Abstract
In this article, the long-time behaviors of weak solutions for the 2D non-autonomous magneto-viscoelastic flows are considered. Unlike the results established by Liu and Liu (Politeh Univ Buchar Sci Bull Ser A Appl Math Phys 81(4):155–166, 2019), utilizing the method of \(\ell \)-trajectories introduced by Málek and Pražák (J Differ Equ 181(2):243–279, 2002), we first justify the existence of finite-dimensional pullback attractors for the process \(\{L(t,\tau )\}_{t\ge \tau }\) in the \(\ell \)-trajectories space \(X_{\ell }\). Then we obtain the corresponding finite-dimensional pullback attractors for the process \(\{U(t,\tau )\}_{t\ge \tau }\) in the original phase space \(\mathbb {H}\).
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1 Introduction
Magneto-viscoelastic flow is a class of important and complex non-Newtonian fluid, which has a broad application prospect in technological applications. The magneto-viscoelastic model has received extensive interest in the past years. A general magneto-viscoelastic model describing magnetoelastic materials was established by Forster in [11], which is based on an energetic variational approach (see, e.g., [14]). Since the magnetoelastic materials are extremely affected by the phenomenon of converting applied changes of the magnetic field and vice versa, they can be regarded as smart materials. For instance, the various magnetic materials can be found in sensors to measure the torque of a force, and can also be used in magnetic actuators and generators for ultrasonic sounds (see, e.g., [4, 5, 11, 13, 17, 33]).
Forster [11] established the following incompressible magneto-viscoelastic fluid model:
System (1.1) consists of the incompressible Navier–Stokes equations coupled with balance equations for the deformation gradient F and the magnetization M, where the magnetization M is a simplification of the Landau–Lifshitz–Gilbert equations with convection (see, e.g., [3, 11]). In technological applications, we sometimes need to investigate the perturbations of the external magnetic field to the magneto-viscoelastic fluid. That is, the magneto-viscoelastic fluid is exposed to an external effective magnetic field \(H_{\textrm{ext}}\). Considering the coupling of magnetic and elastic effects, the study of magnetoelastic materials has attracted more attention from various technological applications and the view of mathematical modeling (see e.g., [2, 9, 17, 18, 34, 42]).
Note that if \(M=0\), system (1.1) is a model for incompressible viscoelastic flows (see e.g., [21, 24]). If \(F=0\), it reduces to the simplified Ericksen–Leslie system for incompressible liquid crystal flows (see e.g., [25, 26]). If \(M=F=0\), it translates into the standard Navier–Stokes equations (see e.g., [23, 38]).
In this paper, taking \(H_{\textrm{ext}}=0\), then we can obtain the following simplified 2D incompressible magneto-viscoelastic fluid model with non-autonomous external force term g(x, t):
in \(\Omega \times [\tau , T]\), where \(\Omega \) is a bounded regular domain with smooth boundary, \(\tau \in \mathbb {R}, \tau \le T\), \(v(x,t): \Omega \times [\tau , T]\rightarrow \mathbb {R}^{2}\) is the velocity of the fluid, \(p=p(x,t)\) is the scalar pressure, and \(F: \Omega \times [\tau , T]\rightarrow \mathbb {R}^{2\times 2}\) is the deformation gradient, \(M: \Omega \times [\tau , T]\rightarrow \mathbb {R}^{3}\) is the magnetization vector. \(\mu , \kappa >0\) are viscosity coefficients, \(\gamma >0\) stands for the parameter that controls the strength of penalization on the deviation of |M| from 1. \(g=g(x,t)\) is time dependent external force term.
System (1.2) is given the Dirichlet boundary conditions for v, F and the Neumann boundary condition for M
and the initial conditions
where I is the \(2\times 2\) identity matrix and \(\textbf{n}\) is the unit outward normal vector.
Let \(h: \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) be a function and \(G(r)=\int _{0}^{r}h(s)\textrm{d}s\) be a potential function as follows
and define the basic energy
It is well known that the long-time asymptotic behavior of dynamical systems is one of the most important problems for nonlinear dissipative evolution systems. Until now, one effective way to deal with this problem for a given evolution system is to study the existence and structure of its attractor. In particular, the non-autonomous evolution systems, which can well describe the intrinsic properties of many natural phenomena, are rather more complicated than autonomous ones. In the past decades, many scholars have focused on the research of more general non-autonomous differential equations. The pullback attractors can well describe the pullback asymptotic behavior of non-autonomous dynamical systems (see [6, 10, 20, 32, 37] and references therein), which is a minimal family of compact invariant sets under the process and pullback attracts any bounded subset of the phase space.
In this paper, we aim to establish the existence of pullback attractors with finite fractal dimension in the original phase space \(\mathbb {H}\) for the magneto-viscoelastic system (1.2)–(1.5). For the autonomous case of system (1.2)–(1.5), i.e., when the non-autonomous external force term \(g=0\), the solution operator defines a semigroup. In [28], the authors proved the existence of global attractors for the autonomous case of system (1.2)–(1.5). However, to the best of our knowledge, there are no results about the existence of the pullback attractors with finite fractal dimension for the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in the phase space. As Liu and Liu in [28] pointed out, since the strong coupling nonlinear terms and the Neumann boundary conditions for problem (1.2)–(1.5), it is difficult to justify the smooth property of the difference of two solutions and the differentiability of the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) on the pullback attractors. Unlike the results obtained in [28], to overcome these difficulties, we utilize the novel idea of the method of \(\ell \)-trajectories in [1, 27, 30]. This novel method is based on an observation that the limit behavior of solutions for a given dynamical system in an original phase space can be equivalently captured by the limit behavior of \(\ell \)-trajectories space (see [30] for more details). By virtue of this method, many scholars have studied a large class of nonlinear dissipation problems, especially for the problems of lack of good regularity properties or uniqueness of solutions (see [7, 12, 19, 22, 29, 35, 41, 43] and references therein). Furthermore, our method can also be used to establish the existence of global attractor with finite fractal dimension for the autonomous case of system (1.2)–(1.5).
