of random subsets of X. The continuous cocycle is said to be D-pullback asymptotically compact in X, if for P-a.s. !2⌦,any ˆ B={B(!):!2⌦}2D,any sequence tn!+1and any sequence xn2B(✓tn!),the sequence {(tn,✓ tn!)xn}1 n=1 has a convergent subsequence in X. Definition 2.9. (see [9, 17]) Let Dbe a collection of random subsets of X. A random set ˆ A={A(!):!2⌦}of Xis called a D-random attractor for , if the following conditions are satisfied, for P-a.e. !2⌦, (i) A(!)is compact and !7! d(x, A(!)) is measurable for any x2X, (ii) ˆ Ais invariant, i.e., (t, !)A(!)=A(✓t!)for any t0, (iii) ˆ Apullback attracts every member of D, i.e., for every ˆ B={B(!):!2⌦}2D, lim t!+1d((t, ✓t!)B(✓t!),A(!)) = 0, where dis the Hausdor↵semi-metric given by d(Y,Z)=sup y2Y inf z2ZkyzkXfor any Y,Z ⇢X. In what follows, we state the definition of weak solutions to problem (2.2). Definition 2.10. Assume that f2L2(D),(ut0,' 0(t0),⌘t0)2H:= H⇥L2(D)⇥M for any fixed t02R, let T>t 0be any fixed time. The function (u, #,⌘t)is called a weak solution of problem (2.2) on the time interval [t0,T], if for P-a.s. !2⌦, u2L1(t0,T;H)\L2(t0,T;V)\H1(t0,T;V⇤), #2L1(t0,T;L2(D)) \L2(t0,T;H1(D)) \H1(t0,T;(H1(D))⇤), ⌘t2L1(t0,T;M), and it satisfies P-a.s. !2⌦, h@tu, ⇣i+hB(u, u),⇣i+⌦@yu1,@ y⇣1↵+⌦@xu2,@ x⇣2↵=⌦#+z,⇣2↵, h@t#,⇠i+hr#,r⇠i+hu·r#,⇠i+Z+1 0⌦µ(r)r⌘t(r),r⇠↵dr =hf,⇠ihu·rz,⇠i+hz,⇠i+↵hz,⇠i, @t⌘t+@s⌘t, M=(#, )M for any ⇣=(⇣1,⇣2)2V,⇠2H1(D)and 2M. 8
3 The existence of a random attractor In this section, we will prove the existence of a random attractor for the two-dimensional stochastic partial dissipative Boussinesq system with memory and additive noise (2.2). 3.1 The well-posedness of problem (2.2) The well-posedness result for problem (2.2) can be obtained by the standard Faedo-Galerkin methods ([32]). Here, we only state it as follows. Theorem 3.1. Assume (A1)-(A2) hold and f2L2(D). Then, for any !2⌦and any (ut0,' 0(t0),⌘t0)2H, there exists a unique weak solution (u, #,⌘t)to problem (2.2) in the sense of Definition 2.10, defined on [t0,T]. By Theorem 3.1, we can define a mapping :R+⇥⌦⇥H!H by (tt0,✓ t0!)(ut0(x),' t0(x),⌘t0(x)) =(u(t, t0;!),'(t, t0;!),⌘t(t0;!)) :=(u(t, t0;!),#(t, t0;!)+¯z(✓t!),⌘t(t0;!)) for any tt0,where(u(t, t0;!),#(t, t0;!),⌘t(t0;!)) is the weak solution of problem (2.2) with initial data (u(t0,t 0;!),#(t0,t 0;!),⌘t0(t0;!)) = (ut0(x),' 0(t0)¯z(✓t0!),⌘t0(x)) 2H. That is, a family of mappings :R+⇥⌦⇥H!Hsatisfies (1) (0,!)istheidentityonH, (2) (t+s, !)=(t, ✓s!)(s, !)forallt, s 2R+, (3) (t, !): H!His continuous for all t2R+. 3.2 The existence of a random attractor In this subsection, we will prove the existence of a random attractor for problem (2.2). To this end, let Dbe the class of all families ˆ D={D(!):!2⌦}of nonempty subsets of H such that lim t!