## 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.

## 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 *L*_{2}-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 *L*_{2}-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 *L*_{2}-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)(2.1)for given *ν*>0, where *Ω* is a bounded open subset of ^{3} with smooth boundary.

First, we recall standard notation and basic results following Temam (1979). In particular, we use the usual function spaceswhere cl_{X} denotes the closure in the space *X*. The spaces *H* and *V* are separable Hilbert spaces and *V*⊂*H*⊂*V*^{*} with dense and compact injections when *H* is identified with its dual *H*^{*}. Let us denote *H*_{w} 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*, *w*∈*V*,defines a trilinear continuous form on *V* with *b*(*u*, *v*, *v*)=0 when *u*∈*V* and . For *u*, *v*∈*V*, let *B*(*u*, *v*) denote the element of *V*^{*} defined by 〈*B*(*u*, *v*), *w*〉=*b*(*u*, *v*, *w*) for all *w*∈*V*.

We say that the function *u* is a *weak solution* of the NSE (2.1) iffor all *T*>*τ* and(2.2)in the sense of distributions on (*τ*, *T*).

If *f*∈*L*^{2}(*τ*, *T*, *V*^{*}), for all *T*>*τ*, and if *u* satisfies equation (2.2), thenfor all *T*>*τ*. In particular, the initial condition *u*(*τ*)=*u*_{τ} makes sense for any *u*_{τ}∈*H* and equation (2.2) with *u*(*τ*)=*u*_{τ} is equivalent tofor all *t*≥*τ* and *w*∈*V*.

Let *A*: *V*→*V*^{*} 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*)=(*H*^{2}(*Ω*))^{3}∩*V*, where the injection *D*(*A*)⊂*V* is dense and continuous. Moreover, ‖Au‖_{H} is a norm equivalent to that induced by (*H*^{2}(*Ω*))^{3}. The NSE (2.1) can then be rewritten as(2.3)

It is well known (Temam 1979) that if *f*∈*L*^{2}(*τ*, *T*, *V*^{*}), for all *T*>*τ*, and if *u*_{τ}∈*H*, then the NSE have at least one weak globally defined solution which satisfies the *energy inequality:*(2.4)whereNow let us write

and denote the corresponding attainability set for *t*≥*τ* byThe first of our main results is the following version of the Kneser property for the weak solutions of the NSE.

*Let f*∈*L*^{∞}(*τ*, *T*; *H*) *for all T*>*τ*. *Then, for all t*≥*τ and u*_{τ}∈*H*, *the attainability set K*_{t}(*u*_{τ}) *is compact and connected with respect to the weak topology on H*.

The proof that *K*_{τ}(*t*) is weakly compact for any *t*≥*τ* 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 {*u*_{n}}⊂*C*([*τ*, *T*], *H*_{w}), the convergence *u*_{n}→*u* in *C*([*τ*, *T*], *H*_{w}) will mean that for any *v*∈*H*, sup_{t∈[τ,} _{T]}|(*u*_{n}(*t*)−*u*(*t*), *v*)|→0, as *n*→∞. We note that *u*_{n}→*u* in *C*([*τ*, *T*], *H*_{w}) if and only if *u*∈*C*([*τ*, *T*], *H*_{w}) and for *t*_{n}→*t*_{0}, *t*_{n}, *t*_{0}∈[*τ*, *T*], we have *u*_{n}(*t*_{n})→*u*(*t*_{0}) weakly in *H*.

*Let* *strongly in H and let* . *Then there exists a subsequence* *and u*(.)∈_{τ}(*u*_{τ}) *such that* *in C*([*τ*, *T*], *H*_{w}) *for all T*>*τ*.

