Journal of Differential Equations 181, 243–279 (2002) doi:10.1006/jdeq.2001.4087, available online at http://www.idealibrary.com on LargeTime Behavior via the Method of ‘-Trajectories1 Josef M!alek and Dalibor Pra$za! k Mathematical Institute, Charles University, Sokolovska! 83, 18675 Prague 8, Czech Republic E-mail: malek@karlin.mff.cuni.cz, prazak@karlin.mff.cuni.cz Received August 19, 1999; revised June 11, 2001 The method of ‘-trajectories is presented in a general setting as an alternative approach to the study of the large-time behavior of nonlinear evolutionary systems It can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear Here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics In both cases we prove the existence of a finite-dimensional global attractor and the existence of an exponential attractor # 2002 Elsevier Science (USA) Key Words: large time behavior; global attractor; exponential attractor; finite fractal dimension; ‘-trajectory; fluids with shear-dependent viscosity; power-law fluids INTRODUCTION The aim of this paper is to present a promising and powerful tool for dealing with the large-time behavior of nonlinear dissipative systems This new approach, called the method of ‘-trajectories, is based on an observation that the limit behavior of solutions to a dynamical system in an original phase space can be equivalently captured by the limit behavior of ‘-trajectories; these are (continuous) parts of solution trajectories that are parametrized by time from an interval of the length ‘; ‘ > 0: In this paper we focus on systems of partial differential equations of a parabolic type and apply the method of ‘-trajectories to (1) an abstract nonlinear dissipative equation of the first order, (2) the system of equations describing the motion of a class of non-Newtonian incompressible fluids (fluids with shear-dependent viscosity) This research was supported by Grants MSM 113200007 and GACR 201/00/0768 243 0022-0396/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved 244 MA´LEK AND PRAZˇA´K In both cases we prove the existence of a global attractor with finite fractal dimension and the existence of an exponential attractor Most of the results are new The alternative way of describing the dynamics, which we are going to explain, allows us on one hand to weaken the requirements on the regularity of the solution and on the other hand enables us to formulate the results for a broader class of nonlinearities and, even more, to treat the problems where the leading elliptic operator is nonlinear The reader can compare our results with the theories presented in [1, 4, 7, 11, 24], for example Before explaining the main features of the method, we want to emphasize that the method of ‘-trajectories is applicable not only to dissipative equations of parabolic type: The same approach can be successfully applied to hyperbolic type problems As an example, the question of the finite-dimensionality of the global attractor for the wave equation with nonlinear damping is addressed in the forthcoming paper [20], where the author proves that the fractal dimension of a global attractor for the damped wave equation is finite provided that the damping function is strictly increasing of the polynomial growth – a result which to our knowledge does not have an analogy using other methods, such as that of the Lyapunov exponents, for example And last, but not least, it turns out that the dynamical system of ‘-trajectories is also the natural description for studying the dynamics of nonlinear dissipative systems with bounded delay; see [19] Now, let us explain the essence of the method of ‘-trajectories