+1e3c1 2t[D(✓t!)] = 0 for any !2⌦, where c1:= min{1 2,},1is the first eigenvalue of the Stokes operator A and [D(!)] = sup{kukH+k#kL2(D)+k⌘tkM:(u, #,⌘t)2D(!)}. Obviously, the universe of fixed bounded sets is contained in D, so the results that hold for the tempered universe also hold for the universe of fixed bounded sets. 9
We derive from Poincar´e’s inequality and Young’s inequality that ZD (u·r¯z(✓s!)) #dx kuk4kr#k2k¯z(✓s!)k4Ck¯z(✓s!)k2 4kruk2 2+1 2kr#k2 2, then we arrive at 1 2 d ds ✓k#(s, t;!)k2 2+Z+1 0 µ(r)kr⌘s(r)k2 2dr◆+1 2kr#k2 2+ 2Z+1 0 µ(r)kr⌘s(r)k2 2dr kfk2 2+Ck#k2 2+Ck¯z(✓s!)k2 4kruk2 2+k¯z(✓s!)k2 2+↵2k¯z(✓s!)k2 2.(3.12) It follows from inequalities (3.11)-(3.12) that d ds ✓kru(s, t;!)k2 2+k#(s, t;!)k2 2+Z+1 0 µ(r)kr⌘s(r)k2 2dr◆ +kuk2 2+kr#k2 2+Z+1 0 µ(r)kr⌘s(r)k2 2dr .k¯z(✓s!)k2 4kruk2 2+k#k2 2+kfk2 2+k¯z(✓s!)k2 2+k¯z(✓s!)k2 2, then we infer from the classical Gronwall inequality that there exists a positive constant M1(!)suchthatforany!2⌦, sup s2[0,T ]kru(s, t;!)k2 2+sup s2[0,T ]k#(s, t;!)k2 2+sup s2[0,T ]k⌘s(t;!)k2 M +ZT 0ku(s)k2 2ds +ZT 0kr#(s)k2 2ds +ZT 0k⌘sk2 Mds .✓ku0k2 V+k#0k2 2+k⌘0k2 M+ZT 0 1+k¯z(✓s!)k2 2+k¯z(✓s!)k2 2ds◆eRT 0(1+k¯z(✓s!)k2 4)ds M1(!), where we used the fact that t!¯z(✓t!) is continuous from Rinto D(A↵⇤)andt!¯z(✓t!) is continuous from Rinto D(A↵⇤1). In order to prove (3.8)-(3.10), it only remains to show that @tu2L2(0,T;H).To do this, for any v2Hwith kvkH1, we have h@tu, vi=hB(u, u),vi+D@2 yu1,@2 xu2>,vE+⌦#+¯z(✓s!),v2↵. Since |hB(u, u),vi|Ckuk4kruk4kvk2.kukVkukZ and D@2 yu1,@2 xu2>,vE ⌦@2 yu1,v1↵+⌦@2 xu2,v2↵ .ku1k2kvk2+ku2k2kvk2 kuk2, 16
then we have [email protected]+kuk2+k#k2+k¯z(✓s!)k2, which entails that k@tuk2 L2(0,T ;H).kuk2 L1(0,T ;V)+1 ZT 0kuk2 Zds +ZT 0k#k2 2+k¯z(✓s!)k2 2ds 1+M1(!)2. In what follows, we will prove the asymptotic compactness of the cocycle associated with problem (2.2). Theorem 3.4. Assume (A1)-(A2) hold and f2L2(D). The cocycle corresponding to problem (2.2) is D-pullback asymptotically compact in H. Proof. Let ˆ B0={ˆ B0(!):!2⌦}be the D-random absorbing set in Hestablished in Theorem 3.2, for any !2⌦,any sequence {tn}1 n=1 with tn!+1and any sequence (utn,# tn,⌘tn)2ˆ B0(✓tn!), we will prove that the sequence {(tn,✓ tn!)(utn,# tn,⌘tn)}1 n=1 ={(un(0,tn;!),# n(0,tn;!)+¯z(!),⌘0 n(tn;!))}1 n=1 possesses a convergent subsequence in H. For any T0andanys2Rwith sT+tn0,denote by (un(sT),# n(sT)+¯z(✓sT!),⌘sT n)=(sT+tn,✓ tn!)(utn,# tn,⌘tn), then by the definition of cocycle and random absorbing set, we have (un(sT),# n(sT)+¯z(✓sT!),⌘sT n)2ˆ B0(✓sT!) and (un(s),# n(s)+¯z(✓s!),⌘s n):=(T,✓sT!)