On convergence of trajectory attractors of 3D Navier Stokes-α model as α approaches 0

Transcription

1 On convergence of rajecory aracors of 3D Navier Sokes-α model as α approaches V.V.Chepyzhov, E.S.Tii, and M.I.Vishik Insiue for Informaion Transmission Problems Russian Academy of Sciences, Bolshoy Kareniy 19 Moscow , GSP-4, Russia Deparmen of Mahemaics and Deparmen of Mechanical and Aerospace Engineering, Universiy of California, Irvine, CA , USA, Also: Deparmen of Compuer Science and Applied Mahemaics, Weizmann Insiue of Science, Rehovo 761, Israel, Absrac We sudy he relaions beween he long-ime dynamics of he Navier Sokes-α model and he exac 3D Navier Sokes sysem. We prove ha bounded ses of soluions of he Navier Sokes-α model converge o he rajecory aracor A of he 3D Navier Sokes sysem as ime ends o infiniy and α approaches zero. In paricular, we show ha he rajecory aracor A α of he Navier Sokes-α model converges o he rajecory aracor A of he 3D Navier Sokes sysem when α. We also consruc he minimal limi A min ( A ) of he rajecory aracor A α as α and we prove ha he se A min is conneced and sricly invarian. Inroducion (Dae: January 17, 27) In his paper, we sudy he connecion beween he long-ime dynamics of soluions of he Lagrange averaged Navier Sokes-α model (N. S.-α model) and he exac 3D Navier Sokes sysem (3D N. S. sysem) wih periodic boundary condiions. The Navier Sokes-α model (also known as he viscous 3D Camassa Holm sysem) under he consideraion was inroduced in he works [1] [6] (see also [7] and he references 1

2 herein). This model is a regularized approximaion of he 3D Navier Sokes sysem depending on a small parameer α, where, in some erms, he unknown velociy vecorfuncion v is replaced by a smooher vecor funcion u which are relaed by he ellipic sysem v = u α 2 u (see Sec.2). For α =, he model is reduced o he exac 3D N. S. sysem. Since he uniqueness heorem for he global weak soluions (or he global exisence of srong soluions) of he iniial-value problem of he 3D Navier Sokes sysem is no proved ye, he known heory of global aracors of infinie dimensional dynamical sysems (making a good showing in he sudy of he 2D N. S. sysem and oher imporan evoluion equaions of mahemaical physics, see, e.g., [8] [14]) is no applicable o he 3D N. S. sysem. I was demonsraed analyically and numerically in many works ha he menioned above N. S.-α model gives a good approximaion in he sudy of many problems relaed o he urbulen flows (see [1] [4], [7, 15, 16]). In paricular, i was found ha he explici seady analyical soluions of he N. S.-α model compare successfully wih empirical and numerical experimenal daa for a wide range of Reynolds numbers in urbulen channel and pipe flows (see [1] [3]). Along he same lines i is worh menioning ha oher approximae α-models for he 3D N. S. sysem also demonsrae good fi wih empirical daa: Clark-α model [17], Leray-α model [18], Modified-Leray-α model [19], simplified Bardina-α model [2] and some oher models. Closed problems relaed o he regularizaion of he 3D N. S. sysem were also considered in he works of Lions [21] and Ladyzhenskaya [22]. In [6], he Cauchy problem for he 3D Navier Sokes-α model was sudied, he global exisence and uniqueness of weak soluions were esablished, he smoohing propery of soluions was proved, and he global aracor for his sysem was consruced. Besides, upper bounds for he dimension of he global aracor (he number of degree of freedom) were found in erms of he relevan physical parameers and some oher urbulence relaed feaures and characerisics (such as specra and boundary layer) were discussed (see also [23, 24]). The heory of rajecory aracors for evoluion parial differenial equaions was developed in [14, 25, 26, 27] wih an emphasis on equaions for which he uniqueness heorem of soluions of he corresponding iniial-value problem is no proved ye, e.g. for he 3D N. S. sysem (see also [13, 28]). In he presen paper, we sudy he connecion beween he soluions of he Navier Sokes-α model and he exac 3D Navier Sokes sysem as α. Our main heorem saes ha bounded (in he corresponding norm) families of soluions {u α (x, )} of he N. S.-α models converge o he rajecory aracor A of 3D Navier Sokes sysem as α and. In paricular, he rajecory aracors A α of he N. S.-α model converges o A as α. In [29, 3], analogous heorems were proved for he Leray-α model. This paper consiss of inroducion, five secions, and an appendix. In Sec. 1, we recall he definiion of he rajecory aracor A of he exac 3D N. S. sysem. In Sec. 2, we consider he N. S.-α model (he viscous Camassa Holm equaions). Following [6], we formulae he main properies of his model. In Sec. 3 and 4, we prove he convergence of he rajecory aracor A α of he N. S.-α model o he rajecory aracor A of he exac 3D N. S. sysem as α. I urns ou ha in order o esablish his convergence i is very fruiful o sudy he 2

