On the existence and uniqueness of solutions to 2D g-benard problem in unbounded domains

The existence and long-time behavior of solutions in terms of existence of an attractor for the 2D B´enard problem have been studied in [3] in the autonomous case and in [1] in the non-autonomous case. The 2D g-Navier-Stokes equations and its relationship with the 3D Navier-Stokes equations in the thin domain Ω g was introduced by Roh in [12]. Since then there have been many works devoted to studying mathematical questions related to these equations. In particular, the existence and long-time behavior of solutions to 2D g-Navier-Stokes equations have been studied extensively, in the both autonomous and non-autonomous cases, see e.g. [2, 5, 6, 7, 10, 13, 14]. The existence of time-periodic solutions to g-Navier-Stokes and g-Kelvin-Voight equations was also studied more recently in [4]. For the 2D g-B´enard problem, in [8] Hitherto, M. Ozl¨uk and M. Kaya considered ¨ Boussinesq equations in the bounded domain Ωg = {(y1, y2, y3) ∈ R3 : (y1, y2) ∈ Ω2, 0 < y3 < g}, where Ω2 is a bounded region in the plane and g = g(y1, y2) is a smooth function defined on Ω2. They proved the existence and uniqueness of weak solutions and derived upper bounds for the number of determining modes. More recently, in [9] M. Ozl¨uk and M. Kaya investigated ¨ the existence, uniqueness of strong solutions, and the continuous dependence of the solutions on the viscosity parameter for problem (1.1) in the non-autonomous case and the function g to be periodic with period 1 in the x1 and x2 directions.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2020-0025 Natural Science, 2020, Volume 65, Issue 6, pp. 23-31 This paper is available online at ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO 2D G-B ´ENARD PROBLEM IN UNBOUNDED DOMAINS Tran Quang Thinh1 and Le Thi Thuy2 1Faculty of Basic Sciences, Nam Dinh University of Technology Education 2Faculty of Mathematics, Electric Power University Abstract. We consider the 2D g-Be´nard problem in domains satisfying the Poincare´ inequality with homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of global weak solutions. The obtained results particularly extend previous results for 2D g-Navier-Stokes equations and 2D Be´nard problem. 1. Introduction Let Ω be a (not necessarily bounded) domain in R2 with boundary Γ. We consider the following two-dimensional (2D) g-Be´nard problem  ∂u ∂t + (u · ∇)u− ν∆u+∇p = ξθ + f1, x ∈ Ω, t > 0, ∇ · (gu) = 0, x ∈ Ω, t > 0, ∂θ ∂t + (u · ∇)θ − κ∆θ − 2κ g (∇g · ∇)θ − κ∆g g θ = f2, x ∈ Ω, t > 0, u = 0, x ∈ Γ, t > 0, θ = 0, x ∈ Γ, t > 0, u(x, 0) = u0(x), x ∈ Ω, θ(x, 0) = θ0(x), x ∈ Ω, (1.1) where u ≡ u(x, t) = (u1, u2) is the unknown velocity vector, θ ≡ θ(x, t) is the temperature, p ≡ p(x, t) is the unknown pressure, f1 is the external force function, f2 is the heat source function, ν > 0 is the kinematic viscosity coefficient, ξ is a constant vector, κ > 0 is thermal diffusivity, u0 is the initial velocity and θ0 is the initial temperature. As derived and mentioned in [8], 2D g-Be´nard problem arises in a natural way when we study the standard 3D Be´nard problem on the thin domain Ωg = Ω × (0, g). Here the g-Be´nard problem is a couple system which consists of g-Navier-Stokes equations and the advection-diffusion heat equation in order to model convection in a fluid. Moreover, when g ≡ const we get the usual Be´nard problem, and when θ ≡ 0we get the g-Navier-Stokes equations. In what follows, we will list some related results. Received June 5, 2020. Revised June 19, 2020. Accepted June 26, 2020 Contact Le Thi Thuy, e-mail address: thuylt@epu.edu.vn 23 Tran Quang Thinh and Le Thi Thuy The existence and long-time behavior of solutions in terms of existence of an attractor for the 2D Be´nard problem have been studied in [3] in the autonomous case and in [1] in the non-autonomous case. The 2D g-Navier-Stokes equations and its relationship with the 3D Navier-Stokes equations in the thin domain Ωg was introduced by Roh in [12]. Since then there have been many works devoted to studying mathematical questions related to these equations. In particular, the existence and long-time behavior of solutions to 2D g-Navier-Stokes equations have been studied extensively, in the both autonomous and non-autonomous cases, see e.g. [2, 5, 6, 7, 10, 13, 14]. The existence of time-periodic solutions to g-Navier-Stokes and g-Kelvin-Voight equations was also studied more recently in [4]. For the 2D g-Be´nard problem, in [8] Hitherto, M. ¨Ozlu¨k and M. Kaya considered Boussinesq equations in the bounded domain Ωg = {(y1, y2, y3) ∈ R3 : (y1, y2) ∈ Ω2, 0 < y3 < g}, where Ω2 is a bounded region in the plane and g = g(y1, y2) is a smooth function defined on Ω2. They proved the existence and uniqueness of weak solutions and derived upper bounds for the number of determining modes. More recently, in [9] M. ¨Ozlu¨k and M. Kaya investigated the existence, uniqueness of strong solutions, and the continuous dependence of the solutions on the viscosity parameter for problem (1.1) in the non-autonomous case and the function g to be periodic with period 1 in the x1 and x2 directions. In this paper we will study the existence and uniqueness of weak solutions to 2D g-Be´nard problem in domains that are not necessarily bounded but satisfy the Poincare´ inequality. To do this, we assume that the domain Ω and functions f1, f2, g satisfy the following hypotheses: (Ω) Ω is an arbitrary (not necessarily bounded) domain in R2 satisfying the Poincare´ type inequality ∫ Ω φ2gdx ≤ 1 λ1 ∫ Ω |∇φ|2gdx, for all φ ∈ C∞0 (Ω); (1.2) (F) f1 ∈ L 2(0, T ;Hg), f2 ∈ L 2(0, T ;L2(Ω, g)); (G) g ∈W 1,∞(Ω) such that 0 < m0 ≤ g(x) ≤M0 for all x = (x1, x2) ∈ Ω, and |∇g|2∞ < m20λ1, (1.3) where λ1 > 0 is the constant in the inequality (1.2). The paper is organized as follows. In Section 2, for convenience of the reader, we recall the functional setting of the 2D g-Be´nard problem. Section 3 is devoted to proving the existence and uniqueness of global weak solutions to the problem by combining the Galerkin method and the compactness lemma. The results obtained here extend and improve some previous results for 2D Be´nard problem in [3] and 2D g-Navier-Stokes equations in [6]. 24 On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains 2. Preliminaries Let L2(Ω, g) = (L2(Ω, g))2 and H10(Ω, g) = (H10 (Ω, g))2 be endowed with the usual inner products and associated norms. We define V1 = {u ∈ (C ∞ 0 (Ω, g)) 2 : ∇ · (gu) = 0}, Hg = the closure of V1 in L2(Ω, g), Vg = the closure of V1 in H10(Ω, g), V ′g = the dual space of Vg, V2 = {θ ∈ C ∞ 0 (Ω, g)}, Wg = the closure of V2 in H10 (Ω, g), W ′g = the dual space of Wg, V = Vg ×Wg, H = Hg × L 2(Ω, g). The inner products and norms in Vg, Hg are given by (u, v)g = ∫ Ω u · vgdx, u, v ∈ Hg, and ((u, v))g = ∫ Ω 2∑ i,j=1 ∇uj · ∇vigdx, u, v ∈ Vg, and norms |u|2g = (u, u)g , ‖u‖2g = ((u, u))g . The norms | · |g and ‖ · ‖g are equivalent to the usual ones in L2(Ω, g) and H10(Ω, g). We also use ‖ · ‖∗ for the norm in V ′g , and 〈·, ·〉 for duality pairing between Vg and V ′g . The inclusions Vg ⊂ Hg ≡ H ′ g ⊂ V ′ g , Wg ⊂ L 2(Ω, g) ⊂W ′g are valid where each space is dense in the following one and the injections are continuous. By the Riesz representation theorem, it is possible to write 〈f, u〉g = (f, u)g,∀f ∈ Hg,∀u ∈ Vg. Also, we define the orthogonal projection Pg as Pg: Hg → Hg and P˜g as P˜g: L2(Ω, g) → Wg. By taking into account the following equality − 1 g (∇ · g∇u) = −∆u− 1 g (∇g · ∇)u, we define the g-Laplace operator and g-Stokes operator as −∆gu = − 1 g (∇ · g∇u) and Agu = Pg[−∆gu], respectively. Since the operators Ag and Pg are self-adjoint, using integration by parts we have 〈Agu, u〉g = 〈Pg[− 1 g (∇ · g∇)u], u〉g = ∫ Ω (∇u · ∇u)gdx = (∇u,∇u)g. 25 Tran Quang Thinh and Le Thi Thuy Therefore, for u ∈ Vg, we can write |A1/2g u|g = |∇u|g = ‖u‖g . Next, since the functional τ ∈Wg 7→ (∇θ,∇τ)g ∈ R is a continuous linear mapping on Wg, we can define a continuous linear mapping A˜g on W ′g such that ∀τ ∈Wg, 〈A˜gθ, τ〉g = (∇θ,∇τ)g, for all θ ∈Wg. We denote the bilinear operator Bg(u, v) = Pg[(u · ∇)v] and the trilinear form bg(u, v, w) = 2∑ i,j=1 ∫ Ω ui ∂vj ∂xi wjgdx, where u, v, w lie in appropriate subspaces of Vg. Then, one obtains that bg(u, v, w) = −bg(u,w, v), which