The L P -Regularity of Attractor for Three-Dimensional (3D) Navier

Abstract

This paper is concerned with dynamics for the three-dimensional non-autonomous incompressible Navier–Stokes equations with nonlinear damping in a bounded domain. Based on the norm-to-weak uniform attractor theory constructed by our research, the $L^{p}$ -regularity of uniform attractor for the considered model has been achieved via the tail estimate technique and localization approach.

Keywords

Navier–Stokes equations norm-to-weak uniform attractor nonlinear damping

1. Introduction

The three-dimensional (3D) incompressible Navier–Stokes equations describe the conservation law of momentum and mass of a viscous fluid, which attracts much attention of mathematicians, such as the global well-posedness and long-time dynamic systems for the so-called Leray–Hopf weak solutions, see Chepyzhov and Vishik (2001), Cheskidov and Lu (2014), Hopf (1951), Leray (1934), Robinson (2013), Robinson et al. (2016), Temam (1997), Zelik (2015), and more related literature. However, the regularity and dynamics have not been solved yet, which leads several mathematicians to propose approximated systems instead of Navier–Stokes models for investigation, such as the Ladyzhenskaya models, Navier–Stokes–Voigt model, the Navier–Stokes- $α$ equations, and Navier–Stokes fluid flow with nonlinear damping and some others.

This paper is concerned with the $L^{p}$ -regularity of uniform attractors for 3D incompressible non-autonomous Navier–Stokes equations with nonlinear damping, which describes various physical situations such as porous media flows, drag, or friction effects, as well as some other mechanisms, the damping coming from resistance to the motion of the fluid flow. The model can be written as the following initial value system:

\begin{aligned} {\begin{cases} \partial_{t} u - ν Δ u + (u \cdot \nabla) u + α | u |^{β - 1} u + \nabla p = g (t, x), \\ \nabla \cdot u = 0, \\ u(t,x) |_{\partial Ω} = 0, \\ u (τ, x) = u_{τ}, x \in Ω, \end{cases} \end{aligned}

(1.1)

in a bounded domain

Ω \subset R^{3}

with smooth boundary

\partial Ω

for

t \geq τ \in R

, where the function

u (t, x)

is the velocity vector field,

p

denotes the pressure,

ν > 0

is the kinematic viscosity, the length scale

α

is a characteristic parameter of elasticity for the fluid and

β

is a fixed positive parameter.

The global existence of weak and strong solutions has been achieved in Cai and Jiu (2008). Based on the results of Cai and Jiu (2008), the literature, Song and Hou (2011, 2015), established the existence of global and uniform attractors for autonomous and non-autonomous systems of (1.1), as the Gromov–Hausdorff stability of attractors in Tao et al. (2024), which inspired the research on regularity of uniform attractors in this presented paper. For more results about the asymptotic dynamic systems for (1.1), the literature, Jia et al. (2011) and Li et al. (2018), also presented the existence of attractors by using different topologies. To our best knowledge, the regularity of attractors in Banach space but not Hilbert case has not been investigated yet, which needs some new theory of uniform attractors in this research.

The main results and features can be stated as follows: (1)

The long-time behavior for autonomous dissipative partial differential equations has been investigated, within the theoretical framework of the global attractor, in the 1980s; see, for example, Temam (1997) for more details. The global attractor $A_{g}$ for the semigroup ${S (t)}$ associated with the problem satisfies, in the case of autonomous systems: (i) it is invariant under the semigroup ${S (t)}$ ; (ii) the compact set $A_{g}$ attracts all bounded subsets in an appropriate phase space. However, due to the loss of invariance for non-autonomous systems, the theoretical framework of global attractors is invalid. Haraux (1988) used minimality in place of invariance, leading to the uniform global attractor. Considering non-autonomous external forces, such as almost periodic or translation compact, uniform attractors were then proposed, see Chepyzhov and Vishik (2001). One natural question is the sharp condition of the external forces ensuring the existence of uniform attractors. Lu (2006), Lu et al. (2005), and Ma and Zhong (2007) used some new function classes, which are weaker than translation compact external forces and derived the same results. However, the differences between uniform attractors with respect to different non-autonomous external forces are still not clear. Zelik (2015) summarized the above approaches and showed the differences and similarities. This constitutes one inspiration of our paper, that is, we assume that the generic external forces $g (t, x)$ of problem (1.1) have less compactness, the norm-to-weak uniform attractors have been achieved from a theoretical viewpoint, which is the basic theory for our application to system (1.1).

(2)

For the increasing of damping in (1.1) for the critical case, using the framework developed in Section 3, we present the norm-to-weak uniform attractor $A_{σ}^{nw}$ , which is one of the key results in our paper. The bootstrap technique and tail estimates are the essential steps for achieving the uniformly asymptotic compactness for norm-to-weak continuous processes, that is, the $L^{p}$ -regularity of uniform attractors.

The structure of this presented paper is arranged as follows. Section 2 gives some preparation for our research. Then the norm-to-weak uniform attractors theory is stated in Section 3, which has been applied to (1.1) for achieving $L^{p}$ -regularity of uniform attractor in Section 4.

2. Preliminary and Well-Posedness

Let us give some notations and working spaces for preparation. Denoting by $L^{p} (Ω)$ ( $1 \leq p \leq + \infty$ ) and $H^{m} (Ω)$ ( $m \in Z$ ) the usual Banach spaces, we then introduce the spaces $L^{p} = L^{p} (Ω) = (L^{p} (Ω))^{3}$ and $H^{m} = H^{m} (Ω) = (H^{m} (Ω))^{3}$ . Let $E := {u | u \in (C_{0}^{\infty} (Ω))^{3}, div u = 0}$ , $H$ be the closure of $E$ in the $(L^{2} (Ω))^{3}$ topology; $‖ \cdot ‖_{H}$ and $(\cdot, \cdot)$ denote the norm and inner product in $H$ , respectively, that is,

‖ u ‖_{H} = (\int_{Ω} | u |^{2} d x)^{1 / 2}, (u, v) = \sum_{j = 1}^{3} \int_{Ω} u_{j} (x) v_{j} (x) d x, \forall u, v \in H .

Furthermore,

V

is the closure of

E

in the

(H^{1} (Ω))^{3}

topology,

‖ \cdot ‖_{V}

and

((\cdot, \cdot))

denoting the norm and inner product in

V

, respectively, that is,

‖ u ‖_{V}^{2} = ((u, u)), ((u, v)) = \sum_{i, j = 1}^{3} \int_{Ω} \frac{\partial u_{j}}{\partial x_{i}} \frac{\partial v_{j}}{\partial x_{i}} d x, \forall u, v \in V .