Let us assume that *T*>*τ*. It follows from the energy inequality (2.4) that each *u*_{n} satisfies(2.5)where *λ*_{1} is the first eigenvalue of the Stokes operator *A*, so the sequence {*u*_{n}} is uniformly bounded in *L*^{2}(*τ*, *T*; *V*)∩*L*^{∞}(*τ*, *T*; *H*). Hence, there exists a subsequence such that(2.6)In addition, in a standard way (Temam 1979), we obtain that(2.7)and, by the compactness lemma (Lions 1969), we have(2.8)(2.9)(Throughout the proof, we will retain the originally indexing for subsequences). From the inequality (refer to proposition 9.2, Robinson 2001), we conclude that the d/d*t*(*u*_{n}) are uniformly bounded in *L*^{4/3}(*τ*,*T*;*V*^{*}). HenceIt 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 *H*⊂*V*^{*} implies that {*u*_{n}(*t*)} is precompact in *V*^{*} for any *t*. Moreover, it is obvious that the *u*_{n}: [*τ*, *T*]→*V*^{*} are an equicontinuous family of functions. Hence, by the Ascoli–Arzelà theorem, *u*_{n}→*u* in *C*([*τ*, *T*], *V*^{*}). Using equation (2.5) again, *u*∈*C*([*τ*, *T*], *H*_{w}) and a standard contradiction argument, we obtain that *u*_{n}→*u* in *C*([*τ*, *T*], *H*_{w}). 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 strongly in *H*, we obtain that *u* satisfies (2.4). Hence, *u*(.)∈_{τ}*u*_{τ}. ▪

## 3. The globally modified Navier–Stokes equations

The globally modified Navier–Stokes equations (GMNSE)(3.1)where *F*_{N}: ^{+}→^{+} is defined bywere 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 *B*_{N}(*u*, v) replaced by *B*_{N}(*u*, v), the element of *V*^{*} defined by 〈*B*(*u*, *v*), *w*〉=*b*_{N}(*u*, *v*, *w*), for all *w*∈*V* given *u*, *v*∈*V*, whereIt was shown in Caraballo *et al.* (2006) that(3.2)(3.3)If *f*∈*L*^{2}(*τ*, *T*; *H*), for all *T*>*τ* and *u*_{τ}∈*V*, then there exists a unique strong solution of the GMNSE (3.1) such that *u*∈*C*([*τ*, *T*], *V*)∩*L*^{2}(*τ*, *T*; *D*(*A*)), for all *T*>0. In addition, if *u*_{τ}∈*H f*∈*L*^{2}(*τ*, *T*; *H*) for all *T*>*τ* and *f*∈*L*^{∞}(*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 satisfiesSince *u*(.)∈*L*^{2}(0, *T*; *V*), it follows from equation (3.2) that d/d*t*(*u*)∈*L*^{2}(0, *T*; *V*^{*}) and hence that every weak solution belongs to *C*([*τ*, *T*], *H*) for all *T*>*τ* with d/d*t*(‖*u*‖)^{2}=2〈d*u*/d*t*, *u*〉 (Temam 1979, p. 260). It is then easy to prove that every weak solution of the GMNSE satisfies the energy inequality:(3.4)In fact, the strict equality holds.

Analogously to the NSE, we writeand denote the corresponding attainability set for *t*≥*τ* by

*Let* *weakly in H and f*^{n}→*f weakly in L*^{2}(*τ*, *T*; *H*) *for all T*>*τ and let* . *Then, there exists a subsequence* *and a* *such that* *in C*([*τ*, *T*], *H*_{w}) *for all T*>*τ*. *Moreover*, *in C*([*t*_{0}, *T*], *H*) *for all T*>*t*_{0}>*τ*.

*On the other hand, let* *strongly in H and f*^{n}→*f weakly in L*^{2}(*τ*, *T*; *H*) *for all T*>*τ and let* . *Then, there exists a subsequence* *and a* *such that* *in C*([*τ*, *T*], *H*) *for all T*>*τ*.

It follows from equation (3.4) that(3.5)Thus, (3.5) implies that *u*^{n}(.) is bounded in *L*^{2}(*τ*, *T*; *V*)∩*L*^{∞}(*τ*, *T*; *H*). In addition, using equation (3.2), we see that the d/d*t*(*u*_{n}(.)) are bounded in *L*^{2}(*τ*, *T*; *V*^{*}). The property in *C*([*τ*, *T*], *H*_{w}) is then proved following the same lines as in lemma 2.1 and the fact is proved in the same way as in theorem 7 of Caraballo *et al.* (2006).

