The attractors for the regularized Bénard problem with fractional Laplacian

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Abstract

In this article, we will study the existence of the global and exponential attractors for the regularized Bénard equations with fractional Laplacian in the three-dimensional case. This system depends on three parameters β, γ and δ, which affect the regularity of the solution.

Introduction

In this article, we will study the long term dynamical behavior of the regularized Bénard problem with fractional Laplacian for autonomous case on the three-dimensional periodic domain Ω=[0,2π]3. At first, we give the following Bénard problem with fractional laplacian.{ut+u·u+μ(Δ)βu+p+ξθ=f,xΩ,t0,θt+u·θ+κ(Δ)γθ=g,xΩ,t0,·u=0,xΩ,t0,u(x,0)=u0(x),θ(x,0)=θ0(x),xΩ,where the vector field u=u(x,t)=(u1(x,t),u2(x,t),u3(x,t)) is velocity of the fluid, θ=θ(x,t) is a scalar function, μ > 0 represents the fluid viscosity, κ > 0 is the diffusivity, ξ ∈ R3 is a vector, p=p(x,t) represents the pressure, 0 < β, γ < 1 and f=f(x),g=g(x) are given. Notice that the system (1.1) is the Navier-Stokes system when θ=0,g=0,β=1. When Ω ⊂ R2, the term ξθ is on the right hand side and ξ=e2,f=0, the above system is the two-dimensional Boussinesq equations with fractional Laplacian. We write Λ=(Δ)1/2, and we haveΛf^(k)=|k|f^(k),where the Fourier transform f^ of a tempered distribution f(x) on the periodic domain Ω is defined asf^(k)=1(2π)3Ωf(x)eik·xdx,and |k|=k12+k22+k32. We define the fractional Laplacian Λsf for s ∈ R by its Fourier seriesΛsf:=kZ3|k|sf^(k)eik·x.For any tempered distribution f on and s ∈ R, we define the normfHs2=ΛsfL22=kZ3|k|2s|f^(k)|2.

The asymptotic behavior of solutions for the two-dimensional incompressible Boussinesq problem was studied by many authors, such as, Huang and Huo in [13] not only proved the global well-posedness of strong solutions, but also given a proof of the existence of the global attractor for the 2D Boussinesq system in periodic domain Ω=[0,2π]2 with two parameters α,β(12,1). Based on [13], the finite dimensions and determining modes of the global attractor for 2D Boussinesq equations with fractional Laplacian were studied in detail by them in [14]. The second equation of (1.1) is called Quasi-Geostrophic equation and the existence of the global attractor for the dissipative Quasi-Geostrophic equations for 2D case was proved by Ning Ju in [16].

The Bénard system consists of Navier-Stokes system for the velocity field and convection diffusion equation for the temperature, many authors studied Bénard system and its related problems. For instance, the authors in [11] take advantage of a maximum principle for the temperature equation to prove the existence of the global attractor for the Bénard problem under several different boundary conditions. In [9], the three authors studied the existence of global solution and the existence and finite dimensionality of the global attractor with non-homogeneous boundary condition, non-homogeneous volume forces and heat source. The regularized Bénard problem was investigated in detail by Kaya and Okay Celebi in [17], and the correspondingly original system is as follows:{ut+u·uμΔu+p+ξθ=f,θt+u·θκΔθ=g,·u=0.

The authors used the averaging technique to (1.2), applying the operator Iα2Δ, and then they obtained the following correspondingly regularized system{utα2ΔutμΔ(uα2Δu)+u·u+p+ξ(θα2Δθ)=f,θtα2ΔθtκΔ(θα2Δθ)+u·θ=g.

They proved the global well-posedness by using Galerkin approximation, the existence of global attractor, and obtained the fractal dimension of the global attractor is finite.

The authors in [2] studied the Rayleigh-Bénard problem with a large aspect ratio. Not only the existence and regularity for the global strong solution under the conditions of the smallness of the forcing but also the existence of the strong attractor was proved by Birnir and Svanstedt in [2]. In [22], the authors studied the global well-posedness and regularity of solutions for a family of incompressible 3D Navier-Stokes-alpha-like models that employ fractional Laplacian operators, they made a very detailed study for the value range of two parameters θ1 and θ2.

It is well known that the averaging technique successfully used for Navier-Stokes equations [1], [18], [25], one can refer to [3], [17], [20] for more details. So we would like to use the same method to the system (1.1) in this paper. Using averaging technique to the system (1.1), we obtain{wt+·(ww¯)+μ(Δ)βw+q+ξη=f¯,ηt+·(wη¯)+κ(Δ)γη=g¯,·w=0,w(x,0)=u¯0(x),η(x,0)=θ¯0(x),where (w, η) and q denote the approximation to (u¯,θ¯) and p¯, respectively.

