Abstract
In this paper, we show the existence of pullback attractors for the nonautonomous NavierStokesVoight equations by using contractive functions, which is more simple than the weak continuous method to establish the uniformly asymptotical compactness in H^{1 }and H^{2}.
2010 Mathematics Subject Classification: 35D05; 35M10
Keywords:
NavierStokesVoight equations; processes; contractive functions; uniform attractors1 Introduction
Let Ω ⊂ R^{3 }be a bounded domain with sufficiently smooth boundary ∂Ω. We consider the nonautonomous 3D NavierStokesVoight (NSV) equations that govern the motion of a KleinVoight linear viscoelastic incompressible fluid:
Here u = u(t, x) = (u_{1}(t, x), u_{2}(t, x), u_{3}(t, x)) is the velocity vector field, p is the pressure, ν > 0 is the kinematic viscosity, and the length scale α is a characterizing parameter of the elasticity of the fluid.
When α = 0, the above system reduce to the wellknown 3D incompressible NavierStokes system:
For the wellposedness of 3D incompressible NavierStokes equations, in 1934, Leray [13] derived the existence of weak solution by weak convergence method; Hopf [4] improved Leray's result and obtained the familiar LerayHopf weak solution in 1951. Since the 3D NavierStokes equations lack appropriate priori estimate and the strong nonlinear property, the existence of strong solution remains open. For the infinitedimensional dynamical systems, Sell [5] constructed the semiflow generated by the weak solution which lacks the global regularity and obtained the existence of global attractor of the 3D incompressible NavierStokes equations on any bounded smooth domain. Chepyzhov and Vishik [6] investigated the trajectory attractors for 3D nonautonomous incompressible NavierStokes system which is based on the works of Leray and Hopf. Using the weak convergence topology of the space H (see below for the definition), Kapustyan and Valero [7] proved the existence of a weak attractor in both autonomous and nonautonomous cases and gave a existence result of strong attractors. Kapustyan, Kasyanov and Valero [8] considered a revised 3D incompressible NavierStokes equations generated by an optimal control problem and proved the existence of pullback attractors by constructing a dynamical multivalued process.
However, the infinitedimensional systems for 3D incompressible NavierStokes equations have not yet completely resolved, so many mathematicians pay attention to this challenging problem. In this regard, Kalantarov and Titi [9] investigated the NavierStokesVoight equations as an inviscid regularization of the 3D incompressible NavierStokes equations, and further obtained the existence of global attractors for NavierStokesVoight equations. Recently, Qin, Yang and Liu [10] showed the existence of uniform attractors by uniform condition(C) and weak continuous method to obtain uniformly asymptotical compactness in H^{1 }and H^{2}, Yue and Zhong [11] investigated the attractors for autonomous and nonautonomous 3D NavierStokesVoight equations in different methods. More details about the infinitedimensional dynamics systems, we can refer to [1227].
Using the contractive functions, we have in this paper established the uniformly asymptotical compactness of the processes {U(t, τ)}(t ≥ τ, τ ∈ R) to obtain the existence of the uniform attractor of the 3D nonautonomous NSV equations.
Main difficulties we encountered are as follows:
(1) how to obtain a contractive function,
(2) how to deduce the uniformly asymptotical compactness from a contractive function,
(3) how to obtain the convergence of contractive function.
2 Main results
Notations: Throughout this paper, we set R_{τ }= [τ, +∞), τ ∈ R. C stands for a generic positive constant, depending on Ω, but independent of t. L^{p}(Ω)(1 ≤ p ≤ +∞) is the generic Lebesgue space, H^{s}(Ω) is the general Sobolev space. We set , H, V, W is the closure of the set E in the topology of (L^{2}(Ω))^{3}, (H^{1}(Ω))^{3}, (H^{2}(Ω))^{3 }respectively. "⇀" stands for weak convergence of sequence.
Let be the hull of f_{0 }as a symbol space:
for all , where denotes the closure in the topology of .
Under the assumptions of the initial data, the problem (1.1)(1.4) has a global solution u ∈ C([τ, +∞), V). U_{f}(t, τ, u_{τ}): V → V denotes the processes generated by the global solutions and satisfies
Let {T(s)} be the translation semigroup on Σ, we see that the family of processes {U_{f}(t, τ)} (f ∈ Σ) satisfies the translation identity if
Next, we recall a simple method to derive uniformly asymptotical compactness which can be found in [28].
Definition 2.1 Let X be a Banach space and B be a bounded subset of X, Σ be a symbol space. We call a function ϕ(·,·;·,·) defined on (X × X) × (Σ × Σ) to be a contractive function on B × B if for any sequence and any {g_{n}} ⊂ Σ, there are subsequences and such that
We denote the set of all contractive functions on B × B by Contr(B, Σ).
Lemma 2.2 Let {U_{f}(t, τ)}(f ∈ Σ) be a family of processes satisfying the translation identity (2.5) and (2.6) on Banach space X and has a bounded uniform (w.r.t f ∈ Σ) absorbing set B_{0 }⊂ X. Moreover, assume that for any ε > 0, there exist T = T(B_{0}, ε) and ϕ_{T }∈ Contr(B_{0}, Σ) such that
Then {U_{f}(t, τ)} (f ∈ Σ) is uniformly (w.r.t. f ∈ Σ) asymptotically compact in X.
Theorem 2.3 Assume that f ∈ Σ ⊆ L^{2}(R, H), u_{τ }∈ V, then the problem (1.1)(1.4) possesses uniform attractors in V.
