The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier–Stokes equations

P.E Kloeden , J Valero

Abstract

The attainability set of the weak solutions of the three-dimensional Navier–Stokes equations which satisfy an energy inequality is shown to be a weakly compact and weakly connected subset of the space H, i.e. the Kneser property holds in the weak topology for such weak solutions. The proof of weak connectedness uses the strong connectedness of the attainability set of the weak solutions of the globally modified Navier–Stokes equations, which is first proved. The weak connectedness of the weak global attractor of the three-dimensional Navier–Stokes equations is also established.

Keywords:

1. Introduction

The three-dimensional Navier–Stokes equations (NSE) are an intriguing system of partial differential equations. They have been intensively investigated for many years (e.g. Ladyzhenskaya 1983, 1987, 1994; Raugel & Sell 1993; Sell 1996; Bondarevski 1997; Chepyzhov & Vishik 1997; Flandoli & Schmalfuss 1999; Ball 2000; Birnir & Svanstedt 2004; Cutland & Keisler 2004; Kapustyan 2005; Cheskidov & Foias 2006), but some very basic issues on their solvability remain unresolved. For example, although weak solutions are known to exist for all future time for each initial condition in the function space H (i.e. the space of divergence-free velocity fields with the L2-norm), it is not known if there is a unique weak solution. Ordinary differential equations without uniqueness (e.g. with a continuous but non-Lipschitz vector field) are well understood and a classical result of Kneser says that their attainability set is a continuum, i.e. compact and connected set. This is often called the Kneser property and is known to hold for certain classes of parabolic and hyperbolic partial differential equations also (e.g. Kaminogo 2001; Ball 2004; Valero 2005). In this paper, we will show that the Kneser property holds for the weak solutions satisfying the energy inequality of the three-dimensional NSE on a bounded, smooth domain, provided we use the weak topology of the function space H.

In our proof, we make extensive use of the properties of the globally modified Navier–Stokes equations (GMNSE), which were introduced recently by Caraballo et al. (2006). These include a factor multiplying the nonlinear term of the NSE which depends on the reciprocal of the L2-norm of the gradient of the velocity field when it exceeds a prescribed value; hence many estimates are very similar to those for the two-dimensional NSE. In particular, the GMNSE have a unique global strong solution for each initial condition in the function space V (i.e. the space of gradients of divergent-free velocity fields with the L2-norm) as well as global weak solutions for each initial condition in the function space H, which instantaneously become strong solutions, but are possibly non-unique.

The main result is stated in the next section after background material on the NSE has been reviewed and the proof of weak compactness of the attainability sets is given. Then, in §3, the GMNSE are introduced and the Kneser property in the strong topology of the space H is proved. This is then used in §4 to complete the proof of the main result, specifically to prove the weak connectedness of the attainability set of the weak solutions of the NSE. In conclusion, the results are used in the final section to show that the weak global attractor in H of the set-valued dynamical system generated by the weak solutions of the NSE is connected in the weak topology of H.

2. Preliminaries

We consider the Navier–Stokes equations (NSE)Embedded Image(2.1)for given ν>0, where Ω is a bounded open subset of Embedded Image3 with smooth boundary.

First, we recall standard notation and basic results following Temam (1979). In particular, we use the usual function spacesEmbedded Imagewhere clX denotes the closure in the space X. The spaces H and V are separable Hilbert spaces and VHV* with dense and compact injections when H is identified with its dual H*. Let us denote Hw for the space H endowed with the weak topology. Let (., .), ‖.‖H and (., .), ‖.‖V denote the inner product and norm in H and V, respectively, and let 〈., .〉 denote pairing between V and V*. For u, v, wV,Embedded Imagedefines a trilinear continuous form on V with b(u, v, v)=0 when uV and Embedded Image. For u, vV, let B(u, v) denote the element of V* defined by 〈B(u, v), w〉=b(u, v, w) for all wV.

We say that the function u is a weak solution of the NSE (2.1) ifEmbedded Imagefor all T>τ andEmbedded Image(2.2)in the sense of distributions on (τ, T).

If fL2(τ, T, V*), for all T>τ, and if u satisfies equation (2.2), thenEmbedded Imagefor all T>τ. In particular, the initial condition u(τ)=uτ makes sense for any uτH and equation (2.2) with u(τ)=uτ is equivalent toEmbedded Imagefor all tτ and wV.