(un(sT),# n(sT)+¯z(✓sT!),⌘sT n). For any n1andanys0,let vn(s)=un(sT), then it follows from Theorem 3.2, Lemma 3.3 and the Aubin-Lions compactness lemma that there exists a subsequence of {vn}1 n=1 (still denote by {vn}1 n=1), such that {un(0)}1 n=1 = {vn(T)}1 n=1 is convergent in Hand {vn}1 n=1 is convergent in L2(0,T;V).For any n, m 1, define unm(s):=vn(s)vm(s)=un(sT)um(sT), #nm(s):=#n(sT)#m(sT),⌘ s nm := ⌘sT n⌘sT m. 17
Then it is clear that (unm(s),# nm(s),⌘s nm) satisfies the following equation: @s#nm #nm +un·r#nm +unm ·r#mZ+1 0 µ(r)⌘s nm(r)dr =unm ·r¯z(✓sT!). We infer from H¨older’s inequality and Young’s inequality that 1 2 d ds(k#nm(s)k2 2+k⌘s nmk2 M)+kr#nmk2 2+ 2k⌘s nmk2 M ZD ((unm ·r)#m)·#nm dx ZD (unm ·r¯z(✓sT!)) #nm dx Ckunmk2 4k#mk2 4+1 2kr#nmk2 2+Ckunmk2 4k¯z(✓sT!)k2 4, which implies that d ds(k#nm(s)k2 2+k⌘s nmk2 M)+kr#nmk2 2+k⌘s nmk2 M Ckunmk2 4k#mk2 4+Ckunmk2 4k¯z(✓sT!)k2 4, then we derive from Poincar´e’s inequality and the classical Gronwall inequality that, for any s2[0,T], k#nm(s)k2 2+k⌘s nmk2 M .(k#nm(0)k2 2+k⌘0 nmk2 M)ec1T +ZT 0kunmk2 4k#mk2 4+kunmk2 4k¯z(✓sT!)k2 4ec1(Ts)ds (k#nm(0)k2 2+k⌘0 nmk2 M)ec1T +kunmkL1(0,T ;L2(D))k#mkL1(0,T ;L2(D))krunmkL2(0,T ;L2(D))kr#mkL2(0,T ;L2(D)) +sup s2[0,T ]k¯z(✓sT!)k2 4kunmkL2(0,T ;L2(D))krunmkL2(0,T ;L2(D)), where c1=min{1 2,}.Due to the fact that (un(sT),# n(sT),⌘sT n)2ˆ B0(✓sT!) for any s2Rand T0, it follows that for any ✏>0,there exists a T2>0 such that (k#nm(0)k2 2+k⌘0 nmk2 M)ec1T2<✏ 2 for any n, m 1. Since {vn}1 n=1 ={un(·T2)}1 n=1 is convergent in L2(0,T 2;V),we obtain krunmkL2(0,T2;L2(D)) !0 18
as n, m !1.Therefore, we infer from Lemma 3.3 and the continuity of t7! ¯z(✓t!)fromR into D(A↵⇤), that for the given ✏as above, there exists a natural number N1suchthat for any n, m N, kunmkL1(0,T2;L2(D))k#mkL1(0,T2;L2(D))krunmkL2(0,T2;L2(D))kr#mkL2(0,T2;L2(D)) +sup s2[0,T2]k¯z(✓sT2!)k2 4kunmkL2(0,T2;L2(D))krunmkL2(0,T2;L2(D)) <✏ 2. Therefore, we conclude that for any ✏>0,there exists a natural number N1,such that for any n, m N, we have k#nm(T2)k2 2+k⌘T2 nmk2 M<✏, which implies that {(#(0,tn;!),⌘0(tn;!))}1 n=1 is a Cauchy sequence in L2(D)⇥M.Therefore, the sequence {(tn,✓ tn!)(utn,# tn,⌘tn)}1 n=1 ={(un(0,tn;!),# n(0,tn;!)+¯z(!),⌘0 n(tn;!))}1 n=1 has a convergent subsequence in H. From Theorem 3.2, Theorem 3.4 and the abstract theory about random attractor proposed in [17], we immediately conclude the following result. Theorem 3.5. Assume that (A1)-(A2) hold and f2L2(D). The cocycle corresponding to the two-dimensional stochastic partial dissipative Boussinesq system with memory and additive noise (2.2) has a random attractor ˆ A={A(!):!2⌦}in H. 4 The upper semi-continuity