Existence, degenerate regularity and limit behavior of trajectory statistical solution for the 3D incompressible micropolar fluids flows with damping term

Existence, degenerate regularity and limit behavior of trajectory statistical solution for the 3D incompressible micropolar fluids flows with damping term∗ Caidi Zhaoa† ,Binbin Miaoa‡ ,Tom´as Caraballob§ aDepartment of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P. R. China bDepartmento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain April 26, 2023 Abstract This article studies the existence, degenerate regularity and limit behavior of the trajectory statistical solution for a three-dimensional incompressible micropolar fluids flows with a damping term. The authors first prove the existence of the trajectory attractor and use it to construct the trajectory statistical solution. Then they establish that the trajectory statistical solution possesses partial regularity provided that the associated Grashof number is small enough. Finally, they investigate the limiting behavior of the trajectory statistical solution as the damping term vanishes. Keywords: Micropolar fluids flows; Trajectory statistical solution; Trajectory attractor; Degenerate regularity; Grashof number. 1 Introduction We study the following equations ut−(ν+κ)∆u+ (u· ∇)u+σ|u|β−1u+∇p= 2κ∇ × ω+f, (1.1) ∇ · u= 0,(1.2) ωt−γ∆ω+ (u· ∇)ω−η∇divω+ 4κω = 2κ∇ × u+g, (1.3) ∗Supported by NSF of China with No.11971356, 11271290, by NSF of Zhejiang Province with No.LY17A010011 and Ministerio de Ciencia e Innovaci´on (Spain) with No. PID2021-122991NB-C21. †Corresponding author E-mail: zhao[email protected] or zhao[email protected] ‡E-mail:bin[email protected] §E-mail: [email protected] 1 with the initial and boundary conditions u(x, t)t=0 =u0, ω(x, t)t=0 =ω0,(1.4) u(x, t)∂Ω×(0,∞)= 0, ω(x, t)∂Ω×(0,∞)= 0,(1.5) where (x, t)∈Ω×R+, and Ω is a bounded domain in R3with smooth boundary ∂Ω. In equations (1.1)-(1.3), the unknown functions u= (u1(x, t), u2(x, t), u3(x, t)), ω = (ω1(x, t), ω2(x, t), ω3(x, t)), p =p(x, t), denote, respectively, the velocity vector, the angular velocity of rotation of particles, and the pressure of the fluid, the functions f= (f1(x), f2(x), f3(x)) and g= (g1(x), g2(x), g3(x)) denote the external force and the angular momentum, respectively, and the positive constant σis the damping coefficient. In addition, the parameters ν, γ,κ,ηare positive constants. Equations (1.1)-(1.3) describe the motion of micropolar fluids [24], which were first introduced to describe the micro-rotational motion and rotational inertia of fluids. In physics, micropolar fluids can represent fluids composed of rigid, randomly oriented (or spherical particles) suspended in a viscous medium. The theory of micropolar fluids was first introduced by Eringen in [13]. If κ=σ= 0, ω=g= (0,0,0) in (1.1) and (1.3), then equations (1.1)-(1.3) turn to be the classical incompressible Navier-Stokes equations. There are some references studying the incompressible micropolar fluids. For the case that there is no damping term, the global well-posedness and the existence of uniform attractor of two-dimensional (2D) micropolar fluids equations was proved in [7, 12]; initial and boundary-value problem for 2D micropolar equations with only angular velocity dissipation was investigated in [22]; the regular criteria of weak solutions for the 3D case was investigated in [10,11,20]; the existence and homogenization of trajectory statistical solution for the 3D case was established in [34, 42]. When equation (1.1) contains a damping term, the global existence of strong solution to the 3D case was verified in [35] for β= 3 and 4σ(ν+κ)>1 or β > 3, and the existence of global attractor was proved in [33] with the assumption β∈(3,5). The original motivation of this article is to investigate the existence of the trajectory statistical solution and its limiting behavior for equations (1.1)-(1.3) when β∈(1,3]. We are interested in the probability distribution of solutions within the temporal-spatial space. The invariant measure and