In the sequel, we make the following assumption.
Hypothesis. Assume that the external force \(g\in L_{loc}^{2}(\mathbb {R}; \textbf{H})\) satisfies
where \(\textbf{H}\) is given later.
In this paper, we obtain the main results as follows.
Theorem 1.1
Assuming that \((A_{1})\) holds, then the following assertions are true:
-
(i)
There exists a pullback attractor \(\hat{\mathcal {A}}=\{\mathcal {A}(t):t\in \mathbb {R}\}=\{e_{1}(\mathcal {A}_{\ell }(t-\ell )):t\in \mathbb {R}\}\) for the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in \(\mathbb {H}\), where \(\mathcal {A}_{\ell }(t-\ell )\) is the section of pullback attractor \(\hat{\mathcal {A}}_{\ell }=\{\mathcal {A}_{\ell }(t):t\in \mathbb {R}\}\) established in Theorem 3.3 for the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in \(X_{\ell }\);
-
(ii)
The pullback attractor for the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in \(\mathbb {H}\) has finite fractal dimension.
The rest of this article is arranged as follows. In the second part, we mainly introduce some basic notations, some useful results and global well-posedness for problem (1.2)–(1.5). In the third part, using the method of \(\ell \)-trajectories in [27, 30, 41], we first define a process \(\{L(t,\tau )\}_{t\ge \tau }\) on the \(\ell \)-trajectories space \(X_{\ell }\) with the topology of \(L^{2}(\tau ,\tau +\ell ; \mathbb {H})\) induced by the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by system (1.2)–(1.5), then we prove the existence of pullback attractor \(\hat{\mathcal {A}}_{\ell }\) for the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in \(X_{\ell }\). In addition, analyzing the smoothing property of the process \(\{L(t,\tau )\}_{t\ge \tau }\), we justify that the pullback attractor \(\hat{\mathcal {A}}_{\ell }\) for the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) has a finite fractal dimension. Finally, using a Lipschitz continuous operator on the pullback attractor \(\hat{\mathcal {A}}_{\ell }\), we obtain that the corresponding finite-dimensional pullback attractor \(\hat{\mathcal {A}}\) for the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) in \(\mathbb {H}\).
2 Preliminaries
In this section, we will present some notations and useful results that are used all through paper. Let C be a positive constant, which may vary in different situations. Specifically, we also use \(C_0,\) \(C_1,\) \(C(\cdot )\) to emphasize certain dependence. As usual, for any \(1\le p<\infty \) and \(k\in \mathbb {N}\), we denote the Lebesgue space and Sobolev space by \(L^p(\Omega )\) and \(W^{k,p}(\Omega )\) endowed with norms \(\Vert \cdot \Vert _{L^p}\) and \(\Vert \cdot \Vert _{W^{k,p}}\), respectively, where
and we also denote the space \(W_{0}^{k,p}(\Omega )\) as completions of \(C_0^{\infty }(\Omega )\) in norms of \(W^{k,p}(\Omega )\).
In particular, \(H^k(\Omega )=W^{k,2}(\Omega ), H_{0}^k(\Omega )=W_{0}^{k,2}(\Omega )\) with \(k\in \mathbb {N}\) and \(p=2\), \(H^{-k}(\Omega )\) is the dual space of \(H_{0}^k(\Omega )\). The Lebesgue space \(L^p(0,t; X)\) consists of all those functions u that take values in X for almost every \(s\in (0,t)\), which satisfy \(\left( \int _{0}^{t}\Vert u(s)\Vert _{X}^{p}\textrm{d}s\right) ^{\frac{1}{p}}<+\infty \), for any \(1\le p\le \infty \). For the sake of conciseness, we do not distinguish functional space when scalar-valued or vector-valued functions are involved. To deal with problem (1.2)–(1.5) in a proper setting, we also introduce some function spaces. Let
where \(C_0^{\infty }(\Omega )\) is the space of any smooth functions v which are zero outside of some compact support depending on v. Denote the closure of \(\mathcal {V}\) by \(\textbf{H}\) and \(\textbf{V}\) with respect to the \(L^{2}(\Omega )\)-norm and \(H^{1}(\Omega )\)-norm, respectively. Let
and the phase space
Let \(\textbf{H}'\) and \(\textbf{V}'\) be the dual spaces of \(\textbf{H}\) and \(\textbf{V}\), respectively, and the injections \(\textbf{V}\hookrightarrow \textbf{H}\equiv \textbf{H}'\hookrightarrow \textbf{V}'\) are dense and continuous. \(\Vert \cdot \Vert _{\textbf{V}'}\) and \(<\cdot ,\cdot>\) stand for the norm in \(\textbf{V}'\) and the duality product between \(\textbf{V}\) and \(\textbf{V}'\)(also \(\textbf{H}\) and its dual space), respectively.
For simplicity, Einstein summation convention is used in our paper. We denote \(a\cdot b=\sum \limits _{i=1}^n a_{i}b_{i}:=a_{i}b_{i},\ (a\otimes b)_{ij}=a_{i}b_{j}\) for the vectors a and b, \(A:B=A_{ij}B_{ij}\), \(\nabla A\vdots \nabla B=\partial _{k}A_{ij}\partial _{k}B_{ij}\) for the matrices \(A=(A_{ij})\), \(B=(B_{ij})\).
