The uniform asymptotic behavior of solutions for 2D g-Navier-Stokes equations with nonlinear dampness and its dimensions

1.   Introduction

It is well-known that the Navier-Stokes equations are the typical evolution equations and widely used in the field of science and engineering. The attractors of Navier-Stokes equations are studied by many scholars in the fields of dynamical systems for a long time (see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]} and reference therein). Especially in recent years, there are many research achievements on g-Navier-Stokes equation. In ^[16,17,18], Roh deduced the 2D g-Navier-Stokes equations from 3D Navier-Stokes equations on thin region. It can be viewed as a perturbation of the usual Navier-Stokes equations. Bae et al. studied the well-posedness of weak solution for the 2D g-Navier-Stokes equations. Kwak et al. researched the global attractor and its fractal dimension of 2D g-Navier-Stokes equations in ^[19]. In ^{[20,21,22,23,24]}, Jiang et al. studied global and the pullback attractor for g-Navier-Stokes equation. Moreover, the long-time behavior for 2D non-autonomous g-Navier-Stokes equations and the stability of solutions to stochastic 2D g-Navier-Stokes equations were studied by Anh in ^[25,26], The stationary solutions and its pullback attractor are researched in ^[27]. On the basis of the above research, we have studied the long time properties for g-Navier-Stokes equation with weakly dampness and time delay in ^[28] recently.

In this manuscript, the uniform attractor of the g-Navier-Stokes equations with nonlinear dampness is researched. Its usual form is as follows:

In (1.1), we can see that $u(t, x)\in {\mathrm{R}}^2$ and $p(t, x)\in {\mathrm{R}}$ denote the velocity and pressure respectively. $\nu > 0$ is the viscosity coefficient, $c|u|^{\beta-1}u$ denotes nonlinear dampness. $c > 0$ and $\beta\geq 1$ are positive constant. $f = f(x, t)$ is the external force term, $0 < m_{0}\leq g = g(x_{1}, x_{2})\leq M_{0}$ and $g = g(x_{1}, x_{2})$ is a suitable smooth function, Let $c = 0$ and $g = 1$, the Eq (1.1) will become the usual 2D Navier-Stokes equations.

This manuscript is organized as follows. In Section 2, we recall some basic results of 2D g-Navier-Stokes equations, then we give the concept about process families and uniform attractor. In Section 3, the global well-posedness of weak solutions for 2D g-Navier-Stokes equations with nonlinear dampness is studied. In Section 4, by the energy equation method, the existence of the uniform attractor of 2D g-Navier-Stokes equation with nonlinear dampness is proved on the unbounded domain. In Section 5, the dimension estimation of the uniform attractor in the quasi-periodic case is obtained.

2.   Preliminaries

We assume $\Omega$ is a smooth unbounded domian of ${\mathrm{R}}^2$, Let $L^{2}(g) = (L^{2}(\Omega))^{2}$ and we denote $(u, v) = \int_{\Omega}{u\cdot \nu gdx}$ and $|\cdot| = (\cdot, \cdot)^{1/2}, \; u, v\in L^{2}(g).$ Let $H^{1}_{0}(g) = (H^{1}_{0}(\Omega))^{2}$, Set

and $||\cdot|| = ((\cdot, \cdot))^{1/2}, u = (u_{1}, u_{2}), v = (v_{1}, v_{2})\in H^{1}_{0}(g).$ We denote $D(\Omega)$ be the space of $\mathcal{C}^{\infty}$ functions with compact support contained in $\Omega$. So we have the following spaces

where $H_g$ and $V_g$ endowed with the inner product and norm of $L^{2}(g)$ and $H_{0}^{1}(g)$ respectively.

We assume that there exists $\lambda_{1} > 0$, such that

This Poincar$\acute{e}$-type inequality imposes some restrictions on the geometry of the domain $\Omega$.