statistical solution for evolution equations have been extensively studied. One can refer, for deterministic equations, to [4–6, 14–19, 23, 25–27, 31, 32, 38, 39, 43, 46, 49, 50] the well-posed systems and to [2, 21, 40, 42, 44, 45] the ill-posed systems, and to [48] the invariant sample measure and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations. For the impulsive differential equations, we can consult [28, 29, 47] for the 2 existence of the statistical solutions. Especially, Zhao, Li and Caraballo [41] established, via the approach of trajectory attractor, some sufficient conditions ensuring the existence of trajectory statistical solution for general evolution equations, including those systems which possess global weak solutions but without a known result of global uniqueness. Recently, the abstract result of [41] was applied to equations (1.1)-(1.3) without damping term to construct the trajectory statistical solution [42]. The first result of this article is the existence of trajectory statistical solution for the 3D micropolar fluids with a damping term. We assume that β∈(1,3]. In this case we can obtain the existence of a weak solution to problem (1.1)-(1.5) corresponding to each initial value. However, it is not known whether the weak solution is unique or not. Therefore, here we cannot use the approach of classical semigroup [33] to investigate the asymptotic behavior of the solution for equations (1.1)-(1.3) because of the possible non-uniqueness of the weak solution. We will first prove the existence of the trajectory attractor Atr σfor equations (1.1)-(1.3) via the natural translation semigroup. Then we use the abstract results [41, Theorem2.1] to obtain the existence of the trajectory statistical solution µσ,wσ, hereinafter wσis an element within Atr σand σis the constant from equation (1.1). The second goal of this article is to investigate the partial degenerate regularity of the trajectory statistical solution µσ,wσ. The regularity of the trajectory statistical solution means that it is supported by a set in which all weak solutions are strong solutions. When constructing µσ,wσ, we observe that it is supported by the trajectory attractor Atr σ. Thus, we naturally consider the degenerate regularity of the trajectory statistical solution via investigating the degenerate regularity of the trajectory attractor. Notice that µσ,wσis a probability measure defined on the trajectory space of equations (1.1)- (1.3) and Atr σitself consists of weak, bounded and complete trajectories of equations (1.1)-(1.3). According to these facts, we use the following form of generalized Grashof number G:= [λ+ 1 δ(λ νkfk2+λ γkgk2)]1/2,(1.6) to discuss the partial degenerate regularity of µσ,wσ. We will prove that if the Grashof number Ggiven by (1.6) is properly small, then the trajectory attractor Atr σdegenerates to a single complete trajectory, that is Atr σ={˜w(t)∈ Kσ:t⩾0}, where ˜w(t) is the unique complete and bounded weak solution of equations (1.1)-(1.3) and Kσis the kernel of equations (1.1)-(1.3). Further, we will prove that µσ, ˜wσpossesses partial degenerate regularity of the Lusin type in the following sense: ∀ > 0, there is a subset R+()⊂R+ with Lebesgue measure mes(R+()< , such that µσ, ˜wσis regular on R+\R+(). The third result of this article is to prove that the trajectory statistical solution µσ,wσconverges to µ0,w as σ→0+, where µ0,w is the trajectory statistical solution of 3 the following classical micropolar fluids flows equations ut−(ν+κ)∆u+ (u· ∇)u+∇p= 2κ∇ × ω+f, (1.7) ∇ · u= 0,(1.8) ωt−γ∆ω+ (u· ∇)ω−η∇divω+ 4κω = 2κ∇ × u+g, (1.9) which have been constructed in [42]. To this end, we prove that the solution of equations (1.1)-(1.3) converges to that of equations (1.7)-(1.9), and that Atr σconverges to Atr 0as σ→0+, where Atr 0is the trajectory attractor of equations (1.7)-(1.9). In [3], Bronzi and Rosa proved the convergence of statistical solutions of the 3D Navier-Stokes-α model as αvanishes. The main tools used in [3] are topological analysis and measure theory. Here we investigate the