ATTRACTORS FOR NONCOMPACT

ATTRACTORS FOR NONCOMPACT NONAUTONOMOUS SYSTEMS VIA ENERGY EQUATIONS Ioana Moise1 , Ricardo Rosa2 and Xiaoming Wang3,4 October 9, 2001 Abstract. An extension to the nonautonomous case of the energy equation method for proving the existence of attractors for noncompact systems is presented. A suitable generalization of the asymptotic compactness property to the nonautonomous case, termed uniform asymptotic compactness, is given, and conditions on the energy equation associated with an abstract class of equations that assure the uniform asymptotic compactness are obtained. This general formulation is then applied to a nonautonomous Navier-Stokes system on an infinite channel past an obstacle, with time-dependent forcing and boundary conditions, and to a nonautonomous, weakly damped, forced Korteweg-de Vries equation on the real line.

1991 Mathematics Subject Classification. 35B40, 35Q30, 35Q53, 76A05, 76D05. Key words and phrases. Uniform attractors, Uniform asymptotic compactness, Energy equation, Nonautonomous equations, Navier-Stokes equations, Korteweg-de-Vries equation. 1 Department of Mathematics, University of Texas at Austin, Austin, TX 78712. 2 Departamento de Matem´ atica Aplicada, Instituto de Matem´ atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brazil. 3 Department of Mathematics, Iowa State University, Ames, IA 50011. 4 Courant Institute, New York University, New York, NY 10012. Typeset by AMS-TEX 1

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I. MOISE, R. ROSA AND X. WANG

§1. Introduction We present an extension to the nonautonomous case of a previous result by the authors [MRW] on the energy method for proving the existence of compact attractors for noncompact systems. Nonautonomous systems are of great importance in applications to natural sciences. Various phenomena can be modeled by nonlinear evolutionary differential equations, and in several cases one does not model the evolution of all the relevant quantities for a given phenomenon. The contributions of the neglected quantities might then be modeled as an external force, which often is time-dependent, usually periodic or quasi-periodic due to seasonal regimes. If one is interested in the “permanent regime,” it is usually necessary to take into account various kinds of dissipation mechanisms, like friction and thermal diffusion. In this case, the nonautonomous equation modeling a given phenomenon is usually wellposed and may be considered as dynamical systems on appropriate phase spaces. The energy being dissipated, it is also likely that a certain relatively small region of the phase space attracts all the orbits of the system. If the system is autonomous, this situation might be described, for instance, by the existence of the so-called global attractor, which is a compact set which attracts all the orbits of the dynamical system, uniformly on bounded sets. In the case of nonautonomous systems the proper extension of the notion of a global attractor is the so-called uniform attractor [Ha,Hr,CV] (see Section 2.2). There are many references on attractors and uniform attractors for nonlinear evolutionary partial differential equations. Let us only mention the extensive works of A. V. Babin and M. I. Vishik [BV], V. V. Chepyzhov and M. I. Vishik [CV], J. K. Hale [Ha], D. Henry [He], A. Haraux [Hr], O. A. Ladyzhenskaya [La1], and R. Temam [T1]. The conditions for the existence of the uniform attractor usually parallel those for the existence of the global attractor for autonomous systems. This follows naturally from the framework of V. V. Chepyzhov and M. I. Vishik [CV], where nonautonomous systems are “lifted” to autonomous systems by “expanding” the phase space. Using this framework the existence of the uniform attractor relies on some compactness property of the solution operator associated with the system. The simplest, and also the strongest, form of compactness is when the solution operator itself is a compact operator for some positive time. This compactness property is usually available for parabolic systems on a bounded domain. When this form of compactness is not available, the system is termed as noncompact. Typical examples of noncompact systems are parabolic equations on unbounded domains and hyperbolic equations on either bounded or unbounded domains. For those types of systems the compactness property usually available (when available) is in some asymptotic sense. As in the autonomous case, one approach for dealing with noncompact nonautonomous systems is to decompose the solution operator into two parts: a (uniformly) compact part and a part which decays to zero as time goes to infinity (see, for instance, V. V. Chepyzhov and M. I. Vishik [CV], J. K. Hale [Ha], A. Haraux [Hr], J. M. Ghidaglia and R. Temam [GT], and A. Eden, V. Kalantarov, and A. Miranville [EKM]). This approach is suitable for many applications, but in several situations it may be difficult to find such a decomposition. Another approach is based on the use of energy equations in direct connection with the concept of asymptotic compactness as defined in [La2, A, R1,

ATTRACTORS FOR NONCOMPACT NONAUTONOMOUS SYSTEMS

3

T1, SY] in the autonomous case. One of our aims here is to introduce a corresponding concept of asymptotic compactness in the case of nonautonomous systems, which we term as uniform asymptotic compactness. Then, we adapt to the nonautonomous case the energy method approach for proving this uniform asymptotic compactness property. We also apply this method to two examples in which the decomposition mentioned above is either not at hand or requires much more work. The energy equation method for proving the asymptotic compactness (in the autonomous case) was initiated by J. Ball [B] in the context of a weakly damped, driven semilinear wave equation. This technique was applied to a weakly damped, driven KdV equation by J. M. Ghidaglia [G1, G2], and, subsequently, by several other authors in different contexts [W, R1, MR]. This technique was later presented in a systematic way and in a general abstract framework by I. Moise, R. Rosa, and X. Wang [MRW]. Our aim is to proceed in a similar way, extending this technique to nonautonomous systems and providing a general abstract framework for its application. The article is organized in the following way: In Section 2 we briefly review some basic concepts on the dynamical system approach to autonomous and nonautonomous evolution equations. We pay particular attention to the asymptotic compactness property, its role in the existence of the global attractor, and its appropriate extension to nonautonomous systems. Then, in Section 3 we state our main abstract theorem on the existence of the uniform attractor via the energy equation method. Finally, in Section 4 we present two applications of our results to physically interesting problems: a simplified case of uniform flow past an obstacle in the plane with time-dependent volume forces and boundary conditions, and a weakly damped, driven (with time almost-periodic forcing term) KdV equation on the whole real line. §2. Asymptotic Compactness In this section we first review some basic concepts on the dynamical system approach to autonomous evolution equations, i.e. nonlinear groups and semigroups, with particular attention given to the asymptotic compactness property and its role in the existence of the global attractor. Then, we recall the framework for non-autonomous systems as presented by V. V. Chepyzhov and M. I. Vishik [CV] and we discuss the extension of the asymptotic compactness property to the case of families of semiprocesses; in doing so, we arrive at the concept of uniform asymptotic compactness linked to the existence of the uniform attractor. As we will see in the following sections, the existence of the uniform attractor for weakly and even strongly dissipative equations can be naturally and easily obtained using energy equations to directly prove the uniform asymptotic compactness property. §2.1. The Autonomous Case. Let the phase space E be a complete metric space and let {S(t)}t≥0 be a semigroup of continuous (nonlinear) operators in E, i.e. (

S(t + s) = S(t) ◦ S(s), S(0) = I = Identity in E,

∀t, s ≥ 0,

(2.1)

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and

(

For each t ≥ 0, S(t) is a continuous (nonlinear) operator from E into itself.

(2.2)

In what follows, a semigroup will always mean a semigroup of continuous operators as defined by (2.1) and (2.2). For a set B ⊂ E, we define the ω-limit set of B by \ [ ω(B) = S(t)B. (2.3) s≥0 t≥s

The following characterization of an ω-limit set is very useful: w ∈ ω(B) ⇐⇒ ∃{wj }j∈N ⊂ B, ∃{tj }j∈N ⊂ R+ such that tj → +∞ and S(tj )wj → w in E.

