A Comparison Between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

A comparison between two theories for multi-valued semiflows and their asymptotic behaviour Caraballo T. ([email protected]) and Mar´ın-Rubio P. ([email protected]) Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain Robinson J.C. ([email protected]) Mathematics Institute, University of Warwick, Coventry, CV4 7AL (UK) Abstract. This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball [5] to treat equations whose solutions may not be unique, and that due to Melnik & Valero [25] tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Ball’s theory to differential inclusions. Keywords: generalized and multi-valued semiflows, partial differential equations without uniqueness, differential inclusions Mathematics Subject Classification (2000): 35R70, 37L, 35B41, 34G20, 34G25 1. Introduction The concept of the attractor has proved an extremely useful tool for studying the asymptotic behaviour of solutions of a wide variety of dynamical systems: deterministic systems in both the autonomous (e.g. Babin & Vishik [4], Hale [13], Ladyzhenskaya [22], Temam [31]) and non-autonomous cases (e.g. Chepyzhov & Vishik [9], Kloeden & Schmalfuss [20], Kloeden & Stonier [21]), and the stochastic flows generated by stochastic differential equations (e.g. Arnold [1], Crauel & Flandoli [11], Crauel et al. [10]). An important step in all these theories is the development of a general abstract framework in which to express the underlying dynamics of the problem (semiflows S(t), processes S(t, s), and cocycles ϕ(t, ω), respectively). When one is interested in studying the asymptotic behaviour of solutions to multi-valued problems such as those arising in control theory and viability theory, or from equations without uniqueness or differential inclusions, we can still expect the attractor to be useful. The development of a theory of “multi-valued semiflows” is a necessary first step in the study of attractors for such problems. c °2003 Kluwer Academic Publishers. Printed in the Netherlands. svan516revised.tex; 26/02/2003; 8:23; p.1 2Caraballo, Mar´ın-Rubio & Robinson In the 1960s and 1970s a number of papers appeared that treated this problem for differential inclusions on locally compact spaces (Bridgland [7], Bushaw [8], Kloeden [18], Roxin [27] & [26], Sell [28], Szeg¨o & Treccani [30]; see Kloeden [19] for a review), while more recently multivalued semiflows on general Banach spaces have been considered by various authors, often in the context of partial differential equations or inclusions (Ball [5], Barbu [6], Elmounir & Simondon [12], Kapustyan [17], Kapustian & Valero [16], Melnik & Valero [25]; for the use of multi-valued systems in numerical analysis see Lamba [23] and Lamba & Stuart [24]). We will be mainly concerned with the recent works of Melnik & Valero [25] and Ball [5] as canonical examples of such theories that also discuss the way one might define an ‘attractor’ for such systems. We take the approach of Kloeden [18] as our prime ‘historical’ example. In order to consider the attractors of partial differential inclusions, in [25] Melnik & Valero define a multi-valued semiflow as a multi-valued mapping GMV :R+×X→2X, where Xis the phase space. Differential equations without uniqueness are the main topic of [5], which presents a theory more in line with previous treatments. Here Ball defines the concept of a generalized semiflow GBon the phase space X, essentially consisting of all possible solutions of the equation. Although he mainly works with this collection GB(to some extent an advantage over the approach of [25] since the idea of an individual solution is a building block of the definition) he also considers the multi-valued map T(t)u0formed from the set of points reached in time tby elements of GB(solutions) which began (at time zero) at u0. The map T(t)u0has very similar properties to the multi-valued semiflow GMV (t, u0) defined by Melnik & Valero. Although Ball’s paper deals primarily with equations without uniqueness, while that of Melnik & Valero concentrates on differential inclusions, in fact they deal with very similar problems in which the dynamics is governed by a collection of possible solutions through each initial condition. We will see below in Proposition 2 that a generalized semiflow can be seen as a particular case of a multi-valued semiflow. A natural question is whether, given a multi-valued map T: [0,+∞)×X→P(X) (as in Melnik & Valero [25]), we can define a generalized semiflow GB consisting of all “solutions” (with some appropriate properties). More