Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor

We consider a non-autonomous two-dimensional Newton-Boussinesq equation with singularly oscillating external forces depending on a small parameter \begin{document}$ \varepsilon $\end{document} . We prove the existence of the uniform attractor \begin{document}$ A^\varepsilon $\end{document} when the Prandtl number \begin{document}$ P_r>1 $\end{document} . Furthermore, under suitable translation-compactness and divergence type condition assumptions on the external forces, we obtain the uniform (with respect to \begin{document}$ \varepsilon $\end{document} ) boundedness of the related uniform attractors \begin{document}$ A^\varepsilon $\end{document} as well as the convergence of the attractor \begin{document}$ A^\varepsilon $\end{document} to the attractor \begin{document}$ A^0 $\end{document} as \begin{document}$ \varepsilon\rightarrow 0^+ $\end{document} .

1. Introduction. In this paper, we consider the following 2D Newton-Boussinesq equation    ∂ t ξ + u∂ x ξ + v∂ y ξ = ∆ξ − Ra Pr ∂ x θ + f 0 (x, y, t) + ε −ρ f 1 ( x ε , y ε , t), ∆Ψ = ξ, u = Ψ y , v = −Ψ x , ∂ t θ + u∂ x θ + v∂ y θ = 1 Pr ∆θ + g 0 (x, y, t) + ε −ρ g 1 ( x ε , y ε , t), where U = (u, v) is the velocity vector of the fluid, θ is the flow temperature, Ψ is the flow function, ξ is the vortex. (x, y) ∈ Ω, Ω ⊆ R 2 is an open bounded domain with sufficiently smooth boundary ∂Ω. The positive constants P r and R a are the Prandtl number and the Rayleigh number, respectively. The Prandtl number is a dimensionless scalar in fluid mechanics. It reflects the mutual influence of the energy and momentum transfer processes in the fluid, and plays an important role in thermal calculations. In fluid mechanics, the Rayleigh number of a fluid is a dimensionless number related to buoyancy-driven convection. When the Rayleigh number of a certain fluid is lower than the critical value, the main form of heat transfer is heat conduction; when the Rayleigh number exceeds the critical value, the main form of heat transfer is convection.
Notice that system (1) can be rewritten as follows: for every (x, y) ∈ Ω and t > 0, Pr ∂θ ∂x = f 0 (x, y, t) + ε −ρ f 1 ( x ε , y ε , t), Ψ = ξ, ∂θ ∂t − 1 Pr θ + J(Ψ, θ) = g 0 (x, y, t) + ε −ρ g 1 ( x ε , y ε , t), with the boundary conditions ξ| ∂Ω = 0, θ| ∂Ω = 0, Ψ| ∂Ω = 0, and the initial conditions ξ(x, y, τ ) = ξ τ (x, y), θ(x, y, τ ) = θ τ (x, y), where the function J is given by The functions g ε (x, y, t) = g 0 (x, y, t) + ε −ρ g 1 ( x ε , y ε , t), 0 < ε ≤ 1, g 0 (x, y, t), ε = 0, represent the external force terms. It is easy to verify that the function J in (5) satisfies: Ω J(u, v)vdxdy = 0, f or all u ∈ H 1 (Ω), v ∈ H 2 (Ω) ∩ H 1 0 (Ω), J(u, v) ≤ C u H 2 v H 2 , f or all u ∈ H 2 (Ω), v ∈ H 2 (Ω), J(u, v) ≤ C u H 3 ∇v , f or all u ∈ H 3 (Ω), v ∈ H 1 (Ω). (10) Throughout this paper, we frequently use the Poincaré inequality u ≤ λ ∇u , ∀u ∈ H 1 0 (Ω), (11) where λ is a positive constant. Natural convection is becoming an interesting and important topic because it is usually encountered in various natural and industrial processes, such as drying, combustion of gas mixtures, chemical reaction, thermal storage system and so on. Bénard convection is one typical natural convection driven by the buoyancy force in a fluid which is simultaneously heated and cooled from below and above, respectively. The Newton-Boussinesq equation describes the Bénard flow. There are some works concerning problem (1). For example, Guo studied the existence and uniqueness of weak solutions of two-dimensional Newton-Boussinesq equation by spectral method and Galerkin method in [15] and [14], respectively. In [13], Fucci et al. proved the existence of a global attractor in L 2 (Ω) × L 2 (Ω) for the Newton-Boussinesq equation defined in a two-dimensional channel by the uniform estimates on the tails of solutions. In [12], Fang et al. considered a class of periodic initial value problems for two-dimensional Newton-Boussinesq equation. Using iterative method, they obtained the local existence of solution, and proved the global existence of solution by the method of a priori estimates. In [23], Song et al. investigated a class of non-autonomous Newton-Boussinesq equation in twodimensional bounded domains. They proved the existence of pullback attractors in L 2 (Ω) × L 2 (Ω) and H 1 (Ω) × H 1 (Ω). In [24], Song et al. studied the existence of H 1 0 (Ω) × H 1 0 (Ω) and H 2 (Ω) × H 2 (Ω)-global attractors in two-dimensional bounded domains. Meanwhile, they obtained the existence of H 1 0 (Ω) × H 1 0 (Ω)-global attractors in two-dimensional channels. In [20], Qiu et al. considered the two-dimensional Newton-Boussinesq equation with the incompressibility condition. They obtained a regularity criterion for the Newton-Boussinesq equation by virtue of the commutator estimate. In [18], Ma et al. investigated the global well-posedness for the 3D Newton-Boussinesq equations in a large class of non-decaying vorticity. With the help of the Fourier analysis and the coupling structure, they established the global-in-time estimate of vorticity in non-Lipschitz vector field. In [35], Wang et al. studied asymptotic autonomy of the kernel sections for Newton-Boussinesq equtions on unbounded zonary domains. They showed that the forward compactness of the kernel sections for the process is a necessary and sufficient condition such that the kernel sections are attracted by the global attractor for the semigroup. And they obtained nonempty, uniformly bounded and forward compact kernel sections for the non-autonomous equation defined on an unbounded zonary domain and perturbed by longtime convergent forces.
Attractor is an important concept in the study of dynamical systems. There are many works concerning this subject, see, e.g., [2,16,17,21,22,32,34,36,37]. Stability of attractors for a dynamical system with some oscillating external forces is also important in natural phenomenon. Indeed, this issue has been considered by some mathematicians and engineers, see [1,3,4,5,7,8,9,10,11,19,27,28,29,30,31,33]. However, as far as we know, there is no paper dealing with the non-autonomous Newton-Boussinesq equation with rapidly oscillating terms. This paper follows the key ideas of the paper [8] devoted to the non-autonomous 2D Navier-Stokes system with singularly oscillating forces. In [8], Chepyzhov and Vishik proposed the divergence conditions assumption for the first time. Later, Tachim applied this method in [27,28,30,31] to discuss the uniform attractors of the solutions of several types of partial differential equations with singular oscillating external force terms. Motivated by the idea of [8], we will investigate the asymptotic behavior of the non-autonomous Newton-Boussinesq equation depending on the small parameter ε, which reflects the rate of oscillations in the term ε −ρ f 1 ( x ε , y ε , t) and ε −ρ g 1 ( x ε , y ε , t) with amplitude of order ε −ρ . Both terms f 0 (x, y, t), g 0 (x, y, t), f 1 ( x ε , y ε , t), g 1 ( x ε , y ε , t) are supposed to be translation bounded in the space L 2 loc (R; L 2 (Ω)). We aim to prove the stability of the uniform attractors A ε associated to problem (2)-(4) as ε → 0 + in L 2 (Ω) × L 2 (Ω). The main purpose of this paper is to show: (1) the uniform (with respect to ε) boundedness of the family A ε in L 2 (Ω) × L 2 (Ω): (2) the convergence of A ε to A 0 as ε → 0 + in the standard Hausdorff semidistance in L 2 (Ω) × L 2 (Ω): lim ε→0 + dist L 2 (Ω)×L 2 (Ω) (A ε , A 0 ) = 0. The outline of this paper is as follows: In the following section, we introduce some preliminary knowledge that will be used in this paper. In section 3, we derive some a priori estimates and prove the existence of uniform attractors in L 2 (Ω) × L 2 (Ω).