It is easy to check that

H

and

V

are Hilbert spaces and denote

W = H^{2} (Ω) \cap V

. Moreover,

V ↪ H \equiv H^{'} ↪ V^{'}

, where

H^{'}

and

V^{'}

are the dual spaces of

H

and

V

, respectively, the injections being dense and continuous. Here,

‖ \cdot ‖_{*}

and

⟨ \cdot, \cdot ⟩

denote the norm in

V^{'}

and the duality product, respectively.

2.1. Some Inequalities

Some inequalities in 3D domains that allow to deal with the convective term will be presented as follows, see Robinson et al. (2016), Temam (1997), and related literature.

Ladyzhenskaya’s inequality:

‖ u ‖_{L^{4}} \leq C ‖ u ‖_{H}^{\frac{1}{4}} ‖ u ‖_{V}^{\frac{3}{4}}, \forall u \in V .

(2.1)

Agmon’s inequality:

\begin{aligned} ‖ u ‖_{L^{\infty}} \leq C ‖ u ‖_{V}^{\frac{1}{2}} ‖ Δ u ‖_{H}^{\frac{1}{2}}, \forall u \in W . \end{aligned}

(2.2)

For the convective term, we define the bilinear and trilinear operators as

B (u, v) := (u \cdot \nabla) v, \forall u, v \in V,

(2.3)

and

\begin{aligned} b (u, v, w) = (B (u, v), w), \forall u, v, w \in V, \end{aligned}

(2.4)

respectively. We write

B (u) = B (u, u)

in short, where

B (u, v)

is a bilinear continuous operator from

V \times V

V^{'}

, and the trilinear operator

b (u, v, w)

satisfies the following properties:

{\begin{cases} b (u, v, v) & = & 0, & \forall u, v, w \in V, \\ b (u, v, w) & = & - b (u, w, v), & \forall u, v, w \in V, \end{cases}

(2.5)

as well as the estimate in the following lemma.

Lemma 2.1.

(See Robinson et al., 2016; Temam, 1997). The bilinear operator $B (u, v)$ and trilinear operator $b (u, v, w)$ satisfy

| b (u, v, w) | \leq C ‖ u ‖_{H}^{\frac{1}{4}} ‖ u ‖_{V}^{\frac{3}{4}} ‖ v ‖_{V} ‖ w ‖_{H}^{\frac{1}{4}} ‖ w ‖_{V}^{\frac{3}{4}}, \forall u, v, w \in V .

(2.6)

2.2. Well-Posedness

The existence of weak and strong solutions can be stated as follows, which is the fundamental preparation for the study of dynamic systems. Cai and Jiu (2008) prove the global well-posedness for problem (1.1) defined on whole space or periodic boundary, which has been extended to non-slip boundary condition in Song and Hou (2011). The strict proof is based on the technique as in Kalantarov and Zelik (2012, 2021), which is omitted here and referred to in another manuscript prepared.

Theorem 2.2 (Existence of global weak solutions)

Suppose that $u_{τ} \in H$ , $g \in L_{loc}^{2} (R; V^{'})$ , and $β > 1$ . Then problem (1.1) possesses at least one weak solution $(u (t, x), p (t, x))$ for any $T > τ$ satisfying $u (t, x) \in L^{\infty} (τ, T; H) \cap L^{2} (τ, T; V) \cap L^{β + 1} (τ, T; L^{β + 1} (Ω))$ , $p \in L^{2} (τ, T; V^{'})$ and the weak form in distribution sense as in Cai and Jiu (2008) and Song and Hou (2011, 2015).

Definition 2.3
We call a pair $(u (t, x), p (t, x))$ is a strong solution if it is weak and satisfies
$u (t, x) \in L^{\infty} (τ, T; V) \cap L^{2} (τ, T; H^{2} (Ω) \cap V) \cap L^{β + 1} (τ, T; L^{β + 1} (Ω)), p (t, x) \in L^{2} (τ, T; L^{2} (Ω)) .$
(2.7)
Theorem 2.4
(Existence of global strong solutions (Song & Hou, 2011, 2015)). Suppose that $u_{τ} \in V \cap L^{β + 1} (Ω)$ and $g \in L_{loc}^{2} (R; H)$ . Then for $β \geq 7 / 2$ , the problem (1.1) has at least one global strong solution $(u (t, x), p (t, x))$ . Moreover, if we assume that $7 / 2 \leq β < 5$ , then the strong solution of problem (1.1) is unique.
3. The Theory of Weak and Strong Uniform Attractors for Abstract Dissipative Equations With Less Regular External Forces

Consider the non-autonomous dissipative system

{\begin{cases} \frac{d u}{d t} = A u + σ (t), \\ u (t = τ, x) = u_{τ}, \end{cases}

(3.1)

with

σ (t) \in E

(

E

is an enveloping topological space). The global solutions in the phase space

Y

generate a family of bi-parameters operators (also called processes)

U_{σ} (t, τ)

satisfying

U_{σ} (τ, τ) = I d and U_{σ} (t, s) U_{σ} (s, τ) = U_{σ} (t, τ)

for

t \geq s \geq τ \in R

Since every trajectory changes with the external forces, the invariance is lost, so the theory of global attractors is no longer valid. Using the translation semigroup $T (s) σ (t) = σ (t + s)$ and translation invariance for external forces to overcome this difficulty, the existence of uniform attractors for (3.1) can be stated as follows.

3.1. The Strong Uniform Attractor of (3.1) With Translation Compact External Forces

If the non-autonomous system is explicit as in (3.1) and the external forces satisfy appropriate conditions such as periodic (quasi or almost periodic, interface) functions, then Chepyzhov and Vishik constructed a new framework to construct the strong uniform attractor, see Chepyzhov and Vishik (2001).