of the random attractor The main objective of this section is to prove the upper semi-continuity of random attractors for problem (2.2) under small viscosity perturbations. First of all, we establish an abstract result about the upper semi-continuity of random attractors. Proposition 4.1. Let >0be a positive constant. Assume that the continuous cocycle ✏0on a Banach space (X, k·k)has a random attractor ˆ A✏0={A✏0(!):!2⌦}and the corresponding cocycle ✏of its perturbed dynamical system possesses a random attractor ˆ A✏={A✏(!):!2⌦}for any ✏2[✏0,✏0+].In addition, assume the following conditions hold: (i) There exists a tempered set ˆ D={D(!):!2⌦}2Dsuch that for any !2⌦, [ |✏✏0|< ˆ A✏(!)⇢D(!). 19
(ii) For any !2⌦,>0and any fixed T>0,there exists >0,such that for any |✏✏0|<, sup u02D(✓T!)k✏(T,✓T!)u0✏0(T,✓T!)u0k<. Then ˆ A✏0and ˆ A✏have the following property of upper semi-continuity, i.e. for any !2⌦, lim ✏!✏0 dist(ˆ A✏(!),ˆ A✏0(!)) = 0. Proof. From the definition of random attractor, we conclude that for any >0andany !2⌦,there exists a time T=T(ˆ D, ),such that dist(✏0(T,✓T!)D(✓T!),ˆ A✏0(!)) < 2. From assumption (ii), we infer that there exists >0,such that for any |✏✏0|<, dist(✏(T,✓T!)D(✓T!), ✏0(T,✓T!)D(✓T!)) sup u02D(✓T!)k✏(T,✓T!)u0✏0(T,✓T!)u0k < 2. Therefore, for any >0,there exists >0,such that for any |✏✏0|<, dist(ˆ A✏(!),ˆ A✏0(!)) = d(✏(T,✓T!)ˆ A✏(✓T!),ˆ A✏0(!)) <. In order to prove the upper semi-continuity of random attractor for problem (2.2), for any ✏>0,let us consider the following viscosity perturbed system of problem (2.2): 8 > > > > > > > > < > > > > > > > > : @tu1 ✏+u✏·ru1 ✏+@xp✏@2 xu1 ✏@2 yu1 ✏=0,(x, t)2Q, @tu2 ✏+u✏·ru2 ✏+@yp@2 xu2 ✏✏@2 yu2 ✏=#✏+z, (x, t)2Q, r·u✏=0,(x, t)2Q, @t#✏#✏+u✏·r#✏R+1 0µ(r)⌘t ✏(r)dr =fu✏·rz+z(t)+↵z, (x, t)2Q, @t⌘t ✏(s)=@s⌘t ✏(s)+#✏(t),(x, t)2Q, s 0, u✏(x, t0)=ut0,# ✏(x, t0)='0(t0)z(t0),⌘ t0 ✏(x, s)=⌘0(s),x2D, s 0. (4.1) We notice that the viscosity terms ✏@2 xu1 ✏and ✏@2 yu2 ✏have no e↵ect on the result of Theorem 3.5. Therefore, it is easy to obtain that for each 0 <✏1, the cocycle ✏corresponding to problem (4.1) has a random attractor ˆ A✏={A✏(!):!2⌦}in H. Now, we are ready to state and prove the main result of this section. 20
Theorem 4.2. Assume (A1)-(A2) hold and f2L2(D). The random attractor ˆ A= {A(!):!2⌦}of problem (2.2) and the random attractor ˆ A✏={A✏(!):!2⌦}of problem (4.1) have the upper semi-continuity property, i.e. for any !2⌦, lim ✏!✏0 dist(ˆ A✏(!),ˆ A✏0(!)) = 0. Proof. Let ˆ D={D(!):!2⌦}be the random absorbing set established in Theorem 3.2, it is easy to deduce from the proof of Theorem 3.2 and the property of random attractor that for any !2⌦, [ 0<✏1 ˆ A✏(!)