Let A: VV* be the linear operator associated with the bilinear form ((u, v))=〈Au, v〉. Then, A is an isomorphism from D(A) onto H and D(A)=(H2(Ω))3V, where the injection D(A)⊂V is dense and continuous. Moreover, ‖Au‖H is a norm equivalent to that induced by (H2(Ω))3. The NSE (2.1) can then be rewritten asEmbedded Image(2.3)

It is well known (Temam 1979) that if fL2(τ, T, V*), for all T>τ, and if uτH, then the NSE have at least one weak globally defined solution which satisfies the energy inequality:Embedded Image(2.4)whereEmbedded ImageNow let us write

Embedded Imageand denote the corresponding attainability set for tτ byEmbedded ImageThe first of our main results is the following version of the Kneser property for the weak solutions of the NSE.

Let fL(τ, T; H) for all T>τ. Then, for all tτ and uτH, the attainability set Kt(uτ) is compact and connected with respect to the weak topology on H.

The proof that Kτ(t) is weakly compact for any tτ is an immediate consequence of the next lemma. The proof of its connectedness is more complicated and requires some preparation. It will be given in §4.

For {un}⊂C([τ, T], Hw), the convergence unu in C([τ, T], Hw) will mean that for any vH, supt∈[τ,T]|(un(t)−u(t), v)|→0, as n→∞. We note that unu in C([τ, T], Hw) if and only if uC([τ, T], Hw) and for tnt0, tn, t0∈[τ, T], we have un(tn)→u(t0) weakly in H.

Let Embedded Image strongly in H and let Embedded Image. Then there exists a subsequence Embedded Image and u(.)∈Embedded Imageτ(uτ) such that Embedded Image in C([τ, T], Hw) for all T>τ.

Let us assume that T>τ. It follows from the energy inequality (2.4) that each un satisfiesEmbedded Image(2.5)where λ1 is the first eigenvalue of the Stokes operator A, so the sequence {un} is uniformly bounded in L2(τ, T; V)∩L(τ, T; H). Hence, there exists a subsequence such thatEmbedded Image(2.6)In addition, in a standard way (Temam 1979), we obtain thatEmbedded Image(2.7)and, by the compactness lemma (Lions 1969), we haveEmbedded Image(2.8)Embedded Image(2.9)(Throughout the proof, we will retain the originally indexing for subsequences). From the inequality Embedded Image (refer to proposition 9.2, Robinson 2001), we conclude that the d/dt(un) are uniformly bounded in L4/3(τ,T;V*). HenceEmbedded ImageIt can then be verified that u is a weak solution of the NSE (2.1) in the same way as in the proof of the existence of weak solutions by the Galerkin method (see Temam 1979 or Lions 1969).

The uniform estimate (2.5) and the compact injection HV* implies that {un(t)} is precompact in V* for any t. Moreover, it is obvious that the un: [τ, T]→V* are an equicontinuous family of functions. Hence, by the Ascoli–Arzelà theorem, unu in C([τ, T], V*). Using equation (2.5) again, uC([τ, T], Hw) and a standard contradiction argument, we obtain that unu in C([τ, T], Hw). The fact that this holds for every T>τ can then be proved by a diagonal procedure.

Finally, by equations (2.6)–(2.9) and Embedded Image strongly in H, we obtain that u satisfies (2.4). Hence, u(.)∈Embedded Imageτuτ. ▪

3. The globally modified Navier–Stokes equations

The globally modified Navier–Stokes equations (GMNSE)Embedded Image(3.1)where FN: Embedded Image+Embedded Image+ is defined byEmbedded Imagewere introduced by Caraballo et al. (2006). Here, N is a fixed positive number.

