Boundary Value Problems volume 2015, Article number: 243 (2015)
Under suitable hypotheses on the continuous delay, distributed delay, and the initial data in this paper, the large-time behavior for the 2D Navier-Stokes-Voigt equations with continuous delay and distributed delay on the Lipschitz domain is studied. The existence of pullback attractors in the non-smooth domain was obtained via verifying some pullback dissipation and asymptotical compactness for the continuous process.
The Navier-Stokes equations essentially constitute a description of hydrodynamical systems, and the well-posedness and large-time behavior of solutions to the Navier-Stokes equations have received very much attention in the understanding of fluid motion and turbulence. For the 2D case, Ladyzhenskaya [1] solved the uniqueness of the global smooth solutions. Based on the well-posedness of the incompressible Navier-Stokes equations (NSE), the infinite dimensional dynamical systems also was investigated. Till now there are many interesting results (the existence of global attractor, uniform attractors, pullback attractors, the structure and dimensions of attractors), the 2D general case can be found in [2–7], and these results were studied in the regular domain. But the well-posedness of global solutions for the Navier-Stokes equations in the non-regular domain (such as the Lifchitz domain) is a difficult problem, and there are fewer corresponding results. For the 2D incompressible Navier-Stokes equation, Brown et al. [8] constructed a stream function which will be used later in our paper to solve the non-homogeneous boundary value problems and gave the existence, dimension of global attractor.
Very recently, using an approximation of the Navier-Stokes equation (such as the Navier-Stokes-Voigt equation (NSVE)) to study the existence of attractor of classical NSE has become a topic generally focused on. The Navier-Stokes-Voigt system is the classical Navier-Stokes system with strong damping which models the dynamics of a Kelvin-Voigt viscoelastic incompressible fluid and was first introduced by Oskolkov [9] as a model of the motion of linear viscoelastic fluids. More results about the well-posedness, the existence of attractor and the infinite dimensional systems, we can see [10, 11]. Also we mention the large-time behavior for the NSV system with delay, especially with both the continuous delay and the distributed delay, which are similar to the memory in elastic system.
The Navier-Stokes equations with delay was studied recently. In 2002, Krasovskii [12] constructed the Navier-Stokes equations with delay and obtained the well-posedness. Barbu and Sritharan [13] established the existence and uniqueness of weak solutions to the Navier-Stokes equations with the forcing term containing delay in 2003. Taniguchi [14] established the existence of absorbing sets of the non-autonomous Navier-Stokes equations with continuous delay in 2005. Caraballo and Real [15–17] studied the Navier-Stokes system with continuous delay and distributed delay and obtained the existence of global and pullback attractors for autonomous and non-autonomous cases, respectively. Marín-Rubio and Real [18] investigated the Navier-Stokes equation with delay on some unbounded domain with the Poincaré inequality and obtained the pullback attractors. Garrido-Atienza and Marín-Rubio [19] also studied the Navier-Stokes equations with delay on an unbounded domain and proved some results on the existence and uniqueness of solutions. In 2014, García-Luengo et al. [20] studied the 2D Navier-Stokes system with the convective term and external force both containing delay and proved the existence of pullback attractors.
In this paper we consider the existence of pullback attractor of the 2D incompressible Navier-Stokes-Voigt equation with continuous delay and distributed delay on the Lipschitz domain,
where \(\Omega\subset\mathbb{R}^{2}\) is a Lipschitz domain, \(\Omega _{\tau }=\Omega\times(\tau,+\infty)\), \(\partial\Omega_{\tau}=\partial \Omega \times(\tau,+\infty)\), \(\Omega_{\tau h}=\Omega\times(\tau-h,\tau)\), \(\tau\in\mathbb{R}\) is the initial time. ν is the kinematic viscosity of the fluid, u is the unknown velocity field of the fluid, p the pressure, and \(\alpha>0\) a length scale parameter characterizing the elasticity of the fluid, \(f(t-\rho(t),u(t-\rho(t)))\) the external force term which contains a memory effect during a fixed interval of time of length \(h>0\), and \(\rho(t)\) a given adequate delay function. Moreover, the inhomogeneous boundary function φ satisfies \(\varphi\in L^{\infty}(\partial\Omega)\), \(\int _{-h}^{0}G(s,u(t+s))\,ds\) is another external force with some hereditary characteristic, and ϕ the initial state of delay in \((-h,0)\) where \(h>0\) is a constant.