generally speaking it is natural to ask whether the two theories are in fact distinct, and if so in what way. The content of the paper is as follows. In Section 2 we recall Ball’s definition of a generalized semiflow, and prove several properties of the multi-valued map T(t) which arises naturally from his definition. svan516revised.tex; 26/02/2003; 8:23; p.2 Multi-valued semiflows and their asymptotic behaviour 3 We compare his definition with the axiomatic approach adopted in the 1960s and 1970s. Then we give Melnik & Valero’s definition of a multi-valued semiflow, and discuss the conditions required in order to construct a generalized semiflow from such a multi-valued semiflow. In Section 3 we give some canonical examples (an ODE and PDE without uniqueness, and an ordinary and a partial differential inclusion) and discuss when these give rise to generalized semiflows. We end by discussing how the existence of global attractors can be proved within the two theories, which was the starting point for our interest in the problem. 2. “Generalized” vs “multi-valued” semiflows Let (X, ρ) be a complete metric space, and denote by 2X,P(X), B(X), C(X), K(X), and Cv(X) the collections of all, nonempty, nonempty bounded, nonempty closed, nonempty compact, and nonempty closed convex subsets of Xrespectively. To measure the distance between sets we will use the Hausdorff metric dH, defined as dH(B, C) = max{dist(B, C),dist(C, B)}(1) where dist(B, C) is the Hausdorff semi-distance, dist(B, C) = sup b∈B inf c∈Cρ(b, c). 2.1. Generalized semiflows: definition We now give Ball’s definition of a generalized semiflow, including in addition the possibility of discrete time generalized semiflows. Note that the definition says nothing a priori about the continuity of the solutions ϕ∈GBin the case Γ = R. However, we will restrict ourselves later to two classes of continuous solutions. In finite-dimensional problems (or more generally when Xis locally compact) we will want to take ϕ continuous from [0,∞) into X, while in infinite-dimensional problems it is more useful to take ϕcontinuous from (0,∞) into X(for more details see Section 2.5 below). DEFINITION 1. Let Γbe Ror Z. A generalized semiflow GBon Xis a family of maps ϕ: Γ+→X(called solutions) satisfying the following hypotheses: (H1) Existence: for each z∈Xthere is at least one ϕ∈GBwith ϕ(0) = z. svan516revised.tex; 26/02/2003; 8:23; p.3 4Caraballo, Mar´ın-Rubio & Robinson (H2) Translates of solutions are solutions: if ϕ∈GBand τ∈Γ+then ϕτ∈GBwhere ϕτ(t) := ϕ(t+τ)for all t∈Γ+. (H3) Concatenation: if ϕ, ψ ∈GBand ψ(0) = ϕ(t)for some t∈Γ+then θ∈GB, where for each τ∈Γ+we define θ(τ) := ½ϕ(τ)for 0≤τ≤t, ψ(τ−t)for t < τ. (H4) Upper semicontinuity with respect to initial data: if {ϕn} ⊂ GB with ϕn(0) →z, then there exists a subsequence {ϕµ}of {ϕn}and ϕ∈GBwith ϕ(0) = zsuch that ϕµ(t)→ϕ(t)for each t∈Γ+. If for each z∈GBthere is exactly one ϕ∈GBwith ϕ(0) = z, then GBis called a semiflow. This definition (including the initially perhaps unintuitive (H4)) arises naturally when one considers solutions of differential equations whose solutions are not unique (see also Sell [28]; it is easy to see (H1– 4) when solutions are unique). Let us consider what is perhaps the simplest such problem (with Γ = R): dy dt=f(y), y(0) = y0,(2) where fis a bounded continuous function from Rninto Rn. It is well known that there exists at least one solution of this problem for each initial condition, but there may exist more than one for a general continuous function f. However, the boundedness assumption does ensure that all solutions exist for all t∈R. Let us denote by D(y0) the set of all classical solutions of (2) restricted to R+: D(y0) = {ϕ∈C1([0,∞); Rn) such that ϕsatisfies (2)}, and set G=[ y0∈Rn D(y0). Since the equation is autonomous, it is not difficult to check that Gforms a generalized semiflow: (H1), (H2) and (H3) are obvious, and (H4) is also relatively straightforward to check. Indeed, the set of all solutions is equicontinuous since for any ϕ∈G |ϕ(t)−ϕ(s)| ≤ kfk∞|t−s|. svan516revised.tex; 26/02/2003; 8:23; p.4 Multi-valued semiflows and their asymptotic behaviour 5 So, if we consider a fixed bounded interval I= [0, T] and a sequence of solutions ϕnwith ϕn(0) convergent, we obtain |ϕn(t)| ≤ |ϕn(0)|+Tkfk∞,for all t∈[0, T], and, consequently, uniformly bounded (on [0, T]) and equicontinuous. Thus we can apply the Arzel`a-Ascoli Theorem to extract a subsequence that converges to a continuous function ϕwhich is itself a solution of (2). There are results due to Barbashin that are closely related to (H4); we will comment on these below in Section 2.6. 