The g-Laplacian operator is defined as follows:

The first equation of (1.1) can be rewritten as follows:

In ^[16], g-orthogonal projection and g-Stokes operator are defined respectively by $P_g:L^{2}(g)\rightarrow H_g$ and $A_g u = -P_g(\frac{1}{g}(\nabla \cdot (g\nabla u))).$ Applying the projection $P_g$ on the Eq (2.2), we have the following weak formulation of (1.1).

where $b_{g}:V_g\times V_g\times V_g\rightarrow \mathrm{R}$ and

we have

Then the formula (2.3) and (2.4) are equivalent to the following functional equations

We denote

From ^[16,17,19], we have

where $B(u) = B(u, u) = P_g(u\cdot \nabla)u$ is defined by

A family of two parametric maps $\{U_f(t, \tau)\} = \{U_f(t, \tau)|t\geq \tau, \tau\in \mathrm{R}\}$ is defined in $H_g$ as follows:

The following concepts and conclusions are given from ^[7]. $\forall\; f\in L^\infty({\mathrm{R}}^+; V_g')$, the translation operator is defined in $L^\infty({\mathrm{R}}^+; V_g')$ as follows.

Obviously

We set $\Sigma = \{T(h)f(x, s) = f(x, s+h), \; \forall\; h\in {\mathrm{R}}\}$, where $T(\cdot)$ is the positive invariant semigroups which act on $\Sigma$ and satify $T(h)\Sigma\subset \Sigma, \; \forall \; h\geq 0$ and

Let $\rho_{\mathcal{F}} > 0$ be constant, $\Sigma\subset \{f\in {L^\infty({\mathrm{R}}^+; V_g')}:{||f||}_{L^\infty({\mathrm{R}}^+; V_g')}\leq \rho_{\mathcal{F}}\}.$ For $\{U_f(t, \tau)\}$ with $f\in \Sigma$, we call the parameter $f$ as the symbols of the process family $\{U_f(t, \tau)\}$, and $\Sigma$ as the symbol space.

Definition 2.1 ^[7] A family of two-parametric maps $\{U(t, \tau)\}$ is called a process in $H_g$, if

(1)$U_f(t, s)U_f(s, \tau) = U_f(t, \tau), \; \; \; \forall\; t\geq s\geq \tau, \; \; \tau\in \mathrm{R},$

(2)$U_f(\tau, \tau) = Id, \; \; \; \; \tau\in \mathrm{R}.$

Let $E$ be the Banach space, ${\mathcal{B}}(E)$ is denoted the set of all bounded sets on $E$, then

Definition 2.2 ^[7] A set $B_0\subset E$ is said to be uniformly absorbing for the family of processes $\{U_f(t, \tau)\}, f\in \Sigma\}$, if for any $\tau \in {\mathrm{R}}$ and each $B\in {\mathcal{B}}(E)$, there exists $t_0 = t_0(\tau, B)\geq \tau$, such that for all $t\geq t_0$,

Definition 2.3 ^[7] A set $P\subset E$ is said uniformly atttracting set of $\{U_f(t, \tau)\}, f\in \Sigma\}$, if for any $\tau \in {\mathrm{R}}$, there is

Definition 2.4 ^[7] A closed set ${\mathcal{A}}_\Sigma\subset E$ is said to be the uniform attractor of the family of processes $\{U_f(t, \tau)\}, \; f\in \Sigma\}$, if

(1) ${\mathcal{A}}_\Sigma\subset E$ is uniformly attractive;

(2) ${\mathcal{A}}_\Sigma\subset E$ is included in any uniformly attracting set ${\mathcal{A}}'$ of $\{U_f(t, \tau)\}, f\in \Sigma\}$, that is ${\mathcal{A}}_\Sigma\subset {\mathcal{A}}'.$

3.   The well-posedness of the solution for 2D g-Navier-Stokes equations with nonlinear dampness in unbounded domain

In the section we will prove the well-posedness of the solution for 2D g-Navier-Stokes equations with nonlinear dampness by the Faedo-Galerkin method.