Inspired by [8, 17], we shall use the background function for the Stokes problem and some pullback dissipation, and asymptotical compactness for the continuous process via the embedding theorem to achieve the pullback attractor. The main features of our present work are summarized as follows:
(1) There are many results as regards the existence of attractors of NS (NSV) equations with delay on the regular domain (such as [15–17]), many conclusions concerning the NS (NSV) equations without delay on the non-regular domain (such as the Lipschitz domain in [8]), but less work about the existence of attractors of NSV equations with continuous delay and distributed delay on the Lipschitz domain.
(2) Since our problem is studied on the Lipschitz domain (not the regular domain), we use Hardy’s inequality and smooth approximation functions to deal with the non-regular boundary. Defining some suitable topology spaces for the solutions and dealing with each delay term to get some a priori estimate on non-smooth domain for the pullback absorbing sets and asymptotical compactness, we conclude to the existence of a pullback attractor for (1.1).
The structure of this paper is the following. In Section 2, some preliminaries are given which will be used in sequel. The existence and uniqueness of solution for our problem are derived in Section 3. In Sections 4 and 5, the existence of pullback attractors for the problem (1.1) is derived in the appropriate topology space.
Denote \(E:=\{u|u\in(C^{\infty}_{0}(\Omega))^{2}, \operatorname {div}u=0\}\), H is the closure of the set E in \((L^{2}(\Omega))^{2}\) topology, \(\vert \cdot \vert \) and \((\cdot,\cdot)\) represent the norm and inner product in H, respectively, i.e.,
V is the closure of the set E in \((H^{1}(\Omega))^{2}\) topology, and \(\Vert \cdot \Vert \) and \(((\cdot,\cdot))\) denote the norm and inner product in V, respectively, i.e.,
P is the Helmholtz-Leray orthogonal projection in \((L^{2}(\Omega))^{2}\) onto the space H, \(A:=-P\Delta\) is the Stokes operator, the sequence \(\{\omega_{j}\}^{\infty}_{j=1}\) is an orthonormal system of eigenfunctions of A, and \(\{\lambda_{j}\}^{\infty}_{j=1}\) (\(0<\lambda_{1}\leq\lambda_{2}\leq\cdots\)) is the eigenvalue of A corresponding to the eigenfunction \(\{\omega_{j}\}^{\infty}_{j=1}\). We can define the power \(A^{s}\) for \(s\in\mathbb{C}\) as follows:
\(D(A^{s})\) is the domain of \(A^{s}\), and we still denote the closure of E in \(D(A^{s})\) by \(D(A^{s})\). The norm of \(D(A^{\frac{s}{2}})\) is written as \(\Vert u\Vert _{s}\), and \(A^{s}\) has the following properties (see [8]):
where V is a Hilbert space, and \(\Vert v\Vert =\vert \nabla v\vert \). Clearly, \(V\hookrightarrow H\equiv H'\hookrightarrow V'\), \(H'\) and \(V'\) are dual spaces of H and V, respectively, where the injection is dense and continuous. The norm \(\Vert \cdot \Vert _{*}\) and \(\langle\cdot \rangle \) denote the norm in \(V'\) and the dual product between V and \(V'\), respectively.
The bilinear form operator and trilinear form operator are defined as follows (see [21]):
where \(B(u,v)\) is a linear continuous operator from V to \(V'\) which maps W into H, and \(b(u,v,w)\) satisfies
and we introduce some useful inequalities, lemmas, and definitions.
