Long-Time Behavior for the Navier-Stokes-Voight Equations with Delay on a Non-smooth Domain

SU Keqinand QIN Yuming

1School of Information Science and Technology,Donghua University,Shanghai201620,China.

2Department of Applied Mathematics,Donghua University,Shanghai201620,China.

Abstract.This paper concerns the long-time behavior for the 2D incompressible Navier-Stokes-Voight equations with distributed delay on a non-smooth domain.Under some assumptions on the initial datum and the delay datum,the existence of compact global attractors is obtained.

Key Words:Navier-Stokes-Voight equations;distributed delay;global attractors.

1 Introduction

In this paper,we discuss the long-time behavior for the 2D Navier-Stokes-Voight equations with a distributed delay on a Lipschitz domain ?

where ?τ=?×(τ,+∞),??τ=??×(τ,+∞),?τh=?×(τ?h,τ).The function u=u(x,t)=(u1(x,t),u2(x,t))is the unknown velocity field of the fluid,p is the pressure,?∈L∞(??),ds is the distributed delay,and h＞0 is a constant.For any t∈(τ,T),we can define u:(τ?h,T)→(L2(?))2,and ut(s)is a function defined on(?h,0)satisfying ut(s)=u(t+s).

Based on the wellposedness of the Navier-Stokes equations,the in finite dimensional dynamical systems have attracted many mathematicians’attentions,which can be found in[1–3],and relating conclusions to the Navier-Stokes-Voightequationscan be seenin[4–6].The delay effect was investigated first for ordinary differential equations in physics,for the results on the Navier-Stokes equations with delays on bounded domain,such as the existence of global solutions and attractors,we can refer to[7–14].

The research on dynamical system for nonlinear dissipative systemon the non-smoothdomains(such as the Lipschitz domain)is a difficult problem,some interesting results can be seen in[15–18],and in this paper we shall use the method of[15]to derive the existence of compact global attractors.

This paper is organized as follows.In Section 2,some preliminaries are given which will be used in the sequel.The existence and uniqueness of solutions for system(1.1)are derived in Section 3.And in the last section,we derive the existence of global attractors in an appropriate topology space.

2 Preliminaries

2.1 Functional spaces

Also,V′is the dual space of V with norm ‖·‖?.Write

with norms ‖u‖CH=supθ∈[?h,0]|u(t+θ)|and ‖u‖CV=supθ∈[?h,0]‖u(t+θ)‖ respectively.Similarly,=L2(?h,0;H).

2.2 Some operators and useful theorems

(1)The Stokes operator.

and D(As)denotes E in D(As)topology with norm ‖u‖2s,and Assatisfies([15])

The Hardy inequality will be used:

(2)The bilinear and trilinear operators.

which had been discussed detailedly in[3],and

Theorem 2.1.(Arzela-Ascoli Theorem)Let X be a compact subset of Rm1,and{fn}is a sequence of continuous functions from X into Rm2.If{fn}is uniformly bounded and equicontinuous,then{fn}has a subsequence that converges uniformly on X.

Proof.See[2].

3 Wellposedness

3.1 Background functions

To reduce the systemto a homogeneousproblem,we considerthe background flow function ψ which solves

and in[15]it was shown that

Let ε∈(0,c·diam(?)),ηε∈(R2)satisfies

and

where

and

3.2 Assumptions

Suppose the external force G in(1.1)satisfies:

(A1)G:[?h,0]×R2→R2is measurable,and G(s,0)=0 for s∈[?h,0];

(A2)? Lg＞0 such thatfor ξ,η ∈CH;

(A3)for any u,v∈C0([τ?h,T];H)and any t＞τ,? Cg＞0,m0≥0(m∈[0,m0])such that

(A4)νλ1＞3Cg.

3.3 Abstract equations with homogeneous boundary

Let v=u?ψ,(1.1)can be rewritten as

3.4 Existence of solutions and uniqueness

Proof.Fix n≥1,we define an approximate solutionto(3.2)and

Multiplying(3.3)by vn,we have

which yields

and in following way we can estimate each term of(3.5).

Using the Hardy inequality,the H?lder inequality and the properties of operators,choosing suitable ε,we can derive(see[15,19,20])

Based on these estimates,(3.5)is reduced to

Integrating(3.7)in t,we obtain

which implies there exists a constant K1＞0 such that

Integrating(3.6),we also derive

and

which means there exists IV＞0 such thatAnd vn(t)∈ L∞(τ,T;V)∩L2(τ?h,T;V)holds naturally,from the Alaoglu compact theorem,we can find a subsequence{vn}such that

In[18],it had been shown that B(vn),R(vn)∈ L2(τ,T;V′).Since vn∈L2(τ,T;V),we derive thatand from

Similar to the proof about uniqueness of solutions in[18],we assume that u1and u2are twosolutionsfor(1.1)withstreamfunctions ψ1and ψ2respectively.Lettingwith divv=0,we get

Write w=u1?u2,we get

which leads to w=0 and the uniqueness of solutions is natural.

Remark 3.1.In general,the regularity of weak solution u(x,t)to problem(1.1)can be extended to D(A)by the similar method since the strong damping.And using the technique as shown in[18],we can derive the continuous dependence on the initial data and delay data.

4 Existence of global attractors

In this dissipative system,the solution generates a continuous semi flow

from the fundamental theory of attractors([1–3]),we also need some dissipation and some compactness for the semi flow to derive the existence of global attractors.

4.1 Dissipation:existence of absorbing sets

Proof.Similar to the proof of Theorem 3.2,we get

and

Integrating(4.2)in t,we have

that is,

Let t＞h,θ∈[?h,0],then

and

which leads to the existence of global absorbing balls in CV.

4.2 Asymptotical compactness and main results

An effective theorem was provided in[5]in 2009,in which a decomposition method for semigroup was demonstrated in detail,and the asymptotical compactness was derived at last.Similar to the proof in[5],we can easily show the asymptotical compactness for the semi flow in CVto the model in this paper,and we also get the following main result.

Theorem 4.2.Assumeˉf∈(L2(?))2,(v0,η)∈V×,and the conditions(A1)～(A4)hold,then the system(3.1)possesses a compact connect global attractorA′in CVwhich contains all equilibriums and unstable manifolds,i.e.,the ω-limit set contains all bounded complete trajectories.

Acknowledgements

This paper is in part supported by NNSF of China with contract numbers 11671075.