particularly implies that bg(u, v, v) = 0. (2.1) Also bg satisfies the inequality |bg(u, v, w)| ≤ c|u| 1/2 g ‖u‖ 1/2 g ‖v‖ 2 g|w| 1/2 g ‖w‖ 1/2 g . (2.2) Similarly, for u ∈ Vg and θ, τ ∈Wg we define B˜g(u, θ) = P˜g[(u · ∇)θ] and b˜g(u, θ, τ) = n∑ i,j=1 ∫ Ω ui(x) ∂θ(x) ∂xj τ(x)gdx. Then, one obtains that b˜g(u, θ, τ) = −b˜g(u, τ, θ), which particularly implies that b˜g(u, θ, θ) = 0. (2.3) And b˜g satisfies the inequality |b˜g(u, θ, τ)| ≤ c|u| 1/2 g ‖u‖ 1/2 g ‖θ‖ 2 g|τ | 1/2 g ‖τ‖ 1/2 g . (2.4) We denote the operators Cgu = Pg [1 g (∇g · ∇)u ] and C˜gθ = P˜g [1 g (∇g · ∇)θ ] such that 〈Cgu, v〉g = bg( ∇g g , u, v), 〈C˜gθ, τ〉g = b˜g( ∇g g , θ, τ). Finally, let D˜gθ = P˜g[ ∆g g θ] such that 〈D˜gθ, τ〉g = −b˜g( ∇g g , θ, τ)− b˜g( ∇g g , τ, θ). 26 On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains Using the above notations, we can rewrite the system (1.1) as abstract evolutionary equations   du dt +Bg(u, u) + νAgu+ νCgu = ξθ + f1, dθ dt + B˜g(u, θ) + κA˜gθ − κC˜gθ − κD˜gθ = f2, u(0) = u0, θ(0) = θ0. 3. Existence and uniqueness of weak solutions Definition 3.1. A pair of functions (u, θ) is called a weak solution of problem (1.1) on the interval (0, T ) if u ∈ L2(0, T ;Vg) and θ ∈ L2(0, T ;Wg) satisfy  d dt (u, v)g + bg(u, u, v) + ν(∇u,∇v)g + νbg( ∇g g , u, v) = (ξθ, v)g +(f1, v)g , d dt (θ, τ)g + b˜g(u, θ, τ) + κ(∇θ,∇τ)g + κb˜g( ∇g g , τ, θ) = (f2, τ)g, (3.1) for all test functions v ∈ Vg and τ ∈Wg. The following theorem is our main result. Theorem 3.1. Let the initial datum (u0, θ0) ∈ H be given, let the external forces f1, f2 satisfy hypothesis (F) and the function g satisfy hypothesis (G). Then there exists a unique weak solution (u, θ) of problem (1.1) on the interval (0, T ). Proof. Existence. We use the standard Galerkin method. Since Vg is separable and V1 is dense in Vg, there exists a sequence {ui}i∈N which forms a complete orthonormal system in Hg and a base for Vg. Similarly, there exists a sequence {θi}i∈N which forms a complete orthonormal system in L2(Ω, g) and a base for Wg. Let m be an arbitrary but fixed positive integer. For each m we define an approximate solution (um(t), θm(t)) of (3.1) for 1 ≤ k ≤ m and t ∈ [0, T ] in the form, u(m)(t) = m∑ j=1 f (m) j (t)uj ; θ (m)(t) = m∑ j=1 g (m) j (t)θj, u(m)(0) = um0 = m∑ j=1 (a0, uj)uj ; θ (m)(0) = θm0 = m∑ j=1 (τ0, θj)θj, d dt (u(m), uk)g + bg(u (m), u(m), uk) + ν((u (m), uk))g + νbg( ∇g g , u(m), uk) = (ξθ (m), uk)g + (f1, uk)g, (3.2) d dt (θ(m), θk)g + b˜g(u (m), θ(m), θk) + κ((θ (m), θk))g + κb˜g( ∇g g , θk, θ (m)) = (f2, θk)g. (3.3) 27 Tran Quang Thinh and Le Thi Thuy This system forms a nonlinear first order system of ordinary differential equations for the functions f (m) j (t) and g (m) j (t) and has a solution on some maximal interval of existence [0, Tm). We multiply (3.2) and (3.3) by f (m)j (t) and g(m)j (t) respectively, then add these equations for k = 1, . . . ,m. Taking into account bg(u(m), u(m), u(m)) = 0 and b˜g(u(m), θ(m), θ(m)) = 0, we get (u′(m)(t), u(m)(t))g + ν‖u (m)(t)‖2g + νbg( ∇g g , u(m)(t), u(m)(t)) = (ξθ(m), u(m)(t))g + (f1, u (m)(t)), (3.4) (θ′(m)(t), θ(m)(t))g + κ‖θ (m)(t)‖2g+κb˜g( ∇g g , θ(m)(t), θ(m)(t)) = (f2, θ (m)(t))g. (3.5) Using (2.2), (2.4), the Schwarz and Young inequalities in (3.4) and (3.5) we obtain d 2dt |u(m)(t)|2g + ν‖u (m)(t)‖2g ≤ ν|∇g|∞ m0λ 1/2 1 ‖u(m)(t)‖2g + ǫν‖u (m)(t)‖2g + ‖ξ‖2 ∞ 2ǫνλ21 ‖θ(m)(t)‖2g + 1 2ǫνλ1 |f1| 2 g, d 2dt |θ(m)(t)|2g + κ‖θ (m)(t)‖2g ≤ κ|∇g|∞ m0λ 1/2 1 ‖θ(m)(t)‖2g + ǫκ‖θ (m)(t)‖2g + 1 4ǫκλ1 |f2| 2 g, so that for ν ′ = 2ν ( 1− |∇g|∞ m0λ 1/2 1 − ǫ ) , κ′ = 2κ ( 1− |∇g|∞ m0λ 1/2 1 − ǫ ) , c′ = ‖ξ‖2 ∞ ǫλ21 we get d dt |u(m)(t)|2g + ν ′‖u(m)(t)‖2g ≤ c′ ν ‖θ(m)(t)‖2g + 1 ǫλ1ν |f1| 2 g, (3.6) d dt |θ(m)(t)|2g + κ ′‖θ(m)(t)‖2g ≤ 1 2ǫλ1κ |f2| 2 g, (3.7) where ǫ > 0 is chosen such that ( 1− |∇g|∞ m0λ 1/2 1 − ǫ ) > 0. Integrating (3.7) and (3.6) from 0 to t, we obtain sup t∈[0,T ] |θ(m)(t)|2g ≤ |θ0| 2 g + T 2ǫλ1κ |f2| 2 g. (3.8) sup t∈[0,T ] |u(m)(t)|2g ≤ |u0| 2 g + c′ νκ′ |θ0| 2 g + c′T 2ǫλ1νκκ′ |f2| 2 g + T ǫλ1ν |f1| 2 g. (3.9) These inequalities imply that the sequences {u(m)}m and {θ(m)}m remain in a bounded set of L∞(0, T ;Hg) and L∞(0, T ;L2(Ω, g)), respectively. We then integrate (3.6) and (3.7) from 0 to T to get |θ(m)(T )|2g + κ ′ ∫ T 0 ‖θ(m)(t)‖2gdt ≤ T 2ǫλ1κ |f2| 2 g, (3.10) 28 On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains |u(m)(T )|2g + ν ′ ∫ T 0 ‖u(m)(t)‖2gdt ≤ c′T 2ǫλ1νκκ′ |f2| 2 g + T ǫλ1ν |f1| 2 g, (3.11) which shows that the sequences {u(m)}m and {θ(m)}m are bounded in L2(0, T ;Vg) and L2(0, T ;Wg), respectively. Due to the estimates (3.8)-(3.11), we assert the existence of elements u ∈ L2(0, T ;Vg) ∩ L ∞(0, T ;Hg), θ ∈ L2(0, T ;Wg) ∩ L ∞(0, T ;L2(Ω, g)), and the subsequences {u(m)}m and {θ(m)}m such that u(m) ⇀ u in L2(0, T ;Vg), θ(m) ⇀ θ in L2(0, T ;Wg), and u(m) ⇀ u weakly-star in L∞(0, T ;Hg), θ(m) ⇀ θ weakly-star in L∞(0, T ;L2(Ω, g)). Applying the Aubin-Lions lemma, we have subsequences {u(m)}m and {θ(m)}m such that u(m) → u in L2(0, T ;Hg), θ(m) → θ in L2(0, T ;L2(Ω, g)). In order to pass to the limit, we consider the scalar functions Ψ1(t) and Ψ2(t) continuously differentiable on [0, T ] and such that Ψ1(T ) = 0 and Ψ2(T ) = 0. We multiply (3.2) and (3.3) by Ψ1(t) and Ψ2(t) respectively and then integrate by parts, − ∫ T 0 (u(m),Ψ′1uk)gdt+ ∫ T 0 bg(u (m), u(m),Ψ1uk)dt + ν ∫ T 0 ((u(m),Ψ1uk))gdt+ ν ∫ T 0 bg( ∇g g , u(m),Ψ1uk)dt = (um0, uk)gΨ1(0) + ∫ T 0 (ξθ(m),Ψ1uk)gdt+ ∫ T 0 (f1, uk)gdt, − ∫ T 0 (θ(m),Ψ′2θk)gdt+ ∫ T 0 b˜g(u (m), θ(m),Ψ2θk)dt+ κ ∫ T 0 ((θ(m),Ψ2θk))gdt + κ ∫ T 0 b˜g( ∇g g , θk,Ψ2θ (m))dt = (θm0, θk)gΨ2(0) + ∫ T 0 (f2,Ψ2θk)gdt. Following the technique given in [15], as m→∞ we obtain − ∫ T 0 (u,Ψ′1v)gdt+ ∫ T 0 bg(u, u,Ψ1v)dt + ν ∫ T 0 ((u,Ψ1v))gdt +ν ∫ T 0 bg( 1 g ∇g, u,Ψ1v)dt = (u0, v)gΨ1(0) + ∫ T 0 (ξθ,Ψ1v)gdt + ∫ T 0 (f1, v)gdt, (3.12) 29 Tran Quang Thinh and Le Thi Thuy − ∫ T 0 (θ,Ψ′2τ)gdt+ ∫ T 0 b˜g(u, θ,Ψ2τ)dt+ κ ∫ T 0 ((θ,Ψ2τ))gdt +κ ∫ T 0 b˜g( ∇g g , τ,Ψ2θ)dt = (θ0, τ)gΨ2(0) + ∫ T 0 (f2,Ψ2τ)gdt. (3.13) The equations (3.12) and (3.13) hold for v and τ which are finite linear combinations of the uk and θk for k = 1, . . . ,m and by continuity (3.12) and (3.13) hold for v ∈ Vg and τ ∈ Hg respectively. Rewriting (3.12) and (3.13) for Ψ1(t),Ψ2(t) ∈ C∞0 (0, T ) we see that (u, θ) satisfy (3.1). Furthermore, applying similar techniques given in [13, 15] it is easy to show that (u, θ) satisfies the initial conditions u(0) = u0 and θ(0) = θ0. Uniqueness. For the uniqueness of weak solutions, let (u1, θ1) and (u2, θ2) be two weak solutions with the same initial conditions. Putting w = u1 − u2 and w˜ = θ1 − θ2. Then we have d dt (w, v)g + bg(u1, u1, v)− bg(u2, u2, v) + ν(∇w,∇v)g + ν(Cgw, v)g = (ξw˜, v)g, d dt (w˜, τ)g + b˜g(u1, θ1, τ)− b˜g(u2, θ2, τ) + κ(∇w˜,∇τ)g + κb˜g( ∇g g , τ, w˜) = 0. Taking v = w(t), τ = w˜(t) and (2.1), (2.3) we obtain 1 2 d dt |w|2g + ν‖w‖ 2 g ≤ |bg(w, u2, w)| + ν|bg( ∇g g ,w,w)| + |(ξw˜, w)g|, 1 2 d dt |w˜|2g + κ‖w˜‖ 2 g+ ≤ |b˜g(w, θ2, w˜)|+ κ|b˜g( ∇g g , w˜, w˜)|. By applying (2.2), (2.4) it then follows by the Cauchy-Schwarz inequality, we have 1 2 d dt |w|2g + ν‖w‖ 2 g ≤ c2 ǫν |w|2g‖u2‖ 2 g + ν|∇g|∞ m0λ 1/2 1 ‖w‖2g + ǫν 2 ‖w‖2g + ‖ξ‖2 ∞ ǫνλ1 |w˜|2, (3.14) 1 2 d dt |w˜|2g + κ‖w˜‖ 2 g ≤ ǫν 2 ‖w‖2g + ǫκ‖w˜‖ 2 g + c4|θ2| 4 g 16ǫ3ν2κλ21 |w˜|2g + κ|∇g|∞ m0λ 1/2 1 ‖w˜‖2g. (3.15) We sum equations (3.14) and (3.15) to obtain d dt (|w|2g + |w˜| 2 g) + 2 ( 1− |∇g|∞ m0λ 1/2 1 − ǫ ) (ν‖w‖2g + κ‖w˜‖ 2 g) ≤ 2c2‖u2‖ 2 g ǫν |w|2g + ( 2‖ξ‖2 ∞ ǫνλ1 + c4‖θ2‖ 4 g 8ǫ3ν2κλ21 ) |w˜|2g, so that for γ = max { 2c2‖u2‖ 2 g ǫν ; 2‖ξ‖2 ∞ ǫνλ1 + c2‖θ‖4g 8ǫ3ν2κλ21 } , we get d dt (|w|2g + |w˜| 2 g) ≤ γ(|w| 2 g + |w˜| 2 g). 30 On the existence and uniqueness of solutions to 2D g-Be´nard problem in unbounded domains Thanks to the Gronwall inequality, we have |w(t)|2g + |w˜(t)| 2 g ≤ ( |w(0)|2g + |w˜(0)| 2 g ) eγt. Hence, the continuous dependence of the weak solution on the initial data in any bounded interval for all t ≥ 0. In particular, the solution is unique. REFERENCES [1] C.T. Anh and D.T. Son, 2013. Pullback attractors for nonautonomous 2D Be´nard problem in some unbounded domains. Math. Methods Appl. Sci., 36, pp. 1664-1684. [2] H. Bae and J. Roh, 2004. Existence of solutions of the g-Navier-Stokes equations, Taiwanese J. Math., 8, pp. 85-102. [3] M. Cabral, R. 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