2.2. Generalized semiflows: properties Let GBbe a generalized semiflow and let E⊂X. Define for t∈Γ+: T(t)E={ϕ(t) : ϕ∈GBwith ϕ(0) ∈E}(3) We now state and prove various properties of the map T(t) (cf. comments at start of Section 3 in Ball [5]). PROPOSITION 2. The map T(t):2X→2Xsatisfies the following properties: (a) {T(t)}t∈Γ+is a semigroup on 2X, i.e. T(0) = Id2Xand T(t+s) = T(t)T(s)for all t, s ∈Γ+, (b) T(t)is monotone with respect to the partial order of set inclusion, i.e., E⊂Fimplies T(t)E⊂T(t)Ffor all t∈Γ+, (c) T(t)xis compact for each x∈X, and (d) if {Kn}n≥1is a sequence of compact subsets of Xsuch that dist(Kn, K)→0as n→ ∞ then dist(T(t)Kn, T(t)K)→0for each t∈Γ+. Note that since dist(a, b) = ρ(a, b) for two points aand b, when GB is a semiflow the result given in (d) reduces to ρ(T(t)xn, T(t)x)→0 if ρ(xn, x)→0 as we would expect. Proof. (a) The first part of this is trivial: T(0)E=Efor any E⊂X because of (H1) and the definition of T(t)E. Now let us check that T(t+s) = T(t)T(s). Fixing E⊂X, if x∈T(t+s)Ethen there exist x0∈Eand a solution ϕ∈GBwith ϕ(0) = x0and ϕ(t+s) = x. Using svan516revised.tex; 26/02/2003; 8:23; p.5 6Caraballo, Mar´ın-Rubio & Robinson (H2) we know that ϕs∈GB, and since ϕs(0) = ϕ(s)∈T(s)Eand ϕs(t) = x, it follows that T(t+s)E⊂T(t)T(s)E. For the opposite inclusion we use (H3): if x∈T(t)T(s)E, then x=ψ(t) with ψ(0) ∈ T(s)Eand this means that there exists another solution ϕ∈GBsuch that ϕ(s) = ψ(0) with ϕ(0) ∈E. If we define for each τ∈Γ+ θ(τ) = ½ϕ(τ) for 0 ≤τ≤s, ψ(τ−s) for s < τ, we have θ∈GBwith θ(0) ∈Eand θ(t+s) = ψ(t) = x, and so T(t)T(s)E⊂T(t+s)E. (b) It is clear that T(t) is monotone by definition. (c) If yn∈T(t)xthen there exist solutions ϕn∈GBwith ϕn(0) = x and ϕn(t) = yn. Since ϕn(0) →x, by (H4) there is a subsequence ϕn0 and a ϕ∈GBsuch that (in particular) ϕn0(t)→ϕ(t), i.e. there exists aϕ∈GBwith ϕ(0) = xand ϕ(t) = y. So yn0→y∈T(t)x. It follows that T(t)xis compact. (d) We consider a fixed value t∈Γ+and prove the result by contradiction. Assuming that dist(T(t)Kn, T(t)K)6→ 0, there exists an ² > 0, a subsequence {Kn0}and elements an0∈T(t)Kn0such that dist(an0, T(t)K)> ² for all n0.(4) But an0=ϕn0(t) with ϕn0(0) ∈Kn0, and so, since dist(Kn, K)→0 with Kcompact, there exists a subsequence {ϕn00 }such that ϕn00 (0) → z∈K. It follows from (H4) that there exist a solution ψ∈GBand a subsequence {ϕn000 }with ϕn000 (t)→ψ(t) and ψ(0) = z∈K. Thus ψ(t)∈T(t)Kwhich contradicts (4), proving the result. Property (d) is just the εdefinition of upper semicontinuity, which is equivalent here to topological upper semicontinuity (u.s.c.) since T(t) has compact values (see Aubin & Cellina [2] (pp. 41 & 45) for details). 2.3. Multi-valued semiflows Melnik & Valero [25] define a multi-valued (semi)flow, or m-(semi)flow, as follows. DEFINITION 3. Let Γbe a nontrivial subgroup of (R,+). The setvalued map GMV : Γ ×X→P(X)is said to be a multi-valued flow (or m-flow) if the following conditions are satisfied: (1) GMV (0,·)is the identity map and (2) defining for any subset Bof X GMV (t, B) = [ x∈B GMV (t, x) svan516revised.tex; 26/02/2003; 8:23; p.6 Multi-valued semiflows and their asymptotic behaviour 7 we have GMV (t+s, x)⊂GMV (t, GMV (s, x)) for all t, s ∈Γand for each x∈X. GMV is called an m-semiflow if we replace Γby Γ+= Γ ∩R+in the definition. A strict inclusion in an m-semiflow means (cf. Proposition 2) that it does not come from a set of solutions with the translation and concatenation properties. Since the translation of a solution is usually a solution, it is most likely that this strict inclusion will arise from a failure of the concatenation property, for example joining two C1 functions in such a way that the resulting function is only C0. A more convincing (but non-autonomous) example arises in the ‘general control systems’ considered by Roxin [27]. Since one would expect the controls available to increase over time due to technological advances, we take {Uj}j≥1to be an increasing sequence of control sets, and denote the states attainable at time n+ 1 from a collection Eof states at time nare Fn(E) = [ x∈E,u∈Un f(x;u). If we define G(n, m)Efor n≥mby G(n, m)E:= (Fm◦Fm◦. . . ◦Fm) | {z } n−mtimes E then it is clear that in general G(n+m, 0)E⊂G(n+m, m)G(m, 0)E, since U0⊂U1⇒(F0◦F0)(E)⊂F1(F0(E)). Nevertheless, for all the applications considered in [25] the m-semiflow is constructed from selected solutions of various differential inclusions; since these satisfy both (H2) and (H3) equality holds in part (2) of Definition 3. However, the abstract definition of an m-semiflow contains no reference