Let *t*_{n}→*t*_{0} with *t*_{n}∈[*τ*,*T*] and *t*_{0}>0. We shall prove that *u*_{n}(*t*_{n})→*u*(*t*_{0}) strongly in *H*. Since *u*_{n}(*t*_{n})→*u*(*t*_{0}) weakly in *H*, we haveThus, if we can show that lim sup‖*u*_{n}(*t*_{n})‖_{H}≤‖*u*(*t*_{0})‖_{H}, then lim sup‖*u*_{n}(*t*_{n})‖_{H}=‖*u*(*t*_{0})‖_{H} and the proof will be finished. Let us writeBy the compactness lemma (Lions 1969), we have *u*_{n}(*t*)→*u*(*t*) for a.a. *t*. Then, *J*_{m}(*t*)→*J*(*t*) for a.a. *t*.

First, we claim that lim sup*J*_{n}(*t*_{n})≤*J*(*t*_{0}). Indeed, let *τ*<*t*_{k}<*t*_{0} be such that *J*_{n}(*t*_{k})→*J*(*t*_{k}). We can assume that *t*_{k}<*t*_{n}. In view of (3.4), *J*_{n}(*t*) is non-increasing, henceSince *u*(*t*) is continuous at *t*_{0}, for any *ϵ*>0, there exist *t*_{k} and *m*_{0}(*t*_{k}) such that *J*_{n}(*t*_{n})−*J*(*t*_{0})≤*ϵ* for all *n*≥*m*_{0}, and the result follows. Since , we have limsup‖*u*_{n}(*t*_{n})‖≤‖*u*(*t*_{0})‖ as claimed.

Since *u*∈*C*([*τ*, *T*], *H*) by a standard contradiction argument, we obtain that *u*_{n}→*u* in *C*([*t*_{0},*T*],*H*) for all *T*>*t*_{0}>*τ*.

If, in addition, in *H*, then we can obtain the property *u*_{n}(*t*_{n})→*u*(*t*_{0}) strongly in *H* also for *t*_{0}=*τ*. We just repeat the same proof, but taking *t*_{k}=*τ*. Hence, *u*_{n}→*u* 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* *is compact in H for any t*≥*τ*.

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

*Suppose that f*∈*L*^{2}(*τ*, *T*; *H*) *for all T*>*τ and that f*∈*L*^{∞}(*τ*, *δ*; *H*) *for some δ*>*τ*. *Then, for each t*≥*τ and u*_{τ}∈*H, the set* *is connected with respect to the strong topology of H*.

The case *t*=*τ* is obvious. Suppose that for some *t*^{*}>*τ*, the set is not connected. Then there exists two non-empty compact sets *A*_{1}, *A*_{2}⊂*H,* such that and *A*_{1}∩*A*_{2}=. Let be such that *u*_{1}(*t*^{*})∈*U*_{1} and *u*_{2}(*t*^{*})∈*U*_{2}, where *U*_{1}, *U*_{2} are disjoint open neighbourhoods of *A*_{1}, *A*_{2}, respectively.

Let *ϵ*_{k}→0+ as *k*→∞. Note that *u*_{i}(*τ*+*ϵ*_{k})∈*V* and for *ρ*∈[0,1] let *u*_{k}(*t*,*ρ*) denote the unique strong solution of the problem(3.6)It is clear by uniqueness that *u*_{k}(*t*, 0)=*u*_{2}(*t*+ϵ_{k}) and *u*_{k}(*t*, 1)=*u*_{1}(*t*+*ϵ*_{k}). Hence, *u*_{k}(*t*^{*}−*ϵ*_{k},0)∈*U*_{2} and *u*_{k}(*t*^{*}−*ϵ*_{k},1)∈*U*_{1}.

Let *w*(*t*)=*u*_{k}(*t*, *ρ*_{1})−*u*_{k}(*t*, *ρ*_{2}) for *ρ*_{1}, *ρ*_{2}∈[0, 1]. As in the proof of theorem 3 in Caraballo *et al.* (2006), we obtainand the Gronwall lemma then implies that(3.7)Moreover, it follows from energy inequality (3.4) thatwhere *λ*_{1} 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 *u*_{m} of the unique solution to (3.6) satisfy(3.8)(3.9)Hence, combining equations (3.8) and (3.9), we obtainwhere the constant *D* does not depend on *ρ*∈[0,1].