(2.4)

A set B ⊂ E is called an absorbing set for the semigroup {S(t)}t≥0 if B “absorbs” all bounded sets of E in finite time, i.e. for every B ⊂ E bounded, there exists a time T = T (B) > 0 such that S(t)B ⊂ B, for all t ≥ T (B). The global attractor of a semigroup {S(t)}t≥0 is defined as the set A ⊂ E which is compact in E, invariant for {S(t)}t≥0 , i.e. S(t)A = A, ∀t ≥ 0, and which attracts all bounded sets of E, i.e. for any B ⊂ E bounded, distE (S(t)B, A) → 0 as t → +∞. (2.5) Here distE is the usual semidistance in E between two sets. One can show that if the global attractor exists, it is unique. Moreover, the global attractor is minimal (with respect to the inclusion relation between subsets of E) among the closed sets that attract all the bounded sets and is maximal (idem) among the bounded, invariant sets. In order to prove the existence of the global attractor one needs some kind of compactness of the semigroup together with the existence of a bounded absorbing set. The compactness that we are interested in is the so-called asymptotic compactness. One says [La2, A, R1, T1, SY] that {S(t)}t≥0 is asymptotically compact in E if the following condition holds: ( If {uj }j∈N ⊂ E is bounded and {tj }j∈N , tj ≥ 0, tj → ∞, (2.8) then {S(tj )uj }j∈N is precompact in E. This condition, together with the existence of a bounded absorbing set, implies the existence of the global attractor. Since this is the result we will be using in the rest of this article, we state it below in the form of a theorem. Theorem 2.1. Let E be a complete metric space, and let {S(t)}t≥0 be a semigroup of continuous (nonlinear) operators in E. If {S(t)}t≥0 possesses a bounded absorbing set B in E and is asymptotically compact in E, then {S(t)}t≥0 possesses the global attractor A = ω(B). Moreover, if t 7→ S(t)u0 is continuous from R+ into E, for any u0 ∈ E, and if B is connected in E, then A is also connected in E. The proof of Theorem 2.1 can be essentially found in [La2, Theorem 3.4].


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One can see from the characterization (2.4) that the condition (2.8) of asymptotic compactness is a natural assumption associated with ω-limit sets. In fact, the asymptotic compactness property alone implies that the ω-limit set of any nonempty, bounded set is nonempty, compact, invariant, and attracts the corresponding bounded set. The further existence of a bounded absorbing set implies then that the ω-limit set of this absorbing set attracts any bounded set. This property is directly related to the asymptotic smoothness property [Ha, Hr] (see the discussion in [MRW]): A semigroup {S(t)}t≥0 is said to be asymptotically smooth if for any nonempty, closed, bounded subset B ⊂ E for which S(t)B ⊂ B, ∀t ≥ 0, there exists a compact set K = K(B) ⊂ B which attracts B in the sense of (2.5). If {S(t)}t≥0 is asymptotically smooth and possesses a bounded absorbing ˜ can be shown to be the global attractor, where B ˜ = ∪t≥t S(t)B set B, then A = ω(B) 0 for t0 such that S(t)B ⊂ B for any t ≥ t0 . §2.2. The Non-Autonomous Case. Let E and Σ be two complete metric spaces and assume, for simplicity, that Σ is bounded. Let {Uσ (t, τ )}t≥τ ≥0,σ∈Σ be a family of semiprocesses in E, i.e. for each σ ∈ Σ, ( Uσ (t, s) ◦ Uσ (s, τ ) = Uσ (t, τ ), ∀t ≥ s ≥ τ ≥ 0, (2.9) Uσ (τ, τ ) = Identity in E, ∀τ ≥ 0. Let {T (s)}s≥0 be a semigroup of continuous operators in Σ and assume the following translation invariance condition: UT (s)σ (t, τ ) = Uσ (t + s, τ + s),

∀σ ∈ Σ,

∀t ≥ τ ≥ 0,

∀s ≥ 0.

(2.10)

The parameter σ is said to be the symbol of the semiprocess {Uσ (t, τ )}t≥τ ≥0 and the set Σ is called the symbol space. For convenience, we will write {Uσ (t, τ )} to stand for the family of semiprocesses {Uσ (t, τ )}t≥τ ≥0,σ∈Σ with the spaces E and Σ understood. The family of semiprocesses {Uσ (t, τ )} is said to be (E × Σ, E)-continuous if for any t and τ , t ≥ τ ≥ 0, the map (u, σ) 7→ Uσ (t, τ )u is continuous from E × Σ into E. Also, a set B ⊂ E is said to be a uniformly absorbing set for the family of semiprocesses if for any τ ≥ 0 and any B ⊂ E bounded, there exists a T = T (τ, B) ≥ τ such that Uσ (t, τ )B ⊂ B for t ≥ T (τ, B) and for all σ ∈ Σ. Thanks to the translation invariance condition (2.10), it suffices to find an absorbing set for τ = 0, whence we can deduce that T (τ, B) = τ + T (0, B), which actually means that for any B ⊂ E bounded, there exists T = T (B) ≥ 0 such that Uσ (t, τ )B ⊂ B for all t ≥ τ ≥ 0 with t − τ ≥ T (B), and for all σ ∈ Σ. Now, a set P ⊂ E is said to be uniformly attracting for {Uσ (t, τ )} if lim sup distE (Uσ (t, τ )B, P ) = 0,

t→+∞ σ∈Σ

∀τ ≥ 0, ∀B ⊂ E bounded.

(2.11)

Finally, a closed set AΣ ⊂ E is said to be the uniform attractor for {Uσ (t, τ )} if it is uniformly attracting and is contained in any other closed uniformly attracting set (minimality property). If it exists, the uniform attractor is unique.

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Note the condition of minimality among the closed uniformly attracting sets in the definition of the uniform attractor instead of an invariance condition similar to the one for the global attractor of a semigroup. In some cases, the uniform attractor may indeed turn out to be invariant, but in general this is not true. For the concepts introduced above and for some of the results below, the reader is referred to the works of V. V. Chepyzhov and M. I. Vishik [CV] and A. Haraux [Hr]. In order to obtain the uniform attractor, one may, for instance, “lift” the problem to the case of semigroups by defining on E × Σ the operator S(t)(u, σ) = (Uσ (t, 0)u, T (t)σ),

∀t ≥ 0,

∀(u, σ) ∈ E × Σ.

(2.12)

Thanks in particular to the translation invariance condition (2.10), it is trivial to deduce that {S(t)}t≥0 is a semigroup (i.e. satisfies (2.1)). Indeed, for t, s ≥ 0 and (u, σ) ∈ E × Σ, S(t + s)(u, σ) = (Uσ (t + s, 0)u, T (t + s)σ) = (Uσ (t + s, s) ◦ Uσ (s, 0)u, T (t)T (s)σ) = (UT (s)σ (t, 0) ◦ Uσ (s, 0)u, T (t)T (s)σ)

(2.13)

= S(t)(Uσ (s, 0)u, T (s)σ) = S(t)S(s)(u, σ). Moreover, {S(t)}t≥0 is a semigroup of continuous operators in E × Σ provided {T (t)}t≥0 is a semigroup of continuous operators in Σ and {Uσ (t, τ )} is (E × Σ, E)-continuous. Now, assume {S(t)}t≥0 has a global attractor A ⊂ E × Σ. Then, A can be represented as A = ω(B) = ∩s≥0 ∪t≥s S(t)B for some bounded absorbing set B ⊂ E × Σ. Let Π1 and Π2 be the canonical projections Π1 : E × Σ → E, Π1 (u, σ) = u, and Π2 : E × Σ → Σ, Π2 (u, σ) = σ. By replacing B by the larger set Π1 B × Σ, if necessary, we can assume B has the form B = B1 × Σ (recall Σ has been assumed to be bounded). Then, from the expression A = ω(B1 × Σ) = ∩s≥0 ∪t≥s S(t)(B1 × Σ), we see that A = {(u, σ) ∈ E × Σ; ∃{(un , σn )}n ⊂ B1 × Σ, ∃{tn }n ⊂ R+ , tn → ∞, S(tn )(un , σn ) → (u, σ)} ⊂ A1 × A2 ,

(2.14)

where A1 = Π1 A = ∩s≥0 ∪σ∈Σ ∪t≥s Uσ (t, 0)B1 = {u ∈ E; ∃{(un , σn )}n ⊂ B1 × Σ, ∃{tn }n ⊂ R+ , tn → ∞, Uσn (tn , 0)un → u}, A2 = Π2 A = ∩s≥0 ∪t≥s T (t)Σ = {σ ∈ Σ; ∃{σn }n ⊂ Σ, ∃{tn }n ⊂ R+ , tn → ∞, T (tn )σn → σ}. (2.15) Note that B1 is a bounded uniformly absorbing set for {Uσ (t, τ )}. It should be clear that A2 is the global attractor for {T (t)}t≥0 . Moreover, if B ⊂ E is bounded, the set B × Σ ⊂ E × Σ is also bounded since Σ is assumed to be bounded,