Next, we also introduce the following some useful operators (see e.g., [1]):
The bilinear form \(a: \textbf{V}\times \textbf{V}\rightarrow \mathbb {R}\) is defined as
Let \(\mathbb {P}\) be the Helmholtz–Leray orthogonal projection operator from \(L^2(\Omega )\) onto \(\textbf{H}\). Then we define the operator \(A: \textbf{V}\rightarrow \textbf{V}'\) by \(Au=-\mathbb {P}\Delta u\), which is the Stokes operator with the domain \(D(A)=H^2(\Omega )\cap \textbf{V}\), endowed with inner product and norm
The mapping \(B: \textbf{V}\times \textbf{V}\rightarrow \textbf{V}'\) is defined by
then
with
Moreover, we shall use the following some identities (see e.g., [28])
Now, applying the projection operator \(\mathbb {P}\) to problem (1.2)–(1.5), we can get the following equivalent functional differential equation
Definition 2.1
Let \((v_{\tau },F_{\tau },M_{\tau })\in \mathbb {H}\) and \(g\in L_{loc}^{2}(\mathbb {R}; \textbf{H})\). The triple (v, F, M) is called a weak solution of problem (1.2)–(1.5), for any \(T\ge \tau , \tau \in \mathbb {R}\), provided that
with \(v_{t}\in L^{2}(\tau ,T; \textbf{V}')\), \(F_{t}\in L^{2}(\tau ,T; H^{-1}(\Omega ))\), \(M_{t}\in L^{2}(\tau ,T; L^{2}(\Omega ))\) such that \(v(x,\tau )=v_{\tau }, F(x,\tau )=F_{\tau }, M(x,\tau )=M_{\tau }\), and if for test functions \(\varphi \in W^{1,\infty }(\tau ,T;\mathbb {R})\) with \(\varphi (T)=0\), \(\psi \in \textbf{V}\), \(\omega \in H_{0}^{1}(\Omega ;\mathbb {R}^{2\times 2})\), \(\phi \in H^{1}(\Omega ;\mathbb {R}^3)\), satisfy
For the given problem (1.2)–(1.5), the global well-posedness of weak solutions for the two dimensional magneto-viscoelastic flows on a bounded smooth domain was obtained in [11, 34]. For convenience, we also briefly present the following results.
Theorem 2.1
[11, 34] Assume that \(g\in L_{loc}^{2}(\mathbb {R}; \textbf{H})\), then for any \((v_{\tau }, F_{\tau }, M_{\tau })\in \mathbb {H}\), there exists a unique weak solution (v(t), F(t), M(t)) to problem (1.2)–(1.5) satisfying the conditions of Definition 2.1 such that \((v(x,\tau ), F(x,\tau ), M(x,\tau ))=(v_{\tau }, F_{\tau }, M_{\tau })\), which depends continuously on the initial data \((v_{\tau }, F_{\tau }, M_{\tau })\) with respect to the norm in \(\mathbb {H}\).
Corollary 2.1
[11, 34] Assume that \(g\in L_{loc}^{2}(\mathbb {R}; \textbf{H})\), the triple \((v_{\tau , m}, F_{\tau , m}, M_{\tau , m})\rightharpoonup (v_{\tau }, F_{\tau }, M_{\tau })\) in \(\mathbb {H}\), and let \(\{(v_{m}(t),F_{m}(t),M_{m}(t))\}_{m\ge 1}\) be a sequence of weak solution for problem (1.2)–(1.5) such that \((v_{m}(\tau ),F_{m}(\tau ),M_{m}(\tau ))=(v_{\tau , m}, F_{\tau , m}, M_{\tau , m})\). For any \(T\ge \tau , \tau \in \mathbb {R}\), if there exists a subsequence of
\(\{(v_{m}(t),F_{m}(t),M_{m}(t))\}_{m\ge 1}\) converging \((*-)\) weakly in the space
\(\left\{ (v(t),F(t),M(t))\in L^{\infty }\left( \tau ,T; \mathbb {H}\right) \bigcap L^{2}(\tau ,T; \textbf{V}\times H^{1}(\Omega ;\mathbb {R}^{2\times 2})\times H^{2}(\Omega ;\mathbb {R}^3)):\ \nonumber \right. \left. (v_{t}, F_{t}, M_{t})\in L^{2}(\tau ,T; \textbf{V}'\times H^{-1}(\Omega )\times L^{2}(\Omega ))\right\} \) to a certain function \((v(t),F(t),M(t))\). Then (v(t), F(t), M(t)) is a weak solution for problem (1.2)–(1.5) with \((v(x,\tau ),F(x,\tau ),M(x,\tau ))=(v_{\tau },F_{\tau },M_{\tau })\).
Next, we also present the following results in the section 3. Let X be a complete metric space with distance \(d_{X}(\cdot ,\cdot )\). We define the Hausdorff semidistance between A and B by
A two-parameter family of mappings \(\{U(t,\tau ),t\ge \tau ,\tau \in \mathbb {R}\}\) is called a continuous process in X. If the two-parameter family of mappings \(\{U(t,\tau ),t\ge \tau ,\tau \in \mathbb {R}\}\) from X to X satisfy
-
(i)
\(U(t,\tau )=U(t,r)U(r,\tau )\), for all \(\tau \le r\le t\),
-
(ii)
\(U(\tau ,\tau )=Id\), for all \(\tau \in \mathbb {R}\),
-
(iii)
\(U(t,\tau )x_{n}\rightarrow U(t,\tau )x\), if \(x_{n}\rightarrow x\) in X.
Let \(\mathcal {D}\) be the family of nonempty sets parameterized with a real parameter \(\hat{B}=\{B(t): B(t)\ne \emptyset , t\in \mathbb {R}\}\) in X such that
where \([B(r)]=\displaystyle {\sup }\{\Vert u\Vert _{X}^{2}:u\in B(r)\}, \alpha _{0}>0\). Similarly, let \(X_{\ell }\) be a \(\ell \)-trajectories space induced by X, and \(\mathcal {D}_{\ell }\) be the family of nonempty sets parameterized with a real parameter \(\hat{\mathcal {B}}_{\ell }=\{\mathcal {B}_{\ell }(t): \mathcal {B}_{\ell }(t)\ne \emptyset , t\in \mathbb {R}\}\) in \(X_{\ell }\) such that
where \([\mathcal {B}_{\ell }(s)]=\displaystyle {\sup }\{\Vert u\Vert _{X_{\ell }}^{2}:u\in \mathcal {B}_{\ell }(s)\}, \alpha _{0}>0\).