Passing to the limit, we conclude that is bounded uniformly with respect to *ρ*∈[0, 1]. Hence, from (3.7), the mapping *ρ*↦*u*_{k}(., *ρ*) from [0, 1] into *C*([*τ*,*T*],*H*) is continuous for each *T*>*τ*. Since *u*_{k}(*t*^{*}−*ϵ*_{k},0)∈*U*_{2} and *u*_{k}(*t*^{*}−*ϵ*_{k}, 1)∈*U*_{1}, there must exist *ρ*_{k} such that *u*_{k}(*t*^{*}−*ϵ*_{k}, *ρ*_{k})∉*U*_{1}∪*U*_{2}. Consider then the sequence of strong solutions *u*_{k}(., *ρ*_{k}) and let *k*→∞. Since the functions *u*_{i}(.) are continuous from [*τ*,+∞) into *H*, it is clear that *u*_{k}(*τ*,*ρ*_{k})=*ρ*_{k}*u*_{1}(*τ*+*ϵ*_{k})+(1−*ρ*_{k})*u*_{2}(*τ*+*ϵ*_{k})→*u*_{τ} strongly in *H* as *k*→∞. In addition, we have *f*(.+*ϵ*_{k})→*f*(.) weakly in *L*^{2}(*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, *u*_{k}(.,*ρ*_{k}) converges in *C*([*τ*,*T*],*H*) for all *T*>*τ* to some weak solution . But thenand we have obtained a contradiction. Hence, 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→2*^{Y}, where *X* and *Y* are Hausdorff topological spaces, is called upper semicontinuous if for all *u*_{0}∈*D*(*F*)={*u*∈*X*: *F*(*u*)≠} and any neighbourhood *O*(*F*(*u*_{0})), there exists a neighbourhood *M* of *u*_{0} such that *F*(*M*)⊂*O*(*F*(*u*_{0})) (Here, 2^{Y} is the set of all subsets of *Y* including the empty set).

The case *t*=*τ* is obvious, so suppose that the set is not weakly connected for some *t*^{*}>*τ*. Then there exists two weakly compact sets *A*_{1}, *A*_{2} of *H* with *A*_{1}∩*A*_{2}=*Ø*, such that . Let *u*_{1}(.), *u*_{2}(.)∈*D*_{τ}(*u*_{τ}) be such that *u*_{1}(*t*^{*})∈*U*_{1} and *u*_{2}(*t*^{*})∈*U*_{2}, where *U*_{1} and *U*_{2} are disjoint weakly open neighbourhoods of *A*_{1} and *A*_{2}, respectively.

Now, for *i*=1, 2 and *γ*≥*τ*, let be equal to {*u*_{i}(*t*)} if *t*∈[*τ*, *γ*] and if *t*≥γ let be the attainability set at time *t* of all globally defined weak solutions of the problem(4.1)Since *f*∈*L*^{∞}(*τ*, *T*; *H*) for all *T*>*τ*, we know that is non-empty. Moreover, it follows from theorem 3.1 that 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*, *γ*≥*τ*.

We shall prove now that the (possibly) multivalued maps are upper semicontinuous as functions from [*τ*,+∞) into *H*_{w} for each fixed *N*≥1 and *t*≥*τ*. We shall omit the index *i* for simplicity of notation.