convergence of trajectory statistical solutions of the 3D incompressible micropolar fluids flows with damping term as the damping term vanishes, via the approach of trajectory attractor. Our result reveals that the trajectory statistical information obtained from the 3D incompressible micropolar fluids flows with damping term are good approximations of the trajectory statistical information of the classical micropolar fluids flows. The article is organized as follows. In the next section we prove the existence of the trajectory attractor and trajectory statistical solution. In Section 3 we prove the degenerate regularity of Lusin type for trajectory statistical solution. Section 4 is devoted to the convergence of the trajectory statistical solution µσ,wσto µ0,w as σ→0+. 2 Existence of trajectory attractor and trajectory statistical solution In this section we first introduce the mathematical setting for problem (1.1)- (1.5) and then prove the existence of the trajectory attractor and trajectory statistical solution. As usual, Lp(Ω) = (Lp(Ω))3and Wm,p(Ω) = (Wm,p(Ω))3stand for the 3D vector Lebesgue space and Sobolev space with norms k·kpand k·km,p, respectively. We denote by Wm,p 0(Ω) the closure of {ϕ:ϕ= (ϕ1, ϕ2, ϕ3)∈(C∞ 0(Ω))3}in Wm,p(Ω) with norm k·km,p. We write Wm,p 0(Ω) = Hm 0,Wm(Ω) = Hmand k·kp=k·kfor p= 2. We also use the following spaces: V={ϕ= (ϕ1, ϕ2, ϕ3)∈(C∞ 0(Ω))3:∇ · ϕ= 0}, H= the closure of Vin L2(Ω) with norm k·kH=k·k and inner product (·,·), V= the closure of Vin H1(Ω) with norm k·kV=k·k1,2, b H=H×L2(Ω) with the inner product (·,·) and norm k·kb H=k·kdefined as (Φ,Ψ) = (ϕ, ψ)+(φ, ξ),Φ=(ϕ, φ),Ψ=(ψ, ξ)∈b H, kΦk= (kϕk2+kφk2)1/2,Φ=(ϕ, φ)∈b H, 4 b V=V×H1 0(Ω) with the norm k·kˆ Vdefined as kΦkb V= (kϕk2 V+kφk2 1,2)1/2,Φ=(ϕ, φ)∈b V . In addition, we use H∗,V∗,b H∗=H∗×L2(Ω) and b V∗=V∗×H−1(Ω) to denote the dual spaces of H,V,b Hand b V, respectively, where L2(Ω)∗=L2(Ω) and H−1(Ω) is the dual space of H1 0(Ω). Then we have V ,→H=H∗,→V∗,b V ,→b H=b H∗,→b V∗and all the embedding is compact. Note that we have used the same notation (·,·) to denote the inner product in the spaces L2(Ω), Hand b H. We will also use the same notation h·,·i to denote the dual pairing between the spaces Vand V∗,b Vand b V∗,Hm 0(Ω) and H−1(Ω) provided that there is no confusion. Define the strong and weak distance in b Hby ds(w, Φ) = kw−Φk,dw(w, Φ) = X j∈Z3 1 2|j| |wj−Φj| 1 + |wj−Φj|, w, Φ∈b H, where wjand Φjare Fourier coefficients of wand Φ, respectively. We next denote by (b H, d•) (•= s or w) the strong or weak metric in b H. Let C([a, b]; b H•) be the space of d•continuous b H-valued functions on [a, b] endowed with the metric dC([a,b]; b H•)(w, Φ) = sup t∈[a,b] d•(w(t),Φ(t)). We also denote by C([a, ∞); b H•) the space of d•continuous b H-valued functions on [a, ∞) endowed with the metric dC([a,∞); b H•)(w, Φ) = X j∈N 1 2j sup{d•(w(t),Φ(t)) : a⩽t⩽a+j} 1 + sup{d•(w(t),Φ(t)) : a⩽t⩽a+j}.(2.1) Next, we introduce some operators. We first define the linear operators A1:V→V∗ and A2:H1 0(Ω) →H−1(Ω) as hA1u, φi= (ν+κ)(∇u, ∇φ),∀u, φ ∈V, hA2ω, ϕi=γ(∇ω, ∇ϕ),∀ω, ϕ ∈H1 0(Ω). It is not difficult to check that A1= (ν+κ)P∆ and A2=γ∆ both with Dirichlet boundary condition, hereinafter Pis the Leray projector from L2(Ω) into H. We can check that D(A1) = V∩H2 0(Ω) and D(A2) = H1 0(Ω) ∩H2 0(Ω). Secondly, we define the following trilinear forms b1(·,·,·) and b2(·,·,·) as b1(u, v, φ) = 3 X j,k=1 ZΩ uj ∂vk ∂xj φkdx, ∀u, v, φ ∈V, b2(u, ω, ϕ) = 3 X j,k=1 ZΩ uj ∂ωk ∂xj ϕdx, ∀u∈V, ω, ϕ ∈H1 0(Ω). 5 We can check that the trilinear forms b1(·,·,·) and b2(·,·,·) are continuous on V×V×V and V×H1 0(Ω) ×H1 0(Ω), respectively. By some simple computations, we have b1(u, v, φ) = −b1(u, φ, v), b1(u, v, v)=0,∀u, v, φ ∈V, b2(u, ω, ϕ) = −b2(u, ϕ, ω), b2(u, ω, ω)=0,∀u∈V, ω, ϕ ∈H1 0(Ω).