Additionally, for the sake of simplicity, we omit some basic definitions and some results in [1, 6, 15, 32] for the given non-autonomous dynamical systems (such as pullback absorbing sets, pullback attractors and fractal dimensions)
3 The Existence of Pullback Attractors
3.1 The Existence of Pullback Attractors in \(X_{\ell }\)
In this section, we first make some priori estimates of solutions to establish the existence of pullback absorbing sets for problem (1.2)–(1.5).
Lemma 3.1
Let (v, F, M) be a weak solution to problem (1.2)–(1.5), the basic energy \(\mathcal {E}(t)\) satisfies
Proof
See the proof of Lemma 2.2 in [28]. \(\square \)
Lemma 3.2
Assuming that \((A_{1})\) holds, then for any bounded subset \(\hat{B}(\tau )\in \mathcal {D}\) and any \(\tau \in \mathbb {R}\), there exists a time \(\tau _{1}=\tau _{1}(\hat{B}(\tau ))\ge 0\) such that for any weak solutions of problem (1.2)–(1.5) with initial data \((v_{\tau }, F_{\tau }, M_{\tau })\in B(\tau )\), we have
and
for any \(t-\tau \ge \tau _{1}\), where \(R_{1}:=1+\frac{C_{0}}{\alpha _{0}}+(\frac{1}{2\nu \lambda _{1}}+\frac{1}{2\nu \lambda _{1}\alpha _{0}})R_{g}; \ R_{2}:=1+\frac{C_{0}\ell }{\alpha _{0}}+(\frac{\ell }{2\nu \lambda _{1}}+\frac{\ell }{2\nu \lambda _{1}\alpha _{0}})R_{g}\).
Proof
Taking \(L^{2}-\)inner product in \(L^{2}(\Omega )\) of (2.2)\(_{4}\) with M, using Hölder’s inequality and Young’s inequality, we have
On the other hand, using Hölder’s inequality, Young’s inequality and (3.2), we have
where \(\alpha _{0}>0\) is given later. Let
Note that
Inserting (3.2), (3.3), (3.5) into (3.4), using Poincaré’s inequality for v and F, we can obtain that
Taking \(\alpha _{0}=\min \{1,\mu \lambda _{1},2\kappa \lambda _{1}\}\), then we have
where \(C_{0}:=\frac{|\Omega |}{2}(\frac{9}{4}+\frac{3\alpha _{0}}{2})\). Multiplying (3.7) by \(e^{\alpha _{0}t}\), we have
Now integrating (3.8) from \(\tau \) to t, we obtain
Note that
where \(R_{g}{:}{=}\sup \limits _{t\in \mathbb {R}}\left( \int _{t-1}^{t}\Vert g(s)\Vert _{L^{2}}^{2}\textrm{d}s\right) <+\infty \).
From (3.9), we conclude that for any \(\hat{B}(\tau )\in \mathcal {D}\), there exists a time \(\tau _{0}=\tau _{0}(\hat{B}(\tau ))>0\) such that for any \((v_{\tau }, F_{\tau }, M_{\tau })\in B(\tau )\),
for any \(t-\tau \ge \tau _{0}\).
Next, integrating (3.8) from \(\tau +\zeta \) to \(t+\zeta \) for any \(\zeta \in (0,\ell )\), we derive that
Combining (3.9) with (3.11), we derive that
where \(\tilde{R}_{g}=\frac{C_{0}}{\alpha _{0}}+(\frac{1}{2\mu \lambda _{1}}+\frac{1}{2\mu \lambda _{1}\alpha _{0}})R_{g}\).
Integrating (3.11) with respect to \(\zeta \) over \((0,\ell )\) and using (3.12), we obtain that
Therefore, for any \(\hat{B}(\tau )\in \mathcal {D}\), there exists a time \(\tau _{1}=\tau _{1}(\hat{B}(\tau ))>\tau _{0}\) such that for any \((v_{\tau }, F_{\tau }, M_{\tau })\in B(\tau )\), we have
for any \(t-\tau \ge \tau _{1}\). \(\square \)
In what follows, we use the method of \(\ell \)-trajectories to construct pullback attractors for system (1.2)–(1.5) in the phase space \(\mathbb {H}\) (see, e.g., [1, 29, 30]).
Let \(C_{\textrm{weak}}([\tau ,\tau +\ell ]; \mathbb {H})\) denote the space of weakly continuous functions from the interval \([\tau ,\tau +\ell ]\) to the Banach space \(\mathbb {H}\) (see [36, 38]). Then we consider the solution \(\textbf{z}(t)=(v(t),F(t),M(t))\in C_{\textrm{weak}}([\tau , \tau +\ell ]; \mathbb {H})\) with the initial data \(\textbf{z}_{\tau }=(v_{\tau },F_{\tau },M_{\tau })\in \mathbb {H}\). Let \(\chi (s,\tau ,\textbf{z}_{\tau })=\textbf{z}|_{s\in [\tau ,\tau +\ell ]}\) denote \(\ell \)-trajectory corresponding to the solution. Then we define the \(\ell \)-trajectories space as follows:
where \(\chi (s,\tau ,\textbf{z}_{\tau })\) is a \(\ell \)-trajectory associated with \(\textbf{z}_{\tau }\).
By Lemma 3.2, the \(\ell \)-trajectories space \(X_{\ell }\) is endowed with the topology of \(L^{2}(\tau ,\tau +\ell ; \mathbb {H})\). Since \(X_{\ell }\subset C_{\textrm{weak}}([\tau , \tau +\ell ]; \mathbb {H})\), it makes sense to deal with the point values of trajectories. From Theorem 2.1, we can define a family of the continuous process \(\{U(t,\tau )\}_{t\ge \tau }\) associated with problem (1.2)–(1.5) in the phase space \(\mathbb {H}\) by
for all \(t\ge \tau \), which is \((\mathbb {H},\mathbb {H})\)-continuous, where \(\textbf{z}(t)\) is the solution of problem (1.2)–(1.5) with initial data \(\textbf{z}(x,\tau )=\textbf{z}_{\tau }\in \mathbb {H}\). Considering the weak solution \(\textbf{z}(t)\) depends continuously on the initial data \(\textbf{z}_{\tau }\), we can infer that the process \(\{U(t,\tau )\}_{t\ge \tau }\) associated with problem (1.2)–(1.5) in \(\mathbb {H}\) is \(\tau \)-continuous.