to solutions per se, so they have to be introduced as an (albeit natural) auxiliary concept. We reproduce here the definition of atrajectory from [25]. DEFINITION 4. The map x(·) : Γ+→Xis said to be a trajectory of the m-semiflow GMV corresponding to the initial condition x0if x(0) = x0and x(t+τ)∈GMV (t, x(τ)) for every t, τ ∈Γ+. svan516revised.tex; 26/02/2003; 8:23; p.7 8Caraballo, Mar´ın-Rubio & Robinson For example, it is easy to check that (H2) implies that the solutions making up Ball’s generalized semiflow are trajectories of the m-semiflow T(t) defined in (3): ϕ(t+τ) = ϕτ(t)∈T(t)ϕ(τ). If we require that GMV (t, B) in fact consists of a union of continuous trajectories of the m-semiflow GMV then Melnik & Valero called GMV a “time-continuous m-semiflow”. This notion of a trajectory, although used only by Melnik & Valero in their discussion of the connectedness of attractors, is extremely useful for us in comparing the two general frameworks and results from the previous literature. However, there are some distinctions between “trajectories” and “solutions”, as we will now explain. 2.4. Solutions and trajectories. As we previously mentioned, given a generalized semiflow GB, it easily follows from Proposition 2 that we can construct an m-semiflow GMV by setting GMV (t, x) = T(t)x t ∈R+, x ∈X. (5) Furthermore, we have a slightly stronger version of property (2) in the definition of an m-semiflow, since we automatically have the equality GMV (t+s, x) = GMV (t, GMV (s, x)) rather than an inclusion (what we will call a “strong m-semiflow”). We also know that GMV (t, x) is upper semicontinuous. However, it is not immediately clear that this m-semiflow cannot have trajectories (in the sense of Definition 4) that are not solutions of the generalized semiflow GB(cf. Szeg¨o & Treccani [30, Obs. 5.2]). We can rule out such spurious solutions using (H3) and (H4) when GBconsists of continuous functions (analogous result have been proved by Szeg¨o & Treccani [30, Th. 5.1] and Barbashin, see [27]). LEMMA 5. Let GBbe a generalized semiflow consisting of functions that are continuous from Jinto X, where J= (0,∞)or [0,∞). Now let GMV be the m-semiflow constructed from GBby (5). If x(t)is a continuous trajectory (on J) of this m-semiflow, i.e. x(t+s)∈GMV (t, x(s)) for all t, s ∈R+then x∈GB. Proof. Consider a sequence ϕn∈GBsuch that ϕn(j2−n) = x(j2−n) for all j= 0,1,2, . . . , n2n. svan516revised.tex; 26/02/2003; 8:23; p.8 Multi-valued semiflows and their asymptotic behaviour 9 The argument for the existence of such functions follows from that for ϕ1: by the definition of GMV (1,·) there exists a ϕ∈GBwith ϕ(0) = x(0) and ϕ(1) = x(1). Similarly there exists a ψ∈GBwith ψ(0) = x(1) and ψ(1) = x(2). So by concatenation there exists a ϕ1∈ GBwith ϕ1(0) = x(0), ϕ1(1) = x(1), ϕ1(2) = x(2). We can continue concatenating in this way to find a ϕn∈GBwith ϕn(j2−n) = x(j2−n). Now we consider the sequence {ϕn} ⊂ GB. Since ϕn(0) = x(0), by (H4) there is a ϕ∈GBand subsequence ϕµsuch that ϕµ(t)→ϕ(t) for each t > 0. Since for any tof the form t=j2−nfor some jand n the value of ϕµ(t) is always x(t) for µlarge enough, it follows from the continuity of ϕand xthat in fact ϕ=x. So x∈GB. However, suppose rather that we construct an m-semiflow directly using a certain class of solutions of some model. In this case there is a priori no reason why there should not be limits of solutions of this m-semiflow that are not solutions themselves. This remark will be important in the applications of Section 3, where in each case we will have to check that the limit of solutions is still a solution: i.e. that convergence of a sequence of solutions ϕnto ϕimplies that ϕis also a solution. We would like to emphasise this point here, i.e. that in general one cannot expect that every set of solutions of a differential problem forms a generalized semiflow: we end this section with a simple example of an ordinary differential inclusion in which the set of all solutions does not satisfy (H4). We take X= [0,∞), and define Fby F(x) = ½0 if x∈[0,1], 1 if x > 1. Consider the ordinary differential inclusion (in fact an ordinary differential equation) dx dt(t)∈F(x(t)),(6) and let Gbe the set of strong solutions of (12); writing fa(t) = aand gb(t) = b+twe have G=[ a∈[0,1] fa(·)∪[ b∈(1,∞) gb(·). It is obvious that Gis an equicontinuous family and satisfies (H1), (H2) and (H3), but the more problematic (H4) does not hold: if we take a sequence bn↓1 the solutions gbn(t) = bn+tconverge to g(t) = 1 + t, but this is not a solution of (12). [This anomaly can be corrected by svan516revised.tex; 26/02/2003; 8:23; p.9 16 Caraballo, Mar´ın-Rubio & Robinson Proof. The continuity in (K4) consists of two parts, dist(T(t)x, T(s)x)→0 as s→t, t ∈[0,∞),(10) and dist(T(s)x, T(t)x)→0 as s→t, t ∈[0,∞).