Similar to the alpha models given in [5], [6], [15], the authors in [20] used the filtering kernel (Iα2Δ)1 which is associated with the Helmholtz operator. Thus, motivated by [17], [20], [22] and the application of filtering kernel (Iα2Δ)1 in studying the 3D Navier-Stokes-α turbulence, such as the viscous Camassa-Holm equations and the simplified Bardina model (see for instance [5], [6], [7], [8]), we apply the operator I+α2(Δ)δ to the system (1.4), and set u=w,θ=η,v=(I+α2(Δ)δ)u=(I+α2(Δ)δ)w,ϕ=(I+α2(Δ)δ)θ=(I+α2(Δ)δ)η,p=(I+α2(Δ)δ)q. Thus, we get the following system{vt+u·u+μ(Δ)βv+p+ξϕ=f,ϕt+u·θ+κ(Δ)γϕ=g,·u=·v=0,u(x,0)=u0,θ(x,0)=θ0,v=u+α2(Δ)δu,ϕ=θ+α2(Δ)δθ,where α > 0 is a length scale. In this way, we have converted the system (1.4) in the form (1.5). In this paper, we consider the model (1.5) which corresponds to the following system{ut+α2(Δ)δut+u·u+μ(Δ)β(u+α2(Δ)δu)+p+ξθ+ξα2(Δ)δθ=f,θt+α2(Δ)δθt+u·θ+κ(Δ)γ(θ+α2(Δ)δθ)=g.

Based on [13], [17], we study the attractors of the system (1.6). Since the term ξα2(Δ)δθ is in the equation about the velocity u, the difficulty of this article is to deal with the couple term of (1.6) between the velocity u and the temperature θ. In order to over come this difficulty, we take appropriate norm, using commutation estimate and energy estimate. Now, we state the main results of this article as the following theorems:

Theorem 1.1

(Existence of a global attractor) Let δmax{3p,1β+3q,1γ+3q,12+32q,2β,2γ} and p, q ∈ [2, ∞], 1p+1q=12. Assume that f, g ∈ L2(Ω), then the solution operator {S(t)}t ≥ 0 of the system (1.6): S(t)(u0,θ0)=(u(t),θ(t)) defines a semigroup in the space Vδ × Vδ for all t > 0. Moreover, the following statements valid:

  • (1)

    for any (u0, θ0) ∈ Vδ × Vδ, tS(t)(u0, θ0) is a continuous function from R+ into Vδ × Vδ;

  • (2)

    for any fixed t > 0, S(t) is a continuous and compact map in Vδ × Vδ;

  • (3)

    {S(t)}t ≥ 0 possesses a global attractor A in the space Vδ × Vδ. The global attractor A is compact and connected in Vδ × Vδ and is the maximal bounded absorbing set and the minimal invariant set in Vδ × Vδ in the sense of the set inclusion relation. A attracts all bounded sets in Vδ × Vδ, i.e. for any B ∈ Vδ × Vδ,limt+dist(S(t)B,A)=0,where dist(A, B) is the Hausdorff semidistance between the two sets A and B, one can refer to Definition 6.1 or [21, 23].

The 3D Bénard equations and their fractional generalizations have attracted considerable attention due to their physical applications and mathematical significance, see [22]. In particular, when β=1,γ=1,δ=1, it becomes the system (1.3) which was considered in [17]. Thus we restrict 0 < β, γ < 1, and the difference from Kaya and Celebi [17] is that we take the operator I+α2(Δ)δ instead of Iα2Δ, where δmax{3p,1β+3q,1γ+3q,12+32q,2β,2γ} and p,q[2,],1p+1q=12.

We know that the global attractor may not be rich enough to describe the nature of the system, since A is strictly invariant, A can easily miss some important trajectories which correspond to a specific transition of states like pattern formations. As a more robust attractor, Eden-Foias-Nicolaenko-Temam in 1994 have founded the new concept of exponential attractors (sometimes called inertial sets) to overcome this difficulty. An attractor E containing the global attractor is called an exponential attractor if E is a compact set of Banach space X with finite fractal dimension and if E attracts all the trajectories at exponential rates. The physical meaning of this bound on the fractal dimension of the exponential attractor is that the long-time behavior of solutions to the regularized Bénard equations can be fully described by a finite number of independent degrees of freedom. To this end, we will prove the existence of exponential attractors.