Based on the above results, we first define the mapping \(b:\mathbb {H}\rightarrow X_{\ell }\) is given by
for each \(s\in [\tau , \tau +\ell ]\) and any \(\textbf{z}_{\tau }=(v_{\tau }, F_{\tau }, M_{\tau })\in \mathbb {H}\).
The second mapping \(e_{\theta }: X_{\ell }\rightarrow \mathbb {H}\) is given by
for any \(\chi (s,\tau ;\textbf{z}_{\tau })\in X_{\ell }\) and \(\theta \in [0,1]\).
Then we can define a new process \(\{L(t,\tau )\}_{t\ge \tau }\) acting on the \(\ell \)-trajectories space induced by process \(\{U(t,\tau )\}_{t\ge \tau }\) as
for any \(\textbf{z}_{\tau }=(v_{\tau }, F_{\tau }, M_{\tau })\in \mathbb {H}.\)
The original phase space and the \(\ell \)-trajectories space defined above satisfy the following commutative diagram (see [30] for more details):
Let
then \(\hat{B}_{0}\in \mathcal {D}\). From Lemma 3.2, we can infer that any weak solutions of problem (1.2)–(1.5) with initial data \((v_{\tau }, F_{\tau },M_{\tau })\in B_{0}\), then there exists a time \(\tau _{0}=\tau _{0}(\hat{B}_{0})\ge 0\) such that \((v(t),F(t),M(t))\in B_{0}\) for any \(t-\tau \ge \tau _{0}\). Thus, we get
for any \(t-\tau \ge \tau _{0}\).
Next, we can define for any \(t\in \mathbb {R}\)
and
In particular, for \(\tau =t\), \(B_{1}(\tau )=\{(v_{\tau },F_{\tau },M_{\tau }):\forall \ (v_{\tau },F_{\tau },M_{\tau })\in B_{0}\},\ \mathcal {B}_{0}^{\ell }(\tau )=\{\chi \in X_{\ell }:e_{0}(\chi )\in B_{1}(\tau )\}\).
From the proof of the bounded absorbing subset of Lemma 3.2, we derive that
and
for any \(t\in \mathbb {R}\) with \(\tau \le t\), and \(\hat{B}_{1}(t)\in \mathcal {D}\).
From Lemma 3.2, we immediately obtain that the following results.
Theorem 3.1
Assuming that \((A_{1})\) holds, then for any \(\hat{\mathcal {B}}_{\ell }(\tau )\in \mathcal {D}_{\ell }\) and any \(\tau \in \mathbb {R}\), there exists a time \(\tau _{1}=\tau _{1}(\hat{\mathcal {B}}_{\ell }(\tau ))\ge 0\) such that for any weak solutions of problem (1.2)–(1.5) with \(\ell \)-trajectory \(\chi (s,\tau ;\textbf{z}_{\tau })\in \mathcal {B}_{\ell }(\tau )\), we have
and
for any \(t-\tau \ge \tau _{1}\), where \(\textbf{z}_{\tau }:=(v_{\tau }, F_{\tau }, M_{\tau })\).
Next, we prove the existence of a compact pullback absorbing set in \(X_{\ell }\) of the process \(\{L(t,\tau )\}_{t\ge \tau }\).
Lemma 3.3
Assuming that \((A_{1})\) holds, then for any \(\tau \in \mathbb {R}\) and \(\hat{\mathcal {B}}_{0}^{\ell }(\tau )\in \mathcal {D}_{\ell }\), there exist times \(\tau _{2}=\tau _{2}(\hat{\mathcal {B}}_{0}^{\ell }(\tau ))>0\) such that for any weak solutions of problem (1.2)–(1.5) with \(\ell \)-trajectory \(\chi \in \mathcal {B}_{0}^{\ell }(\tau )\subset X_{\ell }\), we have
for any \(t-\tau \ge \tau _{2}\), and
for any \(t-\tau \ge \tau _{2}\), where \(R_{5}, R_{6}\) are determined in the following proof.
Proof
From (3.1), we derive that
Integrating (3.15) from \(t-s\) to \(t+\ell \), for any \(t-\tau \ge \tau _{1}+\frac{\ell }{2}, s\in (0,\frac{\ell }{2})\), and considering (3.9), (3.10), we find that
where \(\gamma _{0}=\min \{\mu , 2\kappa , 2\}\).
From (3.16), and taking \(\tau _{2}=\tau _{1}+\frac{\ell }{2}\), we have
for any \(t-\tau \ge \tau _{2}\).
Therefore, we conclude that
for any \(t-\tau \ge \tau _{2}\), where \(R_{3}:=\frac{1}{\gamma _{0}}+\frac{C_{0}}{\gamma _{0}\alpha _{0}}+C(\frac{1}{\gamma _{0}\mu \lambda _{1}}+\frac{1}{\gamma _{0}\alpha _{0}\mu \lambda _{1}})R_{g}\).
From (3.2), (3.7), (3.10), we derive that
where \(R_{4}:=C+(\frac{C}{2\mu \lambda _{1}}+\frac{C}{2\mu \lambda _{1}\alpha _{0}})R_{g}\).
Applying (3.17), (3.19), the interpolation theorem and the Sobolev imbedding theorem (see, e.g., [40]), we have
for any \(t-\tau \ge \tau _{2}\).
So we conclude from (3.14), (3.18), (3.20) that
for any \(t-\tau \ge \tau _{2}\), where \(R_{5}:=C(R_{3}+R_{4}^{3/2}+R_{4})\).