Consider a nonlinear system of differential equations written as an abstract evolutionary problem, u0 ðtÞ¼ F ðuðtÞÞ in X ðt > 0Þ; uð0Þ¼ u0 ; ð0:1Þ where X is an (infinite-dimensional) Banach space, F : X /X is a nonlinear operator, and u0 X : To give a brief characterization of the method of ‘-trajectories, let us assume for a while that solution operators fSt gt50 to (0.1) defined by St u0 ¼ uðtÞ form a semigroup and in addition that fSt gt50 possesses a global attractor A & X : We describe such an arrangement of the dynamics in an equivalent way by introducing two mappings The first mapping b adds to any u0 X the ‘-trajectory that begins at u0 (see Fig 1); i.e., we consider b as a mapping from X into a subset of X‘ ¼ L2 ð0; ‘; X Þ defined by fbðu0 ÞgðtÞ ¼ St u0 ; t ½0; ‘Š: THE ‘-TRAJECTORIES METHOD 245 FIGURE FIGURE The second mapping e assigns to any ‘-trajectory w its end point2 (see Fig 2); i.e., eðwÞ :¼ wð‘Þ: Now, we use b to introduce a new semigroup fLt gt50 acting on the set of ‘-trajectories defined as (see Fig 3) Lt ðbðu0 ÞÞ :¼ bðSt u0 Þ; u0 X : Then, on setting A‘ :¼ bðAÞ ¼ fbðu0 Þ; u0 Ag; Trajectories are supposed to be continuous at least in the weak topology of X ; the value wð‘Þ has then a clear meaning 246 MA´LEK AND PRAZˇA´K FIGURE FIGURE it usually turns out that A‘ is a global attractor related to fLt gt50 : The complete structure is drawn in Fig One might ask: What is the advantage of this alternative viewpoint on A? Clearly, instead of estimating the fractal dimension of A directly, we are given the possibility of estimating the fractal dimension of A‘ in the topology of X‘ ; which is revealed to be an easier task.3 After proving the finiteness of the fractal dimension of A‘ one observes (see Lemma 1.2 below) that if e is Lipschitz (or at least a-Ho lder) continuous then the fractal dimension of A cannot increase (or increases at most 1=a times) Note that the roles of b and e are different While b transfers the dynamics from X into X‘ ; the mapping e is responsible for delivering the properties of A‘ to A: The roles of e and its regularity are more important than the role For example, as shown in this paper, the criterion stated in Lemma 1.3 below is applicable to Lt with ease, while it can be applied to St only in special (quite regular) cases 247 THE ‘-TRAJECTORIES METHOD of b: It can happen that A or fSt gt50 is not defined in X : Such a case can occur when the solution is not unique or when one does not have enough regularity to construct A: Still, it might be possible to introduce fLt gt50 : In this case the attractor A‘ is constructed first, and after evaluating its fractal dimension we set A to be eðA‘ Þ: Not only does the defined set A have properties of the global attractor in X ; but also its fractal dimension is finite (provided e is Lipschitz or Ho lder continuous).4 The paper is organized in the following way: Section recalls the definitions of basic notions and provides general helpful assertions Section presents the method of ‘-trajectories in a general framework In Section we provide a class of evolutionary problems with a nonlinear (monotone) elliptic operator, for which the assumptions of the general scheme are verified directly Models from nonlinear fluid mechanics that include the Smagorinski model of isotropic turbulence and other shear-thickening fluid models in a three-dimensional setting and shear-thinning fluids in two dimensions are studied in Section 4; several new results are obtained both for the space periodic and the Dirichlet problems Conclusions and perspectives are presented in the last section, which also includes bibliographical