(11) Both can be proved by contradiction. If (10) does not hold then there exist a constant ε > 0 and a sequence sn→twith dist(T(t)x, T(sn)x)≥ε. (12) Since T(t)xis compact we can find a zn∈T(t)xsuch that dist(zn, T(sn)x) = dist(T(t)x, T(sn)x), and w.l.o.g. we suppose that zn→z∈T(t)x. Then dist(z, T(sn)x)≥ε. To obtain a contradiction we now find elements yn∈T(sn)xwith yn→z. Observe that since z∈T(t)xthere is a solution ϕ∈GB such that z=ϕ(t). Since solutions are continuous functions of time, yn=ϕ(sn)∈T(sn)xand yn→z. So (10) holds. If (11) does not hold then there exist an ε > 0 and a sequence sn→t such that dist(T(sn)x, T(t)x)≥ε. Since each T(sn)xis compact we can find zn∈T(sn)xsuch that dist(zn, T(t)x)≥ε. Then there are elements ϕn∈GBsuch that ϕn(0) = xand zn= ϕn(sn). W.l.o.g. we can assume using (H4) that there is some ϕ∈GB with ϕ(0) = xsuch that ϕn(t)→ϕ(t) for each t≥0. If Xis locally compact then Theorem 2.3 in [5] (Theorem 9 above) shows that this convergence is in fact uniform on compact subintervals of [0,∞). Thus for all t∈[0,∞) we can deduce that zn−ϕ(t) = [ϕn(sn)−ϕ(sn)] + [ϕ(sn)−ϕ(t)] →0, and so zn→ϕ(t)∈T(t)xand we obtain (11). The upper semicontinuity follows similarly via a contradiction argument. Since (10) and (11) combine to show that dH(T(t)x, T(s)x)→0 as s→t, svan516revised.tex; 26/02/2003; 8:23; p.16 Multi-valued semiflows and their asymptotic behaviour 17 in locally compact spaces the continuity of solutions on [0,∞) plus (H1–4) imply (K4) and (K5). Conversely Barbashin showed (see [27]) that under axioms (K1–5) all trajectories are continuous and forms a compact set provided initial time convergence holds (cf. our Lemma 5), so one can switch between Ball’s theory and that of Kloeden in a consistent way. However, in infinite-dimensional Banach spaces the weaker assumptions of the newer versions of the theory are necessary. We saw in Section 2.5 that a result like Barbashin’s (a generalized semiflow from an m-flow) does not hold in general, assuming only the properties that Melnik & Valero’s require for the ‘attainability sets’ GMV (t, x). In the next section we see what can be done with the set of solutions of particular example problems. 3. Applications In this section we study some examples. In particular we want to check whether or not we can construct generalized semiflows that consist entirely of solutions rather than form the trajectories of the attainability map. In the light of Lemma 5, it is enough to check that the solutions are continuous and form a generalized semiflow by themselves. In this way we avoid the existence of spurious ‘solutions’ in the generalized semiflow. We will see that this problem is significantly more involved for partial differential inclusions that for equations without uniqueness, since for inclusions each solution is associated with a different ‘right-hand side’ and selection theorems are necessary. 3.1. An ODE without uniqueness We have already considered in Section 2.1 the simple example dx dt=f(x), x(0) = x0, with f:Rn→Rna bounded continuous function, and shown that it gives rise to a generalized semiflow. Of course, we could also consider the equation from the point of view of m-semiflows, defining G(t, x) precisely as in (5). In this case it is interesting to note that Kneser’s Theorem on structure of G(t, x) (see Theorem 4.1 in Hartman [14] for example) guarantees that it is closed. Since it is bounded it must also be compact (of course, this also follows from part (c) of Proposition 2). svan516revised.tex; 26/02/2003; 8:23; p.17 18 Caraballo, Mar´ın-Rubio & Robinson 3.2. A PDE without uniqueness The main example that appears in Ball’s paper is the 3D Navier-Stokes equations: ut+ (u· ∇)u=ν∆u− ∇p+f, div u= 0,(13) with boundary condition u|∂Ω= 0. Consider the following spaces: V={u∈C∞ 0(Ω)3; div u= 0}, H= closure of Vin L2(Ω)3, V={u∈H1 0(Ω)3; div u= 0}. Let us denote by Hwthe space Hendowed with its weak topology. It is well known that given u0∈H, there exists at least one weak solution uto the problem (13) such that u∈C([0, T]; Hw)∩L2(0, T;V),du dt∈L1(0, T;V0)∀T > 0. However it is not known whether this solution is unique, so the generalized semiflow framework is brought in to play. The collection GNS of all such weak solutions clearly satisfies (H1) and (H3). What Ball shows in this paper (see Proposition 7.4 in [5]) is that GNS is a generalized semiflow if and only if each weak solution is a continuous function from (0,∞) into H: this is currently an unproved hypothesis. Additional conditions needed to construct the global attractor are shown to be consequences of this same assumption (see Theorem 19 in Section 4 of this paper for details). 