From (2.2), we have
Taking \((\varphi ,\psi )\in L^{\infty }(t,t+\ell ; \textbf{V}\times H_{0}^{1}(\Omega ))\) such that \(\Vert (\varphi ,\psi )\Vert _{L^{\infty }(t,t+\ell ; \textbf{V}\times H_{0}^{1}(\Omega ))}\le 1\), we estimate the time derivatives \(v_{t}, F_{t}\) as follows:
and
Taking the supremum over all \((\varphi ,\psi )\) in (3.23) and (3.24), then we get
The time derivative \(M_{t}\) is estimated from (3.22)\(_{3}\) as follows:
From (3.23)–(3.26), there is a constant \(R_{6}>0\) such that
for any \(t-\tau \ge \tau _{2}\). \(\square \)
Let
endowed with the following norm
for any \(\tau \in \mathbb {R}\).
Then we also define \(\hat{\mathcal {B}}_{1}^{\ell }(t):=\{\mathcal {B}_{1}^{\ell }(t): t\in \mathbb {R}\}\), where
where \(R_{7}\) is a constant that depends on \(R_{5}\) and \(R_{6}\).
From Theorem 3.1 and Lemma 3.3, we conclude that \(L(t,\tau )\mathcal {B}_{0}^{\ell }(\tau )\subset \mathcal {B}_{0}^{\ell }(t)\) for any \(t\ge \tau \) and \(L(t,\tau )\mathcal {B}_{0}^{\ell }(\tau )\subset \mathcal {B}_{1}^{\ell }(t)\) for any \(t-\tau \ge \tau _{2}\). Furthermore, we have
Theorem 3.2
Assuming that \((A_{1})\) holds, then
for any \(t\ge \tau \).
Proof
Similar to the results proved in literature [1], we may just omit it. \(\square \)
Lemma 3.4
Assuming that \((A_{1})\) holds, then for any \(\tau \in \mathbb {R}\), the mapping \(L(t,\tau ): X_{\ell }\rightarrow X_{\ell }\) is Lipschitz continuous on \(B_{0}^{\ell }(\tau )\) for any \(t\ge \tau +\ell \).
Proof
For any fixed \(\tau \in \mathbb {R}\) and any \(\chi _{_{1}}\), \(\chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\), let \(L(t,\tau )\chi _{_{1}}=(v_{1}(t),F_{1}(t),M_{1}(t))\), \(L(t,\tau )\chi _{_{2}}=(v_{2}(t),F_{2}(t),M_{2}(t))\) for any fixed \(t\ge \tau +\ell \) and denote by \((\bar{v}, \bar{F},\bar{M})=(v_{1}-v_{2}, F_{1}-F_{2},M_{1}-M_{2})\). Since \(e_{0}(\chi _{_{1}})\) and \(e_{0}(\chi _{_{2}})\) are uniformly bounded in \(\mathbb {H}\), we can infer from (2.2) that
and
where
For \(I_{1}, I_{4}, I_{8}\), using Hölder’s inequality, Young’s inequality and Sobolev imbedding theorem, we have
and
Next, we estimate the terms \(I_{3}+I_{9}\) and \(I_{2}+I_{5}\). Using the divergence free condition on v and (2.1), we first obtain
From (3.31), (3.35), we find that
Applying results in [34] and Hölder’s inequality, Young’s inequality, Sobolev imbedding theorem, we have
Similarly, we have
For \(I_{6}, I_{7}\), using the following results
and Gagliardo–Nirenberg inequality (see, e.g., [40]), then we have
Substituting the results of (3.32)–(3.40) into (3.29)–(3.31), respectively. Then we obtain
where
Integrating (3.41) from \(\tau +s\) to \(t+s\), for any \(s\in (0,\ell )\), we obtain
From Lemmas 3.2, 3.3, using Gronwall’s lemma, we obtain
where \(N_{\ell }(t,\tau )=exp\left( \int _{\tau }^{t+\ell }\rho (r)\textrm{d}r\right) \) is a finite number depending on \((v_{1,\tau }, F_{1,\tau }, M_{1,\tau })\) and \((v_{2,\tau }, F_{2,\tau }, M_{2,\tau })\).
Integrating (3.43) with respect to s over \((0,\ell )\), we derive that
Therefore,
which implies the mapping \(L(t,\tau ):X_{\ell }\rightarrow X_{\ell }\) is Lipschitz continuous on \(\mathcal {B}_{0}^{\ell }(\tau )\) for all \(t\ge \tau +\ell \). \(\square \)
From Theorems 3.1, 3.2, Lemmas 3.3, 3.4 and Lemma 2.5 in [1], we can infer that \(\hat{\mathcal {B}}_{1}^{\ell }(t):=\{\mathcal {B}_{1}^{\ell }(t): t\in \mathbb {R}\}\) is a family of positive invariant, uniformly pullback absorbing compact subsets of \(X_{\ell }\), where
With a similar method to get the results of the autonomous case from Lemma 2.1 in [1], we can immediately obtain the following result.
Theorem 3.3
Assuming that \((A_{1})\) holds, then the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) possesses a pullback attractor \(\hat{\mathcal {A}}_{\ell }=\{\mathcal {A}_{\ell }(t): t\in \mathbb {R}\}\) in \(X_{\ell }\) and \(e_{1}(\mathcal {A}_{\ell }(t-\ell ))\subset B_{1}(t)\) for any \(t\in \mathbb {R}\), where
for any \(t\in \mathbb {R}\).
Next, we still need to prove the smooth property of the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) to prove the pullback attractor with finite fractal dimension in \(X_{\ell }\).
