H2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains

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Abstract

In this paper, the authors study the long time behavior of a non-Newtonian system in two-dimensional unbounded domains. They prove the existence of H2-compact attractor for the system by showing the corresponding semigroup is asymptotically compact.

Introduction

The motion of an isothermal, incompressible viscous fluid can be described by the systemut+(u·)u+·τ=f,·u=0,where u represents the velocity field of the fluid, τ=(τij) is the stress tensor whose components areτij=pδij−τijv,where p is the pressure, τv=(τijv) is the viscous part of the stress tensor, f is the external body force. If the fluid conforms to the Stokes law, that is, τv depends linearly on e=(eij(u)), the symmetric deformation velocity tensor,τijv=νeij(u),whereeij(u)=12∂ui∂xj+∂uj∂xi,the fluid is called a Newtonian one and the system turns into the well-known Navier–Stokes equations. In particular, if the viscosity of the fluid can be neglected, τv=0, the system is reduced to the Euler equations.

Many fluid materials, for examples, liquid foams, polymeric fluids such as oil in water, blood, etc., do not conform to the Stokes law. Their viscous stress tensors are nonlinear in e=(eij) and may also depend on the derivatives of e. These fluids are called non-Newtonian. A typical model is the bipolar viscous non-Newtonian fluid whose constitutive relations have the formτijv=2μ0(ε+|e|2)−α/2eij−2μ1Δeij,μ01,ε>0,τijk=2μ1∂eij∂xk,k=1,2,where α>0 is a constant and |e|=i,j|eij|21/2.

In this paper, we are going to study the isothermal, incompressible, bipolar, viscous non-Newtonian fluid on a two-dimensional infinite strip Ω=R×(−a,a), , with (1.4), that isut+(u·)u+p=·(2μ0(ε+|e|2)−α/2e−2μ1Δe)+f,x∈Ω,t>0,·u=0,x∈Ω,t>0,u=0,τijknjnk=0,x∈∂Ω,t⩾0,u=u0,x∈Ω,t=0,where the summation convention of repeated indices is used, n=(n1,n2) is the exterior unit normal to the boundary ∂Ω of Ω. We will concentrate our attention on 0<α<1. We refer to [1], [4], [7], [8], [9], [11] for detailed background.

There is series works on the existence and uniqueness, regularity and long time behavior of weak solutions of the above equations (see [2], [3], [5], [6], [10], [13], [16]). In particular, Bloom and Hao [6] proved the existence of the global attractor. Because the domain of the spatial variables is unbounded, the embedding between the usual Sobolev spaces is no longer compact. To recover the compactness, [6] assumed f belongs to certain weighted space. Later on in [16], the authors avoided using the weighted space and proved the existence of global attractor in the H space. In this paper we use the ideas in [12], [14] to show the existence of the global attractor in the V space. The new difficulty comes from the more nonlinear terms in the equation due to the lack of the compactness of the Sobolev embeddings. We use functional analysis to prove the convergence of these nonlinear terms and prove the corresponding semigroup is asymptotically compact.

The main result of this paper is

Theorem 1.1

Let fH. Then the semigroup {S(t)}t⩾0 associated with , , , possesses a global attractor A⊂V satisfying

  • (1)

    (Compactness) A is compact in V;

  • (2)

    (Invariance) S(t)A=A, t⩾0;

  • (3)

    (Attractivity)  B⊂V bounded, limt→∞distV(S(t)B,A)=0,

where distV is the distance in the norm topology of V (see below for the notations).

The rest of the paper is organized as follows. Section 2 is preliminary. We present the functional setting and existence results of solutions. In Section 3 we show the asymptotic compactness property of the semigroup and prove our main result.

Throughout this paper we denote by Lp(Ω),Wm,p(Ω) both the scalar and vector-valued Lebesgue and Sobolev spaces, by ||·||p and ||·||m,p their norms, especially ||·||=||·||2,Hm(Ω)=Wm,2(Ω). That a vector is in some space X means all its components are in X. DenoteV={ϕ=(ϕ12)∈C0(Ω̄),·ϕ=0inΩ,ϕ=0on∂Ω}H is the closure of V in L2(Ω), V is the closure of V in H2(Ω), V′ and H′ the dual spaces of V and H, respectively. If we identify the dual space H′ with H itself, then VH=H′↪V′, with dense and continuous injections. We use “↪” Denote the imbedding between space, “” weakly convergent and “→” strongly convergent, C the generic constant. The summation convention of repeated indices is used in the whole paper.

Section snippets

Preliminaries

Leta(u,v)=Ω∂eij(u)∂xk∂eij(v)∂xkdx,u,v∈V.

Lemma 2.1

Bloom and Hao [5]

There exist positive constants C1 and C2 such thatC1||u||2H2(Ω)∂eij(u)∂xk,∂eij(u)∂xk⩽C2||u||2H2(Ω)∀u∈V.

We see that a(·,·) defines a positive symmetric bilinear form on V. As a consequence of the Lax–Milgram Lemma we obtain an isometry operator AL(V,V′), via〈Au,v〉=a(u,v)∀v∈Vwhere 〈·,·〉 is the dual product between V′ and V.

Let D(A)={ϕV,H}, then D(A) is a Hilbert space and A is also an isometry from D(A) to H. In fact, A=PΔ2, where P is the projector

Asymptotical compactness of S(t)

We assume in the sequel that fH is independent of t. Then S(t) is a semigroup in H. In this section, we prove the asymptotic compactness of S(t). First define 〚·,·〛:D(A)×D(A)↦R as〚u,v〛=2μ1(Au,Av)−C12C2μ1∂eij(u)∂xk,∂eij(v)∂xk∀u,v∈D(A)⊂V.Then we have〚u〛2=〚u,u〛⩾2μ1||Au||2−C12μ1||u||2H2⩾2μ1||Au||2−μ1||Au||21||Au||2andμ1||Au||2⩽〚u〛2⩽2μ1||Au||2.

Let k1=C12/C2, u(t)=S(t)u0,u0V. Multiplying both sides of (2.6) by Au we obtainddt∂eij(u)∂xk,∂eij(u)∂xk+k1μ1∂eij(u)∂xk,∂eij(u)∂xk=

Acknowledgements

The authors express their deep gratitude to the referee for pointing out some mistakes in the earlier proof.

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This work is supported by the National Science Foundation of China under grant number 10001013 and ZheJiang Province Natural Science Foundation under grant number M103043.

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