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and then sup distE (Uσ (t, τ )B, A1 ) σ∈Σ

= sup distE (UT (τ )σ (t − τ, 0)B, A1 ) σ∈Σ

=

sup

distE (Uσ (t − τ, 0)B, A1 )

σ∈T (τ )Σ

≤ sup distE (Uσ (t − τ, 0)B, A1 ) σ∈Σ

≤ sup sup inf dE (Uσ (t − τ, 0)u, a1 ) σ∈Σ u∈B a1 ∈A1

≤ sup sup inf

inf {dE (Uσ (t − τ, 0)u, a1 ) + dΣ (T (t − τ )σ, a2 )}

σ∈Σ u∈B a1 ∈A1 a2 ∈A2

≤

sup

inf

(u,σ)∈B×Σ (a1 ,a2 )∈A

dE×Σ (S(t − τ )(u, σ), (a1 , a2 ))

= distE×Σ (S(t − τ )(B × Σ), A) → 0, as t − τ → +∞, where, for instance, we considered the metric of the sum in E ×Σ. Thus, A1 is a uniformly attracting set for {Uσ (t, τ )}. It is also obvious that A1 is compact. Finally, if P ⊂ E is another closed, uniformly attracting set, then P × A2 is clearly a closed, attracting set for {S(t)}t≥0 . But A is minimal among such sets, so that necessarily A1 = Π1 A ⊂ P. Hence, A1 is the uniform attractor for {Uσ (t, τ )}. Thus, we have proven the following result: Proposition 2.2. Let E and Σ be two complete metric spaces and assume Σ is bounded. Let {Uσ (t, τ )}t≥τ ≥0,σ∈Σ be a (E × Σ, E)-continuous family of semiprocesses in E, and let {T (t)}t≥0 be a semigroup of continuous operators in Σ. Assume the translation invariance condition (2.10) holds. Then, {S(t)}t≥0 given by (2.13) defines a semigroup of continuous operators in E × Σ. Moreover, if {S(t)}t≥0 has a global attractor A ⊂ E × Σ, then A2 = Π2 A = ∩s≥0 ∪t≥s T (t)Σ is the global attractor for {T (t)}t≥0 and A1 = Π1 A = ∩s≥0 ∪σ∈Σ ∪t≥s Uσ (t, 0)B1 is the uniform attractor for {Uσ (t, τ )}, where B1 is an arbitrary bounded uniformly absorbing set for {Uσ (t, τ )}. So we have reduced the problem of finding the uniform attractor of {Uσ (t, τ )} to that of finding the global attractor of the semigroup {S(t)}t≥0 given by (2.12). We need then, besides the existence of an absorbing ball, some kind of compactness. We will say that {Uσ (t, τ )} is uniformly asymptotically compact if for    {u0j }j∈N bounded in E, {σj }j∈N ⊂ Σ, (2.16)   {tj }j∈N ⊂ R+ , tj → +∞ as j → +∞, it follows that {Uσj (tj , 0)u0j }j∈N is precompact in E.

(2.17)

Remark 2.1. Note that if {Uσ (t, τ )} is uniformly asymptotically compact, then thanks to the translation invariance condition (2.10), the set {Uσj (tj , τj )u0j }j is also precompact

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in E provided {u0j } ⊂ E is bounded, {σj } ⊂ Σ, {tj }, {τj } ⊂ R+ , and 0 ≤ tj − τj → ∞ (because Uσj (tj , τj )u0j = UT (τj )σj (tj − τj , 0)u0j ). The above definition of uniform asymptotic compactness is different from the one given by A. Haraux [Hr] and used by V. V. Chepyzhov and M. I. Vishik [CV], but it is in agreement with the corresponding definition for semigroups [La2]. The above definition is also easier to verify directly via energy equations, at least for weakly dissipative systems as shown in the next sections, and leads to the asymptotic compactness for the semigroup {S(t)}t≥0 (provided {T (t)}t≥0 is also asymptotically compact) and to the existence of the uniform attractor (provided there is an absorbing set for {S(t)}t≥0 ) as the theorem below shows. The definition of A. Haraux resembles more the definition of asymptotic smoothness for semigroups recalled in the previous subsection. For this reason, we state it here using a different nomenclature: {Uσ (t, τ )} is said to be globally uniformly asymptotically smooth if there is a compact set K ⊂ E such that for any bounded subset B ⊂ E, lim sup distE (Uσ (t, τ )B, K) = 0,

t→∞ σ∈Σ

∀τ ≥ 0.

(2.18)

Note that the above definition is stronger than that of asymptotic smoothness for semigroups since it includes the existence of a bounded uniformly absorbing set (any neighborhood of K; note that K is independent of B). Accordingly, we can define the notion of uniform asymptotic smoothness by allowing K to depend on B. Then, if a family of processes is uniformly asymptotically compact and possesses a bounded uniformly absorbing set, it is not difficult to see that this family is globally uniformly asymptotically smooth. Moreover, similarly to the case of semigroups, both concepts of uniform asymptotic compactness and uniform asymptotic smoothness are in fact equivalent if the translation invariance relation (2.10) holds (see Remark 2.3 below). For strongly dissipative systems on bounded domains, a compact uniformly absorbing set is usually available, in which case the uniform asymptotic compactness, or equivalently the uniform asymptotic smoothness, follows immediately. However, for weakly dissipative systems or systems on unbounded domains, the uniform asymptotic compactness can be relatively easily obtained directly using energy equations as we will see in the next sections. We now state the main result of this subsection. Theorem 2.3. Let E and Σ be two complete metric spaces and assume Σ is bounded. Let {Uσ (t, τ )}t≥τ ≥0,σ∈Σ be a (E × Σ, E)-continuous family of semiprocesses in E that possesses a bounded uniformly absorbing set B and which is uniformly asymptotically compact. Let {T (t)}t≥0 be an asymptotically compact semigroup of continuous operators in Σ. Assume the translation invariance condition (2.10) holds. Then, {Uσ (t, τ )} possesses the uniform attractor A1 = Π1 A = ∩s≥0 ∪σ∈Σ ∪t≥s Uσ (t, 0)B and {T (t)} has the global attractor A2 = Π2 A = ∩s≥0 ∪t≥s T (t)Σ. Proof. In view of Propositions 2.2 and Theorem 2.1, we only need to show that the semigroup {S(t)}t≥0 given by (2.12) possesses a bounded absorbing ball in E × Σ and


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is asymptotically compact. But these two facts are trivial to verify and their proofs are omitted. Remark 2.2. Of course, with proper modifications, Theorem 2.3 and Proposition 2.2 can be extended to the case in which Σ is not bounded but {T (t)} possesses a bounded absorbing set in Σ. These results, as well as the remark below, can also be properly modified to apply to the cases in which {Uσ (t, τ )} and {T (t)} are defined on [τ0 , +∞), for τ0 ∈ R, or on all R. Remark 2.3. If a suitable semigroup {T (t)}t≥0 as described in Theorem 2.3 is not available, we can still obtain the uniform attractor for a family of semiprocess {Uσ (t, τ )} with σ in an arbitrary set Σ. This can be achieved by assuming simply that {Uσ (t, τ )} is globally uniformly asymptotically smooth [CV, Theorem 5.2], or equivalently that {Uσ (t, τ )} is uniformly asymptotically smooth and possesses a bounded uniformly absorbing set B. In this case, the uniform attractor is obtained as def

AΣ = ω(B) = ∪τ ≥0 ∩s≥τ ∪σ∈Σ ∪t≥s Uσ (t, τ )B,

(2.19)

The uniform asymptotic compactness defined in (2.16)-(2.17), however, is not satisfactory anymore, but it can be properly generalized to cope with the lack of a suitable semigroup {T (t)}t≥0 . Its generalization is based on the following characterization of ω(B) (defined as in (2.19)) for any B ⊂ E :  ∃{τn } ⊂ R+ , ∃{un,` } ⊂ B, ∃{σn,` } ⊂ Σ, and      ∃{tn,` } ⊂ R+ such that tn,` ≥ τn , (2.20) w ∈ ω(B) ⇐⇒ tn,` → +∞ as ` → +∞, and      w = lim lim Uσ (tn,` , τn )un,` . n,` n→+∞ `→+∞

Such characterization leads then naturally to the definition of a uniformly asymptotically compact family of semiprocesses {Uσ (t, τ )} as one with the property that for sequences {τn } ⊂ R+ , {un,` } ⊂ E bounded, {σn,` } ⊂ Σ, and {tn,` } ⊂ R+ such that tn,` ≥ τn and tn,` → +∞ as ` → +∞, there is a subsequence {nj } and subsequences {`j,k } for which the limit lim lim Uσnj ,`j,k (tnj ,`j,k , τnj )unj ,`j,k

j→∞ k→∞

exists in E. It can be shown that if {Uσ (t, τ )} is uniformly asymptotically compact in this sense, then for any B ⊂ E bounded, the set ω(B) is nonempty, compact, attracts B in the sense of (2.11), and is contained in any other closed set which attracts B in that same sense. Then, it is easy to see that this definition of uniform asymptotic compactness is equivalent to that of uniform asymptotic smoothness. Finally, if a suitable semigroup {T (t)}t≥0 satisfying the translation invariance condition is available, then this definition of uniform asymptotic compactness can be shown to be equivalent to the one given earlier in (2.16), (2.17). However, it is not our intent to explore this generalization further, because the energy equation approach developed in the next section relies heavily on the translation invariance relation (2.10).