A weak solution of the GMNSE (3.1) is defined in the same way as for the NSE (2.1), but with the operator BN(u, v) replaced by BN(u, v), the element of V* defined by 〈B(u, v), w〉=bN(u, v, w), for all wV given u, vV, whereEmbedded ImageIt was shown in Caraballo et al. (2006) thatEmbedded Image(3.2)Embedded Image(3.3)If fL2(τ, T; H), for all T>τ and uτV, then there exists a unique strong solution of the GMNSE (3.1) such that uC([τ, T], V)∩L2(τ, T; D(A)), for all T>0. In addition, if uτH fL2(τ, T; H) for all T>τ and fL(τ, δ; H) for some δ>τ, then there exists at least one weak globally defined solution of the GMNSE (3.1). Moreover, every weak solution satisfiesEmbedded ImageSince u(.)∈L2(0, T; V), it follows from equation (3.2) that d/dt(u)∈L2(0, T; V*) and hence that every weak solution belongs to C([τ, T], H) for all T>τ with d/dt(‖u‖)2=2〈du/dt, u〉 (Temam 1979, p. 260). It is then easy to prove that every weak solution of the GMNSE satisfies the energy inequality:Embedded Image(3.4)In fact, the strict equality holds.

Analogously to the NSE, we writeEmbedded Imageand denote the corresponding attainability set for tτ byEmbedded Image

Let Embedded Image weakly in H and fnf weakly in L2(τ, T; H) for all T>τ and let Embedded Image. Then, there exists a subsequence Embedded Image and a Embedded Image such that Embedded Image in C([τ, T], Hw) for all T>τ. Moreover, Embedded Image in C([t0, T], H) for all T>t0>τ.

On the other hand, let Embedded Image strongly in H and fnf weakly in L2(τ, T; H) for all T>τ and let Embedded Image. Then, there exists a subsequence Embedded Image and a Embedded Image such that Embedded Image in C([τ, T], H) for all T>τ.

It follows from equation (3.4) thatEmbedded Image(3.5)Thus, (3.5) implies that un(.) is bounded in L2(τ, T; V)∩L(τ, T; H). In addition, using equation (3.2), we see that the d/dt(un(.)) are bounded in L2(τ, T; V*). The property Embedded Image in C([τ, T], Hw) is then proved following the same lines as in lemma 2.1 and the fact Embedded Image is proved in the same way as in theorem 7 of Caraballo et al. (2006).

Let tnt0 with tn∈[τ,T] and t0>0. We shall prove that un(tn)→u(t0) strongly in H. Since un(tn)→u(t0) weakly in H, we haveEmbedded ImageThus, if we can show that lim sup‖un(tn)‖H≤‖u(t0)‖H, then lim sup‖un(tn)‖H=‖u(t0)‖H and the proof will be finished. Let us writeEmbedded ImageBy the compactness lemma (Lions 1969), we have un(t)→u(t) for a.a. t. Then, Jm(t)→J(t) for a.a. t.

First, we claim that lim supJn(tn)≤J(t0). Indeed, let τ<tk<t0 be such that Jn(tk)→J(tk). We can assume that tk<tn. In view of (3.4), Jn(t) is non-increasing, henceEmbedded ImageSince u(t) is continuous at t0, for any ϵ>0, there exist tk and m0(tk) such that Jn(tn)−J(t0)≤ϵ for all nm0, and the result follows. Since Embedded Image, we have limsup‖un(tn)‖≤‖u(t0)‖ as claimed.

Since uC([τ, T], H) by a standard contradiction argument, we obtain that unu in C([t0,T],H) for all T>t0>τ.

If, in addition, Embedded Image in H, then we can obtain the property un(tn)→u(t0) strongly in H also for t0=τ. We just repeat the same proof, but taking tk=τ. Hence, unu in C([τ,T],H).

Finally, the fact that these properties hold for every T>τ can be proved by a diagonal procedure. ▪

From this lemma, we have immediately the strong compactness of the GMNSE attainability sets.

The set Embedded Image is compact in H for any tτ.

The connectedness of the GMNSE attainability sets is given by the following theorem.

Suppose that fL2(τ, T; H) for all T>τ and that fL(τ, δ; H) for some δ>τ. Then, for each tτ and uτH, the set Embedded Image is connected with respect to the strong topology of H.

The case t=τ is obvious. Suppose that for some t*>τ, the set Embedded Image is not connected. Then there exists two non-empty compact sets A1, A2H, such that Embedded Image and A1A2=Embedded Image. Let Embedded Image be such that u1(t*)∈U1 and u2(t*)∈U2, where U1, U2 are disjoint open neighbourhoods of A1, A2, respectively.