In Section 4, we study the boundedness of the uniform attractors when the external forces satisfy a divergence type condition. In Section 5, we investigate the deviation of a solution of the singular system from the solution of the "limiting" system. In Section 6, we investigate the structure of the uniform attractor of the Newton-Boussinesq equation. The last section is devoted to prove the main result of this article, which is the convergence of the uniform attractors of the oscillating Newton-Boussinesq equation to the attractor of the "limiting" system.

2.
Preliminaries. In this section, we introduce some notations and preliminaries, which will be used throughout this paper.
First, we recall some basic concepts and results of uniform attractor theory which can be found in [25,26].
The Hausdorff semidistance in X from one set B 1 to another set B 2 is defined as is the generic Lebesgue space and H s (Ω) is the usual Sobolev space. We denote by · , (·, ·) the norm and scalar product in L 2 (Ω), respectively. We also denote by C a generic constant, which is different from line to line or even in a same line.
where I is the identity operator.
Definition 2.6. The set where B is a bounded subset of E is said to be the uniform (w.r.t. σ ∈ Σ) ω-limit set of B.
In the non-autonomous system, we usual choose Σ = H(σ 0 ) as symbol space of the system, H(σ 0 ) = [{σ 0 (· + h)|h ∈ R}] X for every fixed σ 0 ∈ X, where [ ] denotes the closure of a set in a topological space X. If H(σ 0 ) is compact in X then we call σ 0 ∈ X is translation compact (tr.c.). The translation semigroup {T (r)|r ≥ 0} satisfies (12), (13), that is, In order to clarify the assumptions on the external forces f ε , g ε , we introduce the following notations. Given a Banach space X, we denote by L 2 loc (R; X) the metrizable space of function ϕ(s), s ∈ R with value in X that are locally 2-power integrable in the Bochner sense. It is equipped with the local 2-power mean convergence topology. We will also denote by Hereafter, we assume that We now assume that the function f 1 (·, t) ∈ Z, g 1 (·, t) ∈ Z for almost every t ∈ R, and has finite norms in the space Now, let us recall the known Gagliardo-Nirenberg inequality as follows.
Lemma 2.8. Let Ω = R n or Ω ⊂ R n be a bounded domain with smooth boundary ∂Ω, and u ∈ L q (Ω), D m u ∈ L r (Ω), 1 ≤ q, r ≤ ∞. Then, there exists a constant c, such that, c depends only on (n, m, j, a, q, r).
3. Some a priori estimates and the uniform attractor. In this section, we first derive some a priori estimates on the solutions to (2)-(4). We then use these estimates to construct the bounded uniformly (with respect to τ ∈ R) absorbing sets in L 2 (Ω) × L 2 (Ω) and H 1 0 (Ω) × H 1 0 (Ω). We will need the following lemma, whose proof is given in [8]. Then, We first estimate the boundedness of the solution (ξ(t), θ(t)) in L 2 (Ω) × L 2 (Ω).
Taking the inner product of (2) 3 with θ in L 2 (Ω) and using (8), the Hölder inequality and the Young inequality gives 1 2 Using (11), we have d dt Taking the inner product of (2) 1 with ξ in L 2 (Ω) and using (8) we obtain 1 2 We notice that the following inequalities hold and By (24), (25), using the Poincaré inequality, it follows from (23) that Applying the Gronwall inequality to (26), we have it follows from (27) that Integrating inequality (22) on [τ, t] we have Integrating inequality (26) on [τ, t], and by (19) we get Proposition 1 is proved.
where the constant C is independent of ε.