Definition 3.1
Let $X$ be a Banach space with spatial variable. The space $L_{loc}^{p} (R; X)$ denotes all functions with spatial values in the Banach space $X$ and time variable locally $p$ -power integrable in the Bochner sense with norm ${(\int_{t_{1}}^{t_{2}} ‖ \cdot ‖_{X}^{p} d s)}^{1 / p} < + \infty$ for any time interval $[t_{1}, t_{2}] \subset R$ . (1)
The set $Σ = H (σ_{0})$ is defined as the symbol space of the system,
$H (σ_{0}) = {\bar{{σ_{0} (\cdot + s) | s \in R}}}^{E},$
for every fixed $σ_{0} \in E$ .
(2)
The space $L_{b}^{2} (R; X)$ denotes the translation bounded functions satisfying
$‖ σ ‖_{L_{b}^{2} (R; X)}^{2} = sup_{t \in R} \int_{t}^{t + 1} ‖ σ (s) ‖_{X}^{2} d s < + \infty,$
for $σ \in L_{b}^{2} (R; X)$ .
(3)
We say that $σ \in L_{loc}^{2} (R; X)$ is translation compact in $L_{loc}^{2} (R; X)$ if $H (σ)$ is compact in $L_{loc}^{2} (R; X)$ . The corresponding space is denoted as $L_{c}^{2} (R; X)$ ; $L_{c}^{2, w} (R; X)$ is the weak translation compact space which is defied by $L_{c}^{2} (R; X)$ with weak topology.
(4)
Based on the definition of the symbol space, we can define the skew product flow as
$S (t) = (U_{σ} (t, τ), T (t)) \in Y \times Σ,$
(3.2)
which satisfies
$S (t) (u (s), σ (s)) = (U_{σ (s)} (t, s) u (s), T (t) σ (s)) = (u (t + s), σ (t + s)),$
(3.3)
and possesses the semigroup property for every $(u (s) σ (s)) \in Y \times Σ$ , where the family of processes ${U_{σ} (t, τ)}$ satisfies the translation identity
$U_{σ (t)} (t + s, τ + s) = U_{T (s) σ (t)} (t, τ), \forall σ (t) \in Σ, t \geq τ, τ, s \in R .$
(3.4)

Remark 3.2
If $σ_{0} \in L_{loc}^{2} (R; X)$ is fixed, then for every $σ \in H (σ_{0})$ , we have $‖ σ ‖_{L_{loc}^{2} (R; X)}^{2} \leq C ‖ σ_{0} ‖_{L_{loc}^{2} (R; X)}^{2}$ , where $‖ \cdot ‖_{L_{loc}^{2} (R; X)} = \int_{τ}^{T} ‖ \cdot ‖_{X}^{2} d s$ for arbitrary $t \in [τ, T] \subset R$ .
Theorem 3.3. (See Chepyzhov & Vishik, 2001)

For the non-autonomous system (3.1), we suppose that

(1)
$Σ$ is the symbol space and is invariant under the continuous translation semigroup ${T (h) | h \in R}$ defined above;
(2)
the family of processes ${U_{σ} (t, τ)} (σ \in Σ)$ acting on the phase space $Y$ , which satisfies the translation identity (3.4) is uniformly asymptotically compact and $(Y \times Σ, Y)$ -continuous;
(3)
there exists a uniformly bounded absorbing set $B$ with respect to every symbol $σ \in Σ$ ; and
(4)
the skew product flow is defined as in (3.2): $Π_{1}$ and $Π_{2}$ are the projectors from $Y \times Σ$ into $Y$ and $Σ$ , respectively, defined as $Π_{1} (u, σ) = u$ and $Π_{2} (u, σ) = σ$ .

Then the skew product flow possesses the global attractor $A = ⋃_{σ \in Σ} K_{σ} (τ) \times {σ}$ , which satisfies (a)
invariance: $S (t) A = A$ ,
(b)
$Π_{1} A = A_{Σ} = ⋃_{σ \in Σ} K_{σ} (τ)$ is the uniform attractor for ${U_{σ} (t, τ)}$ , $Π_{2} A = Σ$ ; here $K_{σ} (τ)$ is the section at $t = τ$ of the kernel $K_{σ}$ for the family of processes ${U_{σ} (t, τ)}$ with symbol $σ \in Σ$ and $K_{σ} (τ)$ consists of all bounded complete trajectories for the process.

Proof.
See, for example, Chepyzhov and Vishik (2001).

From the framework by Chepyzhov and Vishik, we see that the external forces $σ (t, x)$ in (3.1) need to be translation compact. One natural question is to search for the optimal hypotheses for the existence of uniform attractors.
Definition 3.4
We call $g \in L_{b}^{p} (R; X)$ normal if there exists an $η > 0$ such that
$sup_{t \in R} \int_{t}^{t + η} ‖ g (s) ‖_{X}^{p} d s \leq ε$
holds for arbitrary $ε > 0$ and $p > 1$ .

Moreover, if we choose the weak topology here, then we call the space as weak normal, which is equivalent to the translation bounded space.
3.2. The Weak Uniform Attractor of (3.1) With Less Compact External Forces

Using weaker conditions on $σ (t)$ , that is, considering less regular classes such as weak translation bounded $L_{b, w}^{p} (R; X)$ or time (space) regular functions, then the existence of the weak uniform attractor $A_{w}$ for (3.1) can be obtained, see Zelik (2015).

Definition 3.5 (Weakly continuous process)

We call the bi-parameters operators ${U_{σ} (t, τ) : Y \to Y}$ a family of weakly continuous processes with $σ \in Σ$ if

(i)
$U_{σ} (τ, τ) = I d$ ;
(ii)
$U_{σ} (t, s) U_{σ} (s, τ) = U_{σ} (t, τ)$ ;
(iii)
for given $t \in R$ and fixed $x_{n} ⇀ x$ in $Y$ , $σ_{n} \to σ$ in $H_{σ} \subset L_{loc, w}^{p} (R; X)$ , there holds $U_{σ_{n}} (t, τ) x_{n} ⇀ U_{σ} (t, τ) x$ in $Y$ as $n \to + \infty$ .

Definition 3.6 (Weak uniform attractor, see Zelik, 2015)

Let $H_{σ}$ be the symbol space in $L_{loc, w}^{p} (R; X)$ . The set $A_{w} \subset Y \times H_{σ}$ is the weak global attractor of the skew product flow $S (t) = (U_{σ} (t, τ), T (t))$ if

(i)
$A_{w}$ is a compact set in $Y \times H_{σ}$ endowed with the weak topology by the embedding $Y \times H_{σ} \subset Y_{w} \times L_{loc, w}^{p} (R; X)$ ;
(ii)
$A_{w}$ is invariant under $S (t)$ ; and
(iii)
$A_{w}$ attracts all bounded sets of $Y \times H_{σ}$ in the weak topology of $Y \times H_{σ}$ for the space $Y_{w} \times L_{loc, w}^{p} (R; X)$ .

