The pullback attractor for the 2D g-Navier-Stokes equation with nonlinear damping and time delay

: In this article, the global well-posedness of weak solutions for 2D non-autonomous g-Navier-Stokes equations on some bounded domains were investigated by the Faedo-Galerkin method. Then the existence of pullback attractors for 2D g-Navier-Stokes equations with nonlinear damping and time delay was obtained using the method of pullback condition (PC)


Introduction
It is well-known that the Navier-Stokes equations are important in fluid mechanics and turbulence.In the last decades, the research of the asymptotic properties of the solution for Navier-Stokes equations has attracted the attention of scholars [1][2][3][4][5].Especially in the past years, the Navier-Stokes equations with nonlinear damping have been studied [6][7][8][9], where the damping comes from the resistance to the motion of the flow.It describes various physical situations such as porous media flow, drag or friction effects and some dissipative mechanisms.In [6], Cai and Jiu considered the following Navier-Stokes equations with damping: where α|u| β−1 u is nonlinear damping and β is damping exponent.For any β ≥ 1, the global weak solutions of the Navier-Stokes equations with damping α|u| β−1 u (α > 0) is obtained, and for any 7  2 ≤ β ≤ 5, the existence and uniqueness of strong solution is proved.Furthermore, the existence and uniqueness of strong solution is proved for any 3 ≤ β ≤ 5 in [7], the L 2 decay of weak solutions with β ≥ 10  3 is studied and the optimal upper bounds of the higher-order derivative of the strong solution is proved in [8].In recent years, Song et al. researched the following non-autonomous 3D Navier-Stokes equation with nonlinear damping: (1.2) The existence of pullback attractors for the 3D Navier-Stokes equations with damping α|u| β−1 u (α > 0, 3 ≤ β ≤ 5) were proved in [9].Furthermore, Baranovskii and Artemov investigated the solvability of the steady-state flow model for low-concentrated aqueous polymer solutions with a damping term in a bounded domain under the no-slip boundary condition in [10].They proved that the obtained solutions of the original problem converged to a solution of the steady-state damped Navier-Stokes system as the relaxation viscosity tends to zero.
The research of the 2D g-Navier-Stokes equations is originated from the 3D Navier-Stokes equations on thin region.Its form is as follows: where g = g(x 1 , x 2 ) is a suitable smooth real-valued function defined on (x 1 , x 2 ) ∈ Ω and Ω is a suitable bounded domain in R 2 .In [11], by the vertical mean operator, the 2D g-Navier-Stokes equations are derived from 3D Navier-Stokes equations.We study the 2D g-Navier-Stokes equations as a small perturbation of the usual Navier-Stokes equations, so we want to understand the Navier-Stokes equations completely by studying the 2D g-Navier-Stokes equations systematically.Therefore, the research on the g-Navier-Stokes equations has theoretical basis and practical significance.
There are many studies on g-Navier-Stokes equations [12][13][14][15][16][17][18], such as in [12], where Roh showed the existence of the global attractors for the periodic boundary conditions and proved the semiflows was robust with respect to g.The existence and uniqueness of solutions of g-Navier-Stokes equations were proved on R 2 for n=2,3 in [13].Moreover, the existence of global solutions and the global attractor for the spatial periodic and Dirichlet boundary conditions were proved and the dimension of the global attractor was estimated in [14].On the other hand, the global attractor of g-Navier-Stokes equations with linear dampness on R 2 were proved.The estimation of the Hausdorff and Fractal dimensions were also obtained in [15].We investigated the existence of pullback attractors for the 2D non-autonomous g-Navier-Stokes equations on some bounded domains in [16].D. T. Quyet proved the existence of pullback attractor in V g for the continuous process in [17].Recently, we discussed the uniform attractor of g-Navier-Stokes equations with weak dampnesss and time delay in [18], and the corresponding equations have the following forms: For the equation with the restriction of the forcing term f belonging to translational compacted function space, we proved the existence of the uniform attractor by the method of asymptotic compactness.However, as far as we know, the pullback attractor of g-Navier-Stokes equations with nonlinear damping α|u| β−1 u and time delay h(t, u t ) have not been studied, so this is the main motivation of our research.
In this article, we will study pullback asymptotic behavior of solution for the g-Navier-Stokes equations which has nonlinear damping and time delay on some bounded domain Ω ⊂ R 2 , and the usual form as follows: where p(x, t) ∈ R and u(x, t) ∈ R 2 denote the pressure and the velocity respectively, ν > 0 and c|u| β−1 u is nonlinear damping, β is the damping exponent, β ≥ 1 and c > 0 are constant, 0 < m 0 ≤ g = g(x 1 , x 2 ) ≤ M 0 , g = g(x 1 , x 2 ) is a real-valued smooth function.When g = 1, Eq (1.5) become the usual two dimensional Navier-Stokes equations with nonlinear damping and time delay.f = f (x, t) is the external force, h(t, u t ) is another external force term with time delay, u t is the function defined by the relation u t (θ) = u(t + θ), ∀θ ∈ (−r, 0), r > 0 is constant.For the 2D g-Navier-Stokes equations can be seen as a small perturbation of the usual Navier-Stokes equations, so the 2D g-Navier-Stokes equations with nonlinear damping and time delay can be used to describe a certain state of fluid affected by external resistance and historical status.The nonlinear damping term c|u| β−1 u in the balance of linear momentum realizes an absorption if c < 0 and a nonlinear source if c > 0.
By the Faedo-Galerkin method in [19,20], we investigate the global well-posedness of weak solutions for 2D non-autonomous g-Navier-Stokes equations with nonlinear damping and time delay in this article.Then, we prove the existence of pullback attractors using θ-cocycle and the method of pullback condition (PC).Compared with [18], the methods and conclusions are completely different, which can be seen as a further study of related issues.On this basis, inspired by [21][22][23], we can further use the pullback attractor to construct the invariant measures and statistical solutions of 2D g-Navier-Stokes equations and study their statistical solution, invariant sample measures and Liouville type theorem in the future.
The outline of the article is as follows.In the next section, we provide basic definitions and results we use in this article.In Section 3, we prove the global well-posedness of weak solutions and the existence of pullback attractors for 2D non-autonomous g-Navier-Stokes equations with nonlinear damping and time delay.In Section 4, we give some relevant conclusion.