Definition 3.1 Let $u_0\in H_g, f\in L_{Loc}^2({\mathrm{R}}; V_g')$, For any $\tau\in {\mathrm{R}}$, $u\in L^\infty(\tau, T; V_g)\cap L^2(\tau, T; V_g)\cap L^{\beta+1}(\tau, T; L^{\beta+1}(\Omega)), \forall T > \tau$ is called a weak solution of problem $(1.1)$ if it fulfils

$\frac{d}{dt}u(t)+\nu A_gu(t)+B(u(t))+c|u|^{\beta-1}u+\nu R(u(t)) = f(x, t)\; \; \; \; {\rm{on}}\; {\mathcal{D}}'(\tau, +\infty; V_g'), u(\tau) = u_0$.

Theorem 3.1 Let $\beta\geq 1$, $f\in L_{Loc}^2({\mathrm{R}}; V_g^{'})$, Then for every $u_\tau\in V_g$, the equations $(1.1)$ have a unique weak solution $u(t) = u(t; \tau, u_\tau)\in L^\infty(\tau, T; V_g)\cap L^2(\tau, T; V_g)\cap L^{\beta+1}(\tau, T; L^{\beta+1}(\Omega))$, and $u(t)$ is continuously depending on the initial value in $V_g$.

Proof. Let $\{w_j\}_{j\geq 1}$ be the eigenfunctions of $-\Delta$ on $\Omega$ with homogeneous Dirichlet boundary conditions, Its corresponding eigenvalues are $0 < \lambda_1\leq\lambda_2\leq \ldots,$ Obviously, $\{w_j\}_{j\geq 1}\subset V_g$ forms a Hilbert basis in $H_g$, given $u_\tau\in V_g$ and $f\in L_{Loc}^2({\mathrm{R}}; V_g^{'})$.

For any positive integer $n\geq 1$, we structure the Galerkin approximate solutions as $u_n(t) = u_n(t; T, u_\tau)$, It has the following form

where $\gamma_{n, j}(t)$ is determined from the initial values of the following system of nonlinear ordinary differential equations.

where $\langle \cdot\rangle$ is dual product of $V_g$ and $V_g^{'}$.

According to the results of the initial value problems of ordinary differential equations, we have that there exists a unique local solution of (3.1). In the following, we prove that the time interval of the solution can be extended to $[\tau, \infty)$.

Using Cauchy's inequality and Young's inequality, we have

where $||\cdot ||_*$ is norm of $V_g^{'}$. We take (3.3) into (3.2) to obtain

That is

By integrating (3.5) from $\tau$ to $t$, we have

For any $T > 0$ and $\beta\geq 1$, we obtain

So $u_n(t)\in L^\infty(\tau, T; V_g)$. Therefore $B(u_n(t))\in L^\infty(\tau, T; V_g^{'}),$ $|u_n(t)|^{\beta-1}u_n(t)\in L^{\beta+1}(\tau, T; L^{\beta+1}(\Omega))$. As a result,

so $\{u_n^{'}(t)\}$ is bounded in $L^2(\tau, T; V_g)$.

Then we deduce that there is a subsequence in $\{u_n(t)\}$, which is still denoted by $\{u_n(t)\}$. We obtain $u_n(t)\in L^2(\tau, T; V_g)$ and $u_n^{'}(t)\in L^2(\tau, T; V_g)$ such that

(i)$u_n(t)\rightarrow u(t)$ is weakly $^*$ convergent in $L^\infty(\tau, T; V_g)$;

(ii)$u_n(t)\rightarrow u(t)$ is weakly convergent in $L^2(\tau, T; V_g)$;

(iii)$|u_n(t)|^{\beta-1}u_n(t)\rightarrow \xi$ is weakly convergent in $L^{\beta+1}(\tau, T; L^{\beta+1}(\Omega))$;

(iv)$u_n^{'}(t)\rightarrow u^{'}(t)$ is weakly convergent in $L^2(\tau, T; V_g)$;

(v)$u_n(t)\rightarrow u(t)$ is strongly convergent in $L^2(\tau, T; H_g)$;

(vi)$u_n(t)\rightarrow u(t), \; \; a\; e\; (x, t)\in \Omega \times [\tau, T]$.