Let ϵk→0+ as k→∞. Note that ui(τ+ϵk)∈V and for ρ∈[0,1] let uk(t,ρ) denote the unique strong solution of the problemEmbedded Image(3.6)It is clear by uniqueness that uk(t, 0)=u2(tk) and uk(t, 1)=u1(t+ϵk). Hence, uk(t*ϵk,0)∈U2 and uk(t*ϵk,1)∈U1.

Let w(t)=uk(t, ρ1)−uk(t, ρ2) for ρ1, ρ2∈[0, 1]. As in the proof of theorem 3 in Caraballo et al. (2006), we obtainEmbedded Imageand the Gronwall lemma then implies thatEmbedded Image(3.7)Moreover, it follows from energy inequality (3.4) thatEmbedded Imagewhere λ1 is the first eigenvalue of the Stokes operator A. Hence,

On the other hand, it is shown in the proof of theorem 7 in Caraballo et al. (2006) that the Galerkin approximations um of the unique solution to (3.6) satisfyEmbedded Image(3.8)Embedded Image(3.9)Hence, combining equations (3.8) and (3.9), we obtainEmbedded Imagewhere the constant D does not depend on ρ∈[0,1].

Passing to the limit, we conclude that Embedded Image is bounded uniformly with respect to ρ∈[0, 1]. Hence, from (3.7), the mapping ρuk(., ρ) from [0, 1] into C([τ,T],H) is continuous for each T>τ. Since uk(t*ϵk,0)∈U2 and uk(t*ϵk, 1)∈U1, there must exist ρk such that uk(t*ϵk, ρk)∉U1U2. Consider then the sequence of strong solutions uk(., ρk) and let k→∞. Since the functions ui(.) are continuous from [τ,+∞) into H, it is clear that uk(τ,ρk)=ρku1(τ+ϵk)+(1−ρk)u2(τ+ϵk)→uτ strongly in H as k→∞. In addition, we have f(.+ϵk)→f(.) weakly in L2(τ, T; H) for all T>τ (in fact, it converges strongly—refer to chapter 4 of Gajewsky et al. 1974). Therefore, lemma 3.1 implies that, up to a subsequence, uk(.,ρk) converges in C([τ,T],H) for all T>τ to some weak solution Embedded Image. But thenEmbedded Imageand we have obtained a contradiction. Hence, Embedded Image must be connected. ▪

4. Proof of theorem 2.1

We now give the proof of the weak connectedness part of the Kneser property for the NSE (2.1). As mentioned earlier, the weak compactness property is an immediate consequence of lemma 2.1.

Recall that a multivalued map F: X→2Y, where X and Y are Hausdorff topological spaces, is called upper semicontinuous if for all u0D(F)={uX: F(u)≠Embedded Image} and any neighbourhood O(F(u0)), there exists a neighbourhood M of u0 such that F(M)⊂O(F(u0)) (Here, 2Y is the set of all subsets of Y including the empty set).

The case t=τ is obvious, so suppose that the set Embedded Image is not weakly connected for some t*>τ. Then there exists two weakly compact sets A1, A2 of H with A1A2=Ø, such that Embedded Image. Let u1(.), u2(.)∈Dτ(uτ) be such that u1(t*)∈U1 and u2(t*)∈U2, where U1 and U2 are disjoint weakly open neighbourhoods of A1 and A2, respectively.

Now, for i=1, 2 and γτ, let Embedded Image be equal to {ui(t)} if t∈[τ, γ] and if t≥γ let Embedded Image be the attainability set at time t of all globally defined weak solutions of the problemEmbedded Image(4.1)Since fL(τ, T; H) for all T>τ, we know that Embedded Image is non-empty. Moreover, it follows from theorem 3.1 that Embedded Image is a connected set of H (with respect to the strong topology and hence also with respect to the weak topology) for all N≥1 and t, γτ.

We shall prove now that the (possibly) multivalued maps Embedded Image are upper semicontinuous as functions from [τ,+∞) into Hw for each fixed N≥1 and tτ. We shall omit the index i for simplicity of notation.