Let *γ*→γ_{0}. Consider first the case where *γ*>*γ*_{0}, i.e. with . If *t* ≤*γ*_{0}<*γ*, then ^{N}(*t*, *γ*)={*u*(*t*)}=^{N}(*t*, *γ*_{0}). On the other hand, if *t*>*γ*_{0}, then we can assume that *t*>*γ*, so that ^{N}(*t*, *γ*) is the set of values attained by the weak solutions *y*(*t*) of (4.1) with *y*(γ)=*u*(γ) and ^{N}(*t*, *γ*_{0}) is the set of values attained by the solutions *y*(*t*) of (4.1), such that *y*(γ_{0})=*u*(γ_{0}). Since *u*∈*C*([*τ*,+∞], *H*_{w}), we have that *u*(γ)→*u*(γ_{0}) weakly in *H*. If *γ*↦^{N}(*t*, *γ*) were not upper semicontinuous at *γ*_{0} as , then there would exist a neighbourhood *O* of *u*^{N}(*t*, *γ*_{0}) in the weak topology and sequences *γ*_{j}>*γ*_{0} with *γ*_{j}→γ_{0} and *ξ*_{j}∈^{N}(*t*, *γ*_{j}) such that *ξ*_{j}∉*O* for all *j*. Clearly, *ξ*_{j}=*y*_{j}(*t*), where *y*_{j}(.) is a weak solution of (4.1) with *y*(γ_{j})=*u*(γ_{j}). Denote . Then is a weak solution of (4.1) with , but with *f*(.) replaced by *f*(.+γ_{j}−*γ*_{0}). As *f*(.+γ_{j}−*γ*_{0})→*f*(.) weakly in *L*^{2}(*γ*_{0}, *T*; *H*) for all *T*>*γ*_{0}, it follows from lemma 3.1 that (up to a subsequence) in *C*([γ_{0}, *T*], *H*_{w}) for all *T*>*γ*_{0}, where is a weak solution of (4.1) with . But then weakly in *H*, and we obtain a contradiction.

Now consider the case where *γ*<*γ*_{0}, i.e. with . If *t<γ*_{0}, then we can assume that *t*<*γ* and hence have ^{N}(*t*, *γ*)={*u*(*t*)}=^{N}(*t*, *γ*_{0}). On the other hand, if *t≥γ*_{0}>*γ*, then we essentially repeat the proof above.

For any fixed *T*>*t*^{*}, we writeand define the family of multivalued functions *Φ*^{N}(*ρ*): [*τ*, *T*]→(*H*) (the set of non-empty subsets of *H*) with *ρ*∈[−1, 1] byThen and . Moreover, the mapping *ρ*↦*Φ*^{N}(*ρ*)(*t*) is upper semicontinuous and has connected values (in the space *H*_{w}) for any fixed *N*≥1 and *t*∈[*τ*, *T*]. (Note that ).

It is easy to see that the set is connected in *H*_{w} for any fixed *N*≥1, and *t*∈[*τ*, *T*]. In particular, is connected. Indeed, define *F*: [−1, 1]→*P*(*H*_{w}) by *F*(*ρ*):=*Φ*^{N}(*ρ*)(*t*) and *A*=[−1, 1]. If *F*(*A*) were not connected, then there would exist open sets *O*_{1} and *O*_{2} in *H*_{w} with *O*_{1}∩*O*_{2}= such that *F*(*A*)∩*O*_{i}≠, *i*=1,2, and *F*(*A*)⊂*O*_{1}∪*O*_{2}. Denote *M*_{i}={*ρ*∈*A*: *F*(*ρ*)⊂*F*(*A*)∩*O*_{i}}. Since *F* has connected values, *M*_{1}∪*M*_{2}=*A*. In addition, *M*_{1}∩*M*_{2}= and *M*_{i}≠ for *i*=1, 2. Since *F* is upper semicontinuous, it is easy to see that *V*_{i}≔{*ρ*∈*A*: *F*(*ρ*)⊂*O*_{i}} are open sets for *i*=1,2. Then *M*_{i}⊂*V*_{i} and *V*_{1}∩*V*_{2}=, which contradicts the fact that *A* is a connected set.

Since *Φ*^{N}(−1)(*t*^{*})={*u*_{1}(*t*^{*})}∈*U*_{1} and *Φ*^{N}(1)(*t*^{*})={*u*_{2}(*t*^{*})}∈*U*_{2}, there exists a *ρ*_{N}∈(−1, 1) such that *Φ*^{N}(*ρ*_{N})(*t*^{*})⊄*U*_{1}∪*U*_{2}. Then there exists *ξ*_{N}∈Φ^{N}(*ρ*_{N})(*t*^{*}) such that *ξ*_{N}∉*U*_{1}∪*U*_{2}. Note that we can choose infinitely many *N* such that the *ξ*_{N} belong to one of or . Suppose, for example, that this is true for . Then there is a function(4.2)such that *u*^{N}: [*γ*(*ρ*_{N}), *T*]→*H* is a weak solution of (4.1) with *γ*=γ(*ρ*_{N}) and *u*^{N}(*t*^{*})=*ξ*_{N}.