Then the projection $Π_{1} A_{w}$ is the weak uniform attractor, $Π_{1} A_{w} = A^{w}$ .
Definition 3.7
(1)
(Space regular functions, see Zelik, 2015) A function $g \in L_{b}^{p} (R; X)$ is space regular if for any $ε > 0$ , there exists a finite-dimensional subspace $X_{ε} \subset X$ , $\dim X_{ε} < + \infty$ , and a function $g_{ε} \in L_{b}^{p} (R; X_{ε})$ such that
$‖ g - g_{ε} ‖_{L_{b}^{p} (R; X)} \leq ε .$

(2)
(Time regular functions, see Zelik, 2015) A function $h \in L_{b}^{p} (R; X)$ is time regular if for any $ε > 0$ , there exists a function $h_{ε} \in C_{b}^{k} (R; X)$ such that
$‖ h - h_{ε} ‖_{L_{b}^{p} (R; X)} \leq ε .$

Remark 3.8. (See Zelik, 2015)

A function $h \in L_{b}^{p} (R; X)$ is normal if and only if it is time and space regular. From Theorem 3.8, we can see that the space or time regular functions are weaker than the normal class.

Lemma 3.9
(1)
Let the function $g \in L_{b}^{p} (R; X)$ be translation bounded and assume that there exists $g_{ε} \in L_{b}^{p} (R; X)$ such that
$‖ g - g_{ε} ‖_{L_{b}^{p} (R; X)} \leq ε .$
(3.5)
Then for any $h \in H_{g}$ , there exists $h_{ε} \in H_{g_{ε}}$ such that
$‖ h - h_{ε} ‖_{L_{b}^{p} (R; X)} \leq ε .$
(3.6)
(2)
Assume that $g \in L_{b}^{p} (R; X)$ is time regular and let $h_{n} \in H_{g}$ be such that $h_{n} \to h$ in $L_{b, w}^{p} (R; X)$ . Then for any $ε > 0$ , there exists a sequence $φ_{n} \in H_{b}^{k} (R; X)$ such that $φ_{n} \to φ$ in $H_{b, w}^{k} (R; X)$ and
$‖ h_{n} - φ_{n} ‖_{L_{b}^{p} (R; X)} \leq ε, ‖ h - φ ‖_{L_{b}^{p} (R; X)} \leq ε .$
(3.7)
(3)
Assume that $g \in L_{b}^{p} (R; X)$ is space regular, $X = H^{k} (Ω)$ $(k \in Z)$ with a bounded domain $Ω \subset R^{n}$ and let $h_{n} \in H_{g}$ be such that $h_{n} \to h$ in $L_{b, w}^{p} (R; X)$ . Then for any $ε > 0$ , there exists a sequence $φ_{n} \in L_{b}^{p} (R; H^{k} (Ω))$ for all $k \in N$ such that $φ_{n} \to φ$ in $L_{b, w}^{p} (R; H^{k} (Ω))$ and
$‖ h_{n} - φ_{n} ‖_{L_{b}^{p} (R; X)} \leq ε, ‖ h - φ ‖_{L_{b}^{p} (R; X)} \leq ε .$
(3.8)

Proof.
See Zelik (2015).
Theorem 3.10
(1)
Assume that the family of processes ${U_{h} (t, τ) : Y \to Y, h \in H_{σ}}$ associated with problem (3.1) is uniformly dissipative, uniformly asymptotically compact and weakly continuous and that the symbol space (hull) is weak or strong as in Lemma 3.9. Then the skew product flow $S (t) : Y \times H_{σ} \to Y \times H_{σ}$ possesses the weak global attractor $A_{w}$ , which can be projected onto the weak uniform attractor $A^{w}$ .
(2)
If the hypotheses of Theorem 3.10 hold and, moreover, the family of processes $U_{h} (t, τ) : Y \to Y$ is uniformly asymptotically compact with respect to the symbol $h \in H_{σ}$ , then the weak uniform attractor in Theorem 3.10 is strong in $Y$ .

Proof.
See Chepyzhov and Vishik (2001).
3.3. The Norm-to-Weak Uniform Attractor of (3.1) With Less Compact External Forces

Since the continuity of the skew product flow determines the phase space and the strong (weak) cases have been presented above, we will use the norm-to-weak continuity of the skew product flow to achieve our abstract result as follows (see the original work by Zhong et al. (2006)).

Definition 3.11 (Norm-to-weak continuous process)

Let the operators ${U_{σ} (t, τ) | t \geq τ}$ be a family of processes on $Y$ with symbol $σ \in Σ$ . We say that this family of operators is norm-to-weak continuous on $Y$ if

(i)
$U_{σ} (τ, τ) = I d$ ;
(ii)
$U_{σ} (t, s) U_{σ} (s, τ) = U_{σ} (t, τ)$ ; and
(iii)
for given $t \in R$ and fixed $x_{n} \to x$ in $Y$ , $σ_{n} \to σ$ in $H_{σ}$ , there holds $U_{σ_{n}} (t, τ) x_{n} ⇀ U_{σ} (t, τ) x$ .

Definition 3.12 (Norm-to-weak uniform attractor)

Let $H_{σ}$ be the symbol space in $L_{loc, w}^{p} (R; X)$ . A set $A_{nw} \subset Y \times H_{σ}$ is the norm-to-weak global attractor of the skew product flow $S (t) = (U_{σ} (t, τ), T (t))$ if

(i)
$A_{nw}$ is a compact set in $Y \times H_{σ}$ endowed with the weak topology via the embedding $Y \times H_{σ} \subset Y_{w} \times L_{loc, w}^{p} (R; X)$ ;
(ii)
$A_{nw}$ is invariant under the norm-to-weak continuous skew product flow $S (t)$ ;
(iii)
$A_{nw}$ attracts all bounded sets of $Y \times H_{σ}$ in the weak topology of $Y \times L_{loc, w}^{p} (R; X)$ .