Let γ→γ0. Consider first the case where γ>γ0, i.e. with Embedded Image. If tγ0<γ, then Embedded ImageN(t, γ)={u(t)}=Embedded ImageN(t, γ0). On the other hand, if t>γ0, then we can assume that t>γ, so that Embedded ImageN(t, γ) is the set of values attained by the weak solutions y(t) of (4.1) with y(γ)=u(γ) and Embedded ImageN(t, γ0) is the set of values attained by the solutions y(t) of (4.1), such that y0)=u0). Since uC([τ,+∞], Hw), we have that u(γ)→u0) weakly in H. If γEmbedded ImageN(t, γ) were not upper semicontinuous at γ0 as Embedded Image, then there would exist a neighbourhood O of uN(t, γ0) in the weak topology and sequences γj>γ0 with γj→γ0 and ξjEmbedded ImageN(t, γj) such that ξj∉O for all j. Clearly, ξj=yj(t), where yj(.) is a weak solution of (4.1) with yj)=uj). Denote Embedded Image. Then Embedded Image is a weak solution of (4.1) with Embedded Image, but with f(.) replaced by f(.+γjγ0). As f(.+γjγ0)→f(.) weakly in L2(γ0, T; H) for all T>γ0, it follows from lemma 3.1 that (up to a subsequence) Embedded Image in C([γ0, T], Hw) for all T>γ0, where Embedded Image is a weak solution of (4.1) with Embedded Image. But then Embedded Image weakly in H, and we obtain a contradiction.

Now consider the case where γ<γ0, i.e. with Embedded Image. If t<γ0, then we can assume that t<γ and hence have Embedded ImageN(t, γ)={u(t)}=Embedded ImageN(t, γ0). On the other hand, if t≥γ0>γ, then we essentially repeat the proof above.

For any fixed T>t*, we writeEmbedded Imageand define the family of multivalued functions ΦN(ρ): [τ, T]→Embedded Image(H) (the set of non-empty subsets of H) with ρ∈[−1, 1] byEmbedded ImageThen Embedded Image and Embedded Image. Moreover, the mapping ρΦN(ρ)(t) is upper semicontinuous and has connected values (in the space Hw) for any fixed N≥1 and t∈[τ, T]. (Note that Embedded Image).

It is easy to see that the set Embedded Image is connected in Hw for any fixed N≥1, and t∈[τ, T]. In particular, Embedded Image is connected. Indeed, define F: [−1, 1]→P(Hw) by F(ρ):=ΦN(ρ)(t) and A=[−1, 1]. If F(A) were not connected, then there would exist open sets O1 and O2 in Hw with O1O2=Embedded Image such that F(A)∩OiEmbedded Image, i=1,2, and F(A)⊂O1O2. Denote Mi={ρA: F(ρ)⊂F(A)∩Oi}. Since F has connected values, M1M2=A. In addition, M1M2=Embedded Image and MiEmbedded Image for i=1, 2. Since F is upper semicontinuous, it is easy to see that Vi≔{ρA: F(ρ)⊂Oi} are open sets for i=1,2. Then MiVi and V1V2=Embedded Image, which contradicts the fact that A is a connected set.

Since ΦN(−1)(t*)={u1(t*)}∈U1 and ΦN(1)(t*)={u2(t*)}∈U2, there exists a ρN∈(−1, 1) such that ΦN(ρN)(t*)⊄U1U2. Then there exists ξN∈ΦN(ρN)(t*) such that ξN∉U1U2. Note that we can choose infinitely many N such that the ξN belong to one of Embedded Image or Embedded Image. Suppose, for example, that this is true for Embedded Image. Then there is a functionEmbedded Image(4.2)such that uN: [γ(ρN), T]→H is a weak solution of (4.1) with γ=γ(ρN) and uN(t*)=ξN.