Theorem 1.2

(Existence of an exponential attractor) Let δmax{3p,1β+3q,1γ+3q,12+32q,2β,2γ} and p, q ∈ [2, ∞], 1p+1q=12. The dynamical system S(t) generated by the system (1.6) on Vδ × Vδ possesses an exponential attractor E contained and bounded in V2δ × V2δ. Precisely,

  • (1)

    E is positively invariant for S(t), that is, S(t)EE for every t ≥ 0;

  • (2)

    E has finite fractal dimension in Vδ × Vδ;

  • (3)

    there exist an increasing function Q: [0, ∞) → [0, ∞) and k > 0 such that for any bounded set B ⊂ Vδ × Vδ there holdsdistVδ×Vδ(S(t)B,E)Q(BVδ×Vδ)ekt.

To be organized, this article is divided into five parts, and the roadmap is as follows. In Section 2, we introduce some standard notations and recall some basic inequalities that will be used frequently in this paper. And then we prove the continuous dependence of solutions for (1.6) with respect to the initial data. In Section 3, we give a priori estimate for (u, θ), proving the existence of the bounded absorbing set in Vδ × Vδ for the system (1.6). In order to exploit the boundedness of uV2δ and θV2δ in Section 5 and also to prove the existence of the exponential attractor, we show that the solution (u, θ) of the system (1.6) has higher regularity in Section 4. A method of semigroup decomposition is presented in Section 5, which is used to prove the existence of the global attractor. In Section 6, at first, for the convenience of the readers, we recall some abstract results with respect to exponential attractor, and then we take advantage of the semigroup decomposition again to prove the existence of exponential attractor.

Section snippets

Preliminaries

In this section, we recall some notations about some function spaces and the classical notations which used throughout this article. For convenience, here and throughout this article, we will not distinguish the notations and norms for the vector and scalar function spaces.

(1) In this paper, C and Ci(i=1,2,) represent some positive constants, and the value of C is various in different line or even in the same line, so is Ci. C(l, m, n) stands for the dependence of the constant on l, m and n.

The bounded absorbing set in Vδ × Vδ

We will give a proof of the existence of the bounded absorbing set in Vδ × Vδ which is a prerequisite for the existence of global attractor for system (1.6) in this section. At first, we recall the basic definition of the global attractor and the theorem of the existence for a global attractor.

Definition 3.1

Let {S(t)}t ≥ 0 is a semigroup which defined on the Banach space X, a subset AX is called a global attractor of semigroup {S(t)}t ≥ 0, if A is compact in X and has the following properties:

  • (a)

    A is

Higher regularity

In this section, our goal is to show that the solution of system (1.6) has higher regularity to prepare for the existence of the exponential attractor in Section 6. To this end, we will prove that u(t) and θ(t) are bounded in the space V2δ.

We start with giving the V2δ a priori estimate for θ. Multiplying (1.6)2 by Λ2δθ in L2 and integrating by parts, it yields12d(ΛδθL22+α2Λ2δθL22)dt+κΛδ+γθL22+κα2Λ2δ+γθL22=g,Λ2δθu·θ,Λ2δθ.The estimates of the two terms on the right hand side for the

The semigroup decomposition

In order to obtain the existence of the global attractor, we need to prove the asymptotic compactness of the solution semigroup. To this end, we take advantage of the method of the semigroup decomposition.

We split the solution of (1.6) asS(t)(u0,θ0)=(L(t)u0+K(t)u0,J(t)θ0+W(t)θ0),where (v1,w1)=(L(t)u0,J(t)θ0) and (v2,w2)=(K(t)u0,W(t)θ0), respectively, solve{t(v1+α2(Δ)δv1)+u·v1+μ(Δ)β(v1+α2(Δ)δv1)+p=0,t(w1+α2(Δ)δw1)+u·w1+κ(Δ)γ(w1+α2(Δ)δw1)=0,(v1(0),w1(0))=(u0,θ0),and{t(v2+α2(Δ)δv2)+u·

The exponential attractor

We will prove the existence of the exponential attractor in this section. At first, we give the definitions of Hausdorff semidistance and fractal dimension.

Definition 6.1

[21] Given a metric space (X, dX), then the Hausdorff semidistance distX(B, C) between two sets B, C ⊂ X is given bydistX(B,C)=supbBinfcCdX(b,c).

Definition 6.2

[21] X is a metric space and K is a compact set, then the fractal dimension of K in X is defined bydimXK=lim supε0logN(ε,K)log(1ε),where N(ε, K) is the smallest numbers of balls of radius ε

Acknowledgments

The authors express their sincere thanks to the referee for his/her valuable comments and suggestions.

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    The first author is supported by “the Fundamental Research Funds for the Central Universities,” NO.NS2020058 and was supported by the National Natural Science Foundation of China under Grant 11501289.

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