Young’s inequality:
The Poincaré inequality:
The Gagliardo-Nirenberg interpolation inequality:
Hardy’s inequality:
Let X and Y be Banach spaces, \(X\subset Y\), we say that X is compactly embedded in Y, written as
provided
\(\Vert x\Vert _{Y}\leq C\Vert x\Vert _{X}\) (\(x\in X\)) for some constant C;
each bounded sequence in X is precompact in Y.
Let \(X=H, V\textit{ or }V'\), then \(\Vert Pu\Vert _{X}\leq \Vert u\Vert _{X}\), and \(Pu\rightarrow u\) in X.
Let \(X\subset\subset H\subset Y\) be Banach spaces, and X is reflective. If \(u_{n}\) is a uniformly bounded sequence in \(L^{2}(\tau ,T;Y)\), and there exists \(p>1\) such that \(\frac{dv_{n}}{dt}\) is uniformly bounded in \(L^{p}(\tau,T;Y)\), then \(u_{n}\) has a strong convergence subsequence in \(L^{2}(\tau,T;H)\).
(The Gronwall inequality)
Let g, h, and y all be locally integrable functions in \((t_{0},+\infty)\) satisfying
and \(\frac{dy}{dt}\) is locally integrable, then we have
See, e.g., [21]. □
(The generalized Arzelà-Ascoli theorem)
Let \(\{ f_{r}(\theta): \gamma\in\Gamma\}\subset C=C([-r,0]; X)\) is equicontinuous, and for all \(\theta\in[-r,0]\), \(\{f_{r}(\theta): \gamma\in\Gamma\}\) is relatively compact in \(C([-r,0];X)\).
See, e.g., [22]. □
Let X be a metric space, the set class \(\{U(t,\tau)\}\) \((-\infty<\tau\leq t<+\infty): X\rightarrow X\) is called a processes in X, if
\(U(\tau,\tau)x=x\), \(\tau\in R\), \(\forall x\in X\);
\(U(t,\tau)=U(t,s)U(s,\tau)\), \(\forall \tau\leq s\leq t\), \(\tau\in R\).
Let \({\mathcal{P}}(X)\) denote all the family of nonempty subsets of X, and \({\mathcal{D}}\) the class of all families \(\hat{D}=\{D(t)|t\in R\}\subset{\mathcal{P}}(X)\).
The processes class \(\{U(\cdot,\cdot)\}\) is said to be pullback \({\mathcal{D}}\)-asymptotically compact if for any \(t\in R\), \(\hat{D}\in{\mathcal{D}}\) and \(\tau_{n}\rightarrow-\infty\), \(x_{n}\in D(\tau_{n})\), the sequence \(\{U(t,\tau_{n})x_{n}\}\) possesses a convergent subsequence.
A family \(B=\{B(t)|t\in R\}\in{\mathcal{P}}(X)\) is said to be pullback \({\mathcal{D}}\)-absorbing if for each \(t\in R\) and \(\hat{D}\in {\mathcal{D}}\), there exists \(\tau_{0}(t,\hat{D})\leq t\) such that
A family \(\hat{A}=\{A(t)|t\in R\}\in{\mathcal{P}}(X)\) is said to be a global pullback \({\mathcal{D}}\)-attractor with respect to the processes \(\{U(\cdot,\cdot)\}\), if
\(A(t)\) is compact for any \(t\in R\);
 is pullback \({\mathcal{D}}\)-attracting, i.e.,
where \(\operatorname {dist}(C_{1},C_{2})\) denotes the Hausdorff semi-distance between \(C_{1}\) and \(C_{2}\) defined as \(\operatorname {dist}(C_{1},C_{2})={\sup_{x\in C_{1}}\inf_{y\in C_{2}}}d(x,y)\) for \(C_{1}, C_{2}\subset X\);
 is invariant, i.e., for all \(-\infty<\tau \leq t<+\infty\), we have \(U(t,\tau)A(\tau)=A(t)\).