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§3. Abstract Energy Equations In this section we consider families of semiprocesses possessing a uniform absorbing set and satisfying some general abstract energy equations. We study under which conditions on the energy equations we can obtain the uniform asymptotic compactness needed for the existence of the uniform attractor. Let E be a reflexive Banach space or a closed subset of such a space, and let Σ be a bounded complete metric space. Consider a (E×Σ, E)-continuous family of semiprocesses {Uσ (t, τ )}t≥τ ≥0,σ∈Σ in E and a semigroup {T (t)}t≥0 of continuous (nonlinear) operators in the symbol space Σ. It is assumed that the translation invariance condition holds, that {T (t)}t≥0 is asymptotically compact, that {Uσ (t, τ )} possesses a bounded uniformly absorbing set, and that the trajectories of the semiprocesses are continuous in E, i.e. t 7−→ Uσ (t, τ )u ∈ C([τ, ∞), E),

∀τ ≥ 0,

∀u ∈ E,

∀σ ∈ Σ.

(3.1)

We also assume that the family of semiprocesses is weakly continuous in the sense that If σj → σ in Σ and uj * u weakly in E, then Uσj (t, τ )uj * Uσ (t, τ )u weakly in E,

∀t ≥ τ ≥ 0.

(3.2)

For the energy equation, we assume that

d (Φ(Uσ (t, 0)u0 ) + Jσ (t, Uσ (t, 0)u0 )) + γ (Φ(Uσ (t, 0)u0 ) + Jσ (t, Uσ (t, 0)u0 )) dt + Lσ (t, Uσ (t, 0)u0 ) = Kσ (t, Uσ (t, 0)u0 ), ∀u0 ∈ E, ∀σ ∈ Σ,

(3.3)

in the distribution sense in R+ , where γ is a positive constant and Φ, Jσ , Kσ , and Lσ are functionals satisfying the following hypotheses: Φ is defined and continuous on E, bounded on bounded subsets of E, and has the property that if {uj }j is bounded in E, {tj }j ⊂ R+ , tj → ∞, {˜ σj }j ⊂ Σ, Uσ˜j (tj , 0)uj * w weakly in E, and lim supj Φ(Uσ˜j (tj , 0)uj ) ≤ Φ(w), then Uσ˜j (tj , 0)uj → w strongly in E;

(3.4)

Jσ is defined and continuous on R+ × E and it is such that if {uj }j is bounded in E, {tj }j ⊂ R+ , tj → ∞, {˜ σj } ⊂ Σ, Uσ˜j (tj , 0)uj * w weakly in E, and σj → σ in Σ, then Jσj (t, Uσ˜j (tj , 0)uj ) → Jσ (t, w), ∀t ≥ 0;

(3.5)


11

Kσ is defined on R+ × ∪t>0 Uσ (t, 0)E and it is such that if {uj }j is bounded in E, {tj }j ⊂ R+ , tj → ∞, {˜ σj } ⊂ Σ, Uσ˜j (tj , 0)uj * w weakly in E, and σj → σ in Σ, then Z lim

j→∞

0

t

e−γ(t−s) Kσj (s, Uσj (s, 0)Uσ˜j (tj , 0)uj )ds Z t e−γ(t−s) Kσ (s, Uσ (s, 0)w)ds, ∀t > 0, =

(3.6)

0

where it is assumed that s 7→ Kσ (s, Uσ (s, 0)u) belongs to L1 (0, t), ∀t > 0, ∀u ∈ E, ∀σ ∈ Σ; and Lσ is defined on R+ × ∪t>0 Uσ (t, 0)E and it is such that if {uj }j is bounded in E, {tj }j ⊂ R+ , tj → ∞, {˜ σj }j ⊂ Σ, Uσ˜j (tj , 0)uj * w weakly in E, and σj → σ in Σ, then Z 0

t

e−γ(t−s) Lσ (s, Uσ (s, 0)w)ds Z t ≤ lim sup e−γ(t−s) Lσj (s, Uσj (s, 0)Uσ˜j (tj , 0)uj )ds, ∀t > 0, j→∞

(3.7)

0

where it is assumed that s 7→ Lσ (s, Uσ (s, 0)u) belongs to L1 (0, t), ∀t > 0, ∀u ∈ E, ∀σ ∈ Σ. Essentially, one thinks of Φ as a term equivalent to some power of the norm in E, with the possible addition of lower order terms. The terms Jσ and Kσ are usually weakly continuous in E, but in some applications such as that in Section 4.2 they may be only asymptotically weakly continuous as described above. The term Lσ is usually absent in hyperbolic equations and comes from a stronger dissipation typical from parabolic equations; it is usually lower weakly semicontinuous in E, but in some applications it may be asymptotically weakly lower semicontinuous as described above. The idea is to proceed as follows: One uses the usual a priori estimates to obtain a weakly convergent subsequence Uσj0 (tj 0 , 0)u0j 0 * w. With the help of the energy equation (3.3), one obtains, under the hypotheses above, the convergence Φ(Uσj0 (tj 0 , 0)u0j 0 ) → Φ(w). From the hypothesis on Φ, one obtains the desired strong convergence. More precisely, we prove the following result: Theorem 3.3. Let E be a reflexive Banach space or a closed convex subset of such a space, and let Σ be a bounded metric space. Let {Uσ (t, τ )}t≥τ ≥0,σ∈Σ be a family of semiprocesses in E which is (E × Σ, E)-continuous and weakly continuous in the sense of (3.2). Let {T (t)}t≥0 be an asymptotically compact semigroup of continuous (nonlinear) operators in Σ. Assume that the translation invariance condition (2.10) holds, that

12


{Uσ (t, τ )} possesses a bounded uniformly absorbing set, and that the trajectories of the semiprocesses are continuous in E. Assume finally that the energy equation (3.3) holds with γ positive and Φ, Jσ , Kσ , and Lσ satisfying (3.4), (3.5), (3.6), and (3.7), respectively. Then, {Uσ (t, τ )}t≥τ ≥0,σ∈Σ possesses a uniform attractor in E. Proof. Thanks to Theorem 2.3, we only need to prove the uniform asymptotic compactness of {Uσ (t, τ )}. For that purpose, let {u0j }j be bounded in E, {σj }j ⊂ Σ, and {tj }j ⊂ R+ with tj → +∞. Since {T (t)}t≥0 is asymptotically compact and Σ is bounded, it follows that T (tj 0 )σj 0 → σ in Σ,

(3.8)

for some σ ∈ Σ and some subsequence {j 0 }. Similarly, by using a diagonalization process and passing to a further subsequence if necessary, we have that T (tj 0 − T )σj 0 → σT in Σ,

∀T ∈ N

(3.9)

for some σT ∈ Σ. Also, due to the existence of a bounded uniformly absorbing set B ⊂ E, it follows that {Uσj0 (tj 0 , 0)u0j 0 }j 0 is bounded in E, which is reflexive (or a closed, convex subset of a reflexive Banach space) so that Uσj0 (tj 0 , 0)u0j 0 * w weakly in E,

(3.10)

for some w ∈ coB, ¯ and similarly Uσj0 (tj 0 − T, 0)u0j 0 * wT weakly in E,

∀T ∈ N,

(3.11)

with wT ∈ coB ¯ (passing to a further subsequence if necessary). By the translation invariance condition, Uσj0 (tj 0 , 0)u0j 0 = Uσj0 (tj 0 − T + T, tj 0 − T )Uσj0 (tj 0 − T, 0)u0j 0 = UT (tj0 −T )σj0 (T, 0)Uσj0 (tj 0 − T, 0)u0j 0 , so that by (3.9), (3.11), and the weak continuity (3.2), it follows that Uσj0 (tj 0 , 0)u0j 0 * UσT (T, 0)wT weakly in E,

∀T ∈ N.