Now, the function u1(t) satisfies the energy inequality (2.4) and the weak solutions of the system (4.1) satisfy (3.4). It is then easy to see that uN(t) also satisfies the energy inequality (2.4). The case tγ(ρN) is trivial, so consider the case ts≥γ(ρN). ThenEmbedded ImageFinally, for tγ(ρN)>sτ, we haveEmbedded ImageIt then follows thatEmbedded Imageso that uN is a bounded sequence in L(τ, T; H)∩L2(0, T; V). In addition, by the standard inequality (refer proposition 9.2 in Robinson 2001)Embedded Imageand the inequality (3.3) for BN we see that the mappingEmbedded Imageis bounded in the space L4/3(τ, T; V*). Since uN satisfies the evolution equationEmbedded Imageit follows that the duN/dt are bounded in L4/3(τ, T; V*). Then, by the compactness theorem (Lions 1969), there exists a subsequence Nj→∞ and a uL(τ, T; H)∩L2(0, T; V) such thatEmbedded Image(4.3)Embedded Image(4.4)Embedded Image(4.5)We have to show that u is a weak solution of the NSE (2.1) on [τ, T]. For this, it is enough to check thatEmbedded Imageholds for all t∈[τ, T], wV. This is done in the same way as for the convergence to a weak solution of Galerkin approximations for the NSE. The only difference arises with the nonlinear term b, so that we shall describe this part. In particular, we have to show thatEmbedded Image(4.6)for all t∈[τ, T] and wD(A). Note thatEmbedded ImagewhereEmbedded Image

Embedded Image in Lp(τ, T; Embedded Image) as Nj→∞ for all p>1.

It is proved in lemma 12 of Caraballo et al. (2006) that Embedded Image in Lp(τ,T;Embedded Image) as Nj→∞ for all p>1. The result then follows fromEmbedded Image ▪

Arguing as in the proof of the existence of weak solutions for the NSE (e.g. Lions 1969), it can be shown thatEmbedded Imagefor all t∈[τ, T] and wD(A). ThenEmbedded Imageand the second integral converges to 0. For the first integral, we use the inequalityEmbedded Imagelemma 4.1, and the estimateEmbedded Imagewhich follows fromEmbedded Image(refer to proposition 9.2 in Robinson 2001). Hence, the first integral above also converges to 0, which gives the limit (4.6) for all wD(A). Finally, a density argument gives the limit (4.6) for all wV.

It follows that u(.) is a weak solution of the NSE (2.1) defined on [τ, T]. In fact, the function (4.2) can be defined also in [τ, 2T], [τ, 3T], etc., and by a standard diagonal argument we obtain that u(.) is a globally defined weak solution. In addition, since uN(.) satisfies the energy inequality (2.4), it is easy to show that u(.) also satisfies this inequality. Hence, u(.)∈Embedded Imageτ(uτ).

Finally, we have to show thatEmbedded Image(4.7)For any t∈[τ, T], the uniform estimate Embedded Image and the compact embedding HV* imply that the sequence Embedded Image is precompact in V*. In addition, the sequence {duNj/dt} is bounded in L4/3(τ, T; V*), so by the Ascoli–Arzelà theorem, the sequence Embedded Image is precompact in C([τ, T], V*). Then, passing to a subsequence, we haveEmbedded ImageFrom this and the estimate Embedded Image, we obtain (4.7) by a standard argument.

Hence, u(t*)∉U1U2 and Embedded Image, which is a contradiction. Thus, Embedded Image must be connected with respect to the weak topology of H, which completes the proof of theorem 2.1.

5. Weak connectedness of the weak attractor

We now consider the autonomous case, i.e. fH does not depend on t, and we set τ=0. Our aim now is to prove that the global weak attractor of NSE (2.1) (which was proved to exist in Kapustyan (2005)) is connected with respect to the weak topology of H.

It is standard to show, using the Galerkin approximations of the weak solution, the existence of at least one weak solution of the NSE satisfying the energy inequalityEmbedded Image(5.1)where V0 is the energy functional (see equation (2.4)), and the inequalityEmbedded Image(5.2)whereEmbedded ImageWe denoteEmbedded ImageEmbedded Imagewhere RR0. We note that these sets are non-empty.

We first need to prove the weak connectedness of the set Embedded Image for all t≥0.

The set Embedded Image is compact and connected with respect to the weak topology in H for all RR0, t≥0 and u0H.

In order to prove the weak compactness of Embedded Image, we argue in a similar way as in the proof of lemma 2.1. The only difference is that the limit function u has to satisfy (5.1) and (5.2) for all ts and a.a. s>0. Since (up to a subsequence) (2.6)–(2.9) hold, it is clear that equation (5.1) is satisfied for u. In addition,Embedded Imagefor ts and a.a. s>0, so that equation (5.2) holds.

The proof of the fact that Embedded Image is connected is rather similar to that of theorem 2.1, so we only sketch it here.