We claim that \(A(t)=\overline{{\bigcup_{\hat{D}\in {\mathcal{D}}}}\Lambda(\hat{D},t)}\), \(t\in R\), where \(\Lambda(\hat{D},t)\) is defined as
Let the process \(\{U(t,\tau)\}\) be continuous and pullback \({\mathcal{D}}\)-asymptotically compact, and let there exist \(\hat{B}\in{\mathcal{D}}\) which is pullback \({\mathcal{D}}\)-absorbing with respect to \(\{U(t,\tau)\}\). Then the family \(\hat{A}=\{A(t)|t\in R\}\subset{\mathcal{P}}(X)\), \(A(t)=\Lambda(\hat{B},t)\), \(t\in R\) is a global pullback \({\mathcal {D}}\)-attractor which is minimal in the sense that if \(\hat{C}=\{ C(t)|t\in R\}\subset{\mathcal{P}}(X)\) is closed and \({\lim_{\tau\rightarrow-\infty}} \operatorname {dist}(U(t,\tau)B(\tau),C(t))=0\), then \(A(t)\subset C(t)\).
See, e.g., [23]. □
First we introduce a stream function ψ which solves the Stokes system (see [8])
and ψ satisfies
It follows that
Let \(\varepsilon\in(0,c\cdot \operatorname {diam}(\Omega))\) be a constant to be determined later, and \(\eta_{\varepsilon}\in C_{0}^{\infty}(\mathbb {R}^{2})\) such that
and
where \(\eta_{\varepsilon}\) is in the form \(h(\frac{\rho (x)}{\varepsilon })\), h is a standard bump function, and \(\rho\in C^{\infty}\) is a regularized distance bump function to ∂Ω.
We also know
and
Assume ψ satisfies (3.7)-(3.9), then we have
where
For any \(t\in(\tau,T)\), we define \(u:(\tau-h,T)\rightarrow (L^{2}(\Omega))^{2}\), and \(u_{t}\) is a function defined on \((-h,0)\) satisfying \(u_{t}=u(t+s)\), \(s\in(-h,0)\). Let
be three Banach spaces with the norms
respectively, and
The problem (1.1) can be written as the abstract form
where \(f_{\rho}(u)=f(t-\rho(t),u(t-\rho(t)))\), \(g(t,u_{t})=\int _{-h}^{0}G(t,s,u(t+s))\,ds\), and it satisfies
\(\forall \xi\in C_{H}\), \(t\in\mathbb{R}\mapsto g(t,\xi) \in(L^{2}(\Omega))^{2} \) is measurable and \(g(t,0)=0\), \(\forall t\in\mathbb{R}\);
there exists \(L_{g}>0\) such that for all \(t\in\mathbb{R}\), \(\xi ,\eta \in C_{H}\),
\(\exists m_{0}\geq0\), \(C_{g}>0: \forall m\in[0,m_{0}]\), \(\tau\leq t\), \(u,v\in C^{0}([\tau-h,t];H)\),
\(\rho: [0,\infty)\rightarrow[0,h]\), \(\vert \frac{d\rho}{dt} \vert \leq M <1\);
\(f(t,u)\) satisfies the Lipschitz condition with respect to u: \(\exists L(\beta)>0\) such that
\(\exists a>0, b>0\) such that \(\vert f(t,u)\vert ^{2}\leq a\vert u\vert ^{2}+b\);
\(\nu>\frac{3C_{g}}{\lambda_{1}}\);
under the conditions (a)-(g), \(\exists K_{1}>0\), and let
Let \(v=u-\psi\), (1.1) can be reduced to the following system:
where \(\bar{f}=f_{\rho}(v+\psi)+\nu F\), \(g(t,v_{t}+\psi)=\int _{-h}^{0}G(s,v(t+s)+\psi)\,ds\), \(\phi\in L^{2}_{V}\cap L^{\infty}_{H}\).