(3.12)

From (3.10) and (3.12) we deduce that w = UσT (T, 0)wT ,

∀T ∈ N.

Now, we can write Uσj0 (tj 0 , 0)u0j 0 = Uσj0 (tj 0 , tj 0 − T )Uσj0 (tj 0 − T, 0)u0j 0 = UT (tj0 −T )σj0 (T, 0)Uσj0 (tj 0 − T, 0)u0j 0 .

(3.13)


13

Then, since the trajectories of {Uσ (t, τ )}t≥τ ≥0 are continuous in E we can integrate the energy equation (3.3) from 0 to T , with σ = T (t0j − T )σj 0 and u0 = Uσj0 (t0j − T, 0)u0j 0 , to obtain that Φ(Uσj0 (tj 0 , 0)u0j 0 ) + JT (tj0 −T )σj0 (T, Uσj0 (tj 0 , 0)u0j 0 ) Z T + e−γ(T −s) LT (tj0 −T )σj0 (s, UT (tj0 −T )σj0 (s, 0)Uσj0 (tj 0 − T, 0)u0j 0 )ds 0 (3.14) = (Φ(Uσj0 (tj 0 − T, 0)u0j 0 ) + JT (tj0 −T )σj0 (0, Uσj0 (tj 0 − T, 0)u0j 0 ))e−γT Z T + e−γ(T −s) KT (tj0 −T )σj0 (s, UT (tj0 −T )σj0 (s, 0)Uσj0 (tj 0 − T, 0)u0j 0 )ds. 0

Using (3.9), (3.11), and the assumptions (3.5), (3.6), and (3.7), we can pass to the lim sup in (3.14) to find that lim sup(Φ(Uσj0 (tj 0 , 0)u0j 0 )) + JσT (T, w) j 0 →∞

Z +

T

e−γ(T −s) LσT (s, UσT (s, 0)wT )ds

0

≤ (cB + JσT (0, wT ))e−γT Z T + e−γ(T −s) KσT (s, UσT (s, 0)wT )ds,

(3.15)

0

where cB = sup{Φ(v), v ∈ coB} ¯ < ∞. By using again the energy equation now with u0 = wT and σ = σT , we find thanks to (3.13) that Z Φ(w) + JσT (T, w) +

T

e−γ(T −s) LσT (s, UσT (s, 0)wT )ds

0

= (Φ(wT ) + JσT (0, wT ))e−γT Z T + e−γ(T −s) KσT (s, UσT (s, 0)wT )ds.

(3.16)

0

Subtract now (3.16) from (3.15) to obtain that lim sup(Φ(Uσj0 (tj 0 , 0)u0j 0 )) − Φ(w) ≤ 2cB e−γT , j 0 →∞

Let then T → ∞ in (3.17) to see that lim sup Φ(Uσj0 (tj 0 , 0)u0j 0 ) ≤ Φ(w), j 0 →∞

so that by the assumption (3.4) on Φ and (3.10) it follows that Uσj0 (tj 0 , 0)u0j 0 → w strongly in E,

∀T ∈ N.

(3.17)

14


which proves the uniform asymptotic compactness. This completes the proof. In some applications, for example in equations in higher order Sobolev spaces on unbounded domains, where the Sobolev imbeddings are not compact, the functionals J and K in the energy equation (3.3) might not be weakly continuous. This is the case, for instance, with the weakly dissipative Korteweg-deVries equation on the whole line considered in Subsection 4.2, for which the phase space is H 2 (R). In this case however, we can make use of one more energy equality to deduce first the asymptotic compactness with respect to the L2 (R) strong topology, which is then used to show the “asymptotic weak continuity” (see (3.5) and (3.6)) of J and K with respect to the H 2 (R) topology. In view of such applications, we assume we are given another reflexive Banach space F with F ⊃ E, the injections being continuous, and also Φ is defined and continuous on E, bounded on bounded subsets of E, and has the property that if {uj }j bounded in E, {tj }j ⊂ R+ , tj → ∞, {˜ σj }j ⊂ Σ, Uσ˜j (tj , 0)uj * w weakly in E and lim supj Φ(Uσ˜j (tj , 0)uj ) ≤ Φ(w), then Uσ˜j (tj , 0)uj → w strongly in F .

(3.18)

Then, the following lemma holds, and its proof is essentially the same as that for Theorem 3.3. Lemma 3.4. Under the assumptions of Theorem 3.3 except with (3.4) replaced by (3.18), with a reflexive Banach space F ⊃ E with continuous injection, it follows that if {uj }j is bounded in E, {σj }j ⊂ Σ, and {tj } ⊂ R+ , tj → ∞, then Uσj0 (tj 0 , 0)uj 0 → w strongly in F for some w ∈ E and some subsequence {j 0 }. §4. Applications §4.1. Flows Past an Obstacle. In this section we study the long time behavior of a uniform flow past an infinite long cylindrical obstacle. We will assume that the flow is uniform in the direction of the axis of the cylindrical obstacle and the flow approaches U∞ ex farther away from the obstacle. In this respect we can consider a two dimensional flow and assume the obstacle is a disk with radius r (more general obstacle can be treated in exactly the same way). A further simplification is to observe that since the flow is uniform at infinity, we may assume that the flow is in an infinitely long channel with width 2L (L >> r) and the obstacle is located at the center, while the flow at the boundary of the channel is almost the uniform flow at infinity. More precisely we assume that the flow is governed by the following Navier-Stokes equations in Ω = R1 × (−L, L) \ Br (0) (L >> r) : ∂u − ν∆u + (u · ∇)u + ∇p = f in Ω, ∂t

(4.1.1a)

div u = 0 in Ω,

(4.1.1b)

u = u0 at t = 0,

(4.1.1c)


15

u = 0 on ∂Ω2 = ∂Br ,

(4.1.1d)

u = ϕ on ∂Ω1 = {y = ±L},

(4.1.1e)

div u0 = 0,

(4.1.2a)

u02 = 0 on ∂Ω1 , u0 · ~n = 0 on∂Ω2 ,

(4.1.2b)

u0 − U∞ ex ∈ L2 (Ω),

(4.1.2c)

with

ϕ − U∞ ex ∈ Cb1 (R+ ; H2 (R1 × {±L})) and is asymptotically almost periodic, ϕ2 ≡ 0, div f = 0, f2 = 0 at ∂Ω, and f ∈ Cb (R+ ; L2 (Ω)) and is asymptotically almost periodic.

(4.1.2d) (4.1.2e) (4.1.2f)

Remark 4.1.1. The simplest and physically interesting case is f ≡ 0 and ϕ ≡ U∞ ex . The first simplification is to introduce the new variables u e = u − U∞ ex ,

(4.1.3a)

ϕ˜ = ϕ − U∞ ex ,

(4.1.3b)

u ˜0 = u0 − U∞ ex

(4.1.3c)

∂u ˜ − ν∆˜ u + (˜ u · ∇)˜ u + U∞ ∂x u ˜ + ∇p = f, ∂t

(4.1.4a)

div u ˜ = 0,

(4.1.4b)

u e=u e0 at t = 0,

(4.1.4c)

u ˜ = −U∞ ex on ∂Ω2 = ∂Br ,

(4.1.4d)

u ˜ = ϕ˜ on ∂Ω1 = {y = ±L}.

(4.1.4e)

Then u ˜ satisfies the equations

We observe that ϕ˜ ∈ Cb1 (R+ ; H2 (R1 × {±L})) and is asymptotically almost periodic.