3.3. Two examples involving differential inclusions 3.3.1. An ordinary differential inclusion As an illustrative example we recall briefly the case of a simple ordinary differential inclusion (see e.g. Roxin [26] or Aubin & Frankowska [3]). Let us consider an F:Rn→Cv(Rn) such that F(0) is bounded and Fis globally Lipschitz, dH(F(u), F(v)) ≤L|u−v|. Consequently, Fhas bounded convex values (this convexity is important to guarantee that the attainability map has closed values, svan516revised.tex; 26/02/2003; 8:23; p.18 Multi-valued semiflows and their asymptotic behaviour 19 cf. comments towards the end of section 2.4). We are going to show that it is possible to construct a generalized semiflow for the ordinary differential inclusion du dt∈F(u)u(0) = u0.(14) Let us denote by Gthe set of strong or Carath´eodory solutions to (14), that is all maps u: [0, T]→Rnsuch that (i) u(0) = u0, (ii) u(·) is continuous on [0, T], (iii) u(·) is absolutely continuous on any compact subinterval of (0, T), and satisfies (14) a.e. on (0, T). In particular we require a measurable selection h(x) with h(x)∈F(x), so that du dt=h(u(t)). The existence of a continuous (and not merely measurable) selection is guaranteed by the Chebyshev Selection Theorem (see Aubin & Cellina [2], p. 74), and then standard methods can be used to obtain the existence of a solution defined locally in time. If we fix an interval [0, T] then we can show that each solution and its corresponding selection are bounded, thereby obtaining solutions that are global in time. (The proof is standard and will be omitted.) PROPOSITION 12. Under the above assumptions, if uis a strong solution to (14) and h(u)the corresponding selection then the following bounds hold: |u(t)|2≤e(2L+1)t|u(0)|2+C 2L+ 1(e(2L+1)t−1) and |h(u(t))| ≤ C+Lµe(2L+1)t|u(0)|2+C 2L+ 1(e(2L+1)t−1)¶1/2 , where C= diam F(0). Since (H1–3) are straightforward for this example, we concentrate on (H4). Suppose we have a sequence of strong solutions unwith converging initial data. The bound on |h(u(t))|from the above proposition implies that the set of strong solutions of (14) are equicontinuous, which along with the bound on |u(t)|enables the use of the Arzel`a-Ascoli svan516revised.tex; 26/02/2003; 8:23; p.19 20 Caraballo, Mar´ın-Rubio & Robinson Theorem to find a convergent subsequence. But, this is not enough on its own, since we also need to check that the limit is still a solution of the problem. This follows from an application of a selection convergence theorem from Aubin & Cellina [2], p. 60: any convergent subsequence un0→uwith selectors fn0(t)∈F(un0(t)) has a subsequence fn00 * f with f(t)∈F(u(t)). Of course, the same conclusion arises for the non-autonomous case if we include an artificial time s=tas an additional direction in the phase space. 3.4. A partial differential inclusion We now turn to the more involved case of partial differential inclusions, which we describe within an abstract framework. Let Hbe a Hilbert space with norm | · | and inner product (·,·). We consider the following evolution inclusion problem: dy dt∈ −Ay +F(y), t ∈[0, T],(15) y(0) = y0∈H, (16) where A:D(A)⊂H→His a linear (unbounded) operator with Im(Id + A) = Hand compact inverse such that (−Ax, x)≤0 for all x∈D(A), (so D(A) = H) and with e−At an analytic semigroup. (This is a particular case of the theory developed by Melnik & Valero [25] which also covers multi-valued subdifferential operators A:D(A)⊂X→2X with Xa Banach space.) We assume further that Fis convex-valued, F:H→Cv(H), F(0) is bounded, and Fsatisfies the global Lipschitz condition dH(F(u), F(v)) ≤C1|u−v| (this is a strong assumption; in particular it implies that Fhas bounded values). As with ordinary differential inclusions, a strong solution y(·) of (15)- (16) is a continuous function on [0, T] with y(0) = y0,y(·) absolutely continuous on any compact subinterval of (0, T), such that (15) holds a.e. on (0, T). However, it is also useful to define a weaker notion of solution which we term (following the single-valued case) a mild solution. DEFINITION 13. The map y: [0, T]→His called a mild solution of (15)-(16) if it is continuous, y(0) = y0, and there exists a selection svan516revised.tex; 26/02/2003; 8:23; p.20 Multi-valued semiflows and their asymptotic behaviour 21 f∈L1([0, T]; H)of F(that is f(t)∈F(y(t)) a.e. on [0, T]), such that