Let \(v_{0}\in H\), \(\eta\in L_{H}^{2}\), we consider the equivalent abstract system of (3.14)
where \(R(v)=B(v,\psi)+B(\psi,v)\), which is also a linear continuous operator from V into \(V'\) and maps W into H (see [21]).
Let \(u_{\tau},f\in H\), \(\varphi\in L^{\infty }(\partial \Omega)\) and \(\varphi\cdot n=0\) on ∂Ω, u is called a weak solution of the problem (1.1) provided
\(u\in C([\tau-h,T];V)\), \(u(\cdot,\tau)=u_{\tau}\), and \(du/dt\in L^{2}([\tau,T];V')\);
\(\forall v\in C^{\infty}_{0}(\Omega)\) with \(\operatorname {div}v=0\), we get
\(\exists \psi\in C^{2}(\Omega)\cap L^{\infty}(\Omega)\), \(q\in C^{1}(\Omega)\) and \(g\in L^{2}(\Omega)\) such that
where we assume that ψ obtain its boundary values in sense of non-tangential convergence and \(u-\psi\in L^{2}([\tau,T];V)\).
We shall give the main result in this section.
Let \(v_{\tau}\in V\), \(\eta\in L_{H}^{2}\), and the assumptions (a)-(h) hold, then there exists a unique global weak solution of (3.15) which satisfies
and \(\frac{dv}{dt}\) is uniformly bounded in \(L^{2}(\tau,T;V')\).
We first use the standard Faedo-Galerkin method to establish the existence of a solution to (3.15).
Fix \(n\geq1\), we define an approximate solution \(v_{n}\) to (3.15) as \(v_{n}(t)={\sum_{j=0}^{n}}a_{nj}(t)w_{j}\), which satisfies
We also denote \(f_{n}=f(t,v_{n}(t)+\psi)\), \(f_{n\rho}=f(t-\rho (t),v_{n}(t-\rho (t))+\psi)\), and \(g_{n}=g(t,v_{n}(t)+\psi)\).
Multiplying (3.16) by \(v_{n}\), we have
and
We estimate each term on the right side of (3.18) in the following.
Using Hardy’s inequality, we obtain
and choose suitable ε such that
By the Young inequality, the Hölder inequality, Hardy’s inequality, the Cauchy inequality, and the property of the trilinear operator, we derive
Combining (3.19)-(3.23), we conclude
i.e.,
where
Choosing suitable \(m>0 \) such that \(\nu>\frac{3C_{g}}{\lambda _{1}}+\frac {m}{\lambda_{1}}+m\alpha^{2}+\frac{6ae^{mh}}{\nu\lambda^{2}_{1}(1-M)}\), we have
Integrating (3.25) over \([\tau,t]\), we derive
which implies
Integrating (3.24) over \([t,t+1]\), we obtain
and
That is,
which means \(v_{n}(t)\) is uniformly bounded in \(L^{\infty}(\tau ,T;V)\cap L^{2}(\tau,T;V)\). Using the Alaoglu compact theorem, we can find a subsequence (still written as \(v_{n}\) without confusion) such that
i.e., \(v\in L^{\infty}(\tau,T;V)\cap L^{2}(\tau,T;V)\).
Next, we prove that \(\frac{dv_{n}}{dt}\) is uniformly bounded in \(L^{2}(\tau,T;V')\). Since
and \(v_{n}\in L^{2}(\tau,T;V)\), we derive that \(-\nu Av_{n},\alpha^{2} v_{nt},g_{n}\in L^{2}(\tau,T;V')\), and
Similarly, we have
Since \(B(\psi)\in L^{2}(\tau,T;V')\), we conclude that \(\frac {dv_{n}}{dt}\) is uniformly bounded in \(L^{2}(\tau,T;V')\). By the compact embedding theorem, we also have
□
Let \(u_{\tau},f\in H\), \(\varphi\in L^{\infty}(\partial\Omega)\), and \(\varphi\cdot n=0\) on ∂Ω. Then (1.1) has a unique weak solution.