(4.1.5)

Note that u e, u ˜0 and ϕ e decay nicely near infinity. However the boundary condition is not homogeneous and thus we apply a modified Hopf’s technique (see [H, MW1, MW2, TW, T1]) to homogenize the boundary condition. More specifically we choose 1 ρi ∈ C ∞ ([0, 1]), supp ρi ⊂ [0, ], 2

(4.1.6a)

16


Z

1

ρ1 (s)ds = 0, ρ1 (0) = 1,

(4.1.6b)

0

ρ2 (0) = 1, ρ02 (0) = 0, |sρ2 (s)| ≤

ν ν , |sρ02 (s)| ≤ , 1001U∞ 1001U∞

(4.1.6c)

and we define, for ε < 1,  Z L−y L s   − ϕ˜1 (x, L; t) ρ1 ( )ds, for < y < L,   Lε 2  0  Z L+y Ψ1 (x, y; t) = s L  ϕ˜1 (x, −L; t) ρ1 ( )ds, for − L < y < − ,   Lε 2  0   0, otherwise;

2

Ψ (x, y; t) =

  

(4.1.7)

p − U∞ ρ2 (

p x2 + y 2 − 1)y, for r < x2 + y 2 < 2r, r 0, otherwise.

(4.1.8)

We then define φi (x, y; t) = curl Ψi = (∂y Ψi , −∂x Ψi ).

(4.1.9)

Observe that φ1 matches ϕ˜ at y = ±L and φ2 matches −U∞ ex at ∂Br . If we set → H = v ∈ L2 (Ω), div v = 0, v · − n = 0 on ∂Ω ,

(4.1.10)

V = v ∈ H10 (Ω), div v = 0, v = 0 on ∂Ω ,

(4.1.11)

V 0 = the dual of V,

(4.1.12)

→ where − n denotes the unit outward normal at ∂Ω, we have that v=u ˜ − φ, where φ = φ1 − φ2

(4.1.13)

satisfies the equation ∂v − ν∆v + (v · ∇)v + (v · ∇)φ + (φ · ∇)v + U∞ ∂x v + ∇p ∂t ∂φ1 = + f + ν∆φ + −(φ · ∇)φ − U∞ ∂x φ ∂t =F (ε, ν, U∞ , r, L, t), v∈V

(4.1.14a)

for t > 0,

(4.1.14b)

v = v0 at t = 0,

(4.1.14c)

v0 = u0 − U∞ ex − φ1 |t=0 − φ2 ∈ H.

(4.1.14d)

where


17

It is easy to check that for fixed ε, ν, U∞ , r, and L, the right-hand-side of (4.1.14a), namely F (ε, ν, U∞ , r, L, t), belongs to Cb (R+ ; L2 (Ω)), and is asymptotically almost periodic in Cb (R+ ; L2 (Ω)), φ is almost periodic in Cb (R+ ; H2 (Ω)), thanks to our construction of φ1 and φ2 . We say that v is a weak solution of (4.1.14) if v ∈ L∞ (0, T ; H) ∩ L2 (0, T ; V ), d (v, w) + ν(∇v, ∇w) + b(v, v, w) + b(v, φ1 + φ2 , w)+ dt b(φ1 + φ2 , v, w) + b(U∞ ex , v, w) = (F, w), ∀w ∈ V,

(4.1.15a)

(4.1.15b)

in the distributional sense and v(0) = v0 ,

(4.1.15c)

where the trilinear term b : H10 × H10 × H10 → R is defined by Z ∂vj 2 b(u, v, w) = Σi,j=1 ui wj dx. ∂xi Ω

(4.1.15d)

The well-posedness of (4.1.15) can be derived using a standard Faedo-Galerkin approach (see for instance [T2, Chapter 3]). It can be viewed as a family of semiprocesses on H with the symbol space Σ defined as Σ = Σ1 × Σ2 = {φ(· + τ )}τ ≥0 × {F (· + τ )}τ ≥0 , endowed with the product norm of the supremum norm on Cb (R+ ; H2 (Ω)) (for φ) and the supremum norm of Cb (R+ ; L2 (Ω)) (for F ). Σ is a compact space by our assumptions on ϕ and f , and the explicit construction of φ and F . For each v0 ∈ H and σ = (σ 1 , σ 2 ) ∈ Σ, t > τ , Uσ (t, τ )v0 is the solution to dv + νAv + B(v, v) + B(v, σ 1 ) + B(σ 1 , v) + B(U∞ ex , v) = P σ 2 in V 0 , for t > τ, dt (4.1.16a) v(τ ) = v0 , (4.1.16b) where A : V → V 0 is the Stokes operator defined by < Av, w >= (∇v, ∇w),

∀v, w ∈ V,

(4.1.16c)

and B(u, v) is a bilinear operator H10 × H10 → V 0 defined by < B(u, v), w >= b(u, v, w),

∀u, v ∈ H10 , ∀w ∈ V,

(4.1.16d)

and P is the Leray-Hopf projection from L2 (Ω) onto H. We can also define on Σ the semigroup {T (s)}s≥0 given by T (s)σ = (T (s)σ)(t) = σ(t + s), ∀t ≥ 0, ∀s ≥ 0, ∀σ ∈ Σ. Since Σ is compact, the semigroup {T (s)}s≥0 is continuous and compact and in particular asymptotically compact. It is then obvious that this family of semiprocesses satisfies the translation invariance property (2.10).

18


Our goal in this section is to show that (4.1.16) possesses a uniform attractor in H using Theorem 3.3. The special case of time independent external forcing was considered in our earlier work [MRW]. It is easy to verify that the family of semiprocesses {Uσ (t, τ )} is (H × Σ, E) continuous and all trajectories are continuous in H. Moreover, for v0 ∈ H, σ ∈ Σ and T > τ, there exists a constant κ > 0, such that for v(t) = Uσ (t, τ )v0 we have kvkL∞ (τ,T ;H) ≤ κ,

(4.1.17a)

kvkL2 (τ,T ;V ) ≤ κ,

(4.1.17b)

kv 0 kL2 (τ,T,V 0 ) ≤ κ,

(4.1.17c)

v ∈ C([τ, ∞); H),

(4.1.17d)

where κ = κ(ν, T, ε, |v0 |, |σ|Σ , r, L, U∞ ). This immediately implies that we have the following energy equation: Z 1 d 2 2 |v| + ν|∇v| + (v · ∇)σ 1 · v = (σ 2 , v). (4.1.18) 2 dt Ω A closer investigation into the well-posedness proof reveals that the solution set is compact in the sense that if {vn (t) = Uσn (t, τ )v0n , n ≥ 1} is a family of solutions on [τ, T ] satisfying estimates (4.1.17) for a κ independent of n, then there exists a subsequence {vn0 , n0 ≥ 1}, σ∞ ∈ Σ, and v0∞ ∈ H, v∞ (t) = Uσ∞ (t, τ )v0∞ , such that vn0 * v∞ weakly-star in L∞ (τ, T ; H),

(4.1.19a)

vn0 * v∞ weakly in L2 (τ, T ; V ),

(4.1.19b)

0 vn0 0 * v∞ weakly in L2 (τ, T ; V 0 ).

(4.1.19c)

For a proof the reader is referred to [T2, Chapter 3] or [R1] for more details. This actually implies the weak continuity of the family of semiprocesses. Indeed, let v0n * v0∞ be a weakly convergent subsequence in H, σn → σ∞ in Σ, then vn (t) = Uσn (t, τ )v0n satisfies (4.1.17) with a constant κ independent of n. Notice that each subsequence of {vn , n ≥ 1} contains a subsubsequence which converges to some v∞ in the sense of (4.1.19). It is easy to check that v∞ (τ ) = v0∞ using a test function of the form ζ(t) with ζ(τ ) = 1 since Z T Z T 0 (vn0 (t), ζ(t)w) = − (vn0 (t), ζ 0 (t)w) − (v0n , w), τ

τ

↓ Z

T 0 (v∞ (t), ζ(t)w) = −

τ

↓ Z

↓

T

(v∞ (t), ζ 0 (t)w) − (v0∞ , w).