Now, the function *u*_{1}(*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 *u*^{N}(*t*) also satisfies the energy inequality (2.4). The case *t*≤*γ*(*ρ*_{N}) is trivial, so consider the case *t*≥*s≥γ*(*ρ*_{N}). ThenFinally, for *t*≥*γ*(*ρ*_{N})>*s*≥*τ*, we haveIt then follows thatso that *u*^{N} is a bounded sequence in *L*^{∞}(*τ*, *T*; *H*)∩*L*^{2}(0, *T*; *V*). In addition, by the standard inequality (refer proposition 9.2 in Robinson 2001)and the inequality (3.3) for *B*_{N} we see that the mappingis bounded in the space *L*^{4/3}(*τ*, *T*; *V*^{*}). Since *u*^{N} satisfies the evolution equationit follows that the d*u*^{N}/d*t* are bounded in *L*^{4/3}(*τ*, *T*; *V*^{*}). Then, by the compactness theorem (Lions 1969), there exists a subsequence *N*_{j}→∞ and a *u*∈*L*^{∞}(*τ*, *T*; *H*)∩*L*^{2}(0, *T*; *V*) such that(4.3)(4.4)(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 thatholds for all *t*∈[*τ*, *T*], *w*∈*V*. 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 that(4.6)for all *t*∈[*τ*, *T*] and *w*∈*D*(*A*). Note thatwhere

*in L*^{p}(*τ*, *T*; ) *as N*_{j}→∞ *for all p*>1.

It is proved in lemma 12 of Caraballo *et al.* (2006) that in *L*^{p}(*τ*,*T*;) as *N*_{j}→∞ for all *p*>1. The result then follows from ▪

Arguing as in the proof of the existence of weak solutions for the NSE (e.g. Lions 1969), it can be shown thatfor all *t*∈[*τ*, *T*] and *w*∈*D*(*A*). Thenand the second integral converges to 0. For the first integral, we use the inequalitylemma 4.1, and the estimatewhich follows from(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 *w*∈*D*(*A*). Finally, a density argument gives the limit (4.6) for all *w*∈*V*.

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*, 2*T*], [*τ*, 3*T*], etc., and by a standard diagonal argument we obtain that *u*(.) is a globally defined weak solution. In addition, since *u*^{N}(.) satisfies the energy inequality (2.4), it is easy to show that *u*(.) also satisfies this inequality. Hence, *u*(.)∈_{τ}(*u*_{τ}).

Finally, we have to show that(4.7)For any *t*∈[*τ*, *T*], the uniform estimate and the compact embedding *H*⊂*V*^{*} imply that the sequence is precompact in *V*^{*}. In addition, the sequence {d*u*^{N}*j/*d*t*} is bounded in *L*^{4/3}(*τ*, *T*; *V*^{*}), so by the Ascoli–Arzelà theorem, the sequence is precompact in *C*([*τ*, *T*], *V*^{*}). Then, passing to a subsequence, we haveFrom this and the estimate , we obtain (4.7) by a standard argument.

Hence, *u*(*t*^{*})∉*U*_{1}∪*U*_{2} and , which is a contradiction. Thus, 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. *f*∈*H* 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 inequality(5.1)where *V*_{0} is the energy functional (see equation (2.4)), and the inequality(5.2)whereWe denotewhere *R*≥*R*_{0}. We note that these sets are non-empty.

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

*The set* *is compact and connected with respect to the weak topology in H for all R*≥*R*_{0}, *t*≥0 *and u*_{0}∈*H*.

In order to prove the weak compactness of , 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 *t*≥*s* 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,for *t*≥*s* and a.a. *s*>0, so that equation (5.2) holds.