Preliminaries
We define L 2 (g) = (L 2 (Ω)) 2 and Furthermore, H g is endowed with the inner product and norm of L 2 (g), V g is endowed with the inner product and norm of H 1 0 (g), where D(Ω) is the space of C ∞ functions which have compact support contained in Ω, and Let h : R × C H g → (L 2 (Ω)) 2 satisfies the following assumptions: Since the Poincaré inequality holds on Ω: There exists λ 1 > 0 such that then, The g-Laplacian operator is defined as follows: the first equation of (1.5) can be rewritten as follows: A g-orthogonal projection is defined by P g : L 2 (g) → H g and g-Stokes operator with A g u = −P g ( 1 g (∇ • (g∇u))).Applying the projection P g to (1.5), ∀v ∈ V g , ∀t > 0, we obtain ) where b g : , then the formulations (2.4) and (2.5) are equivalent to the following equations: ) where , where C denote positive constants.From [11,12], we have the following inequality: ) From [3,4,16], we have the following concepts and conclusions.
Let Γ be a nonempty set and we define a family {θ t } t∈R of mappings θ t (θ τ γ) = θ t+τ γ for all γ ∈ Γ, t, τ ∈ R, then the operators θ t are called the shift operators.
Let X be a metric space, for any (γ, x) ∈ Γ × X and t, τ ∈ R + , ϕ : R , where θ t is the shift operators.
We denote the metrizable space of function f (s) ∈ X with s ∈ R by L 2 loc (R, X), where X is locally two-power integrable in the Bochner sense.It is equipped with the local two-power mean convergence topology.Lemma 2.1.[16] If H g is Hilbert space and {ω i } i∈N is orthonormal in H g , let f (x, t) ∈ L 2 loc (R; H g ) and there exists a σ > 0, such that for any t ∈ R, where P m : H g → span{ω 1 , . . ., ω n } be an orthogonal projector.