From Lemma 1.3 of ^[29], we can see $\xi = |u|^{\beta-1}u$. Since $\bigcup_{n\in N^+}{span}\{w_1, w_2, \cdots, w_n\}$ is denseness in $V_g$, Taking the limit $n\rightarrow \infty$ on both sides of Eq (3.1), we can obtain that $u$ is a weak solution of $(1.1)$.

In the following, the solution is proved to be unique and continuously dependent on initial values. Let $u_1, u_2$ be two weak solutions of $(1.1)$ corresponding to the initial values $u_{1\tau}, u_{2\tau}\in V_g$, We take $u = u_1-u_2$, From $(2.3)$ we have

Using Hölder inequality and Sobolev embedding theorem, we obtain

we have

where $C_1 > 0$ is any constant.

where $\alpha = \frac{\nu ||\nabla g||_\infty}{2m_0 \lambda_1^{1/2}}$. so

Thus

Let $C$ be a constant and $C = \alpha-2\lambda_1(\nu-C_1-\alpha) > 0$, then

Therefore

So we prove the continuous dependence on the initial value. When $u_{1\tau} = u_{2\tau}$, that is $u_\tau = 0$, then the uniqueness of the solution holds.

4.   The uniform attractor of 2D g-Navier-Stokes equations with nonlinear dampness in unbounded domain

In the following we have that the family of processes $\{U_f(t, \tau)\}, \; f\in \Sigma$ is uniformly bounded $(w.r.t.f\in \Sigma)$ and it has uniform absorbing sets.

Firstly, the existence of uniformly absorbing sets is proved. Taking the inner product of (2.5) with $u$, we have

Then

For $\beta\geq 1$, we obtain

where $\gamma = 1-\frac{2|\nabla g|_\infty}{m_0 \lambda_1^{1/2}} > 0$ for sufficiently small $|\nabla g|_\infty$. Using the Gronwall inequality, we have

and from

we have

So

Integrating (4.1) in s from 0 to t, we have

then we know that the family of processes corresponding to $u$ is uniformly bounded, and

is uniformly absorbing set in $H_g$. Then the following lemma holds.

Lemma 4.1 Let $\Sigma$ be symbolic space, The process family corresponding to Eq $(1.1)$ is uniformly bounded in $L^\infty ({\mathrm{R}}^+; H_g)\cap L^2(\tau, T; V_g)$ and there is a uniform absorbing set in $H_g$.

Lemma 4.2 Let $\tau\geq 0$, $u_{\tau_n}$ be the sequence in $H_g$ that weakly converges to $u_\tau \in H_g$, $f_n \in \Sigma$ is the sequence in $L^\infty({\mathrm{R}}^+; V_g^{'})$ that weakly converges to $f$, then

$(1)$ For $\forall \; t > \tau$, $U_{f_n}(t, \tau)u_{\tau_n}$ is weakly converges to $U_f(t, \tau)u_\tau$ in $H_g$;

$(2)$ For $\forall\; T > \tau$, $U_{f_n}(\cdot, \tau)u_{\tau_n}$ is weakly converges to $U_f(\cdot, \tau)u_\tau$ in $L^2(\tau, T; V_g)$.

The proof is similar to Lemma 3.2 of ^[7], so it is omitted.