Proof. It is similar to the proof of Corollary 2.5 in [31].
Now from (17) and (18) we derive where the constant C 0 depends on λ, P r , R a and the norms f 0 L 2 b (R;L 2 (Ω)) and g 0 L 2 b (R;L 2 (Ω)) , C 1 depends on λ, P r , R a and the norms f 1 L 2 b (R;Z) and g 1 L 2 b (R;Z) . And where the constant C 2 depends on P r , λ and the norm g 0 L 2 b (R;L 2 (Ω)) , C 3 depends on P r , λ and the norm g 1 L 2 b (R;Z) . Now we consider the process corresponding to the problem (2)-(4). More precisely, the mapping U (f ε ,g ε ) (t, τ ) : is the solution to the system (2)-(4).

Proposition 2.
The assumptions are the same as in Proposition 1. Then the solution (ξ(t), θ(t)) satisfying Proof. Taking the inner product of (2) 3 with −(t − τ )∆θ in L 2 (Ω), we have Note that the first term on the right-hand side of (37) is given by We now estimate the first term on the right-hand side of (38). Using the Hölder inequality, the Gagliardo-Nirenberg inequality and the Poincaré inequality, we have Similarly for the second term on the right-hand side of (38) we have Note that (41) Combining (38), (39), (40), (41) with (37), and multiplying the resulting inequality by 2, we get Taking the inner product of (2) 1 with −(t − τ )∆ξ in L 2 (Ω), we have We notice that the first term on the right-hand side of (43) can be written as Similar with (39) and (40), we have and Note that the last two terms on the right-hand side of (43) are bounded by Combining (44), (45), (46), (47) with (43), and multiplying the resulting inequality by 2, we have d dt Adding (42) to (48), we get Then from (49)-(50) we have dψ dt which yields (using the Gronwall lemma) for all t ≥ τ , is a positive increasing function of R 1 ,R 2 ,R 3 ,R 4 and R 5 . It follows from (52)-(54) that Based on Proposition 2, we have the following result.
We now consider the "limiting" system where f 0 , g 0 ∈ L 2 b (R; L 2 (Ω)). We can check that and It follows that where C 4 depends on P r , λ and g 0 L 2 b , C 5 depends on P r , R a , λ, f 0 L 2 b and g 0 L 2 b . This proves that the process {U (f0,g0) (t, τ )} has the uniform (with respect to τ ∈ R) absorbing set and the set B 0,0 is bounded in L 2 (Ω) × L 2 (Ω). Similar to the proof of Proposition 2, we can get . According to (63), when t is large enough, we have is a positive increasing function of R 1 , R 2 , R 3 and R 4 . Thus, there exists an absorbing set B 1,0 in H 1 0 (Ω) × H 1 0 (Ω), i.e., where C 6 , C 7 depend on C 4 , C 5 , f 0 L 2 b and g 0 L 2 b . Therefore the process {U (f0,g0) (t, τ )} is uniform compact. In particular, the process has a compact uniform attractor A 0 such that

Divergence type condition and boundedness of
We assume that f 1 (x, y, t) and g 1 (x, y, t) satisfy the following condition referred to hereafter (see [8]) as the divergence type condition.
Proof. For θ, we have Noting and using the boundary condition on θ, we have Combining (68)-(70), we have By assumptions, we have t+1 t g 0 (s) 2 ds ≤ g 0 It follows from Lemma 3.1 that Note that and Similar with (70), we have (78) Combining (75)-(78) we have By assumptions, we have where C > 0 is independent of ε.
Furthermore, we can easily deduce that So, from (100), (101) and (102), we have where the constants C and r are independent of ε.