The case t=0 is obvious. Suppose then that for some t*>0, the set Embedded Image is not connected. Then there exists two weakly compact sets A1 and A2 of H with A1A2=Embedded Image, such that Embedded Image. Let Embedded Image be such that u1(t*)∈U1 and u2(t*)∈U2, where U1 and U2 are disjoint weakly open neighbourhoods of A1 and A2, respectively.

For i=1, 2 and γ≥0, let Embedded Image be equal to {ui(t)} for t∈[0,γ] and let Embedded Image be the attainability set at time t of all globally defined weak solutions of the GMNSE initial value problem (4.1) for t≥γ. We know that Embedded Image is non-empty and, by theorem 3.1, that Embedded Image is a connected subset of H (with respect to the strong topology and hence also with respect to the weak topology) for all N≥1, t≥0 and γ≥0.

As in the proof of theorem 2.1, we can show that the (possibly) multivalued maps Embedded Image are upper semicontinuous as functions from [0, ∞) into Hw for each fixed N≥1 and t≥0. For any fixed T>t* setEmbedded Imageand define the family of multivalued functions ΦN(ρ): [0, T]→Embedded Image(H) for ρ∈[−1, 1] byEmbedded ImageWe have Embedded Image and Embedded Image. Moreover, the mapping ρΦN(ρ)(t) is upper continuous and has connected values (with respect to the space Hw) for any fixed N≥1, t∈[0,T] (note that Embedded Image). Then, as proved in theorem 2.1, the set Embedded Image is connected in Hw for each fixed N≥1 and t∈[0, T]. In particular, Embedded Image is connected.

Since ΦN(−1)(t*)={u1(t*)}∈U1 and ΦN(1)(t*)={u2(t)}∈U2, there exists ρN∈(−1,1) such that ΦN(ρN)(t*)⊄U1U2. Then there exists ξN∈ΦN(ρN)(t*) such that ξN∉U1U2. Note that we can choose infinitely many N such that the ξN belong to one of Embedded Image or Embedded Image. Suppose, for example, that this is true for Embedded Image. Then, there is a functionEmbedded Imagesuch that uN: [γ(ρN), T]→H is a weak solution of the GMNSE problem (4.1) with γ=γ(ρN) such that uN(t*)=ξN.

Moreover, it follows in the same way as in theorem 2.1 that uN(t) satisfies the energy inequality (5.1). In addition, uN(t) satisfies the inequality (5.2). This is trivial in the case tγ(ρN). If ts≥γ(ρN), then multiplying equation (4.1) by uN and using the property b(u,u,u)=0, we obtainEmbedded Imageand (5.2) follows in a standard way for all tsγ(ρN). Finally, for tγ(ρN)≥s, and a.a. s>0, we haveEmbedded ImageFinally, note that Embedded ImageR2 for all t≥0. For 0≤tγ(ρN), this property is immediate. If tγ(ρN), then using the fact that (5.2) holds for s=γ(ρN), we obtainEmbedded ImageHenceEmbedded ImageIt then follows thatEmbedded Imageso the sequence uN is a bounded in L(τ, T; H)∩L2(τ, T; V). Arguing again as in theorem 2.1, we obtain the existence of a subsequence Embedded Image, which converges to a weak solution u of the NSE in the sense of (4.3)–(4.5).

We now have to show thatEmbedded Image(5.3)For any t∈[0,T], the uniform estimate Embedded Image and the compact embedding HV* imply that the sequence Embedded Image is precompact in V*. In addition, the sequence {duNj/dt} is bounded in L4/3(0,T;V*). Hence, by the Ascoli–Arzelà theorem, the sequence Embedded Image is precompact in C([T,0],V*). Passing to a subsequence, we then obtainEmbedded ImageThe convergence (5.3) is then obtained from this and the estimate Embedded Image by a standard argument.

It follows that u(.) is a weak solution of the NSE (2.1) in the interval [0, T]. In fact, the function (4.2) can also be defined in the intervals [0, 2T], [0, 3T], etc., therefore by a standard diagonal argument, we obtain that u(.) is a globally defined weak solution of the NSE. Moreover, since uN(.) satisfies (5.1) and (5.2) and ‖uN(t)‖≤R, it is easy to show that u(.) also satisfies these properties. Hence Embedded Image.