notes DEFINITIONS AND BASIC LEMMAS In this section we recall several notions from the theory of dynamical systems Let X be (a subset of) a normed space One parameter family of (nonlinear) mappings St : X/X ðt50Þ is called the semigroup provided that Stþs ¼ St Ss for all t; s50 and S0 ¼ I: A typical example is a semigroup formed by the solution operators for a certain evolutionary problem, defined on some suitable space of initial conditions for which there exists a unique global solution The couple ðSt ; XÞ is usually referred to as a dynamical system A set A & X is called a global attractor to ðSt ; XÞ if (i) A is compact in X; (ii) St A ¼ A for all t50; and (iii) for any B & X that is bounded, distX ðSt B; AÞ ! as t ! 1; where distX ðB; AÞ ¼ supb2B inf a2A jjb À ajjX : Note that a dynamical system can have at most one global attractor A set C & X is called positively invariant w.r.t St if for all t50; St C & C; and it is called uniformly absorbing w.r.t St if for any B & X that is bounded there exists t0 ¼ t0 ðBÞ such that St B & C for all t5t0 : An example of such a situation is given in Section 4.1 248 MA´LEK AND PRAZˇA´K Lemma 1.1 Let ðSt ; XÞ be a dynamical system Assume that there exists a compact set B1 & X which is uniformly absorbing and positively invariant w.r.t St : Let moreover St be continuous on B1 : Then ðSt ; XÞ has a global attractor Proof We simply set A to be the o-limit set of B1 ; cf [24] ] Finally, the fractal dimension5 of a compact set C & X; denoted by is defined as dfX ðCÞ; dfX ðCÞ :¼ lim sup e!0 log NeX ðCÞ ; logð1=eÞ where NeX ðCÞ is the minimal number of e-balls (with respect to the metric of X) needed to cover C: Lemma 1.2 Let X; Y be metric spaces and F: X/Y be a-Ho lder continuous on C:6 Then dfY ðFðCÞÞ4 dfX ðCÞ: a In particular, the fractal dimension does not increase under a Lipschitz continuous mapping Proof Since jjFðuÞ À FðvÞjjY 4cjju À vjjaX ; it holds that FðBX ðu; ðe=cÞ1=a ÞÞ & BY ðFðuÞ; eÞ: Thus NeY ðFðCÞÞ4NZX ðCÞ; where Z ¼ ðe=cÞ1=a : Therefore X log NZX ðCÞ log NeY ðFðCÞÞ log Nðe=cÞ1=a ðCÞ ¼ ; logð1=eÞ logð1=eÞ a logð1=ZÞ À log c which leads to the conclusion letting e ! ð) Z ! 0Þ: ] The importance of the notion of a finite fractal dimension is illustrated by a result of Foias and Olson [5]: if C is a compact metric space such that dfC ðCÞ5m2 ; m N; then there exists an injective Lipschitz continuous mapping P : C/Rm such that its inverse is Ho lder continuous In other words, if dfC ðCÞ5m2 ; then C is placed in the graph of a Ho lder continuous mapping that maps the compact subset of Rm onto C: Moreover, if C is a subset of a Hilbert space then P can in addition be an orthogonal projector By this sentence we simply mean that a-Ho lder continuity is considered between metrics of X and Y respectively, though the mapping F can only be defined on a proper subset C of X: THE ‘-TRAJECTORIES METHOD 249 Lemma 1.3 Let X; Y be normed spaces such that Y + + X and C & X be bounded Assume that there exists a mapping L such that LC & C and L: X/Y is Lipschitz continuous on C: Then dfX ðCÞ is finite Proof Let k be a Lipschitz constant of L and N be the number of balls in X of radii 1=4k necessary to cover the unit ball in Y: Let us choose R > and u C such that C & BX ðu; RÞ: Then we have C & LC & BY ðLu; kRÞ & [ BX ðu# i ; R=4Þ; 14i4N where u# i X: We can assume that C \ BX ðu# i ; R=4Þ=|; which leads to the covering C& [ BX ðui ; R=2Þ; 14i4N where ui C: X k Repeating the scheme inductively, we have NR=2 k ðCÞ4N : Now, for any positive e4R there exists an integer k50 such that R=2k 5e > R=2kþ1 : Then log NeX ðCÞ ðk þ 1Þ log N ; logð1=eÞ k log À log R and consequently