τ

Since this is true for each subsequence, we conclude that the whole sequence converges, and hence Z t Z t 0 0 Uσn (t, τ )v0n = vn (t) = vn (s) ds + v0n * v∞ (s) ds + v0∞ = v∞ (t) = Uσ∞ (t, τ )v0∞ τ

τ


19

weakly in V 0 and then in H by density and (4.1.17). This completes the weak continuity proof. Before we apply Theorem 3.3, we need to verify the existence of a bounded absorbing set in H. This can be done via an appropriate choice of ε in (4.1.7) and using (4.1.18). Observe that Z Z (v · ∇)φ2 · v = (v · ∇)v · φ2 Ω

Ω

p !

v x2 + y 2

≤ r p |∇v| − 1 φ2

x2 + y 2 − r 2 ∞ r L (B2r \Br )

2

≤ 4r|∇v| · |U∞ |(|sρ2 (s)|L∞ +

L

(B2r \Br )

4|sρ02 (s)|L∞ )

(thanks to Hardy’s inequality and (4.1.6)-(4.1.9)) ν (thanks to (4.1.6)) ≤ |∇v|2 5 Z Z 1 1 (v · ∇)φ · v = (v · ∇)v · φ Ω Ω ! v v |∇v| + ≤ L − y L2 (1−ε< y 0.

(4.2.12)

The proof of Theorem 4.2.1 follows as in the case of γ = 0 and f = 0. The existence of solutions in L∞ ((τ, T ); H 2 (R)) ∩ C([τ, T ], L2 (R)) and an inequality (≤) in (4.2.6) can


23

be obtained by parabolic regularization [T3,BS,MR]. The uniqueness is straightforward. The equality in (4.2.6) and, as a consequence, the regularity u ∈ C([τ, T ], H 2 (R)) can be obtained by using the time reversibility of the solutions as done in Subsection 4.2.1 for the second grade fluids. Thanks to Theorem 4.2.1, one can define for each σ ∈ Σ and for γ > 0, which is the case of interest for us, a semiprocess {Uσ (t, τ )}t≥τ ≥0 in H 2 (R) by Uσ (t, τ )u0 = u(t), where u = u(t) is the solution of (4.2.1)-(4.2.2) with f = σ. We can also define on Σ the semigroup {T (s)}s≥0 given by T (s)σ = (T (s)σ)(t) = σ(t + s), ∀t ≥ 0, ∀s ≥ 0, ∀σ ∈ Σ. Since Σ is compact, the semigroup {T (s)}s≥0 is compact and in particular asymptotically compact. By the uniqueness of the solutions of (4.2.1)-(4.2.2), assured by Theorem 4.2.1, one can easily deduce the translation invariance condition (2.10). The continuity of the trajectories t → Uσ (t, τ )u0 follows from (4.2.5). Thus, most of the conditions of the Theorem 3.3 hold, and we need to verify the remaining conditions: Proposition 4.2.2. The family of semiprocesses {Uσ (t, τ )} possesses a bounded uniformly absorbing set in H 2 (R). Proof. Since Σ is compact in Cb (R+ , H 2 (R)) and since ||σ(t)||H 2 (R) is bounded uniformly for t > 0 and σ ∈ Σ, the existence of a bounded uniformly absorbing set can be obtained just like in the autonomous space periodic case treated by J. M. Ghidaglia [G1], the differences being that the Agmon inequality has a different constant and the norms of the time-independent forcing term are replaced by the supremum for all t ≥ 0 and all σ ∈ Σ of the corresponding norms of σ(t). We do not develop the details here. Proposition 4.2.3. The family of semiprocesses {Uσ (t, τ )} is (H 2 (R) × Σ, H 2 (R))continuous and is weakly continuous in the sense of (3.2). Proof. For the weak continuity, let σj → σ in Σ and let u0j * u0 weakly in H 2 (R). Fix T and τ, T > τ ≥ 0 (T = τ is obvious). For simplicity, we set uj (t) = Uσj (t, τ )u0j for τ ≤ t ≤ T. Note that {u0j }j is bounded in H 2 (R) since it has a weak limit in that space. Then, thanks to the long-time estimates given by the existence of a uniformly absorbing set (Proposition 4.2.2) and thanks to the local in time estimates given by (4.2.12), it follows that {uj }j is bounded in L∞ (τ, T ; H 2 (R)). (4.2.13) Then, from the equation (4.2.1) itself, {u0j }j is bounded in L∞ (τ, T ; H −1 (R)),

(4.2.14)

where H −1 (R) is the dual of H 1 (R) when we identify L2 (R) with its dual. From (4.2.14) it follows that for 0 < a < T − τ and v ∈ H 1 (R), Z (uj (t + a) − uj (t), v)L2 (R) = ≤

t+a

< u0j (s), v >H −1 (R),H 1 (R) ds

t 0 a||uj ||L∞ (τ,T ;H −1 (R)) ||v||H 1 (R) ,

∀τ ≤ t ≤ T − a,

24


where < ·, · >H −1 (R),H 1 (R) denotes the duality product between the two spaces H −1 (R) and H 1 (R). By taking v = uj (t + a) − uj (t) for each t ∈ [τ, T − a], which is possible since uj ∈ C([τ, T ], H 2 (R)), we find that ||uj (t + a) − uj (t)||2L2 (R) ≤ 2a||u0j ||L∞ (τ,T ;H −1 (R)) ||uj ||L∞ (τ,T ;H 1 (R)) , ∀t ∈ [τ, T − a]. Taking (4.2.13) into account, we obtain ||uj (t + a) − uj (t)||L2 (R) ≤ cT a1/2 ,

∀t ∈ [τ, T − a], ∀a ∈ (0, T − τ ).

(4.2.15)

Now, for each r > 0, consider the sequence {uj,r }j , where uj,r (t) = ρr uj (t), for ρr = ρr (x) = ρ(x/r), with ρ ∈ C ∞ (R), ρ ≥ 0, ρ(ξ) = 1 for |ξ| ≤ 1, and ρ(ξ) = 0 for |ξ| ≥ 2. Thus, from (4.2.13) and (4.2.15), it follows that for each r > 0, the sequence {uj,r }j is equibounded and equicontinuous in C([τ, T ], L2 (−2r, 2r)). Moreover, from (4.2.13) and the fact that each uj is continuous from [τ, T ] to H 2 (R), it follows that for each t ∈ [τ, T ], the set {uj,r (t)}j is bounded in H02 (−2r, 2r), hence precompact in L2 (−2r, 2r). Therefore, we can apply the Arzela-Ascoli Theorem to {uj,r }j to deduce that this sequence is precompact in C([τ, T ], L2 (−2r, 2r)). It is then clear that for each r, the sequence {uj |(−r,r) }j is precompact in C([τ, T ], L2 (−r, r)), where uj |(−r,r) is the restriction of uj in space to (−r, r). Then, by a diagonalization process, we can find a subsequence {uj 0 }j 0 and an element u ˜ ∈ C([τ, T ], L2loc (R)) such that {uj 0 |(−r,r) }j 0 converges to u ˜|(−r,r) in 2 C([τ, T ], L (−r, r)), which is to say that {uj 0 }j 0 converges to u ˜ in the topology of the Frechet space C([τ, T ], L2loc (R)). On the other hand, from (4.2.13) one can also assume that (passing to a further subsequence if necessary) {uj 0 }j 0 converges to u ˜ weakly-star in L∞ (τ, T ; H 2 (R)), which gives in particular that u ˜ ∈ L∞ (τ, T ; H 2 (R)). Thus we have that ( uj 0 → u ˜ weakly-star in L∞ (τ, T ; H 2 (R)) (4.2.16) and strongly in C([τ, T ], L2loc (R)). The convergence (4.2.16) allows us to pass to the limit in the weak form of the equation for uj (the weak form of (4.2.1)-(4.2.2) with f = σj and u0 = u0j ) to find that u ˜ solves (the weak and the strong form of) the equations (4.2.1)-(4.2.2) with f = σ. By the uniqueness of the solutions, we must have u ˜(t) = Uσ (t, τ )u0 . Then, by a contradiction argument, one can deduce that in fact the whole sequence {uj } converges to u ˜ in the sense of (4.2.16), and hence that Uσj (t, τ )u0j → Uσ (t, τ )u0 weakly-star in L∞ (τ, T ; H 2 (R)) and strongly in C([τ, T ], L2loc (R)).