Proofs of the main results
In the section, we will prove the well-posedness of the weak solution for 2D g-Navier-Stokes equations with nonlinear damping and time delay by the Faedo-Galerkin method.Definition 3.1.
), then for every u τ ∈ V g , the Eq (1.5) exist the only weak solution u(t) = u(t; τ, u τ ) ∈ L ∞ (τ, T ; V g ) ∩ L 2 (τ, T ; V g ) ∩ L β+1 (τ, T ; L β+1 (Ω)), and u(t) continuously depends on the initial value in V g .Proof.Let {w j } j≥1 be the eigenfunctions of −∆ on Ω with homogeneous Dirichlet boundary conditions, its corresponding eigenvalues are 0 < λ 1 ≤ λ 2 ≤ . . ., obviously, {w j } j≥1 ⊂ V g forms a Hilbert basis in H g , given u τ ∈ V g and f ∈ L 2 Loc (R; V ′ g ).For every positive integer n ≥ 1, we structure the Galerkin approximate solutions as u n (t) = u n (t; T, u τ ).It has the following form: where γ n, j (t) is determined from the initial values of the following system of nonlinear ordinary differential equations: where ⟨•⟩ is dual product of V g and V ′ g .
According to the results of the initial value problems of ordinary differential equations, there exists a unique local solution to problem (3.1).In the following, we prove that the time interval of the solution can be extended to [τ, ∞).
Using Cauchy's inequality and Young's inequality, we have where We take (3.3) and (3.4) into (3.2) to obtain that is By integrating (3.6) from τ to t, we can obtain For any T > 0 and β ≥ 1, we have and

Since {u
′ n (t)} is bounded in L 2 (τ, T ; V g ), then there exists a subsequence in {u n (t)}, it still denoted by {u n (t)}, we have u n (t) ∈ L 2 (τ, T ; V g ) and u is denseness in V g , taking the limit n → ∞ on both sides of (3.1), we can obtain that u is a weak solution of (1.5).
In the following, the solution is proved to be unique and continuously dependent on initial values.Let u 1 and u 2 be two weak solutions of (1.5) corresponding to the initial values u 1τ , u 2τ ∈ V g , we take Using Hölder inequality and Sobolev embedding theorem, we obtain where C 1 > 0 is any constant. where Since u(s) = 0 for s ≤ 0, we take the maximum in [0, t] for any t ∈ [0, T ], and we obtain We can obtain that the uniqueness of the solution holds after applying the Gronwall inequality.
In the following, we will prove the existence of pullback attractor for (1.5).First, we will prove the existence of pullback absorbing sets.
) be a weak solution of Eq (1.5).Let σ = νλ 1 , for any t ≥ τ, then where Let u(x, t) be a weak solution of Eq (1.5), we obtain 1 2 Hence, where where we have the uniformly absorbing set is a strong solution of (1.5), for any t ≥ τ, then Proof.We suppose u(x, t) be a strong solution of (1.5), multiplying (2.6) by A g u and we have 1 2 Then, 1 2 Since 2c  Proof.The following we will prove that cocycle {ϕ(t, γ, x)} satisfies pullback condition in V g .As (A g ) −1 is continuous compact in H g , we can use spectral theory, there is a sequence as j → ∞, and a family of {ω j } ∞ j=1 of D(A g ), they are orthonormal in H g and A g ω j = λ j ω j , ∀ j ∈ N. We suppose V m = span{ω 1 , ω 2 , . . ., ω m } in V g , P m : V g → V m is orthogonal projector.