(2.2) For every u, v ∈V, the operator B1(u, v) : V×V7→ V∗defined as hB1(u, v), φi=b1(u, v, φ),∀φ∈V, is continuous. Similarly, for every u∈Vand ω∈H1 0(Ω), the operator B2(u, ω) : V×H1 0(Ω) 7→ H−1(Ω) defined via hB2(u, ω), ϕi=b2(u, ω, ϕ),∀ϕ∈H1 0(Ω), is continuous. Using above operators, equations (1.1)-(1.3) can be written as      du dt+A1u+B1(u, u) + σ|u|β−1u= 2κ∇ × ω+Pf, dω dt+A2ω+B2(u, ω)−η∇divω+ 4κω = 2κ∇ × u+g. (2.3) Further, we set for w= (u, ω)∈b Vthat        Aw =A1u, A2ω, Bw =B1(u, u), B2(u, ω), Nσw=σ|u|β−1u, 0, Lw =−2κ∇ × ω, 4κω −2κ∇ × u−η∇divω. (2.4) Using the above notations and setting F= (P f, g), we can write the weak form of problem (1.1)-(1.5) as dw(t) dt+Aw(t) + Bw(t) + Nσw(t) + Lw(t) = F, in D0((0,∞),b V∗),(2.5) w(0) = w0= (u0, ω0).(2.6) To handle with the nonlinear term σ|u|β−1u, we denote by ˆ Lβ+1(Ω) = (u, 0)∈R3×R3:ZΩ |u|β+1dx < +∞. Definition 2.1. A weak solution of problem (2.5)-(2.6) on [0,∞)is an b H-valued function w(t) = (u(t), ω(t)) defined on [0,∞)with w(0) = w0= (u0, ω0), such that          dw dt∈L4/3 loc ([0,∞); ˆ V), w(·)∈C([0,∞); b Hw)∩L2 loc([0,∞); b V)∩Lβ+1([0,∞); ˆ Lβ+1(Ω)) (dw dt,Φ) + hAw, Φi+Bw, Φ+hNσw, Φi+hLw, Φi= (F, Φ),∀Φ∈b V , 6 hold in the distribution sense D0(0,∞), and w(t)satisfies the following energy inequality 1 2kw(t)k2+Zt 0 hAw(s), w(s)ids+Zt 0Nσw(s), w(s)ds+Zt 0 hLw(s), w(s)ids ⩽1 2kw0k2+Zt 0F, w(s)ds in the sense that −1 2Zt 0 kw(s)k2ζ0(s)ds+Zt 0 hAw(s), w(s)iζ(s)ds +Zt 0 hNσw(s), w(s)iζ(s)ds+Zt 0 hLw(s), w(s)iζ(s)ds ⩽Zt 0F, w(s)ζ(s)ds, ∀ζ(·)∈C∞ 0[0, t]with ζ(·)⩾0,∀t⩾0. (2.7) If there exists some interval I⊂R+= [0,∞)such that w(t)∈L∞(I;b V)for the weak solution w(t), then we call that the weak solution w(t)possesses “partial” regularity. Lemma 2.1. Suppose that F∈b H. Then for each w0= (u0, ω0)∈b H, problem (2.5)- (2.6) corresponds at least one weak solution w(t) = w(t;w0). Moreover, there exists a time t∗⩾0such that w(t)∈X,nw∈b H:kwk⩽Ro,∀t⩾t∗,(2.8) where R=2λ δ[λ νkfk2+λ γkgk2]1/2,δ= min{ν+κ, γ}and λis a positive constant depending only on Ω. Proof. Let w0= (u0, ω0)∈b Hbe given. For the existence of a global weak solution w=w(t;w0), one can refer to [24, Theorem 1.6.1, P128] or to [35]. Here we omit the details and establish (2.8). Taking the inner product (·,·) of wwith equation (2.5) gives 1 2 d dtkw(t)k2+ (ν+κ)k∇u(t)k2+γk∇ω(t)k2−2κZΩ ∇ × ω(t)·u(t)dx −ηZΩ ∇divω(t)·ω(t)dx−2κZΩ ∇ × u(t)·ω(t)dx+ 4κZΩ ω2(t)dx+σku(t)kβ+1 Lβ+1(Ω) =ZΩ f·u(t)dx+ZΩ g·ω(t)dx. (2.9) Now using the following Poincar´e’s inequality kϕk2⩽λk∇ϕk2,∀ϕ∈H1 0(Ω), λ is a constant depending only on Ω,(2.10) we have by some direct computations and estimates that                      ZΩ ∇ × ω(t)·u(t)dx=ZΩ ∇ × u(t)·ω(t)dx⩽kω(t)k2+k∇u(t)k2, −ηZΩ ∇divω(t)·ω(t)dx=ηZΩ |divω(t)|2dx, ZΩ ω2(t)dx⩽λk∇ω(t)k2, ZΩ f·u(t)dx+ZΩ g·ω(t)dx⩽ν 2k∇u(t)k2+γ 2k∇ω(t)k2+λ 2νkfk2+λ 2γkgk2. 7 Substituting this into (2.9) gives      d dtkw(t)k2+δ λkw(t)k2+ 2σku(t)kβ+1 Lβ+1(Ω) ⩽λ νkfk2+λ γkgk2, d dtkw(t)k2+δk∇w(t)k2+ 2σku(t)kβ+1 Lβ+1(Ω) ⩽λ νkfk2+λ γkgk2, (2.11) hereinafter δ= min{ν+κ, γ}. Applying Gronwall’s inequality to (2.11)1yields kw(t)k2⩽kw(0)k2e−δt λ+λ δ[λ νkfk2+λ γkgk2].(2.12) We end the proof by setting R=q2λ δ[λ νkfk2+λ γkgk2]. We next select some definitions concerning the trajectory attractor. Define the trajectory space T+ σand kernel Kσof equation (2.5) respectively as T+ σ={w(·) : w(·) is a weak solution of (2.5) and w(t)∈Xfor all t∈R+}, Kσ={w(·) : w(·) is a weak solution of (2.5) and w(t)∈Xfor all t∈R}. We also set F+=Cloc([0,∞); Xw) be the space of continuous functions from [0,∞) to Xw, where Xw= (X, dw) denotes the space Xendowed with the weak distance. Define the weak distance in F+as dF+(w, Φ) = dC([0,∞);Xw)(w, Φ) = X j∈N 1 2j sup{dXw(w(t),Φ(t)) : 0 ⩽t⩽j} 1 + sup{dXw(w(t),Φ(t)) : 0 ⩽t⩽j}, which is compatible with the compact-open topology (denoted by Θ+ loc) of F+. Note that (F+,Θ+ loc) is a Hausdorff topological space. Define the natural translation operator {T(s)}s⩾0on F+as T(s)w(t) = w(t+s)[0,∞), w ∈ F+.(2.13) Because that the trajectory space T+ σincludes all bounded weak solutions of (2.5), we have T(s)T+ σ⊂ T + σfor any s⩾0. For a set P ⊂ F+and some r > 0, we set B(P, r) = {u∈ F+dF+(w, P) = inf Φ∈PdF+(w, Φ) < r}. Definition 2.2. A set P ⊂ F+is said to uniformly attract a set Q⊂ T +if for any  > 0there is a t>0such that T(t)Q⊂ B(P, ),∀t⩾t. A set P ⊂ F+is said to be a trajectory attracting set if it uniformly attracts T+. A set U ⊂ F+is called a trajectory attractor if Uis the minimal compact trajectory attracting set and T(t)U=Ufor all t⩾0. 