⇢ˆ D(!). Let (u, #,⌘t), (u✏,# ✏,⌘t ✏) be the weak solutions for problem (2.2) and (4.1), respectively. Denote by (˜u, ˜ #,˜⌘t)=(uu✏,##✏,⌘t⌘t ✏),then (˜u, ˜ #,˜⌘t) satisfies the following problem: 8 > > > > > > > > < > > > > > > > > : @t˜u1+˜u·ru1+u✏·r˜u1+✏@2 xu1 ✏@2 y˜u1=0,(x, t)2Q, @t˜u2+˜u·ru2+u✏·r˜u2@2 x˜u2+✏@2 yu2 ✏=˜ #,(x, t)2Q, r·˜u=0,(x, t)2Q, @t˜ #˜ #+˜u·r#+u✏·r˜ #R+1 0µ(r)˜⌘t(r)dr =˜u·rz, (x, t)2Q, @t˜⌘t(s)=@s˜⌘t(s)+˜ #(t),(x, t)2Q, s 0, ˜ut0(x)=0,˜ #t0(x)=0,˜⌘t0(x, s)=0,x2D, s 0. (4.2) Multiplying the first and second equation of (4.2) by ˜u1and ˜u2,respectively,thenintegrating the resulting equalities over D, we infer from Lemma 2.3 and Young’s inequality that 1 2 d dsk˜u(s, t0;!)k2 2+1 2kr˜uk2 2 C✏2kru✏k2 2+1+Ckruk2 2k˜uk2 2+1 8kr˜uk2 2+k˜ #k2 2.(4.3) Multiplying now the fourth equation of (4.2) by ˜ #and integrating over D,weobtain 1 2 d ds ✓k˜ #(s, t0;!)k2 2+Z+1 0 µ(r)kr˜⌘s(r)k2 2dr◆+kr˜ #k2 21 2Z+1 0 µ0(r)kr˜⌘s(r)k2 2dr =ZD (˜u·r#)˜ #dxZD (˜u·r¯z(✓s!)) ˜ #dx. Moreover, we infer from the interpolation and Young’s inequalities that ZD (˜u·r#)˜ #dx k˜uk4kr#k2k˜ #k4 Ck˜uk 1 2 2kr˜uk 1 2 2kr#k 1 2 2kr#k 1 2 2k˜ #k 1 2 2kr˜ #k 1 2 2 Ck˜uk2kr˜uk2kr#k2+Ckr#k2k˜ #k2kr˜ #k2 Ckr#k2 2k˜uk2 2+1 16kr˜uk2 2+Ckr#k2 2k˜ #k2 2+1 4kr˜ #k2 2 21
and ZD (˜u·r¯z(✓s!)) ˜ #dx k˜uk4kr¯z(✓s!)k2k˜ #k4 Ck˜uk 1 2 2kr˜uk 1 2 2kr¯z(✓s!)k2kr˜ #k2 Ck˜uk2kr˜uk2kr¯z(✓s!)k2 2+1 8kr˜ #k2 2 Ckr¯z(✓s!)k4 2k˜uk2 2+1 16kr˜uk2 2+1 8kr˜ #k2 2. Then we arrive at 1 2 d ds ✓k˜ #(s, t0;!)k2 2+Z+1 0 µ(r)kr˜⌘s(r)k2 2dr◆+kr˜ #k2 21 2Z+1 0 µ0(r)kr˜⌘s(r)k2 2dr Ckr#k2 2k˜uk2 2+Ckr#k2 2k˜ #k2 2+Ckr¯z(✓s!)k4 2k˜uk2 2+1 8kr˜uk2 2+3 8kr˜ #k2 2.(4.4) We conclude from inequalities (4.3)-(4.4) and the assumption on the memory kernel that d ds ✓k˜u(s, t0;!)k2 2+k˜ #(s, t0;!)k2 2+Z+1 0 µ(r)kr˜⌘s(r)k2 2dr◆ +kr˜uk2 2+kr˜ #k2 2+Z+1 0 µ(r)kr˜⌘t(r)k2 2dr .1+kruk2 2+kr#k2 2+kr¯z(✓s!)k4 2k˜uk2 2+1+kr#k2 2k˜ #k2 2+✏2kru✏k2 2. Thus, we derive from the classical Gronwall inequality that for any tt0, k˜u(t, t0;!)k2 2+k˜ #(t, t0;!)k2 2+k˜⌘tk2 M .✏2Zt t0kru✏(s)k2 2eRt s1+kru(⌧)k2 2+kr#(⌧)k2 2+kr¯z(✓⌧!)k4 2d⌧ds !0 as ✏!0+. Acknowledgement Partial financial support was received from the National Science Foundation of China Grant (11871389), the Fundamental Research Funds for the Central Universities (xzy012022008), Shaanxi Fundamental Science Research Project for Mathematics and Physics (22JSY032) and the Spanish Ministerio de Ciencia e Innovaci´on, Agencia Estatal de Investigaci´on and FEDER Grant PID2021-122991NB-C21. 22