(4.2.17)

Now, from the strong convergence in (4.2.17), we find that for every t such that τ ≤ t ≤ T and for every v ∈ Cc∞ (R) – the space of C ∞ functions with compact support –, (Uσj (t, τ )u0j , v)H 2 (R) = (Uσj (t, τ )u0j , Lv)L2 (R) → (Uσ (t, τ )u0 , Lv)L2 (R) = (Uσ (t, τ )u0 , v)H 2 (R) ,


25

where Lv = v − vxx + vxxxx ∈ Cc∞ (R). Then from (4.2.13) and the density of Cc∞ (R) in H 2 (R), it follows that for every t, τ ≤ t ≤ T and hence for every t ≥ τ and τ ≥ 0, since τ and T are arbitrary, (Uσj (t, τ )u0j , v)H 2 (R) → (Uσ (t, τ )u0 , v)H 2 (R) ,

∀v ∈ H 2 (R), ∀t ≥ τ ≥ 0,

(4.2.18)

which proves the weak continuity of {Uσ (t, τ )} in the sense of (3.2). For the strong continuity, assume σj → σ in Σ and u0j → u0 strongly in H 2 (R). In particular u0j * u0 weakly in H 2 (R), so that the above convergences (4.2.17) and (4.2.18) hold. From the energy equation (4.2.6) for m = 0 it follows that for a given τ ≥ 0, Z t 2 2 −2γ(t−τ ) |uj (t)|L2 (R) = |u0j |L2 (R) e +2 e−2γ(t−s) (σj (s), uj (s))L2 (R) ds, ∀t ≥ τ, τ

(4.2.19) where uj (t) = Uσj (t, τ )u0j and u(t) = Uσ (t, τ )u0 . From the strong convergence σj → σ in Σ, the weak continuity (4.2.18), and the uniform boundedness (4.2.13), together with the strong convergence u0j → u0 in H 2 (R), we can pass to the limit in (4.2.19) to find lim |uj (t)|2L2 (R) = |u0 |2L2 (R) e−2γ(t−τ ) Z t +2 e−2γ(t−s) (σ(s), u(s))L2 (R) ds = |u(t)|2L2 ,

j→∞

(4.2.20) ∀t ≥ τ.

τ

From the weak continuity (4.2.18) and the L2 -norm convergence (4.2.20) it follows, since L2 (R) is a Hilbert space, that uj (t) → u(t) strongly in L2 (R),

∀t ≥ τ.

(4.2.21)

Using interpolation, it follows from (4.2.21) and (4.2.18) that uj (t) → u(t) strongly in H 1 (R),

∀t ≥ τ.

Now, from the energy equation (4.2.6) for m = 2, Z t −2γ(t−τ ) I2 (uj (t)) = I2 (u0j )e + e−2γ(t−s) K2,σj (s, uj (s))ds,

(4.2.22)

∀t ≥ τ.

(4.2.23)

τ

As above, using also (4.2.22), we can pass to the limit in (4.2.23) to find that lim I2 (uj (t)) = I2 (u(t)),

j→∞

∀t ≥ τ.

(4.2.24)

Using (4.2.22) again, it follows from (4.2.24) that ||uj (t)||H 2 (R) → ||u(t)||H 2 (R) ,

∀t ≥ τ.

(4.2.25)

Then, (4.2.25) together with the weak continuity (4.2.18) implies finally that Uσj (t, τ )u0j → Uσ (t, τ )u0 strongly in H 2 (R),

∀t ≥ τ ≥ 0.

(4.2.26)

which proves the (H 2 (R) × Σ, H 2 (R))-continuity of {Uσ (t, τ )}. In order to apply Theorem 3.3, it remains to verify the corresponding conditions (3.4), (3.5) and (3.7) for the energy equation with m = 2. In order to do that, we first need the following lemma:

26


Lemma 4.2.4. Let {uj }j be bounded in H 2 (R), {σj }j ⊂ Σ, and {tj }j ⊂ R+ with tj → ∞. Then, there exist w ∈ H 2 (R) and a subsequence {j 0 } such that Uσj0 (tj 0 , 0)uj → w strongly in H 1 (R). Proof. We apply Lemma 3.4 with the energy equation (4.2.6) for m = 0. In the notations of Lemma 3.4, the terms of this energy equation are Φ(u) = I0 (u) = |u|2L2 , Jσ (t, u) = 0, Z Kσ (t, u) = K0,σ (t, u) = 2

σ(t)udx,

Lσ (t, u) = 0. and obviously F = L2 (R). The hypothesis (3.18) is trivially satisfied. For (3.6), let {uj }j σj }j ⊂ Σ such that Uσ˜j (tj , 0)uj * w weakly be bounded in H 2 (R), {tj }j ⊂ R+ , tj → ∞, {˜ in H 2 (R), and σj → σ in Σ. Since the family of semiprocesses is weakly continuous in the sense of (3.2) (from Proposition 4.2.3), we deduce that Uσj (s, 0)Uσ˜j (tj , 0)uj * Uσ (s, 0)w weakly in H 2 (R),

∀s ≥ 0.

(4.2.27)

The map s 7→ K0,σ (s, Uσ (s, 0)u) belongs to L1 (0, t), ∀t > 0, ∀u ∈ H 2 (R), ∀σ ∈ Σ. Taking into account (4.2.27), the fact that σj → σ in Σ and the form of K0,σ , we have that K0,σj (s, Uσj (s, 0)Uσ˜j (tj , 0)uj ) → K0,σ (s, Uσ (s, 0)w) as j → ∞, for s ∈ (0, t). Moreover, s 7→ K0,σj (s, Uσj (s, 0)Uσ˜j (tj , 0)uj ) is uniformly bounded on R+ thanks to the compactness of Σ and the existence of a bounded uniformly absorbing set for {Uσ (t, τ )}. Then, by the Lebesgue Dominated Convergence Theorem, Z t lim e−2γ(t−s) K0,σj (s, Uσj (s, 0) Uσ˜ j (tj , 0)uj )ds j→∞ 0 Z t = e−2γ(t−s) K0,σ (s, Uσ (s, 0)w)ds, ∀t > 0. 0

Consider now {uj }j bounded in H 2 (R), {σj }j ⊂ Σ, and {tj }j ⊂ R+ with tj → ∞. Since the family of semiprocesses {Uσ (t, τ )} has a bounded uniformly absorbing set in H 2 (R), we deduce that there exists a subsequence {j 0 } such that Uσj0 (tj 0 , 0)uj 0 * w weakly in H 2 (R),

(4.2.28)

for some w ∈ H 2 (R). Now, we can apply Lemma 3.4 to deduce (passing to a further subsequence and then using a contradiction argument) that Uσj0 (tj 0 , 0)uj 0 → w strongly in L2 (R).

(4.2.29)

By interpolation, we finally deduce from (4.2.28) and (4.2.29) that Uσj0 (tj 0 , 0)uj 0 → w strongly in H 1 (R),

(4.2.30)


27

which completes the proof. We now apply Theorem 3.3 with the energy equation (4.2.6) with m = 2. In the notation of the Theorem 3.3, the terms of the energy equation are Z Φ(u) = u2xx dx, Z 5 4 5 2 Jσ (t, u) = J(u) = − ux + u dx, 3 36 Kσ (t, u) = K2,σ (t, u), and Lσ (t, u) = 0. Consider again {uj }j bounded in H 2 (R), {tj }j ⊂ R+ , tj → ∞, and {˜ σj }j ⊂ Σ such that Uσ˜j (tj , 0)uj * w weakly in H 2 (R). From Lemma 4.2.4, there exist a subsequence {j 0 } such that Uσ˜j0 (tj 0 , 0)uj 0 → w strongly in H 1 (R), and, by a contradiction argument, the whole sequence convergences to w strongly in H 1 (R). Then, (3.4) and (3.5) are trivially satisfied, and for (3.6) we use the same arguments as in Lemma 4.2.4. Thus, we can apply Theorem 3.3 to conclude that {Uσ (t, τ )}t≥τ ≥0,σ∈Σ possesses a uniform attractor in H 2 (R) : Theorem 4.2.5. Let γ > 0, f be as in (4.2.3), and Σ be as in (4.2.4). Then, the family {Uσ (t, τ )}t≥t≥0,σ∈Σ of semi-processes in H 2 (R) possesses a uniform attractor A in H 2 (R). Acknowledgements This work was supported in part by the National Science Foundation under the Grant NSF-DMS-9971986, and by the Research Fund of the Indiana University. The second author is partly supported by CNPq, Bras´ılia, Brasil, and by FAPERJ and FUJB, Rio de Janeiro, Brasil. REFERENCES [A] [BV] [B] [BS] [CV] [EKM] [G1]

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