Theorem 2.4 Assume that f ∈ Σ ⊆ L^{2}(R, H), u_{τ }∈ W, then the problem (1.1)(1.4) possesses uniform attractors in W.
3 Proof of Theorem 2.3
In this section, we shall prove Theorem 2.3 by two steps as follows, the first one is to get the existence of an absorbing ball, the second is to prove the asymptotical compactness by means of a contractive function.
From the property of solutions, we can easily derive that the set class {U_{f}(t, τ, u_{τ})} (τ, ≤ t) is a process in V for all τ ≤ t. Moreover, the mapping U_{f}(t, τ, u_{τ}): V → V is continuous.
Lemma 3.1 We assume that and , f^{n }→ f in L^{2}(R, H), then
Proof. From the boundedness of the solutions in corresponding topological spaces, we easily conclude the results. □
Lemma 3.2 Assume f ∈ L^{2}(R, H), u_{τ }∈ V, then there exists a uniform (w.r.t. f ∈ Σ) absorbing set B_{0 }of processes {U_{f}(t, τ, u_{τ})}.
Proof. For all u ∈ V, multiplying both sides of (1.1) with u and noting that ((u·∇)u, u) = 0, we derive
Consequently, for all τ ∈ R, there holds
Consider the property of the functional ⟨·,·⟩ + α^{1}⟨∇·,∇·⟩, we get
and there exists a constant C_{0 }satisfying C_{1 }≤ C_{0 }≤ C_{2}, such that
Setting the radius , we easily get that there exists a constant C > 0 such that
for all u_{τ }∈ V, t ≥ τ.
Setting
then we denote R the nonnegative number given by
and consider the family of closed balls B_{0 }in V defined by
It is straightforward to check that B_{0 }is a uniform absorbing ball for the processes {U_{f}(t, τ, u_{τ})}. □
Lemma 3.3 Under the condition of f ∈ L^{2}(R, H), the process {U_{f}(t, τ, u_{τ})} generated by the global solutions for problem (1.1)(1.4) is uniformly asymptotically compact in V.
Proof. For any initial data , let u^{i}(t, x) be the corresponding solutions to the symbols f^{i }with , that is, u^{i}(t) is the solution of the problem:
Denote
then w(t) satisfies the equivalent abstract equations
where B(u) = (u·∇)u, p has disappeared by the projection operator P.
Setting
Multiplying (3.13) by w and integrating over [s, T] × Ω, we deduce
where τ ≤ s ≤ T. Then we have
Hence,
Integrating (3.18) over [τ, T] with respect to s, we get
If we set
then we have
Since the family of processes has a uniformly bounded absorbing set, we choose T large enough such that
Let u^{n}, u^{m }be the solutions with respect to the initial data and symbols f^{n}(t), f^{m}(t) ∈ Σ, m, n = 1, 2, . . . respectively. Then from Lemma 3.1, we can derive
and
Hence for the above T. By Lemma 2.2 and the property of the functional ⟨·,·⟩ + α^{2 }⟨∇·, ∇·⟩, the conclusion holds. □
Proof of Theorem 2.3 From Lemmas 3.13.3, we can deduce the result easily. □
4 Proof of Theorem 2.4
Similarly to the proof of Theorem 2.3, we easily obtain that the set class {U_{f }(t, τ, u_{τ})} (τ ≤ t) is a process in W for all τ ≤ t. Moreover, the mapping U_{f }(t, τ, u_{τ}): W → W is continuous. If we assume that is a sequence in W and weakly converges to u_{τ }∈ W, f^{n }→ f in L^{2}(R, H), then
Lemma 4.1 Assume f ∈ L^{2}(R, H), then there exists a global uniform (w.r.t. f ∈ Σ) absorbing set B_{0 }of the process {U_{f }(t, τ, u_{τ})}.
Proof. By the FaedoGalerkin method, the standard elliptic operator theory and the Poincaré inequality, we get that u belongs to L^{2}((τ, T); D(A)) ∩ L^{∞}((τ, T); W), then using the Gronwall inequality and similar energy method to the proof of Theorem 3.1 in Qin, Yang and Liu [10], we can deduce the boundedness of u and the existence of absorbing set. □
Lemma 4.2 Under the condition of f ∈ L^{2}(R, H), u_{τ }∈ W, the process {U_{f }(t, τ, u_{τ})} generated by the global solutions for problem (1.1)(1.4) is asymptotically compact in W.
Proof. For any initial data , let u^{i}(t, x) be the corresponding solutions to the symbols f^{i }with , that is, u^{i}(t) is the solution of the problem (3.8)(3.11). Denote A = Δ and w(t) = u^{1}(t)  u^{2}(t), then w(t) satisfies the equivalent abstract equations (3.13)(3.14).
Setting
Multiplying (3.13) by Aw and integrating over [s, T] × Ω, we deduce
where τ ≤ s ≤ T. Then we have
Hence,
Integrating (4.4) over [τ, T] with respect to s, we get
If we set
then we have
Since the family of the processes has a uniformly bounded absorbing set, we choose T large enough such that
Let u^{n}, u^{m }be the solutions with respect to the initial data and symbols f^{n}(t), f^{m}(t) ∈ Σ, m, n = 1, 2, . . . respectively. Then we can obtain
and
Hence for the above T. By Lemma 2.2 and the property of the functional ⟨·, A·⟩ + α^{2}⟨A·, A·⟩, the conclusion holds. □
Proof of Theorem 2.4 From Lemmas 4.14.2, we can deduce the result easily. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The work in part was supported by the NNSF of China (No. 11031003 and 10871040).
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