Then the projection $Π_{1} A_{nw}$ is the norm-to-weak uniform attractor, $Π_{1} A_{nw} = A^{nw}$ .
Theorem 3.13
Assume that the family of processes ${U_{h} (t, τ) : Y \to Y, h \in H_{σ}}$ associated with problem (3.1) is uniformly dissipative and norm-to-weak continuous; the symbol space (hull) can be weak or strong as in Lemma 3.9. Then the skew product flow $S (t) : Y \times H_{σ} \to Y \times H_{σ}$ possesses the norm-to-weak global attractor $A_{nw}$ which can be projected onto the norm-to-weak uniform attractor $A^{nw}$ .
Proof.
Using the framework of Chepyzhov and Vishik (2001), the existence of uniform attractors needs dissipation, continuity, and compactness of the skew product flow $S (t)$ . Since $S (t)$ is norm-to-weak continuous, we take the initial datum $u_{0} \in Y$ with the strong topology and $σ \in H_{g}$ ( $H_{g}$ can be weak or strong). Moreover, we consider the uniform dissipation in $Y$ with the strong norm and the uniformly asymptotic compactness for the skew product flow in $Y_{w} \times H_{g}$ with the weak topology. Then by the framework of Zhong et al. (2006), the global attractor $A_{nw}$ for $S (t)$ can be constructed, together with the projection onto the first component, yielding the uniform attractor $A^{nw} = Π_{1} A_{nw}$ .
4. Main Result: The $L^{p}$ -Regularity of Uniform Attractor for (1.1)

The objective here is to construct the norm-to-weak uniform attractor in $L^{p}$ .

Lemma 4.1. (See Zhong et al., 2006)

Assume that the family of processes ${U_{σ} (τ, t)}$ is norm-to-weak and strongly continuous in $L^{\tilde{p}} (Ω)$ and $L^{\tilde{q}} (Ω)$ , respectively, for $\tilde{p} > \tilde{q} \geq 2$ , and that ${U_{σ} (τ, t)}$ has a uniformly bounded absorbing set in $L^{\tilde{p}} (Ω)$ . Moreover, assume that the processes $U_{σ} (τ, t)$ satisfy

(i)
${U_{σ} (τ, t)}$ has a uniformly bounded absorbing set in $L^{\tilde{q}} (Ω)$ ;
(ii)
for any $ε > 0$ and $B \in B (E)$ , there exists $M = M (ε, B)$ such that $\int_{| U_{σ} (τ, t) u_{τ} | \geq M} | U_{σ} (τ, t) u_{τ} |^{\tilde{p}} d x \leq ε, \forall u_{τ} \in B .$

Then the family of processes possesses the uniform attractor in $L^{\tilde{p}} (Ω)$ .
Theorem 4.2 (The norm-to-weak uniform attractor in $L^{β + 1} (Ω)$ )

Assume that $7 / 2 \leq β \leq 5$ , $u_{τ} \in V = V \cap L^{β + 1} (Ω)$ and $g \in H (g_{0})$ , where $g_{0} (t) \in L_{b}^{(2 β - 1) / (β - 1)} (R; L^{(2 β - 1) / (β - 1)} (Ω)) \subset L_{b}^{2} (R; H)$ is arbitrarily fixed, space regular and satisfies the incompressibility condition. Then since $V \subset L^{β + 1} (Ω)$ , the norm-to-weak continuous processes $U_{g} (t, τ) : V \to L^{β + 1} (Ω)$ generated by the global strong solutions of problem (1.1) possess the norm-to-weak uniform attractor $A_{L^{β + 1}}^{nw}$ in the topology of $L^{β + 1} (Ω)$ .

Proof.

In view of Lemma 4.1, inspiring from Wang et al. (2007), we divide our proof into the following steps.

$∙$ Step 1: Dissipativity of the processes $U_{g} (t, τ)$ in $L^{β} (Ω)$ .

Lemma 4.3

Suppose that the hypotheses of Theorem 4.2 hold. Then the norm-to-weak continuous processes $U_{g} (t, τ) : V \to L^{β + 1} (Ω)$ generated by the global strong solutions possess a uniformly bounded absorbing set $B_{L^{β} (Ω)}$ determined by $‖ u ‖_{L^{β}}^{β} \leq ρ^{β}$ in the space $L^{β} (Ω)$ , where

ρ^{β} = ‖ u_{τ} ‖_{L^{β} (Ω)}^{β} + \frac{β}{2} \frac{1}{1 - e^{- α β / 2}} ‖ g_{0} ‖_{L_{b}^{2} (R; H)}^{2} + ρ_{1}^{1} (‖ u ‖_{V}, ‖ Δ u ‖_{H}, β, | Ω |, α),

(4.1)

and here

ρ_{1}^{1}

is a positive constant.

Proof.

Let $Q = [τ, T] \times Ω$ , $δ > 0$ be an arbitrary constant, denote $D = {\cdot | \cdot \in C_{0}^{\infty} ([τ, T] \times Ω), \nabla \cdot φ = 0}$ and $\tilde{D} = {\cdot | \cdot \in C_{0}^{\infty} (Ω), \nabla \cdot φ = 0}$ , $ψ_{δ} \in D$ with $ψ (T, \cdot) = 0$ be such that $ψ_{δ} \geq 0$ and $\int_{Q} ψ_{δ} (x) d x = 1$ , $ϕ_{δ} \in \tilde{D}$ such that $ϕ_{δ} \geq 0$ and $\int_{Ω} ϕ_{δ} (x) d x = 1$ . By the establishment of $u_{δ} (t, x) = ψ_{δ} * [θ_{m} (ϕ_{δ} * u)]$ and $p_{δ} (t, x) = ψ_{δ} * [θ_{m} (ϕ_{δ} * p)]$ , then we have $\nabla \cdot u_{δ} = 0$ with $u_{δ} |_{\partial Ω} = 0$ . Denoting $u_{δ} = u^{δ}$ and $p_{δ} = p^{δ}$ , considering the regularized equation of (1.1) as

\begin{aligned} {\begin{cases} \partial_{t} u^{δ} - ν Δ u^{δ} + ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)] + α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)] + \nabla p^{δ} = g^{δ} (t), \\ \nabla \cdot u^{δ} = 0, \\ u^{δ} (t = τ) = u_{τ}^{δ}, \end{cases} \end{aligned}

(4.2)

and taking the inner product of (4.2) by

| u^{δ} |^{β - 1} u^{δ}

yields

\begin{aligned} \frac{1}{β} \frac{d}{d t} ‖ u^{δ} (t) ‖_{β}^{β} + ν \int_{Ω} (- Δ) u^{δ} | u^{δ} (t) |^{β - 2} u^{δ} d x + (α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) \\ = - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) + (g^{δ} (t), | u^{δ} |^{β - 1} u^{δ}), \end{aligned}

(4.3)

where

\begin{aligned} (\nabla p^{δ}, | u^{δ} |^{β - 1} u^{δ}) = - \int_{Ω} [p^{δ} | u^{δ} |^{β - 1} (\nabla \cdot u^{δ}) + p^{δ} | u^{δ} |^{β - 3} (\nabla \cdot (u^{δ})^{2}) \cdot u^{δ}] d x = 0. \end{aligned}