The family of stream functions \(\psi_{\varepsilon}\) was constructed in [8] which satisfied \(\psi_{\varepsilon}\in C^{\infty}(\Omega)\). In addition, the solution v of (3.16) is obtained in Theorem 3.1. Let \(u=v+\psi_{\varepsilon}\), it is easy to check that u is the weak solution of (1.1) which satisfies (i), (ii), and (iii).
Suppose that \(u_{1}\) and \(u_{2}\) are two solutions to (1.1) with stream functions \(\psi_{1}\) and \(\psi_{2}\), respectively. Let \(v\in C^{\infty}_{0}(\Omega)\), \(\operatorname {div}v=0\), from the condition (ii) we get
We claim that (3.36) holds for any \(v\in V\). In fact, from the condition (ii), we have
thus we can write \(\langle u_{1}-u_{2},\Delta v\rangle =-(u_{1}-u_{2},v)\) (\(l=1,2\)),
and \(u_{l}\in L^{4}(\Omega\times(\tau,T))\), thus
which implies (3.36) holds for any \(v\in V\).
Let \(v=u_{1}-u_{2}\), we get
which means
Since \(u_{1}\in L^{4}(\Omega\times(\tau,T))\) and \(v(\cdot,\tau )=0\), we derive \(v=0\) which means the uniqueness of solution holds. □
Consider the two solutions \(u(\cdot)\) and \(v(\cdot)\) to problem (1.1) with corresponding initial data \((u_{\tau},\phi_{1})\) and \((v_{\tau},\phi_{2})\), respectively. Let \(w=u-v\), then w satisfies the problem
Since \(B(u,u)-B(v,v)=B(w,u)+B(v,w)\), we obtain the abstract form
Multiplying (3.43) by w, we derive
which implies
Noting
and integrating (3.44) over \([\tau, t]\), we obtain
Since \(u(t)\in L^{\infty}(\tau,T;V)\cap L^{2}(\tau,T;V)\), neglecting the integrating term on left side of (3.45), putting \(s\in (t-h,t)\) instead of t and using the Gronwall inequality to (3.45), we see
Similarly, using the Poincaré inequality, we get
Moreover,
This implies the continuous dependence on the initial data for the solution which generates a continuous process \(\{\tilde{U}(\cdot ,\cdot )\}\).
In this section we shall derive the existence of pullback absorbing sets for the 2D Navier-Stokes-Voigt equations with continuous delay and distributed delay on the Lipschitz domain.
From Theorem 3.1, we obtain the process \(\tilde{U}(\cdot,\tau ;(v_{\tau },\phi))=v_{t}(\cdot;(v_{\tau},\phi))\), where \((v_{\tau}, \phi)\in V\times L_{H}^{2}\), \(V\times L_{H}^{2}\) is a Hilbert space, and the corresponding norm can be defined as
To derive the existence of pullback attractor, we need to prove the existence of pullback absorbing set of \(\tilde{U}(\cdot,\tau ;(v_{\tau },\phi))\) in \(C_{V}\) first of all thus:
Let \((v_{\tau},\eta)\in V\times L_{H}^{2}\), and the assumptions (a)-(h) hold, then there exists a pullback absorbing set in \(C_{V}\) for the system (3.15).