8 To prove the existence of a trajectory attractor for {T(s)}s⩾0in F+, we next establish two lemmas. In the sequel, we will use the notation a.b(also a&b) to mean that a⩽cb (also a⩾cb) for a universal constant c > 0 that only depends on the parameters coming from the problem. Lemma 2.2. Let w= (u, ω)∈L∞([0, T]; b H)∩L2([0, T]; b V)∩Lβ+1([0, T]; ˆ Lβ+1(Ω)) for all T > 0, then t7−→ Aw(t)∈L2([0, T]; b V∗),(2.14) t7−→ Bw(t)∈L4/3([0, T]; b V∗),(2.15) t7−→ Lw(t)∈L2([0, T]; b V∗),(2.16) t7−→ Nσw(t)∈L(β+1)∗([0, T]; ˆ L(β+1)∗(Ω)),(2.17) hereinafter (β+ 1)∗= (β+ 1)/β is the conjugate exponent of β+ 1. Proof. Consider given T > 0. For a.e. t∈[0, T], we see from the definitions of operators A,B,Nσand Lthat Aw(t), B(u(t), w(t)), Lw(t) and Nσw(t) belong to b V∗ for w(t)∈L∞([0, T]; b H)∩L2([0, T]; b V)∩Lβ+1([0, T]; ˆ Lβ+1(Ω)). The measurability of the functions t7−→ Aw(t), t7−→ Bw(t), t7−→ Lw(t), t7−→ Nσw(t) is not difficult to check. Now, for any Φ = (φ, ϕ)∈b Vwe have, by using Cauchy’s inequality, |hAw(t),Φi| .|(∇u(t),∇φ)|+|(∇ω(t),∇ϕ)|.k∇u(t)kk∇φk+k∇ω(t)kk∇ϕk .ku(t)kVkφkV+kω(t)k1,2kϕk1,2 .(ku(t)k2 V+kω(t)k2 1,2)1/2(kφk2 V+kϕk2 1,2)1/2.kw(t)kb VkΦkb V.(2.18) At the same time, by using Gagliardo-Nirenberg’s inequality and the embedding b V ,→ L6(Ω) ×L6(Ω), we have |hBw(t),Φi| .ku(t)k1/2k∇u(t)k1/2(k∇u(t)k+k∇ω(t)k)kΦkb V .kw(t)k1/2kw(t)k3/2 b VkΦkb V,∀Φ∈b V . (2.19) It is also clear that |hLw(t),Φi| .kw(t)kb VkΦkb V,∀Φ∈b V , (2.20) |hNσw(t),Φi| ⩽σZΩ |u|β|φ|dx⩽σk|u|βkL(β+1)∗(Ω)kφkLβ+1(Ω) .ku(t)kβ L(β+1)(Ω)kφkLβ+1(Ω) .kw(t)kβ ˆ Lβ+1(Ω)kΦkˆ Lβ+1(Ω),∀Φ=(φ, ϕ)∈ˆ Lβ+1(Ω). (2.21) 9 From (3.1) we can conclude that each weak solution w(·) within Atr σis “nearly” regular on [ς, ς +ϑ] for each ς∈R+. But we cannot say that each w(·) within Atr σis a.e. regular on [ς, ς +ϑ]. This is much like Lusin’s Theorem describing the relations between measurable functions and continuous functions: each measurable function on an interval Iis “nearly” continuous on I. This “nearly” regular result of the elements within Atr σis stated as follows. Corollary 3.1. For any ς⩾0, denote E(ς) = [ς, ς +ϑ]. Then, for any  > 0and each w∈ Atr σ, there corresponds a subset E,w,ς ⊆E(ς)and a positive constant Csuch that (1) The Lebesgue measure of E,w,ς satisfies mes(E,w,ς)< ; (2) kw(τ)k2 b V⩽C,∀τ∈E(s)\E,w,ς. Proof. Note that the norms k∇wkand kwkb Vare equivalent. For any given  > 0, there is some k∈Nsuch that ϑ/k< . For each w∈ Atr and above , we set E,w,ς ={θ∈[ς, ς +ϑ] : kw(τ)k2 b V⩾M(k)}. Then from (3.1) we have that mes(E,w,ς) = ϑ−mes({θ∈[ς, ς +ϑ] : kw(θ)k2 b V⩽M(k)})⩽ϑ−k−1 k ϑ < . Item (2) obviously holds true. The proof is complete. We next prove that if the Grashof number Ggiven by (1.6) is sufficiently small, then the trajectory attractor Atr σdegenerates to a single bounded complete trajectory. Lemma 3.2. Let that the Grashof number Ggiven by (1.6) satisfy 77c8 1 2δ7 1 G8+c2 22β+2σ2 δ1 (1 + 1 λ)β+1 2Gβ+1 <δ1 λ,(3.7) where c1and c3are positive constants depending only on Ω. Then, the trajectory attractor Atr σgiven by (2.50) degenerates to a single bounded complete trajectory: Atr σ={˜w(t)∈ Kσ:t⩾0}.(3.8) Proof. Let w(t)=(u(t), ω(t)) and Φ(t)=(φ(t), ϕ(t)) lie in Atr σ. Then (2.50) shows that both w(t) and Φ(t) belong to Kσ. Set z(t) = w(t)−Φ(t). Then z(t) satisfies ∂z ∂t +Az +Bw −BΦ + Nσw−NσΦ + Lz = 0, t ∈R.(3.9) Taking the dual pairing h·,·i between zand equation (3.9) yields 1 2 d dtkzk2+hAz, zi+hLz, zi+hBw −BΦ, zi+hNσw−NσΦ, zi= 0.(3.10) 16 Similar to [37, (2.7)], we have hAz, zi+hLz, zi⩾δ1kzkˆ V,where δ1= min{γ, ν}.(3.11) We shall handle the nonlinear terms hBw −BΦ, ziand hNσw−NσΦ, zi. Firstly, by (2.2) and (2.4)2, we obtain hBw −BΦ, zi=hB1(u, u)−B1(φ, φ), u −φi+hB2(u, ω)−B2(φ, ϕ), ω −ϕi =b1(u−φ, φ, u −φ) + b2(u−φ, ϕ, ω −ϕ).