References [1] D. Adhikari, C. S. Cao, H. Shang, J. Wu, X. Xu, and Z. Ye. Global regularity results for the 2D Boussinesq equations with partial dissipation. J. Di↵erential Equations, 260(2):1893–1917, 2016. [2] D. Adhikari, C. S. Cao, and J. H. Wu. The 2D Boussinesq equations with vertical viscosity and vertical di↵usivity. J. Di↵erential Equations, 249(5):1078–1088, 2010. [3] D. Adhikari, C. S. Cao, and J. H. Wu. Global regularity results for the 2D Boussinesq equations with vertical dissipation. J. Di↵erential Equations, 251(6):1637–1655, 2011. [4] L. Arnold, C. Jones, K. Mischaikow, and G. Raugel. Random Dynamical Systems. Springer, Berlin, Germany, 1998. [5] P. W. Bates, K. N. Lu, and B. X. Wang. Random attractors for stochastic reactiondi↵usion equations on unbounded domains. J. Di↵erential Equations, 246(2):845–869, 2009. [6] A. Biswas, C. Foias, and A. Larios. On the attractor for the semi-dissipative Boussinesq equations. Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire, 34(2):381–405, 2017. [7] S. Borini and V. Pata. Uniform attractors for a strongly damped wave equation with linear memory. Asymptot. Anal., 20(3):263–277, 1999. [8] C. S. Cao and J. H. Wu. Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal., 208(3):985–1004, 2013. [9] T. Caraballo and P. E. Kloeden. Non-autonomous attractor for integro-di↵erential evolution equations. Discrete Contin. Dyn. Syst. Ser. S, 2(1):17–36, 2009. [10] T. Caraballo, J. Real, and I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete Contin. Dyn. Syst. Ser. B, 9(3-4):525539, 2008. [11] D. Chae. Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math., 203(2):497–513, 2006. [12] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier. Fluids with anisotropic viscosity. Math. Model. Numer. Anal., 34(2):315–335, 2000. [13] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier. Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations. Oxford Univ. Press, 2006. 23
[14] I. D. Chueshov and M. Scheutzow. Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations. J. Dynam. Di↵erential Equations, 13(2):355–380, 2001. [15] P. Constantin and C. Foias. Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988. [16] H. Crauel, A. Debussche, and F. Flandoli. Random attractors. J. Dynam. Di↵erential Equations, 9:307–341, 1997. [17] H. Crauel and F. Flandoli. Attractors for random dynamical systems. Probab. Theory Related Fields, 100:365–393, 1994. [18] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, 2014. [19] S. Gatti, A. Miranville, V. Pata, and S. Zelik. Attractors for semi-linear equations of viscoelasticity with very low dissipation. Rocky Mountain J. Math., 38:1117–1138, 2008. [20] A. E. Gill. Atmosphere-Ocean Dynamics. Academic Press, London, 1982. [21] C. Giorgi, V. Pata, and A. Marzocchi. Uniform attractors for a non-autonomous semilinear