Let \(D\subset V\times L_{H}^{2}\) be any bounded set, and \((v_{\tau},\phi)\in D\), then there exists a constant \(d>0\) such that
Similar to the proof of Theorem 3.1, we get
Choose a suitable constant \(m>0\) such that \(\nu>\frac{C_{g}}{\lambda _{1}}+\frac{m}{\lambda_{1}}+m\alpha^{2}+\frac{6ae^{mh}}{\nu\lambda _{1}^{2}(1-M)}\) and
Integrating (4.3) over \([\tau,t]\), we have
which implies
Choosing \(\sigma\in[-h,0]\), and substituting t with \(t+\sigma\), we have
Denoting \(v_{t}(\cdot;(v_{\tau},\phi))\) as \(\tilde{U}(\cdot;\tau ,(v_{\tau},\phi))\), we have
Let \(\rho_{V}^{2}=\frac{2K^{2}_{0}}{m}+(\frac{4C_{g}}{m}+\frac {24ae^{mh}}{\nu\lambda_{1}(1-M)}h)C_{4}^{2}\Vert \varphi \Vert ^{2}_{L^{\infty }(\partial \Omega)}\), for any \((v_{\tau},\phi)\in V\times L_{H}^{2}\), when
\(\tilde{U}(\cdot,\cdot)D\subset B_{V}(0,\rho_{V})\), where \(B_{V}(0,\rho _{V})\) is a pullback absorbing ball centered at 0 with radius \(\rho _{V}\) in \(C_{V}\), which completes the proof. □
Assume that the assumptions (a)-(h) hold, and \((v_{\tau},\eta)\in D(A^{\frac{3}{4}})\times L_{H}^{2}\), then there exists a pullback absorbing set in \(C([\tau,T],D(A^{\frac {3}{4}}))\) for the system (3.15).
Let \(D\subset D(A^{3/4})\times L_{H}^{2}\) be any bounded set, and \((v_{\tau},\phi)\in D\), then there exists a constant \(d>0\) such that
Multiplying (3.15) by \(A^{\frac{1}{2}}v\), we have
Next, we shall estimate term by term the right side of (4.7).
Combining (4.7)-(4.13), we conclude
where
and
Choosing a suitable \(m>0\) such that
and integrating (4.15) over \([\tau,t]\), we obtain
It follows that
Integrating (4.14) over \([t,t+1]\), we obtain
and
where
which implies \(v\in L^{\infty}(\tau,T;D(A^{3/4}))\cap L^{2}(\tau,T;D(A^{3/4}))\).
From (4.16), we also have
and
Choosing \(\sigma\in[-h,0]\) and substituting t with \(t+\sigma\), we have
Denoting \(v_{t}(\cdot;(v_{\tau},\phi))\) as \(\tilde{U}(\cdot;\tau ,(v_{\tau},\phi))\), we get
Let \(\rho_{3/4}^{2}=\frac{2K_{3}^{2}+4C_{g}C_{4}^{2}\Vert \varphi \Vert ^{2}_{L^{\infty }(\partial\Omega)}}{m}\) for any \((v_{\tau},\eta)\in D(A^{3/4})\times L_{H}^{2}\), when
\(\tilde{U}(\cdot,\cdot)D\subset B_{3/4}(0,\rho_{3/4})\), where \(B_{3/4}(0,\rho_{3/4})\) is a pullback absorbing ball centered at 0 with radius \(\rho_{3/4}\) in \(C_{3/4}\), which completes the proof. □
The main result in the paper can be stated as follows.
Assume that the assumptions (a)-(h) hold, \((v_{\tau},\eta)\in D(A^{\frac{3}{4}})\times L_{H}^{2}\), then the system (1.1) possesses a pullback attractor \({\mathcal{A}}\).