(3.12) At the same time, from the classical estimation (see [14, (A.26f)]) we see that there is a positive constant c1depending only on Ω such that            |b1(u−φ, φ, u −φ)|⩽c1ku−φk1/4ku−φk7/4 Vkφk1/4kφk3/4 V ⩽77c8 1 8δ7 1 kzk2kΦk2 b Vk∇Φk6+δ1 8kzk2 b V, |b2(u−φ, ϕ, ω −ϕ)|⩽77c8 1 8δ7 1 kzk2kΦk2 b Vk∇Φk6+δ1 8kzk2 b V. (3.13) It then follows from (3.12) and (3.13) that |hBw −BΦ, zi| ⩽77c8 1 4δ7 1 kzk2kΦk2 ˆ V+δ1 4kzk2 ˆ V.(3.14) Secondly, hNσw−NσΦ, zi=hσ(|u|β−1u− |φ|β−1φ), u −φi =σk|u|β−1 2|u−φ|k2+σZΩ|u|β−1− |φ|β−1(u−φ)φdx. (3.15) Set %(y) = |y|β−1,y= (y1, y2, y3)∈R3. Then %(y) is differentiable for y6=0, and ∇%(y)=(|y|β−3 2y1,|y|β−3 2y2,|y|β−3 2y3),|∇%(y)|⩽|y|β−1 2. By the differential mean value theorem we see that there exists θ∈(0,1) such that |u|β−1− |φ|β−1=|%(u)−%(φ)|⩽|∇%(φ+θ(u−φ))||u−φ| ⩽|(1 −θ)φ+θu|β−1 2|u−φ|⩽2β−1 2|φ|β−1 2|u−φ|+ (2θ)β−1 2|u−φ||u|β−1 2.(3.16) Taking (3.15)-(3.16) into account and using H¨older’s and Cauchy’s inequalities, we 17 arrive at σZΩ|u|β−1− |φ|β−1(u−φ)φdx ⩾−2β−1 2σZΩ |φ|β+1 2|u−φ|2dx−2β−1 2σZΩ |φ||u|β−1 2|u−φ|2dx ⩾−2β−1 2σkφk β+1 2 L 3(β+1) 2(Ω) ku−φkku−φkL6(Ω) −2β−1 2σkφkL6(Ω)kuk β−1 2 L3(β−1)(Ω)ku−φkL6(Ω)ku−φk ⩾−2β−1 2σkφk β+1 2 L 3(β+1) 2(Ω) ku−φkku−φkV−2β−1 2c2σkΦkb Vkwk β−1 2 b Vkzkkzkb V ⩾−c2 22βσ2 δ1kΦkβ+1 b V+kΦk2 b Vkwkβ−1 b Vkzk2−δ1 4kzk2 b V,(3.17) where we have also used Cauchy’s inequality, the embedding V ,→L6(Ω), V ,→ L3(β−1)(Ω) and V ,→L3(β+1) 2(Ω), and c2is a positive embedding constant depending only on Ω. It then follows from (3.10), (3.11), (3.14) and (3.17) that d dtkzk2+δ1kzk2 b V⩽kzk277c8 1 2δ7 1 kΦk2k∇Φk6+c2 22β+1σ2 δ1kΦkβ+1 b V+kΦk2 b Vkwkβ−1 b V. (3.18) For any τ∈Rwith τ < t, we integrate (3.18) over [τ, t] and use (2.10) to obtain kz(t)k2− kz(τ)k2 ⩽Zt τ77c8 1 2δ7 1 kΦk2k∇Φk6+c2 22β+1σ2 δ1kΦkβ+1 b V+kΦk2 b Vkwkβ−1 b V−δ1 λkz(θ)k2dξ. (3.19) Since w(t) and Φ(t) are bounded complete trajectories of equation (2.5), we obtain from (2.11) that                kΦ(t)k2+δ1 λZt τ kΦ(θ)k2dθ⩽kΦ(τ)k2+ (λ νkfk2+λ γkgk2)(t−τ), kw(t)k2+δ1 λZt τ k∇w(θ)k2dθ⩽kw(τ)k2+ (λ νkfk2+λ γkgk2)(t−τ), kΦ(t)k2+δ1Zt τ k∇Φ(θ)k2dθ⩽kΦ(τ)k2+ (λ νkfk2+λ γkgk2)(t−τ). (3.20) Therefore,                lim sup τ→−∞ 1 t−τZt τ kΦ(θ)k2dθ⩽λ δ1 (λ νkfk2+λ γkgk2), lim sup τ→−∞ 1 t−τZt τ k∇w(θ)k2dθ⩽λ δ1 (λ νkfk2+λ γkgk2), lim sup τ→−∞ 1 t−τZt τ k∇Φ(θ)k2dθ⩽1 δ1 (λ νkfk2+λ γkgk2), 18 which implies for s∈(−∞, t] that                kΦ(s)k2⩽lim sup τ→−∞ 1 t−τZt τ kΦ(θ)k2dθ⩽λ δ1 (λ νkfk2+λ γkgk2), k∇w(s)k2⩽lim sup τ→−∞ 1 t−τZt τ k∇w(θ)k2dθ⩽λ δ1 (λ νkfk2+λ γkgk2), k∇Φ(s)k2⩽lim sup τ→−∞ 1 t−τZt τ k∇Φ(θ)k2dθ⩽1 δ1 (λ νkfk2+λ γkgk2). (3.21) Now if (3.7) holds true, we infer from (3.21) for θ∈[τ, t] that 77c8 1 2δ7 1 kΦk2k∇Φk6+c2 22β+1σ2 δ1kΦkβ+1 b V+kΦk2 b Vkwkβ−1 b V−δ1 λ ⩽77c8 1 2δ7 1 kΦ(θ)kk∇Φ(θ)k6+c2 22β+1σ2 δ1 (1 + 1 λ)β+1 2k∇Φ(θ)kβ+1 +c2 22β+1σ2 δ1 (1 + 1 λ)β+1 2k∇Φ(θ)k2k∇wkβ−1−δ1 λ ⩽77c8 1 2δ7 1 G8+c2 22β+2σ2 δ1 (1 + 1 λ)β+1 2Gβ+1 −δ1 λ:= α < 0.(3.22) Inserting (3.22) into (3.19) and then using Gronwall’s inequality, we have kz(t)k2⩽kz(τ)k2eα(t−τ), t > τ. Letting τ→ −∞ gives w(θ) = Φ(θ) for θ∈(−∞, t]. Particularly, w(θ) = Φ(θ) for θ∈[0, t]. The proof of Lemma 3.2 is complete. Recall that the partial regularity of the trajectory statistical solution means that it is carried by a set in which all weak solutions are “partially” regular. Combining Corollary 3.1 and Lemma 3.2, we conclude that the statistical solution µσ, ˜wis “nearly” regular (but not a.e. regular) on each interval R+. This Lusin type degenerate regularity result of the statistical solution is stated as follows. Theorem 3.1. Let F∈b Hand assumption (3.7) hold. Then, the trajectory statistical solution µσ, ˜wguaranteed by Theorem 2.2 and Lemma 3.2 possesses the following Lusin’s type degenerate regularity: for any  > 0, there exists a subset R+()⊂R+such that mes (R+()) <  and µσ, ˜wis regular on R+\R+(). Proof. Let ˜w(t) be the unique complete and bounded weak solution from Lemma 3.2. For each j∈Z+={0,1,2,· · · } and for any  > 0, we see from Corollary 3.1 and Lemma 3.2 that there exists an interval E(, j)⊂Ij= [jϑ, (j+ 1)ϑ] and a positive constant C,j such that mes (E(, j)) < 22jand k˜w(θ)k2 b V⩽C,j,∀θ∈Ij\E(, j).(3.23) 19 Set R+() = S j∈Z+ E(, j). We obviously have mes (R+()) ⩽X j∈Z+ mes(E(, j)) ⩽X j∈Z+  22j⩽. At the same time, we have R+=S j∈Z+ Ij, and by (3.23), k˜w(θ)k2 b V<+∞,∀θ∈R+\R+(). This ends the proof. 