dfX ðCÞ4 log N : log ] Furthermore, following [4], we introduce the concept of the exponential attractor Under the assumptions of Lemma 1.1 we say that the set E & B1 is an exponential attractor w.r.t the dynamical system ðSt ; B1 Þ if (i) E is compact, (ii) E is positively invariant w.r.t St ; (iii) dfX ðEÞ is finite, (iv) there exist c1 ; c2 > such that distX ðSt B1 ; EÞ4c1 eÀc2 t for all t50: Note that necessarily A & E; so the basic idea behind the exponential attractor is to enlarge the global attractor so that the rate of convergence becomes exponential, yet keep the ‘‘good’’ properties (i)–(iii) The following lemma resumes a criterion on the existence of the exponential attractor obtained in [4] Lemma 1.4 Let X be a Hilbert space Let B1 & X satisfy the assumptions of Lemma 1.1 Assume that there exists t > such that 250 MA´LEK AND PRAZˇA´K ðP1Þ St : X/X is Lipschitz continuous on B1 ; ðP2Þ there exist W ð0; 1=4Þ and a finite-dimensional orthogonal projector P : X/X such that for all x1 ; x2 B1 there holds either pffiffiffi jjSt x1 À St x2 jjX 2jjP ðSt x1 À St x2 ÞjjX or jjSt x1 À St x2 jjX 4Wjjx1 À x2 jjX ; and ðP3Þ the mapping G: X  ½0; tŠ/X defined by Gðx; tÞ :¼ St x is on B1  ½0; tŠ Ho lder continuous Then the dynamical system ðSt ; B1 Þ possesses an exponential attractor Proof ] See [4, Chaps and 3] We will also use this elementary lemma Lemma 1.5 Let X; Y be normed spaces such that Y ++ X; let moreover X be a Hilbert space Then for a given e > there exists a finite-dimensional subspace Xn & X such that, denoting by P the ortho-projector to Xn ; jjðI À P ÞujjX 4ejjujjY for any u Y: Proof Without loss of generality we assume that u S ¼ fv Y; jjvjjY ¼ 1g: But the set S & X is compact, and denoting by u1 ; ; un its e-net, we see that the space Xn spanned by u1 ; ; un has the desired property ] Finally, for the reader’s convenience we formulate the celebrated so-called Aubin–Lions lemma as it plays an important role in our paper Lemma 1.6 Let p1 ð1; 1Š; p2 ½1; 1Þ: Let X be a Banach space and Y ; Z be separable and reflexive Banach spaces such that Y ++ X + Z: Then for any t ð0; 1Þ; fu Lp1 ð0; t; Y Þ; u0 Lp2 ð0; t; ZÞg ++ Lp1 ð0; t; X Þ: Proof See [23], for example ] GENERAL SCHEME The method of ‘-trajectories can be used for various purposes in studying the large-time behavior of dynamical systems Depending on the purpose, THE ‘-TRAJECTORIES METHOD 251 one needs certain assumptions to be satisfied In order to make this presentation transparent we divide the general scheme, and correspondingly also the assumptions, into several subsections that emphasize their specific role The titles of these subsections are: (1) Dynamical system on the set of ‘-trajectories, (2) A‘ – Attractor in the set of ‘-trajectories, (3) Finite fractal dimension of A‘ ; (4) A – Attractor in the original space, (5) Finite fractal dimension of A; (6) E‘ – Exponential attractor in the set of ‘-trajectories, (7) E – Exponential attractor in the original space (1) Dynamical System on the Set of ‘-Trajectories The first assumptions concern the existence and uniqueness of the solution to (0.1) Let ðX ; jj Á jjX Þ; ðY ; jj Á jjY Þ; and ðZ; jj Á jjZ Þ be three Banach spaces, X being reflexive and separable, such that Y ++ X and X + Z: ð2:1Þ For p1 ½2; 1Þ; p2 ½1; 1Þ; and t > fixed we denote Xt :¼ L2 ð0; t; X Þ; Yt :¼ fu Lp1 ð0; t; Y Þ; u0 Lp2 ð0; t; ZÞg: Then Lemma 1.6 implies Yt ++ Xt : ð2:2Þ The space Cð½0; tŠ; Xw Þ is defined as Cð½0; tŠ; Xw Þ :¼ fu L1 ð0; t; X Þ; huðÁÞ; jiX ; X n Cð½0; tŠÞ for all j X n g: From now on, by solution to (0.1) on the interval ½0; T Š with initial condition u0 – or by solution, for short – we