The proof of the fact that 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 is not connected. Then there exists two weakly compact sets *A*_{1} and *A*_{2} of *H* with *A*_{1}∩*A*_{2}=, such that . Let be such that *u*_{1}(*t*^{*})∈*U*_{1} and *u*_{2}(*t*^{*})∈*U*_{2}, where *U*_{1} and *U*_{2} are disjoint weakly open neighbourhoods of *A*_{1} and *A*_{2}, respectively.

For *i*=1, 2 and *γ*≥0, let be equal to {*u*_{i}(*t*)} for *t*∈[0,*γ*] and let 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 is non-empty and, by theorem 3.1, that 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 are upper semicontinuous as functions from [0, ∞) into *H*_{w} for each fixed *N*≥1 and *t*≥0. For any fixed *T*>*t*^{*} setand define the family of multivalued functions *Φ*^{N}(*ρ*): [0, *T*]→(*H*) for *ρ*∈[−1, 1] byWe have and . Moreover, the mapping *ρ*↦*Φ*^{N}(*ρ*)(*t*) is upper continuous and has connected values (with respect to the space *H*_{w}) for any fixed *N*≥1, *t*∈[0,*T*] (note that ). Then, as proved in theorem 2.1, the set is connected in *H*_{w} for each fixed *N*≥1 and *t*∈[0, *T*]. In particular, is connected.

Since *Φ*^{N}(−1)(*t*^{*})={*u*_{1}(*t*^{*})}∈*U*_{1} and *Φ*^{N}(1)(*t*^{*})={*u*_{2}(*t*)}∈*U*_{2}, there exists *ρ*_{N}∈(−1,1) such that *Φ*^{N}(*ρ*_{N})(*t*^{*})⊄*U*_{1}∪*U*_{2}. Then there exists *ξ*_{N}∈Φ^{N}(*ρ*_{N})(*t*^{*}) such that *ξ*_{N}∉*U*_{1}∪*U*_{2}. Note that we can choose infinitely many *N* such that the *ξ*_{N} belong to one of or . Suppose, for example, that this is true for . Then, there is a functionsuch that *u*^{N}: [*γ*(*ρ*_{N}), *T*]→*H* is a weak solution of the GMNSE problem (4.1) with *γ*=γ(*ρ*_{N}) such that *u*^{N}(*t*^{*})=*ξ*_{N}.

Moreover, it follows in the same way as in theorem 2.1 that *u*^{N}(*t*) satisfies the energy inequality (5.1). In addition, *u*^{N}(*t*) satisfies the inequality (5.2). This is trivial in the case *t*≤*γ*(*ρ*_{N}). If *t*≥*s≥γ*(*ρ*_{N}), then multiplying equation (4.1) by *u*^{N} and using the property *b*(*u*,*u*,*u*)=0, we obtainand (5.2) follows in a standard way for all *t*≥*s*≥*γ*(*ρ*_{N}). Finally, for *t*≥*γ*(*ρ*_{N})≥*s*, and a.a. *s*>0, we haveFinally, note that ≤*R*^{2} for all *t*≥0. For 0≤*t*≤*γ*(*ρ*_{N}), this property is immediate. If *t*≥*γ*(*ρ*_{N}), then using the fact that (5.2) holds for *s*=*γ*(*ρ*_{N}), we obtainHenceIt then follows thatso the sequence *u*^{N} is a bounded in *L*^{∞}(*τ*, *T*; *H*)∩*L*^{2}(*τ*, *T*; *V*). Arguing again as in theorem 2.1, we obtain the existence of a subsequence , which converges to a weak solution *u* of the NSE in the sense of (4.3)–(4.5).

We now have to show that(5.3)For any *t*∈[0,*T*], the uniform estimate and the compact embedding *H*⊂*V*^{*} imply that the sequence is precompact in *V*^{*}. In addition, the sequence {d*u*^{Nj}/d*t*} is bounded in *L*^{4/3}(0,*T*;*V*^{*}). Hence, by the Ascoli–Arzelà theorem, the sequence is precompact in *C*([*T*,0],*V*^{*}). Passing to a subsequence, we then obtainThe convergence (5.3) is then obtained from this and the estimate 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, 2*T*], [0, 3*T*], etc., therefore by a standard diagonal argument, we obtain that *u*(.) is a globally defined weak solution of the NSE. Moreover, since *u*^{N}(.) satisfies (5.1) and (5.2) and ‖*u*^{N}(*t*)‖≤*R*, it is easy to show that *u*(.) also satisfies these properties. Hence .