As we know, when $u_{\tau_n}$ is bounded in $H_g$, $f_n\subset \Sigma$, $t_n\rightarrow +\infty$. If $\{U_{f_n}(t_n, \tau)u_{\tau_n}\}$ is precompact in $H_g$, then the family of processes $\{U_f(t, \tau)\}, \; f\in \Sigma$ is asymptotically compact. So we construct an energy functional $[\cdot, \cdot]:V_g\times V_g\rightarrow {\mathrm{R}}$ as follows:

Obviously $[\cdot, \cdot]$ is bilinear and symmetric, and

Let $|\nabla g|_\infty$ be sufficiently small in (4.2), such that $\frac{|\nabla g|_\infty}{m_0 \lambda_1^{1/2}} < \frac{1}{4}$. Hence

Since

Given $u = u(t) = U_f(t, \tau)u_\tau, \; \; u_\tau\in H_g, \; \; t\geq \tau\geq 0$, Then we have

That is $\forall\; u_\tau\in H_g, \; t\geq \tau \geq 0$, we obtain

Lemma 4.3 Let $\{U_f(t, \tau)\}_{f\in \Sigma}$ is the family of processes of Eq $(1.1)$, then $\{U_f(t, \tau)\}_{f\in \Sigma}$ is uniformly asymptotically compact.

Proof. Let $B\subset H_g$ is bounded, $u_{\tau_n}\in B, \; f_n\in \Sigma$ and $t_n\in {\mathcal{R}}^+$ is satisfied $t_n\rightarrow +\infty(n\rightarrow +\infty)$. From Lemma 4.1, we have a constant $M(B, \tau) > \tau$ and

There exists sufficiently large $t_n\geq M(B, \tau)$, such that $U_{f_n}(t_n, \tau)B\subset B_0$. then $\{U_{f_n}(t_n, \tau)u_{\tau_n}\}$ is weakly precompact in $H_g$. For $w\in B_0\subset H_g$, we can deduce that $U_{f_{n^{s}}}(t_{n^{s}}, \tau)u_{\tau_{n^{s}}}$ is weakly convergent to $w$ in $H_g$. Similarly $\forall\; T > 0$ and $t_{n^{s}}\geq T+M(B, \tau)$, we obtain $U_{f_{n^{s}}}(t_{n^{s}}-T, \tau)u_{\tau_{n^{s}}}\in B_0$. The same to $w_T\in B_0$, we can take $n^{s}$, $\forall T > 0$, so we have $u_{t_{n^{s}}} = U_{f_{n^{s}}}(t_{n^{s}}-T, \tau)u_{\tau_{n^{s}}}$ is weakly convergent to $w_T$ in $H_g$. According to the definition of process and translation operator, we have

Let $g_{T, n^{s}} = T(t_{n^{s}}-T)f_{n^{s}}$, we denote $\lim_{n^{s}H_w}$ as weak limit in $H_g$, then

thus

Now we will prove

$\forall T > 0$, we have $w^k = U_{g_T^k}(T, 0)w_T$. When $t_{n^{s}}\geq T+M(B, \tau)$, we obtain

Obviously

From Lemma 4.2, we obtain

So

For

thus

From $w^k = U_{g_T^s}(s, 0)w_T$, we have

$\forall T > 0$, we have

From $w = U_{g_{T}}(T, 0)w_T$, by the Lemma 3.3 of ^[7] and Lemma 4.2, we can obtain $w^k\rightarrow w$ in $H_g$. So there exists any sufficiently small $\varepsilon > 0$, such that $|w^k|^2\leq |w|^2+\varepsilon$. Since

When $\varepsilon\rightarrow 0$, $T\rightarrow \infty$, we have

Let $B\subset H_g$ be any bounded set, we have

and $v\in \omega_{\tau, \Sigma}(B)$ iff there exists a sequence $v_n\in B, \; \; f_n\in \Sigma, \; \; t_n\in [\tau, +\infty)$. When $n\rightarrow \infty$, we have $t_n\rightarrow +\infty$ and $U_{f_n}(t_n, \tau)v_n\rightarrow v$ in $H_g$. When $\{U_f(t, \tau)\}, \; f\in \Sigma\}$ is uniformly asymptotically compact, $t\rightarrow +\infty$, we have

We will obtain the minimization of the uniform attractor in the following.