heat equation with memory. Quart. Appl. Math., 58(4):661683, 2000. [22] J. F. He, S. Ma, and C. Y. Sun. Well-posedness and attractors for a 2D Boussinesq system with partial dissipation. J. Di↵erential Equations, 319:1–40, 2022. [23] T. Hmidi, S. Keraani, and F. Rousset. Global well-posedness for a Boussinesq-NavierStokes system with critical dissipation. J. Di↵erential Equations, 249(9):2147–2174, 2010. [24] T. Hmidi, S. Keraani, and F. Rousset. Global well-posedness for Euler-Boussinesq system with critical dissipation. Comm. Partial Di↵erential Equations, 36(3):420–445, 2011. [25] P. E. Kloeden and J. A. Langa. Flattening, squeezing and the existence of random attractors. Proc. A, 463(2077):163–181, 2007. [26] A. Larios, E. Lunasin, and E. S. Titi. Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat di↵usion. J. Di↵erential Equations, 255(9):2636–2654, 2013. [27] F. Li and B. You. Random attractor for the stochastic Cahn-Hilliard-Navier-Stokes system with small additive noise. Stoch. Anal. Appl., 36(3):546–559, 2018. [28] A. J. Majda. Introduction to PDEs and Waves for the Atmosphere and Ocean.vol.9, American Mathematical Soc., 2003. 24
[29] A. J. Majda and M. J. Grote. Model dynamics and vertical collapse in decaying strongly stratified flows. Phys. Fluids, 9(10):2932–2940, 1997. [30] T. Nguyen Duong. Existence and long-time behavior of solutions to the velocityvorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory. Math. Methods Appl. Sci., 45(17):11635–11658, 2022. [31] J. Pedlosky. Geophysical Fluid Dyanmics. Springer, New York, 1987. [32] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, Springer, New York, 1997. [33] J. H. Xu and T. Caraballo. Long time behavior of stochastic nonlocal partial di↵erential equations and Wong-Zakai approximations. SIAM J. Math. Anal., 54:2792–2844, 2002. [34] J. H. Xu, T. Caraballo, and J. Valero. Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local di↵usion. J. Di↵erential Equations, 327(9):418–447, 2022. [35] J. H. Xu, T. Caraballo, and J. Valero. Asymptotic behavior of nonlocal partial di↵erential equations with long time memory. Discrete Contin. Dyn. Syst. Ser. S, 15(10):3059– 3078, 2022. [36] D. D. Yang, Z. Chen, and T. Caraballo. Dynamics of a globally modified Navier-Stokes model with double delay. Z. Angew. Math. Phys., 73(5):216, 2022. [37] S. Yang, T. Caraballo, and Y. Li. Dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay on unbounded domains. Appl. Math. Optim., 89(1):11,43, 2024. [38] B. You. The existence of a random attractor for the three dimensional damped NavierStokes equations with additive noise. Stoch. Anal. Appl., 35(4):691–700, 2017. 25