ysatisfies y(t) = e−Aty0+Zt 0 e−A(t−s)f(s) ds(17) for a.e. t∈[0, T](i.e. y(t)is a mild solution of dy/dt=−Ay +f(t)). Before showing how we can define a generalized semiflow using this definition, we will discuss its relationship to that given in Melnik & Valero’s paper. There they make use of the much more general theory of partial differential inclusions in which the linear operator Acan also be multi-valued (full details of this can be found in Chapter III of Barbu [6]). In order to deal with the fact that Ais multi-valued one needs to introduce a generalisation of the notion of a mild solution, which they term an “integral solution”. In our setting, an integral solution of (17) is a continuous function y: [0, T]→Hwith y(0) = y0such that for all u∈D(A) |y(t)−u|2≤ |y(s)−u|2+ 2 Zt s (f(τ) + Au, y(τ)−u) dτ, t ≥s. (18) An integral solution of (15)-(16) is a function y(t) for which there exists a selection f∈L1([0, T]; H) of Fsuch that yis an integral solution of (17). The two notions of solutions can be shown to be equivalent for our example: first, note that any strong solution is an integral solution (this is straightforward). Now we show that any integral solution is also a mild solution (the argument is due to Valero, personal communication): take a sequence of strong solutions un=I(un 0)fnwith fn→fand un 0→ u0in L1(0, T;H) and Hrespectively (the existence of such sequence is guaranteed by Corollary 2.2 in Barbu [6], Chapter III). Then un→u in C([0, T]; H). The contraction property of e−At leads to |e−Atun 0−e−Atu0| ≤ |un 0−u0| → 0, Zt 0 |e−A(t−s)fn(s)−e−A(t−s)f(s)|ds≤Zt 0 |fn(s)−f(s)|ds→0. Now, since the strong solutions unare mild solutions, we can take limits in order to show that integral solutions of the problem (15) are mild solutions as well: un(t)=e−Atun 0+Rt 0e−A(t−s)fn(s) ds ↓ ↓ ↓ u(t) = e−Atu0+Rt 0e−A(t−s)f(s) ds. Since there is always a unique integral solution and a unique mild solution of (17) it follows that the definitions are equivalent. svan516revised.tex; 26/02/2003; 8:23; p.21 22 Caraballo, Mar´ın-Rubio & Robinson Under our assumptions, and using Definition 13, there exists at least one mild solution of (15)-(16), cf. [25], and we take Gto be the set of all such mild solutions. For this collection (A1) is easily seen to be true, and we now show that both parts of assumption (A20) hold. Although we will not apply Theorem 7, rather checking (H4) directly, we will still need (A1) and (A20) in order to do this. We first check the equicontinuity property (recall that we only require equicontinuity on compact subintervals of (0,∞)). PROPOSITION 14. For each 0< θ < 1all solutions of (15) satisfy |u(t)−u(s)| ≤ K[²,T](θ, |u0|)£|t−s|θ+|t−s|¤for all t, s ∈[², T]. To prove this proposition, we need the following lemma from Henry [15]: LEMMA 15. We have the following two estimates: there exists λ > 0 such that for any 0≤α < 1 kAαe−Atkop ≤Cαt−αe−λt (19) and, for any 0< α < 1 |(e−At −I)x| ≤ C0 αtα|Aαx|.(20) The first observation is that the strong condition on Fgives a uniform bound on all solutions. LEMMA 16. If u(t)is a solution of (15) and f(t)∈F(u(t)) then |f(t)| ≤ M(T, |u0|)for all t∈[0, T]. Proof. Note that the Lipschitz assumption on Fimplies that |F(u)| ≤ C1|u|+ diam[F(0)] ≡C0+C1|u|. In particular, since every solution of (15) satisfies du dt+Au =f(t) with f(t)∈F(u(t)), we have 1 2 d dt|u|2+|A1/2u|2≤C0|u|+C1|u|2 and so d dt|u|2≤C2 0+ (1 + 2C1)|u|2, from which the result follows. svan516revised.tex; 26/02/2003; 8:23; p.22 Multi-valued semiflows and their asymptotic behaviour 23 We can now prove the equicontinuity result. Proof. Essentially we follow the work in Henry [15]. We consider u(t+h)−u(t) with t≥²; using the integral expression for the solution, we have u(t+h)−u(t) = (e−Ah −I)e−Atx0+Zt 0 (e−Ah −I)e−A(t−s)f(s) ds +Zt+h t e−A(t+h−s)f(s) ds, and so |u(t+h)−u(t)| ≤ C0 θhθ|Aθe−Atx0|+Zt 0 C0 θhθ|Aθe−A(t−s)f(s)|ds+Zt+h t |f(s)|ds ≤C0 θhθCθe−λtt−θ|x0|+C0 θhθCθZt 0 (t−s)−θ|f(s)|ds+hM(T, |u0|) ≤K(θ, |x0|, ², T)hθ+hM(T, |u0|) which provides the equicontinuity property needed for (A2). As for the second part of (A2) (compactness) we show that the solutions are bounded in D(Aα), which is compactly embedded in H. The proof is almost straightforward, since |Aαu(t)|≤kAαe−Atkop|u0|+Zt 0 kAαe−A(t−s)kopM(|u0|, s) ds ≤Cαt−αe−λt|u0|+CαM(|u0|, t)Zt 0 (t−s)−αe−λ(t−s)ds ≤K(t, |u0|). We now need to check that sequences of solutions have a subsequence that converges to a limit that is itself a solution. First use the Arzel`aAscoli to find a (diagonal) subsequence such that unconverges uniformly to uon any compact subinterval (0,∞) while (relabeling