4 Convergence of the trajectory statistical solution to that of the 3D incompressible micropolar fluids flows Let wσ∈ Atr σand wσ→win the topology Θ+ loc as σ→0+, we will prove that wis a bounded complete trajectory of the classical 3D incompressible micropolar fluids flows (see equation (4.1) below), and that the trajectory statistical solution µσ,wσ converges to the trajectory statistical solution µ0,w of equation (4.1) as σ→0+. Firstly, we select some known results concerning the following 3D incompressible micropolar flows (see e.g. [42]):              ut−(ν+κ)∆u+ (u· ∇)u+∇p= 2κ∇ × ω+f, ∇ · u= 0, ωt−γ∆ω+ (u· ∇)ω−η∇divω+ 4κω = 2κ∇ × u+g, u(x, t)t=0 =u0, u(x, t)∂Ω×(0,∞)= 0, ω(x, t)t=0 =ω0, ω(x, t)∂Ω×(0,∞)= 0. (4.1) Excluding the pressure p, we can write, using the notations as in Section 2, the weak form of problem (4.1) as ∂w ∂t +Aw +Bw +Lw =F, in D0((0,∞); b V∗),(4.2) w(0) = w0= (u0, ω0).(4.3) Definition 4.1. ( [42]) A weak solution to problem (4.2)-(4.3) on [0,∞)is an b H-valued function w(t)=(u(t), ω(t)) defined on [0,∞)with w(0) = w0= (u0, ω0), such that          dw dt∈L4/3 loc ([0,∞); V∗), w(·)∈C([0,∞); b Hw)∩L2 loc([0,∞); b V) (dw dt,Φ) + hAw, Φi+Bw, Φ+hLw, Φi= (F, Φ),∀Φ∈b V , holds in the distribution sense D0(0,∞), and w(t)satisfies the following energy inequality 1 2kw(t)k2+Zt 0 hAw(s), w(s)ids+Zt 0Lw(s), w(s)ds⩽1 2kw0k2+Zt 0F, w(s)ds 20 in the sense that −1 2Zt 0 kw(s)k2ξ0(s)ds+Zt 0 hAw(s), w(s)iξ(s)ds+Zt 0 hLw(s), w(s)iξ(s)ds ⩽Zt 0F, w(s)ξ(s)ds, ∀ξ(·)∈C∞ 0([0, t]) with ξ(t)⩾0,∀t⩾0.(4.4) Lemma 4.1. ( [42]) Let w0= (u0, ω0)∈b Hand suppose F∈b H. Then, problem (4.2)- (4.3) possesses at least one weak solution w(t) = w(t;w0)corresponding to the initial value w0. Moreover, there exists a time t0 ∗⩾0yielding w(t)∈X, ∀t⩾t0 ∗,(4.5) where Xis given by (2.8). The trajectory space T+ 0and kernel K0of equation (4.2) is defined respectively as T+ 0={w(·) : w(·) is a weak solution of (4.2) and w(t)∈Xfor all t∈R+}, K0={w(·) : w(·) is a weak solution of (4.2) and w(t)∈Xfor all t∈R}. Lemma 4.2. ( [42]) Let F∈b H. (a) The translation semigroup {T(t)}t⩾0defined by (2.13) possesses a trajectory attractor Atr 0⊂ T + 0with respect to the topology Θ+ loc: Atr 0= Π+K0={u(·)|[0,∞)u∈ K0}⊂T+ 0.(4.6) (b) For a given generalized Banach limit LIMt→+∞and for each w∈ T + 0, there corresponds a unique Borel probability measure µ0,w on F+such that ZF+ ζ(z)dµ0,w(z) = LIMt→+∞ 1 tZt 0 ζ(T(s)w)ds, ∀ζ∈C(F+).(4.7) Moreover, µ0,w is supported by Atr 0and is a trajectory statistical solution for equation (4.2), and µ0,w is invariant under the acting of {T(t)}t⩾0in the following sense ZF+ ζ(T(t)z)dµ0,w(z) = ZF+ ζ(z)dµ0,w(z),∀t⩾0,∀ζ∈C(F+).(4.8) We now prove the convergence of solution of equation (2.5) to that of equation (4.2) as σ→0+. Lemma 4.3. Let a sequence wσn(·)∈ T + σn,n∈N, satisfy the following conditions: (I) σn→0+as n→ ∞;(II) wσn(·)→w(·)in the topology Θ+ loc as n→ ∞. Then, w∈ T + 0. 21 Proof. We first prove that wsatisfies (4.2). In fact, since wσn(·)∈ T + σn,wσn(·) satisfies for each T > 0 that ∂wσn ∂t +Awσn+Bwσn+Nσnwσn+Lwσn=F, in D0(0, T;b V∗).(4.9) From (2.35) we infer that both the nonlinear terms Bwσnand Nσnwσnare uniformly (with respect to n∈N) bounded in L4/3([0, T]; b V∗). Hence, by (2.36) and condition (II), we have as σn→0+that                ∂wσn ∂t *∂w ∂t weakly in L4/3(0, T;b V∗), Awσn* Aw weakly in L2(0, T;b V∗), Bwσn* Bw weakly in L4/3(0, T;b V∗), Nσnwσn*0 weakly in L2(0, T;b V∗), Lwσn* Lw weakly in L4/3(0, T;b V∗). (4.10) (4.9) and (4.10) imply ∂w ∂t +Aw +Bw +Lw =F, in D0(0, T;b V∗).(4.11) The fact that w(·) meets (4.4) can be proved by using (2.49) and (4.10). By (4.11) and the same derivations of Lemma 2.1, we conclude that w(·) satisfies (4.5). The proof is complete. Next we set Bσ={wσ|wσ∈ T + σ,kwσkBσ⩽c(R),for all wσ∈ Bσ},where c(R) is a positive constant which depends only on Rand can be chosen for our purpose, and kwσkBσ= sup t⩾0 kwσ(t)k+ sup t⩾0 (Zt+1 t kwσ(θ)k2 b Vdθ)1/2+ sup t⩾0 (Zt+1 t k∂twσ(θ)k3/4 b V∗dθ)3/4. Lemma 4.4. Let σ∈(0,1]. Then T(t)Bσ→ Atr 0in the topology Θ+ loc as t →+∞and σ →0+,(4.12) where (4.12) is interpreted in following sense: for {wσ}0<σ⩽1with wσ∈Bσ, there is some w∈ Atr 0such that T(t)wσ→win the topology Θ+ loc as t→+∞and σ→0+. Proof. We prove (4.14) by contradiction. Assume that there is a neighbourhood O(Atr 0) of Atr 0in the topology Θ+ loc and two sequences σn→0+,tn→+∞as n→ ∞, such that T(tn)Bσn6⊂ O(Atr 0).