Then u(t*)∉U1U2 and Embedded Image, which is a contradiction. Hence, Embedded Image must be connected with respect to the weak topology of H. ▪

Let BR={uH: ‖uHR} and let Embedded Image(BR) denote the set of all non-empty subsets of BR. Since BR is a bounded, closed and convex subset of the separable Hilbert space H, we can consider BR as a complete metric space endowed with the metric ρw in which the convergence is equivalent to the weak convergence in H. We will denote this space by XR and note that it is a connected space since the set BR is connected (in both the strong and weak topologies of H).

For any RR0, we define the (possibly) multivalued mapping GR: Embedded Image+×BREmbedded Image(BR) byEmbedded Image(5.4)This mapping is a strict multivalued semiflow, i.e. GR(0, .)=Id and GR(t+s, u0)=GR(t, GR(s, u0)) for all u0H and t, sEmbedded Image+. This result is proved in Kapustyan et al. (in press) for the Boussinesq system of equations, but the proof for the Navier–Stokes system is quite similar. We recall that the set Embedded ImageR is said to be a global attractor of GR if

  1. it is negatively semi-invariant, i.e. Embedded ImageRGR(t, Embedded ImageR) for all t≥0,

  2. it is attracting, i.e.

Embedded Image(5.5)where dist(C, A)=supcC infaAρw(c, a) is the Hausdorff semidistance based on the metric ρw, and

  1. Embedded ImageR is minimal, i.e. Embedded ImageRY for any closed set YXR satisfying (5.5).

In addition, the global attractor Embedded ImageR is said to be invariant if Embedded ImageR=GR(t, Embedded ImageR) for all t≥0.

The existence of such an attractor for the NSE was at first proved in Kapustyan (2005), who also showed that Embedded Image for all RR0, i.e. the attractor does not depend on R. (The same result is obtained for the Boussinesq system in Kapustyan et al. (in press)). In addition, since GR is a strict semiflow, the attractor is invariant (refer to Remark 8 in Melnik & Valero (1998)).

Our aim now is to prove that Embedded ImageR is a connected set with respect to Hw. First, we recall an abstract theorem on connectedness proved in Melnik & Valero (1998), which requires the next definition.

The multivalued semiflow GR: Embedded Image+×XREmbedded Image(XR) is said to be time-continuous ifEmbedded Image

The following result is a consequence of theorem 5 in Melnik & Valero (1998).

Suppose that GR(t, .): XREmbedded Image(XR), where Embedded Image(XR) is the set of all closed subsets of XR, is upper semicontinuous and there exists a compact set KXR such thatEmbedded Image(5.6)Suppose also that GR is a strict time-continuous semiflow with connected values. Then GR has a global attractor Embedded Image and Embedded Image is connected in XR.

We observe that since XR is endowed with the weak topology of the space H, we will call Embedded Image a weak global attractor rather than a global attractor.

We note that the weak solutions u: [0, ∞)→Hw of the NSE (2.1) are continuous, so GR is time-continuous in our case. We can thus apply theorem 5.2 to the NSE.

The global weak attractor Embedded ImageR of the NSE is connected in XR. Hence, it is connected with respect to the weak topology of H.

We saw above that GR is a time-continuous strict multivalued semiflow. In addition, in view of theorem 5.1, GR has connected values and equation (5.6) is trivially satisfied with K=BR. Finally, it follows in the same way as in Kapustyan et al. (in press) for the Boussinesq system that GR has closed values, i.e. GR(t,u0)∈Embedded Image(XR), and that GR(t,.):XREmbedded Image(XR) is upper semicontinuous. The result then follows from theorem 5.2. ▪

Acknowledgments

This work has been mainly supported by the DAAD and Ministerio de Educación y Ciencia (Spain), grant HA2005-0082. Also, it has been partially supported by the Ministerio de Educación y Ciencia (Spain) and FEDER (Fondo Europeo de Desarrollo Regional), grants MTM2005-01412, MTM2005-03868, and by the Consejería de Cultura y Educación (Comunidad Autónoma de Murcia) grant 00684/PI/04. We wish to express our thanks to the referees for their several useful comments and suggestions.

Footnotes

    • Received November 9, 2006.
    • Accepted February 7, 2007.

References

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