mean a function u Cð½0; T Š; Xw Þ \ YT satisfying (0.1) in some weak sense we have chosen; we assume that this weak formulation makes sense for a considered class of functions We require that: (A1) For any u0 X and arbitrary T > there exists (not necessarily unique) u Cð½0; T Š; Xw Þ \ YT ; a solution to (0.1) on ½0; T Š with uð0Þ ¼ u0 ; 252 MA´LEK AND PRAZˇA´K moreover, for any solution the estimates of jjujjYT are uniform with respect to jjuð0ÞjjX : (A2) There exists a bounded set B0 & X with the following properties: if u is an arbitrary solution to (0.1) with initial condition u0 X then (i) there exists t0 ¼ t0 ðjju0 jjX Þ such that uðtÞ B0 for all t5t0 and (ii) if u0 B0 then uðtÞ B0 for all t50: Now, let ‘ > be an arbitrary fixed number By the ‘-trajectory we mean any solution on the time interval ½0; ‘Š: The set of all ‘-trajectories is denoted by X‘ and equipped with the topology of X‘ ¼ L2 ð0; ‘; X Þ: Note that since X‘ & Cð½0; ‘Š; Xw Þ; it makes sense to talk about the point values of trajectories On the other hand, it is not clear whether X‘ is closed in X‘ and hence X‘ in general is not a complete metric space Since we not require uniqueness of the solution, it is possible that more than one trajectory will start from a point u0 X : We will impose a weaker condition: (A3) Each ‘-trajectory has among all solutions unique continuation In other words, from an end point of an ‘-trajectory there starts at most one solution Combined with the assumption (A1) about the global existence of solutions this in particular implies that if w X‘ and T > ‘ then there exists a unique u which is a solution to (0.1) on ½0; T Š such that w ¼ uj½0;‘Š :7 Using (A3), we can define the semigroup Lt on X‘ by fLt wgðtÞ :¼ uðt þ tÞ; t ½0; ‘Š; where u is the unique solution on ½0; ‘ þ tŠ such that uj½0;‘Š ¼ w: (2) A‘ – Attractor in the Set of ‘-Trajectories We define B0‘ as the set of all ‘-trajectories starting at any point of B0 from (A2) In symbols, B0‘ :¼ fw X‘ ; wð0Þ B0 g: Observe that owing to (A2), B0‘ is positively invariant w.r.t Lt : We add two more assumptions: (A4) For all t > 0; Lt : X‘ /X‘ is continuous on B0‘ ; (A5) For some t > 0; Lt ðB0‘ Þ X‘ & B0‘ : This assumption is suited to cover the situation in fluid mechanics where inner points of trajectories belong to a better space than X ; hence solutions starting from them are more regular and consequently unique even in the wider class of weak solutions Similarly, (A3) would be satisfied if we work with equations containing terms delayed (in time) by ‘; at most; cf [19] ¼ u ðsÞ ds t1 þs jt2 À t1 X Z 1À j q 1 q q [...]...253 THE ‘ -TRAJECTORIES METHOD The assumption (A5) represents the crucial step in overcoming the problem of incompleteness of X‘ ; since it asserts that the closure (¼ completion) of Lt ðB0‘ Þ remains in X‘ : Yet the assumption (A5) is naturally fulfilled provided that B0 is (weakly) closed and we have the ‘‘compactness’’ of solutions; cf (E2) in Section 3 or the proof of Theorem 4.1 in Section... ð2:3Þ The observation that B1‘ is in fact a compact subset of X‘ is a key step in the proof of the following theorem Theorem 2.1 Let (A1)–(A5) hold Then the dynamical system ðLt ; X‘ Þ possesses a global attractor A‘ : Proof Consider a set B1‘ & X‘ defined in (2.3) Clearly, B1‘ is closed, and by (A1), (A2) it is bounded in Y‘ ++ X‘ ; and hence compact Moreover, by the continuity of Lt – cf (A4) – and the. .. for t5t0 ; t0 depending on C only By these considerations, the assumptions of Lemma 1.1 with St ¼ Lt ; X ¼ X‘ ; and B1 ¼ B1‘ are satisfied and the existence of the global attractor follows ] (3) Fractal Dimension of A‘ The assumption