Lemma 3.5
Assuming that \((A_{1})\) holds, then there exists a constant \(\kappa _{1}>0\) such that for any fixed \(\tau \in \mathbb {R}\) and any \(t\ge \tau +\ell \),
where \(\chi _{_{1}}\) and \(\chi _{_{2}}\) are two \(\ell \)-trajectories in \(\mathcal {B}_{0}^{\ell }(\tau )\), \(\kappa _{1}, C_{0}(\tau ,t,\ell )\) and \(C_{1}(\tau ,t,\ell ))\) are given in (3.52), (3.66), respectively.
Proof
From (3.28) and Lemma 2.5 in [1], we can infer that \(\mathfrak {X}_{\ell }\subset \subset X_{\ell }\). For any fixed \(\tau \in \mathbb {R}\) and any \(\chi _{_{1}}, \chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\), let \(L(t,\tau )\chi _{_{1}}=(v_{1}(t),F_{1}(t),M_{1}(t))\), \(L(t,\tau )\chi _{_{2}}=(v_{2}(t),F_{2}(t),M_{2}(t))\) and \((\bar{v}, \bar{F}, \bar{M})=(v_{1}-v_{2}, F_{1}-F_{2}, M_{1}-M_{2})\) for any \(t\ge \tau +\ell \).
For any \(t\ge \tau +\ell \), integrating (3.41) from \(t-s\) to \(t+\ell \) with \(s\in [0,\frac{\ell }{2}]\), we obtain
where \(C_{0}=\min \{\mu , \kappa , 1\}>0\).
Similar to the proof of Lemma 3.4 in [1], using Gronwall’s lemma to (3.46), we obtain
where \(K_{\ell }(t,\tau ):=\int _{\tau +\frac{\ell }{2}}^{t+\ell }\rho (r)\textrm{d}r+1\).
For any \(t\ge \tau +\ell \), integrating (3.41) from \(\tau +s\) to \(t-s\) with \(s\in [0,\frac{\ell }{2}]\), we have
Applying Gronwall’s Lemma to (3.48), we get
Considering (3.47) and (3.49), we have
where \(N_{\ell }(t,\tau ):=exp\left( \int _{\tau }^{t+\ell }\rho (r)\textrm{d}r\right) \).
Integrating (3.50) with respect to s over \((0,\frac{\ell }{2})\), we get
Noticing that \(K_{\ell }(t,\tau ), N_{\ell }(t,\tau )\) are bounded for any fixed \(t\in [\tau +\ell , +\infty )\), we can infer that
Therefore,
for any \(\chi _{_{1}}, \chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\) and any \(t\ge \tau +\ell \).
Taking the difference of (2.2) solved by \((v_{1}, F_{1}, M_{1}), (v_{2}, F_{2}, M_{2})\), we have
Taking \((\varphi ,\psi )\in L^{\infty }(t,t+\ell ; \textbf{V}\times H_{0}^{1}(\Omega ))\) such that \(\Vert (\varphi ,\psi )\Vert _{L^{\infty }(t,t+\ell ; \textbf{V}\times H_{0}^{1}(\Omega ))}\le 1\), we estimate the time derivatives \(\bar{v}_{t}, \bar{F}_{t}\) as follows:
where
Let us estimate the terms on the right hand sides of (3.55), (3.55) one by one. From Lemma 3.3, using Hölder’s, Young’s, Sobolev’s and Poincaré’s inequalities, we derive that
and
From (2.1), we conclude that
and
Plugging (3.57)–(3.63) into (3.55), (3.55) and taking the supremum over all \((\varphi ,\psi )\), we conclude that
Similar to the above estimates, we can estimate the time derivative \(\bar{M}_{t}\) as follows:
We obtain from (3.52), (3.64) and (3.65) that
Therefore,
for any \(\chi _{_{1}}, \chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\) and any \(t\ge \tau +\ell \). \(\square \)
Combining with Lemma 2.2 in [1], Theorem 3.3 and Lemma 3.5, we can obtain that the following result.
Theorem 3.4
Assuming that \((A_{1})\) holds, then the fractal dimension of a pullback attractor \(\hat{\mathcal {A}}_{\ell }=\{\mathcal {A}_{\ell }(t): t\in \mathbb {R}\}\) in \(X_{\ell }\) of the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) obtained in Theorem 3.3 is finite.
Proof
Since each section \(\mathcal {A}_{\ell }(t)\) of pullback attractor \(\hat{\mathcal {A}}_{\ell }=\{\mathcal {A}_{\ell }(t): t\in \mathbb {R}\}\) is bounded in \(X_{\ell }\), there exist a positive constant R and some \(\chi _{_{0}}\in \mathcal {A}_{\ell }(t)\) such that
From Lemma 3.5, we can infer that there is a constant \(C_{2}=C_{2}(t,\ell )>0\) such that
On the other hand, from Theorem 3.3 and (3.69), we derive that
It follows from (3.69) and \(\mathfrak {X}_{\ell }\subset \subset X_{\ell }\), then there exist a finite number N of balls with the same radius \(\frac{R}{2}\) centered in \(\xi _{1}, \xi _{2}, \cdot \cdot \cdot , \xi _{N}\in X_{\ell }\) such that
Considering (3.69)–(3.71) and Theorem 3.3, we get
Since each ball \(B_{\mathfrak {X}_{\ell }}(L(t,\tau )\xi _{i};\frac{R}{2})\) can be covered by N balls with the same radius \(\frac{R}{2^{2}}\) centered in \(\xi _{i1}, \xi _{i2}, \cdot \cdot \cdot , \xi _{iN}\in X_{\ell }\), we have
Repeatedly, we can obtain that
For any \(\varepsilon >0\), we can choose some positive integer n sufficiently large such that \(\frac{R}{2^{n}}\le \varepsilon <\frac{R}{2^{n-1}}\); then we can infer from (3.74) that
This completes the proof of Theorem 3.4.
\(\square \)
3.2 The Existence of Pullback Attractors in \(\mathbb {H}\)
In order to obtain that the process \(\{U(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5) has a pullback attractor with finite fractal dimension in the original phase space \(\mathbb {H}\), we also need to prove the following results.