Theorem 3.1 guarantees that the process \(\{\tilde {U}(\cdot ,\cdot)\}\) of the system (3.15) is continuous. By Theorem 4.1 and Theorem 4.2 we show that the system (3.15) possesses two pullback absorbing balls \(B_{V}(0,\rho_{V})\) and \(B_{3/4}(0,\rho_{3/4})\) in \(C_{V}\) and \(C_{D(A^{3/4})}\), respectively. Since \(V \hookrightarrow \hookrightarrow D(A^{3/4})\) and \(\{\tilde{U}(\cdot,\cdot)\}\) is equicontinuous, by the generalized Arzelà-Ascoli theorem we can show that the process \(\{\tilde{U}(\cdot,\cdot)\}\) is asymptotically compact in \(C_{V}\). From the fundamental existence theory of pullback attractors (see, e.g., [21, 22]), the process \(\{\tilde {U}(\cdot,\cdot)\}\) generated by the system (3.15), which is equivalent to (1.1), possesses a pullback attractor in \(C_{V}\). □
Ladyzhenskaya, OA: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics. Am. Math. Soc., Providence (2001)
Hale, JK: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc., Providence (1988)
Ladyzhenskaya, OA: Attractors for Semigroup and Evolution Equations. Cambridge University Press, Cambridge (1991)
Lu, S: Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces. J. Differ. Equ. 230, 196-212 (2006)
Lu, S, Wu, H, Zhong, C: Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces. Discrete Contin. Dyn. Syst. 13(3), 701-719 (2005)
Rosa, R: The global attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal. TMA 32(1), 71-85 (1998)
Brown, RM, Perry, PA, Shen, Z: On the dimension of the attractor of the non-homogeneous Navier-Stokes equations in non-smooth domains. Indiana Univ. Math. J. 49(1), 1-34 (2000)
Oskolkov, AP: The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Nauč. Semin. POMI 38, 98-136 (1973)
Luengo, JG, Marín-Rubio, P, Real, J: Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations. Nonlinearity 25, 905-930 (2012)
Çelebi, AO, Kalantarov, VK, Polat, M: Global attractors for 2D Navier-Stokes-Voigt equations in an unbounded domain. Appl. Anal. 88(3), 381-392 (2009)
Krasovskii, NN: Stability of Motion. Stanford University Press, Stanford (2002)
Barbu, V, Sritharan, SS: Navier-Stokes equations with hereditary viscosity. Z. Angew. Math. Phys. 54, 449-461 (2003)
Taniguchi, T: The exponential behavior of Navier-Stokes equations with time delay external force. Discrete Contin. Dyn. Syst. 12(5), 997-1018 (2005)
Caraballo, T, Real, J: Navier-Stokes equations with delays. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 457, 2441-2453 (2001)
Caraballo, T, Real, J: Asymptotic behavior for two-dimensional Navier-Stokes equations with delays. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 459, 3181-3194 (2003)
Caraballo, T, Real, J: Attractors for 2D Navier-Stokes models with delays. J. Differ. Equ. 205, 271-297 (2004)
Marín-Rubio, P, Real, J: Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. Nonlinear Anal. 67, 2784-2799 (2007)
Garrido-Atienza, MJ, Marín-Rubio, P: Navier-Stokes equations with delays on unbounded domains. Nonlinear Anal. 64, 1100-1118 (2006)
García-Luengo, J, Marín-Rubio, P, Planas, G: Attractors for a double time-delayed 2D-Navier-Stokes model. Discrete Contin. Dyn. Syst. 34(10), 4085-4105 (2014)
Temam, R: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, Berlin (1997)
Robinson, JC: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)
Cai, X, Jiu, Q: Weak and strong solutions for the incompressible Navier-Stokes equations with damping. J. Math. Anal. Appl. 343, 799-809 (2008)
This work was supported by the NSFC of Education Department in Henan Province (No. 14B110029, 15A110033).
The authors thank the referees by his/her comments, which led to improvements in the presentation of this paper.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Su, K., Zhao, M. & Cao, J. Pullback attractors of 2D Navier-Stokes-Voigt equations with delay on a non-smooth domain. Bound Value Probl 2015, 243 (2015). https://doi.org/10.1186/s13661-015-0505-3
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DOI: https://doi.org/10.1186/s13661-015-0505-3