By the Hölder and Young inequalities, using the hypotheses and denoting $C_{1} = C_{1} (| Ω |, α, β)$ , we obtain

\begin{aligned} \int_{Ω} g^{δ} (t) \cdot | u^{δ} |^{β - 1} u^{δ} d x \leq & ‖ g^{δ} (t) ‖_{H} ‖ u^{δ} ‖_{L^{2 β - 2} (Ω)}^{β - 1} \\ \leq & \frac{1}{2} ‖ g^{δ} (t) ‖_{H}^{2} + \frac{α}{4} ‖ u^{δ} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} + C_{1} . \end{aligned}

(4.4)

Define a smooth cut-off function $θ_{m} (x)$ as

\begin{aligned} θ_{m} (x) = {\begin{cases} 0, dist (x, \partial Ω) \leq \frac{1}{m}, \\ smooth\ function,\ otherwise, \\ 1, dist (x, \partial Ω) \geq \frac{2}{m} . \end{cases} \end{aligned}

(4.5)

Using the properties of the trilinear operator and combining with (2.2), we have

‖ u ‖_{L^{\infty} (Ω)} \leq C ‖ u ‖_{V}^{\frac{1}{2}} ‖ Δ u ‖_{H}^{\frac{1}{2}}, ‖ u^{δ} ‖_{L^{\infty} (Ω)} \leq C ‖ u^{δ} ‖_{V}^{\frac{1}{2}} ‖ Δ u^{δ} ‖_{H}^{\frac{1}{2}},

(4.6)

hence

\begin{aligned} | - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) | \\ = | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) | \\ + | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) | \\ + | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) |, \end{aligned}

(4.7)

with

\begin{aligned} | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \\ \leq | (ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \\ + | (ψ_{δ} * [θ_{\infty} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) |, \end{aligned}

(4.8)

and

\begin{aligned} | (ψ_{δ} * [θ_{\infty} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \\ \leq ‖ u^{δ} ‖_{L^{\infty} (Ω)} ‖ u^{δ} ‖_{V} ‖ u^{δ} ‖_{L^{2 β - 2} (Ω)}^{β - 1} \\ \leq C ‖ u^{δ} ‖_{V}^{\frac{3}{2}} ‖ Δ u^{δ} ‖_{H}^{\frac{1}{2}} ‖ u^{δ} ‖_{L^{2 β - 2} (Ω)}^{β - 1} \\ \leq C ‖ u^{δ} ‖_{V}^{6} ‖ Δ u^{δ} ‖_{H}^{2} + C (\int_{Ω} | u^{δ} |^{2 β - 2} d x)^{4 / 3 (β - 1)} \\ \leq C ‖ u^{δ} ‖_{V}^{6} ‖ Δ u^{δ} ‖_{H}^{2} + \frac{α}{4} ‖ u^{δ} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} + C_{2}, \end{aligned}

(4.9)

and

\begin{aligned} | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \\ | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \\ | (ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \end{aligned}

(4.10)

δ \to 0

and

m \to + \infty

, where

C_{2} = C_{2} (| Ω |, α, β)

By the boundedness of $u^{δ}$ yields

\begin{aligned} | - (α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) | \\ = | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) | \\ + | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) | \\ + | (α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], | u^{δ} |^{β - 1} u^{δ}) |, \end{aligned}

(4.11)

with

\begin{aligned} | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \\ | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \\ | (α ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \to 0, \end{aligned}

(4.12)

δ \to 0

and

m \to + \infty

, and

| (α ψ_{δ} * [θ_{\infty} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], | u^{δ} |^{β - 1} u^{δ}) | \leq α \int_{Ω} | u^{δ} |^{2 β - 1} d x .

(4.13)

Integrating by parts in the second term on the left-hand side of (4.3), we find

ν \int_{Ω} (- Δ) u^{δ} | u^{δ} |^{p - 2} u d x = ν \int_{Ω} | u^{δ} |^{p - 2} | \nabla u^{δ} |^{2} d x \geq 0.

(4.14)

By interpolation, it follows that

\begin{aligned} ‖ u^{δ} ‖_{L^{β} (Ω)}^{β} \leq | Ω |^{(β - 1) / (2 β - 1)} ‖ u^{δ} ‖_{L^{2 β - 1} (Ω)}^{β} \leq & \frac{β - 1}{2 β - 1} | Ω |^{(2 β - 1) / β} + \frac{β}{2 β - 1} ‖ u^{δ} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} \\ \leq & C_{3} (| Ω |, β) + ‖ u^{δ} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} . \end{aligned}

(4.15)

Combining (4.3)–(4.15), passing to the limit

δ \to 0

, then we conclude that

\frac{d}{d t} ‖ u (t) ‖_{L^{β} (Ω)}^{β} + ν β \int_{Ω} | u |^{p - 2} | \nabla u |^{2} d x + \frac{α β}{2} ‖ u ‖_{L^{β} (Ω)}^{β} \leq C ‖ u ‖_{V}^{6} ‖ Δ u ‖_{H}^{2} + \frac{β}{2} ‖ g (t) ‖_{H}^{2} + \hat{C},

(4.16)

where

\hat{C} = C_{1} + C_{2} + C_{3} > 0

. Then by the uniform Gronwall inequality, from the uniform boundedness of

u

L^{\infty} (τ, T; V) \cap L^{2} (τ, T; W)

and neglecting the second term on the left-hand side of (4.16), it is easy to achieve our lemma by choosing an appropriate

ρ

as in (4.1), which means the existence of a uniformly bounded absorbing set in

L^{β} (Ω)

$∙$ Step 2: Uniformly asymptotic compactness of the processes in $L^{β} (Ω)$ with $7 / 2 \leq β \leq 5$ .

By the compact embedding $V ↪↪ L^{β} (Ω)$ , which is uniform with respect to the symbol, we can obtain the uniformly asymptotic compactness for the family of processes. $∙$ Step 3: Uniformly asymptotic compactness of the processes in $L^{β + 1} (Ω)$ containing the critical case.

Lemma 4.4

Suppose that the processes $U_{g} (τ, t)$ of problem (1.1) possess a bounded absorbing set $B_{L^{β} (Ω)}$ in $L^{β} (Ω)$ and all hypotheses in Theorem 4.2 hold. Then for any $ε > 0$ , there exists a positive constant $M > 0$ such that we have the tail estimate $\int_{| U_{g} (τ, t) u_{τ} | \geq M} | U_{g} (τ, t) u_{τ} |^{β + 1} d x \leq ε, \forall u_{τ} \in B_{L^{β} (Ω)},$ which implies that the tail estimate holds also for the critical case.