the same) fn* f weakly in L1((0, T); H). Then uis a mild solution of du dt=−Au +f(t). However, while (A20) only requires convergence on compact subintervals of the open interval (0,∞) (which we have just shown), we require that u(t) is a mild solution of (15), and by Definition 13 this requires in addition that u(t) should be continuous on the interval [0,∞) and svan516revised.tex; 26/02/2003; 8:23; p.23 24 Caraballo, Mar´ın-Rubio & Robinson satisfies (17). But these properties follow since, passing to the limit in the expressions for un,u(t) satisfies u(t) = e−Atz+Zt 0 e−A(t−s)f(s) ds, from whence u(t)→zas t→0 and the function is indeed continuous on [0,∞). It remains to check that f(t)∈F(u(t)), but this follows once more using the same selection convergence theorem from Aubin & Cellina [2], p. 60, that we used in the ordinary differential inclusion case. Then uis a mild solution of du dt=−Au +f(t). Therefore we have shown (H4), and the set of all mild solutions of (15) forms a generalized semiflow. 4. Relations between the two theories of attractors Our interest in these two theories arose from their application to the investigation of the long-time behaviour of solutions. Indeed, both the papers in which the two abstract frameworks discussed here were developed generalise the notion of global attractor from single-valued dynamical systems to multi-valued evolutions (whether they come from systems without uniqueness or from differential inclusions). We now discuss the analogies and differences between the attractor results provided by both theories. An initial remark is that both frameworks approach the theory of attractors in a similar way, Melnik & Valero giving all their definitions using the m-semiflow GMV , and Ball using the equivalent T(t) derived from the collection GBof solutions (see (3)). In what follows we use G(t, ·) for either GMV (t, ·) or T(t)·when no confusion can arise. The important concepts are as follows: DEFINITION 17. (a) It is said that Aattracts Bif limt→∞ dist(G(t, B), A) = 0. (b) The semiflow Gis called eventually bounded if for any bounded set B, there exists a sufficiently large constant τ=τ(B)such that γ+ τ(B)is bounded, where, as usual in dynamical systems, γ+ τ(B) denotes the set of all points reached at any time greater than τby solutions beginning in B:∪t≥τG(t, B). svan516revised.tex; 26/02/2003; 8:23; p.24 Multi-valued semiflows and their asymptotic behaviour 25 (c) The ω-limit set of Mis defined as the set of limits of all converging sequences {ξn}where ξn∈G(tn, M). As in the single-valued case this is the same as the intersection of all γ+ t(M)with t∈Γ+. (d) The semiflow is called point dissipative if there is a bounded set B0such that all solutions are attracted by B0. (The solutions will be “absorbed” by any neighbourhood of B0.) (e) The semiflow is asymptotically upper semicompact if for any bounded set Bsuch that for some T(B)∈Γ+, γ+ T(B)(B)∈B(X), any sequence ξn∈G(tn, B)with tn→ ∞ is precompact in X. (f) The semiflow is asymptotically compact if for any sequence of solutions ϕnwith {ϕn(0)}bounded, and any sequence tn→ ∞, the set {ϕn(tn)}is precompact. Obviously asymptotically compact is equivalent to asymptotically upper semicompact plus eventually bounded. The definition of “an attractor” in both papers is similar, and is essentially a compact, invariant set that attracts all bounded sets. However, in the light of certain applications (see Remark 4 in [25]) Melnik & Valero initially require the attractor only to be negatively semi-invariant, i.e. A ⊂ G(t, A) for all t∈Γ+: THEOREM 18. (cf. [25], Theorem 3 and Remark 8) Let GMV be a pointwise dissipative and asymptotically upper semicompact m-semiflow, and suppose that GMV (t, ·) : X→P(X)has closed graph. If for any bounded Bthere exists a T(B)such that γ+ T(B)(B)is bounded, then GMV has a compact global attractor Awhich is minimal among all the closed sets attracting each B∈B(X). We note here that the condition that GMV has closed graph is automatically satisfied if GMV (t, ·) : X→C(X) is upper semicontinuous for any t∈Γ+(see Aubin & Cellina [2], for example). The negatively semi-invariant attractor of Theorem 18 becomes fully invariant whenever we have GMV (t1+t2, x) = GMV (t1, GMV (t2, x)) for all x∈X (what we called a strong m-semiflow). The result from [5] is, of course, similar. THEOREM 19. (cf. [5], Theorem 3.3) A generalized semiflow GB has a global attractor if and only if GBis point dissipative and asymptotically compact. The global attractor Ais unique and is given by A=[ B∈B(X) ω(B). svan516revised.tex; 26/02/2003; 8:23; p.25