Lemma 4.4 Let $\{U_f(t, \tau)\}, \; f\in \Sigma$ is any the family of processes, $B_0$ is uniformly absorbing set, ${\mathcal{A}}_\Sigma = \omega_{0, \Sigma}(B_0)$. then ${\mathcal{A}}_\Sigma$ is contained in any uniform absorbing set of $\{U_f(t, \tau)\}, \; f\in \Sigma$.

Proof. $\forall \; \tau > 0, \; \forall\; B\subset H_g$, Suppose there is another bounded closed set $P\subset H_g$ which satisfies

where ${\mathcal{A}}_\Sigma$ is not contained in the $P$. We deduce there is at least one $v\in {\mathcal{A}}_\Sigma$ and $v\notin P.$ Since $v\in {\mathcal{A}}_\Sigma = \omega_{0, {\mathcal{F}}}(B_0)$, From the definition of the uniform $\omega$ limit set, there is a sequence $v_n\in B, \; \; f_n\in \Sigma, \; \; t_n\in [\tau, +\infty)$, as $n\rightarrow \infty$, we have $t_n\rightarrow +\infty$, then $U_{f_n}(t_n, 0)v_n\rightarrow v$ is obtained in $H_g$. Given $\tilde{v}_n = U_{f_n}(t_n, 0)v_n$, when $n\rightarrow +\infty$, There must be $U_{f_n}(t_n, \tau)\tilde{v}_n\rightarrow v$. Let $\tilde{v}_n\in B$, then we obtain $v\in P$, It is contradiction, so ${\mathcal{A}}_\Sigma\subset P$.

Theorem 4.1 Let $\{U_f(t, \tau), \; f\in \Sigma\}$ is a family of processes of Eq $(1.1)$, Then the process family has a unique compact uniform attractor ${\mathcal{A}}_\Sigma\subset H_g$. where ${\mathcal{A}}_\Sigma = \omega_{0, \Sigma}(B_0)$, $B_0$ is any uniform absorbing set corresponding to a family of processes.

5.   The dimension estimation of the uniform attractor in the quasi-periodical case

When $f(x, t) = f(x, w_1(t), w_2(t), \ldots, w_k(t))$ is a quasi-periodic function, That is, there exists a set of rational independent real numbers $\alpha_1, \ldots, \alpha_k$ which satisfies $f(x, \alpha_1 t, \ldots, \alpha_i t+2\pi, \ldots, \alpha_k t) = f(x, \alpha_1 t, \ldots, \alpha_i t, \ldots, \alpha_k t)\; \; \; (1\leq i\leq k).$ Here $w_1(t+\alpha_1) = w_1(t), w_2(t+\alpha_2) = w_2(t), \ldots, w_k(t+\alpha_k) = w_k(t)$ and $\alpha_1, \alpha_2, \ldots, \alpha_k$ are rational independent.

Let $\alpha t = (\alpha_1 t, \ldots, \alpha_k t)$, $\alpha = (\alpha_1, \ldots, \alpha_k)$, $w(t) = (w_1(t), \ldots, w_k(t)) = [\alpha t+w_0] = (\alpha t+w_0)mod(2\pi)^k$, $w_0 = (w_{01}, \ldots, w_{0k})\in T^k = [0, 2\pi]^k, F(x, w(t)) = f(x, t).$ we can obtain the following conclusion.

Theorem 5.1 Let ${\mathcal{A}}$ is the uniform attractor of $(1.1)$, then its Hausdorff and Fractal dimensions are estimated as follows:

where

Proof. We transform the Eq $(1.1)$ into the following forms of autonomous systems by semigroup $S(t)(u_0, w_0) = (U_{w_0}(t, 0)u_0, T_1(t)w_0)$,

Let $y(t) = (u(x, t), w(t))^T$, $M(y(t)) = (\nu \Delta_g u-B(u, u)-\nu (\frac{\nabla g}{g}\cdot \nabla)u-c|u|^{\beta-1}u-\nabla p+F(x, w(t)), \alpha)^T$.