Lemma 3.6
Assuming that \((A_{1})\) holds, then the mapping \(e_{1}\): \(\mathcal {B}_{0}^{\ell }(\tau -\ell )\rightarrow B_{1}(\tau )=e_{1}(\mathcal {B}_{0}^{\ell }(\tau -\ell ))\) is Lipschitz continuous for any fixed \(\tau \in \mathbb {R}\). That is, for any two \(\ell \)-trajectories \(\chi _{_{1}},\chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\), there exists a positive constant C dependent on \(\ell \) such that
Proof
For any fixed \(\tau \in \mathbb {R}\) and any \(\chi _{_{1}}, \chi _{_{2}}\in \mathcal {B}_{0}^{\ell }(\tau )\), let \(L(t,\tau )\chi _{_{1}}=(v_{1}(t),F_{1}(t),M_{1}(t))\), \(L(t,\tau )\chi _{_{2}}=(v_{2}(t),F_{2}(t),M_{2}(t))\) for any \(t\ge \tau +\ell \). We use \((\bar{v}, \bar{F}, \bar{M})=(v_{1}-v_{2}, F_{1}-F_{2}, M_{1}-M_{2})\) to denote the difference of two solutions for problem (1.2)–(1.5).
For any fixed \(\tau \in \mathbb {R}\) and any \(s\in (0,\ell )\), using Gronwall’s lemma for (3.41), we get
Integrating (3.77) with respect to s over \((0,\ell )\), we can conclude that
where \(C(\ell ,\tau ):=\frac{1}{\ell }N_{\ell }(\tau ), N_{\ell }(\tau ):=exp\left( \int _{\tau }^{\tau +\ell }\rho (r)\textrm{d}r\right) \) is a finite number depending on \((v_{1,\tau }, F_{1,\tau }, M_{1,\tau })\) and \((v_{2,\tau }, F_{2,\tau }, M_{2,\tau })\).
Finally, we can infer from (3.78) that the mapping \(e_{1}\): \(\mathcal {B}_{0}^{\ell }(\tau -\ell )\rightarrow B_{1}(\tau )\) is Lipschitz continuous. The proof is complete. \(\square \)
According to the above results, we now give the proof of Theorem 1.1 as follows:
Proof of Theorem 1.1
From Lemma 2.3 in [1], Theorem 3.4 and Lemma 3.6, we can infer that for any \(t\in \mathbb {R}\), the sections \(\mathcal {A}(t)\) of the pullback attractor \(\hat{\mathcal {A}}\) are compact and their fractal dimensions are uniformly finite. From the invariance of \(\hat{\mathcal {A}}_{\ell }\): \(L(t-\ell ,s-\ell )\mathcal {A}_{\ell }(s-\ell )=\mathcal {A}_{\ell }(t-\ell )\) for any \(t\ge s\), we derive that
for any \(t\ge s\).
For any bounded subset B of \(\mathbb {H}\), from the definition of \(B_{1}(t)\) and \(\mathcal {B}_{0}^{\ell }(t)\), then there exists some time \(\tilde{\tau }=\tilde{\tau }(B)>0\) such that
for any \(t-\tau \ge \tilde{\tau }\).
Therefore, we only need to prove that
Otherwise, there exist some sequence \(\{(v_{n},F_{n},M_{n})\}_{n\ge 1}\subset B_{1}(\tau _{n})\), a positive constant \(\varepsilon _{0}\) and some \(\tilde{\tau }_{n_{0}}>0\) such that for any \(t-\tau _{n}\ge \tilde{\tau }_{n_{0}}\), we have
On the other hand, from the definition of \(B_{1}\), we can infer that there exists a sequence \(\{\chi _{n}\}_{n\ge 1}\subset \mathcal {B}_{0}^{\ell }(\tau _{n})\) such that
Since \(\{\chi _{n}\}_{n\ge 1}\) is bounded in \(X_{\ell }\) and \(\hat{\mathcal {A}}_{\ell }\) is a pullback attractor in \(X_{\ell }\) of the process \(\{L(t,\tau )\}_{t\ge \tau }\) generated by problem (1.2)–(1.5), there exist a subsequence \(\{\chi _{n_{j}}\}_{n_{j}\ge 1}\) of \(\{\chi _{n}\}_{n\ge 1}\) and a subsequence \(\{\tau _{n_{j}}\}_{n_{j}\ge 1}\) of \(\{\tau _{n}\}_{n\ge 1}\) such that
Using the continuity of \(e_{1}\), we have
The contradiction to (3.81) completes the proof. \(\square \)
Remark 3.1
For system (1.2)–(1.5), if we consider the external force term \(\varepsilon g(x,t)\) depending on a small parameter \(\varepsilon \in (0, 1]\) as a small perturbation to the autonomous system, then we obtain a continuous process \(U^{\varepsilon }(\cdot ,\cdot )\) driven by the non-autonomous dynamical system. Since the upper semicontinuity implies some stability for the attractors of the systems under some perturbations (see [8, 16, 31, 32, 39] and references therein). It is also interesting and important to consider the relationship between the pullback attractors \(\hat{\mathcal {A}}^{\varepsilon }=\{\mathcal {A}^{\varepsilon }(t):t\in \mathbb {R}\}\) for the perturbed system (1.2)–(1.5) with \(\varepsilon \in (0,1]\) and the global attractor \(\mathcal {A}^{0}\) for the unperturbed system (1.2)–(1.5) with \(\varepsilon =0\) (see, e.g., [28]).
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Acknowledgements
J. Shen was supported by NSFC (Nos. 11831012, 12090013, 12071317). The authors also wish to thank the referees and the editor for their valuable comments and suggestions.
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Ai, C., Shen, J. Finite Fractal Dimensional Pullback Attractors for a Class of 2D Magneto-Viscoelastic Flows. Bull. Malays. Math. Sci. Soc. 47, 17 (2024). https://doi.org/10.1007/s40840-023-01606-y
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DOI: https://doi.org/10.1007/s40840-023-01606-y