which leads to the finiteness of the fractal dimension and which is also a key step in constructing the exponential attractor reads (A6) There exists a space W‘ with W‘ ++... þ t and Bt & StÀðt0 þtÞ B1 for t sufficiently large (5) Finite Fractal Dimension of A If we strengthen (A7) and require that (A8) e: X‘ /X is a-Ho lder continuous on B1‘ ; we come to the following assertion Theorem 2.4 Let (A1)–(A6) and (A8) hold Then the fractal dimension of A in X is finite and 1 dfX ðAÞ4 dfX‘ ðA‘ Þ: a Proof The proof is a consequence of Theorem 2.2, (A8), and Lemma 1.2, where we take... subsequence if necessary – that Ltn wn ! w 2 A‘ : But by the continuity of e; Stn un ¼ eðLtn wn Þ ! eðwÞ 2 A; which contradicts (2.5) ] Remark 2.1 The set A is also a global attractor to the dynamics of (0.1) on the whole space X in the following sense: if B & X is bounded and Bt denotes the set of all values of all solutions to (0.1), starting from B; at time t; then distX ðBt ; AÞ ! 0 as t ! 1: Indeed, by (A2),... obtained A from A‘ via the mapping e: X‘ /X we obtain E as an image of E‘ : We put E :¼ eðE‘ Þ: ð2:6Þ Theorem 2.6 Let (A1)–(A6), (A8)–(A10) hold Then E defined in (2.6) is an exponential attractor to the dynamical system ðSt ; B1 Þ: Proof It immediately follows from the facts that E‘ is an exponential attractor w.r.t ðLt ; B1‘ Þ and the mapping e is Ho lder continuous THE ‘ -TRAJECTORIES METHOD 257 We conclude... Proof Since A is a continuous image of a compact set, it is compact Also, since Lt ðA‘ Þ ¼ A‘ ; we have St ðAÞ ¼ St ðeðA‘ ÞÞ ¼ eðLt ðA‘ ÞÞ ¼ eðA‘ Þ ¼ A: To verify the attracting property of A; we proceed by contradiction: let there exist sequences un 2 B1 ; tn ! 1; and a d > 0 such that distX ðStn un ; AÞ5d: ð2:5Þ THE ‘ -TRAJECTORIES METHOD 255 By the definition of B1 there exists a sequence fwn g & B1‘... see that all assumptions of the lemma are fulfilled ] (4) A – Attractor in the Original Space Now, we introduce a mapping e: X‘ /X which to a given ‘-trajectory assigns its end point In symbols, eðwÞ ¼ wð‘Þ: In this manner we construct a one-way bridge between the set X‘ on one side and the space X on the other side Note that the definition of e is meaningful since, due to (A1), trajectories are weakly... (A9)–(A10) hold Then ðLt ; B1‘ Þ possesses an exponential attractor E‘ : MA´LEK AND PRAZˇA´K 256 Proof We will apply Lemma 1.4 with X ¼ X‘ ; St ¼ Lt and B1 ¼ B1‘ : Note that since X is the Hilbert space, X‘ is the Hilbert space as well Let us verify the assumptions (P1)–(P3) We fix a t > 0 for which (A6) holds Then (P1) follows from (A6) or from (A9) Next, let k be the Lipschitz constant of the mapping... whenever it holds that a þp b4c ffiffiffi þ d then necessarily either a4c or b4d; which is just (P2) with W ¼ ð4 2ÞÀ1 : Finally, (A9), (A10) imply (P3) since jjGðw1 ; t1 Þ À Gðw2 ; t2 ÞjjX‘ ¼ jjLt1 w1 À Lt2 w2 jjX‘ 4 jjLt1 w1 À Lt1 w2 jjX‘ þ jjLt1 w2 À Lt2 w2 jjX‘ 4 c#ðjt1 À t2 jb þ jjw1 À w2 jjX‘ Þ: The proof of Theorem 2.5 is complete ] (7) E – Exponential Attractor in the Original Space Analogously to subsection ... vd Þ and P are the velocity and the pressure; the initial velocity v0 and the external body force f are given: f is time-independent and v0 is at xi L-periodic and divergence-free MA´LEK AND. .. existence and uniqueness of the solution to (0.1) Let ðX ; jj Á jjX Þ; ðY ; jj Á jjY Þ; and ðZ; jj Á jjZ Þ be three Banach spaces, X being reflexive and separable, such that Y ++ X and X + Z:... exists c > and b ð0; 1Š such that for all w B1‘ and t1 ; t2 ½0; tŠ it holds that jjLt1 w À Lt2 wjjX‘ 4cjt1 À t2 jb : Theorem 2.5 Let X be a Hilbert space and let assumptions (A1)–(A6) and (A9)–(A10)