(4.13) Then there is a solution wσn∈Bσnsuch that zσn(t) := T(tn)wσn(t)6∈ O(Atr 0).(4.14) 22 Notice that zσn(t) is a solution of equation (2.5) with σ=σnon the interval [−tn,+∞) and zσn(t) is the backward time shift of the solution wσn(t) by tn. By the definition of Bσ, we have that sup t⩾−tn kzσn(t)k+ sup t⩾−tn (Zt+1 t kzσn(θ)k2 b Vdθ)1/2 + sup t⩾−tn (Zt+1 t k∂θzσn(θ)k3/4 b V∗dθ)3/4⩽c(R).(4.15) For every T > 0, we consider σnwith the index nsuch that tn⩾T. Then (4.15) implies that we can extract a subsequence (still denote by) {zσn(·)}and a function z(t), t∈(−T, T), such that      zσn* z weakly star in L∞([0, T]; b H) as n→ ∞, zσn* z weakly in L2([0, T]; b V) as n→ ∞, zσn*dz dtweakly in L4/3([0, T]; b V∗) as n→ ∞. (4.16) Using the standard diagonal procedure, we can construct a function z(t), t∈R, such that zσn(·)→z(·) in the topology Θ+ loc as n→ ∞.(4.17) By Lemma 4.3 we see that z(·)∈ T tr 0. This fact, together with (4.15) and (4.17), implies that z∈ K0. Now, by (4.6) we have Π+z∈Π+K0=Atr 0. Eq. (4.17) shows that Π+zσn→Π+zin the topology Θ+ loc as n → ∞. Therefore, we have for large enough nthat Π+zσn∈ O(Π+z)⊂ O(Atr 0), which contradicts (4.13). The proof is complete. Lemma 4.4 implies the convergence of the trajectory attractor Atr σto Atr 0as σ→0+. The convergence of this type was investigated in [36] for the convective Brinkman- Forchheimer equations. At this stage, we can state and prove the main results of this section. Theorem 4.1. Suppose that F∈b H. Let Atr σand Atr 0be the trajectory attractors guaranteed by Theorem 2.1 and Lemma 2.2(I), respectively. Let that wσ∈ Atr σand wσ→win the topology Θ+ loc as σ→0+, and that µσ,wσand µ0,w be the trajectory statistical solutions guaranteed by Theorem 2.2 and Lemma 4.2(b), respectively. Then Atr σ→ Atr 0in the topology Θ+ loc as σ →0+,(4.18) lim σ→0+ZF+ ζ(z)dµσ,wσ(z) = ZF+ ζ(z)dµ0,w(z),∀ζ∈C(F+).(4.19) 23 Proof. From (2.8), (2.11), (2.35) and the definition k·kBσ, we see that the family {Atr σ}0<σ⩽1is uniformly (with respect to σ∈(0,1]) bounded when we choose c(R) large enough. Applying Lemma 4.4 with Bσ=Atr σand the invariant property of Atr σ, we obtain by letting t→+∞that Atr σ=T(t)Atr σ→ Atr 0in the topology Θ+ loc as σ→0+, that is, (4.18) is proved. Now, let wσ∈ Atr σsuch that wσ→win the topology Θ+ loc as σ→0+. By Lemma 4.3 we see that w∈ Atr 0. Then, Theorem 2.2 and Lemma 4.2(b) show that there exist trajectory statistical solutions µσ,wσand µ0,w corresponding respectively to wσand w, and that for a given generalized Banach limit LIMt→+∞and for any ζ∈C(F+) there holds        ZF+ ζ(z)dµσ,wσ(z) = LIMt→+∞ 1 tZt 0 ζ(T(s)wσ)ds, ZF+ ζ(z)dµ0,w(z) = LIMt→+∞ 1 tZt 0 ζ(T(s)w)ds. (4.20) Since that LIMt→+∞is a linear continuous functional, ζ∈C(F+) and that wσ→win the topology Θ+ loc as σ→0+, by (4.20) we have lim σ→0+ZF+ ζ(z)dµσ,wσ(z) = lim σ→0+LIMt→+∞ 1 tZt 0 ζ(T(s)wσ)ds =LIMt→+∞ 1 tZt 0 ζ(T(s) lim σ→0+wσ)ds =LIMt→+∞ 1 tZt 0 ζ(T(s)w)ds =ZF+ ζ(z)dµ0,w(z), where we have also used the continuity of the translation semigroup T(·) on F+with respect to the topology Θ+ loc (cf. [8]). The proof is complete. Acknowledgements The authors warmly thank the anonymous referee for his/her careful reading of the article and many pertinent remarks that lead to various improvements to this article. References [1] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis,A Hithhiker’s Guide, third editon, Springer-Verlag, 2006. [2] A. Bronzi, C. F. Mondaini, R. Rosa, Trajectory statistical solutions for threedimensional Navier-Stokes-like systems, SIAM J. Math. Anal.,46(2014), 1893- 1921. 24 [3] A. Bronzi, R. 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