Proof.

Considering the regularized equation of (1.1),

\begin{aligned} {\begin{cases} \partial_{t} u^{δ} - ν Δ u^{δ} + ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)] + α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)] + \nabla p^{δ} = g^{δ} (t), \\ \nabla \cdot u^{δ} = 0, \\ u^{δ} (t = τ) = u_{τ}^{δ}, \end{cases} \end{aligned}

(4.17)

and defining

(u - M)_{+}

as the positive part of

(u - M)

\begin{aligned} (u - M)_{+} = {\begin{cases} u - M & u \geq M, \\ 0 & u < M, \end{cases} \end{aligned}

denoting

(u - M)_{+}^{δ}

as the regularized function for

(u - M)_{+}

, which satisfies

\nabla \cdot (u - M)_{+}^{δ} = 0

, noting that

\begin{aligned} | - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], ((u - M)_{+}^{δ})^{β}) | \\ \leq | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)], ((u - M)_{+}^{δ})^{β}) | \\ + | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], ((u - M)_{+}^{δ})^{β}) | \\ + | (ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], ((u - M)_{+}^{δ})^{β}) | \\ + | (ψ_{δ} * [θ_{\infty} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], ((u - M)_{+}^{δ})^{β}) |, \end{aligned}

(4.18)

with

\begin{aligned} | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)], ((u - M)_{+}^{δ})^{β}) | \to 0, \\ | (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u^{δ} \cdot \nabla) u)] - (ψ_{δ} * [θ_{m} ϕ_{δ} * ((u \cdot \nabla) u)], ((u - M)_{+}^{δ})^{β}) | \to 0, \\ | (ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * ((u^{δ} \cdot \nabla) u^{δ})], ((u - M)_{+}^{δ})^{β}) | \to 0, \end{aligned}

δ \to 0

and

m \to + \infty

In addition,

\begin{aligned} | - (α ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], ((u - M)_{+}^{δ})^{β}) | \\ = | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)], ((u - M)_{+}^{δ})^{β}) | \\ + | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], ((u - M)_{+}^{δ})^{β}) | \\ + | (α ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], ((u - M)_{+}^{δ})^{β}) | \\ + | (α ψ_{δ} * [θ_{\infty} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], ((u - M)_{+}^{δ})^{β}) |, \end{aligned}

(4.19)

with

\begin{aligned} | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)], ((u - M)_{+}^{δ})^{β}) | \to 0, \\ | α (ψ_{δ} * [θ_{m} ϕ_{δ} * (| u^{δ} |^{β - 1} u)] - ψ_{δ} * [θ_{m} ϕ_{δ} * (| u |^{β - 1} u)], ((u - M)_{+}^{δ})^{β}) | \to 0, \\ | (α ψ_{δ} * [(θ_{m} - θ_{\infty}) ϕ_{δ} * (| u^{δ} |^{β - 1} u^{δ})], ((u - M)_{+}^{δ})^{β}) | \to 0, \end{aligned}

δ \to 0

and

m \to + \infty

Then using a similar technique and letting $δ \to 0$ and $m \to + \infty$ , we conclude that

\begin{aligned} \frac{1}{β + 1} \int_{Ω} \frac{d u}{d t} (u - M)_{+}^{β + 1} d x + \int_{Ω} ν (- Δ) u (u - M)_{+}^{β} d x \\ + \int_{Ω} B (u, u) (u - M)_{+}^{β} d x + \int_{Ω} α | u |^{β - 1} u (u - M)_{+}^{β} d x \\ = \int_{Ω} g (t) (u - M)_{+}^{β} d x . \end{aligned}

(4.20)

Noting that

\int_{Ω} ν (- Δ) u (u - M)_{+}^{β} = \int_{Ω} ν β (u - M)_{+}^{β - 1} | \nabla u |^{2} d x > 0

and if

u

is large enough, there exists

M_{1} = M_{1} (p) > 0

such that

u - M_{1} >> M

, yielding

\begin{aligned} \int_{Ω} α | u |^{β - 1} u (u - M)_{+}^{β} d x \\ = \frac{1}{2} \int_{Ω} α | u |^{β - 1} (u - M + M) (u - M)_{+}^{β} d x + \frac{1}{2} \int_{Ω} α | u |^{β - 1} u (u - M)_{+}^{β} d x \\ \geq \frac{α}{2} M_{1}^{β - 1} ‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} + \frac{α}{2} M M_{1}^{β - 1} ‖ (u - M)_{+} ‖_{L^{β} (Ω)}^{β} \\ + \frac{α}{2} M_{1} \int_{Ω} | u |^{β - 1} (u - M)_{+}^{β} d x \\ \geq \frac{α}{2} M_{1}^{β - 1} ‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} + \frac{α}{2} M M_{1}^{β - 1} ‖ (u - M)_{+} ‖_{L^{β} (Ω)}^{β} \\ + \frac{α}{2} M_{1} ‖ (u - M)_{+} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} . \end{aligned}

(4.21)

By the Young and Hölder inequalities, using the properties of the trilinear operator and the compact embedding

V ↪↪ L^{\hat{β}} (Ω)

with

\hat{β} \in [1, 6)

in the special case

V ↪↪ L^{(4 β - 2) / β} (Ω)

since

β \in [7 / 2, 5]

, we have

\begin{aligned} - \int_{Ω} B (u, u) (u - M)_{+}^{β} d x \\ \leq \int_{Ω} | u | | \nabla u | (u - M)_{+}^{(β - 1) / 2} (u - M)_{+}^{(β + 1) / 2} d x \\ \leq ν β \int_{Ω} (u - M)_{+}^{β - 1} | \nabla u |^{2} d x + \frac{4}{ν β} \int_{Ω} (u - M)_{+}^{β + 1} | u |^{2} d x \\ \leq ν β \int_{Ω} (u - M)_{+}^{β - 1} | \nabla u |^{2} d x + \frac{α M_{1}}{4} ‖ (u - M)_{+} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} \\ + \frac{C_{α, ν, β}}{M_{1}^{q_{β}}} ‖ u ‖_{L^{(4 β - 2) / β - 2} (Ω)}^{2} \\ \leq ν β \int_{Ω} (u - M)_{+}^{β - 1} | \nabla u |^{2} d x + \frac{α M_{1}}{4} ‖ (u - M)_{+} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} \\ + \frac{C_{α, ν, β, λ_{1}}}{M_{1}^{q_{β}}} ‖ u ‖_{V}^{[2 (5 β - 7)] / (5 β - 2)} + \frac{C_{α, ν, β, λ_{1}}}{M_{1}^{{\hat{q}}_{β}}} ‖ u ‖_{W}^{2}, \end{aligned}