Then we can write the Eqs (5.1) and (5.2) as follows,

$\forall {y_0}\in {\mathcal{A}}$, where $y(t) = (u(t), w(t))^T$ is the solution of Eqs (5.1) and (5.2) and $y_0$ as initial value. The linearized equation of (5.1) in $y(t)$ is

In the equation of (5.6), $z(t) = (v(t), w(t))^T, \; \; \mu(t) = (\mu_1(t), \cdots, \mu_k(t)), \; \; z_0 = (v_0, \mu_0)^T\in H_g\times T^k$,

while

Let

then

$b$ is any positive constant, Let $(M(y(t))z, z) = (M_1 v, v)+(M_2 \mu, \mu)$,

$I_k$ is identity operator in ${\mathrm{R}}^k$. Thus operator

is block operator, $(v_1, 0), \cdots, (v_{n-k}, 0)$ are respectively solution of $(2.7)$ with $(\xi_1, 0), \cdots, (\xi_{n-k}, 0)$ as the initial value, where $\xi_1, \cdots, \xi_{n-k}$ is linearly independent basis in $H_g$, $\phi_1, \cdots, \phi_{n-k}$ is unit orthogonal basis of $span\{v_1, \cdots, v_{n-k}\}$, ${\tilde{\mu}}_{n-k+1}, \cdots, {\tilde{\mu}}_n$ is unit orthogonal basis of ${\mathrm{R}}^k$, then $(\phi_1, 0), \cdots, (\phi_{n-k}, 0)$, $(0, {\tilde{\mu}}_{n-k+1}), \cdots, (0, {\tilde{\mu}}_n)$ is unit orthogonal basis of $H_g\times {\mathrm{R}}^k$. Let $\theta_i = (\phi_i, 0)\; (i = 1, \cdots, n-k), v_i = (0, \widetilde{\mu_i}), \; (n-k+1\leq i\leq n).$

We make $m_1 = 1-\frac{|\nabla g|_\infty}{m_0\lambda_1^{1/2}}-\frac{c|u|^{\beta-1}}{\lambda_1}$, then

We take $b = \frac{\nu m_1\lambda_1}{2G}$, then

Since

That is

We take $m_2 = 1-\frac{2|\nabla g|_\infty}{m_0 \lambda_1^{1/2}}-\frac{|u|^{\beta-1}}{\lambda_1}$, then

So $||u||^2\leq \frac{|f|^2}{\nu^2 \lambda_1 m_2}$. therefore

Let

where $[\cdot]_*$ denotes trunc, then we have $q_{n_0} < 0$. so

We take another

then $q_{n_1} < 0$, and $\max_{1\leq j\leq {n_1-1}}\frac{(q_j)_+}{q_{n_1}} < 1$, thus

6.   Conclusions

In this paper, using a priori estimates of the solutions and the energy equation method, we show how to control the nonlinear dampness and obtain the uniform attractor of the g-Navier-Stokes equation on unbounded domain. Meanwhile, the dimension of the uniform attractor is estimated in the quasi-periodic case. The methods in this paper can bring some inspiration for the research of 3D Navier-Stokes equations in the future.

From a theoretical point of view, it is important to analysis the connection between Navier-Stokes equations and g-Navier-Stokes equations. So it is of great significance to study the dynamics for the g-Navier-Stokes equations. To obtain more research results for the study of g-Navier-Stokes equations in the next research, we may continue the research in this line, extending the case of Lebesgue space $L^2$ to the case of $L^{2, \lambda}$, for suitable $0 < \lambda < 2$. On the other hand, we may consider that the pullback asymptotic behavior of solutions for 2D g-Navier-Stokes equations with nonlinear dampness on the unbounded domain.

Acknowledgments

The author would like to thank the referees for the helpful suggestions. This work is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11971378).

Conflict of interest

The authors declare this work does not have any conflicts of interest.