Then *u*(*t*^{*})∉*U*_{1}∪*U*_{2} and , which is a contradiction. Hence, must be connected with respect to the weak topology of *H*. ▪

Let *B*_{R}={*u*∈*H*: ‖*u*‖_{H}≤*R*} and let (*B*_{R}) denote the set of all non-empty subsets of *B*_{R}. Since *B*_{R} is a bounded, closed and convex subset of the separable Hilbert space *H*, we can consider *B*_{R} as a complete metric space endowed with the metric *ρ*_{w} in which the convergence is equivalent to the weak convergence in *H*. We will denote this space by *X*_{R} and note that it is a connected space since the set *B*_{R} is connected (in both the strong and weak topologies of *H*).

For any *R*≥*R*_{0}, we define the (possibly) multivalued mapping *G*_{R}: ^{+}×*B*_{R}→(*B*_{R}) by(5.4)This mapping is a strict multivalued semiflow, i.e. *G*_{R}(0, .)=*Id* and *G*_{R}(*t*+*s*, *u*_{0})=*G*_{R}(*t*, *G*_{R}(*s*, *u*_{0})) for all *u*_{0}∈*H* and *t*, *s*∈^{+}. 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 _{R} is said to be a global attractor of *G*_{R} if

it is negatively semi-invariant, i.e.

_{R}⊂*G*_{R}(*t*,_{R}) for all*t*≥0,it is attracting, i.e.

(5.5)where dist(*C*, *A*)=sup_{c∈C} inf_{a∈A}*ρ*_{w}(*c*, *a*) is the Hausdorff semidistance based on the metric *ρ*_{w}, and

_{R}is minimal, i.e._{R}⊂*Y*for any closed set*Y*⊂*X*_{R}satisfying (5.5).

In addition, the global attractor _{R} is said to be invariant if _{R}=*G*_{R}(*t*, _{R}) for all *t*≥0.

The existence of such an attractor for the NSE was at first proved in Kapustyan (2005), who also showed that for all *R*≥*R*_{0}, 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 *G*_{R} is a strict semiflow, the attractor is invariant (refer to Remark 8 in Melnik & Valero (1998)).

Our aim now is to prove that _{R} is a connected set with respect to *H*_{w}. First, we recall an abstract theorem on connectedness proved in Melnik & Valero (1998), which requires the next definition.

The multivalued semiflow *G*_{R}: _{+}×*X*_{R}→(*X*_{R}) is said to be time-continuous if

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

Suppose that *G*_{R}(*t*, .): *X*_{R}→(*X*_{R}), *where* (*X*_{R}) *is the set of all closed subsets of X*_{R}, *is upper semicontinuous and there exists a compact set K*⊂*X*_{R} *such that*(5.6)*Suppose also that G*_{R} *is a strict time-continuous semiflow with connected values. Then G*_{R} *has a global attractor* *and* *is connected in X*_{R}.

We observe that since *X*_{R} is endowed with the weak topology of the space *H*, we will call a weak global attractor rather than a global attractor.

We note that the weak solutions *u*: [0, ∞)→*H*_{w} of the NSE (2.1) are continuous, so *G*_{R} is time-continuous in our case. We can thus apply theorem 5.2 to the NSE.

*The global weak attractor* _{R} *of the NSE is connected in X*_{R}. *Hence, it is connected with respect to the weak topology of H*.

We saw above that *G*_{R} is a time-continuous strict multivalued semiflow. In addition, in view of theorem 5.1, *G*_{R} has connected values and equation (5.6) is trivially satisfied with *K*=*B*_{R}. Finally, it follows in the same way as in Kapustyan *et al.* (in press) for the Boussinesq system that *G*_{R} has closed values, i.e. *G*_{R}(*t*,*u*_{0})∈(*X*_{R}), and that *G*_{R}(*t*,.):*X*_{R}→(*X*_{R}) 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.

- © 2007 The Royal Society