(4.22)

where

{\hat{q}}_{β} > 0

and

q_{β} > 0

C_{α, ν, β}

and

C_{α, ν, β, λ_{1}}

are constants which depend on

β

α

ν

, and

λ_{1}

Next, let ${\hat{q}}_{β}$ and ${\hat{C}}_{α, β}$ be positive constants as above. Using the hypotheses on the external force $g (t)$ and employing the Young and Hölder inequalities, we obtain

\int_{Ω} g (t) (u - M)_{+}^{β} d x \leq \frac{{\hat{C}}_{α, β}}{M_{1}^{{\hat{q}}_{β}}} \int_{Ω} | g |^{(2 β - 1) / (β - 1)} d x + \frac{α M_{1}}{4} ‖ (u - M)_{+} ‖_{L^{2 β - 1} (Ω)}^{2 β - 1} .

(4.23)

Combining estimates (4.18)–(4.23), we can achieve

\begin{aligned} \frac{d}{d t} ‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} + \frac{α}{2} (β + 1) M_{1}^{β - 1} ‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} \\ \leq \frac{{\hat{C}}_{α, β} (β + 1)}{M_{1}^{{\hat{q}}_{β}}} ‖ g (t) ‖_{L^{(2 β - 1) / (β - 1)} (Ω)}^{(2 β - 1) / (β - 1)} + \frac{C_{α, ν, β, λ_{1}}}{M_{1}^{q_{β}}} ‖ u ‖_{V}^{[2 (5 β - 7)] / (5 β - 2)} + \frac{C_{α, ν, β, λ_{1}}}{M_{1}^{{\hat{q}}_{β}}} ‖ u ‖_{W}^{2} . \end{aligned}

Since the external force

g

is assumed to be space regular, then for any

ε > 0

, there exists a sequence

φ \in L_{b}^{(2 β - 1) / (β - 1)} (R; V)

such that

‖ g - φ ‖_{L_{b}^{(2 β - 1) / (β - 1)} (R; L^{(2 β - 1) / (β - 1)} (Ω))} \leq ε .

By the uniform Gronwall inequality, this yields

\begin{aligned} ‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} & \leq ‖ u_{τ} ‖_{L^{β + 1} (Ω)}^{β + 1} e^{- (α / 2) (β + 1) M_{1}^{β - 1} (t - τ)} \\ + \frac{{\hat{C}}_{α, β} (β + 1)}{M_{1}^{{\hat{q}}_{β}}} (‖ φ (t) ‖_{L_{b}^{(2 β - 1) / (β - 1)} (R; L^{(2 β - 1) / (β - 1)} (Ω))} + {\hat{C}}_{ε}) \\ + \frac{C_{α, ν, β, λ_{1}} (β + 1)}{M_{1}^{q_{β}}} ‖ u ‖_{L^{\infty} (τ, T; V)}^{[2 (5 β - 7)] / (5 β - 2)} + \frac{C_{α, ν, β, λ_{1}} (β + 1)}{M_{1}^{{\hat{q}}_{β}}} ‖ u ‖_{L^{2} (τ, T; W)}^{2} . \end{aligned}

(4.24)

For arbitrary

ε > 0

, choosing a large enough

M_{1}

, we can achieve

‖ (u - M)_{+} ‖_{L^{β + 1} (Ω)}^{β + 1} \leq ε

for

u > M_{1}

Next, we choose $(u + M)_{-}$ as test function. Using the procedure above again, there exists a constant $M_{2}$ such that $‖ (u + M)_{-} ‖_{L^{β + 1} (Ω)}^{β + 1} \leq ε$ for $- u < - M_{2}$ .

In conclusion, choosing $M = max {M_{1}, M_{2}}$ , then we end up with $\int_{Ω (| U_{g} (τ, t) u_{τ} | \geq M)} | u |^{β + 1} d x \leq ε,$ which finishes the proof of Lemma 4.4.

$∙$ Step 4: End of the proof. By virtue of Lemmas 4.3 and 4.4, based on Step 2, and using Lemma 4.1, since the processes are norm-to-weak continuous, we can finish our proof. $◻$

Remark 4.5

Our results are based on the existence of a global solution, which is studied in Cai and Jiu (2008) and Song and Hou (2011, 2015). However, the proof by Galerkin’s method is not strict and needs more technique to overcome the difficulty of convective term, nonlinear damping, and pressure. The strict proof for a periodic case is proved in Yang et al. (2025).

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Postgraduate Education Reform and Quality Improvement Project of Henan Province (No. YJS2023JC23), the international communication project from the Chinese Educational Ministry (No. HZKY20220270), the international communication project from the Department of Science and Technology in Henan province (No. 242102521039), Workshop of Outstanding Foreign Scientists in Henan Province (No. GZS2024007) and Xingjie Yan was partly supported by the Fundamental Research Funds for the Central Universities (Grant No. 2024KYJD2001).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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The L P -Regularity of Attractor for Three-Dimensional (3D) Navier–Stokes Equations With Damping

Abstract

Keywords

1. Introduction

2.1. Some Inequalities

Theorem 2.2 (Existence of global weak solutions)

Definition 3.5 (Weakly continuous process)

Definition 3.11 (Norm-to-weak continuous process)

(i) U σ ( τ , τ ) = I d ; (ii) U σ ( t , s ) U σ ( s , τ ) = U σ ( t , τ ) ; and (iii) for given t ∈ R and fixed x n → x in Y , σ n → σ in H σ , there holds U σ n ( t , τ ) x n ⇀ U σ ( t , τ ) x . Definition 3.12 (Norm-to-weak uniform attractor)

Lemma 4.1. (See Zhong et al., 2006)

Footnotes

Funding

Declaration of Conflicting Interests

References

(i)
$U_{σ} (τ, τ) = I d$ ;
(ii)
$U_{σ} (t, s) U_{σ} (s, τ) = U_{σ} (t, τ)$ ; and
(iii)
for given $t \in R$ and fixed $x_{n} \to x$ in $Y$ , $σ_{n} \to σ$ in $H_{σ}$ , there holds $U_{σ_{n}} (t, τ) x_{n} ⇀ U_{σ} (t, τ) x$ .

Definition 3.12 (Norm-to-weak uniform attractor)