3 equaion o which he funcion w α () = (1 α 2 ) 1/2 u α () is saisfied. Here, u α () is he smooher velociy field of he soluion of he N. S.-α model. The main heorem of Sec. 3 saes ha if a sequence of soluions w αn () of he menioned above equaion converges o he limi w() as α n and n in he space Θ loc (see Sec. 1), hen w() is a Leray Hopf weak soluion of he exac 3D N. S. sysem. Using mosly his heorem in Sec. 4, we prove he convergence of he rajecory aracors A α o he rajecory aracor A in he space Θ loc as α. In Sec. 5, we esablish he exisence of he minimal limi A min ( A ) of he rajecory aracors A α as α, i.e., A α A min (α ), where A min is he smalles closed subse of A saisfying his limi relaion (See Sec. 5). We prove ha he se A min is conneced and sricly invarian wih respec o he ranslaion semigroup. These properies of he minimal limi A min make i a very useful objec in he sudy of various models ha approximae he 3D N. S. sysem. We noe ha he quesion of he connecedness of he rajecory aracor A of he 3D N. S. sysem remains open. Now, he hypohesis also arises ha, o differen α-models of he 3D N. S. sysem (Camassa Holm, Leray-α, Clark-α, simplified Bardina-α, ec.), differen minimal limis of heir rajecory aracors A α as α may correspond. This work was parly suppored by he Russian Foundaion of Basic Researches (projec no and ), Civilian Research & Developmen Foundaion (Gran RUM MO-5), and he Russian Science Suppor Foundaion. E.S.T. was also suppored in par by he NSF gran no. DMS 24794, he ISF gran no.12/6, and he BSF gran no D Navier Sokes sysem and is rajecory aracor We consider he auonomous 3D N. S. sysem wih periodic boundary condiions v ν v 3 v j xj v p = g(x), j=1 v =, x T 3 := [R mod 2πL] 3,, which is equivalen o he nonlinear nonlocal funcional differenial equaion v ν v P 3 v j xj v = g(x), v =. (1.1) j=1 Here v = v(x, ) = (v 1 (x, ), v 2 (x, ), v 3 (x, )) is he unknown vecor funcion describing he moion of he fluid in T 3, P is he Leray Helmholz orhogonal projecor, g(x) = (g 1 (x), g 2 (x), g 3 (x)) is a given exernal force wih zero mean in x-variable, i.e., g(x)dx =, and g = P g. We assume ha v(x, ) is a periodic funcion in T 3 x = (x 1, x 2, x 3 ) T 3 wih zero mean, i.e., v(x, )dx =, and v = P v. T 3 We se V = {φ(x) = (φ 1 (x), φ 2 (x), φ 3 (x)), x T 3 φ j (x) are rigonomerical polynomial wih period 2πL in each x i, i, j = 1, 2, 3, such ha φ = and φ(x)dx = }. T 3 We denoe by H and V he closure of he se V in he norms H =: and 3

4 H 1 =: of he spaces L 2 (T 3 ) 3 and H 1 (T 3 ) 3, respecively (see, e.g., [11, 31]). Then he Leray Helmholz projecor P : L 2 (T 3 ) 3 H. We define also he space D(A) = {v H v H}, where A = P is he Sokes operaor wih domain D(A). Recall ha, in he periodic case, A = and he norm Av =: v D(A) on D(A) is equivalen o he norm induced by H 2 (T 3 ) 3. The operaor A is self-adjoin, posiive, and has a compac resolven. We denoe by ((u, v)) := (A 1/2 u, A 1/2 v) = ( u, v), u := A 1/2 u, u, v V, he scalar produc and he norm in V, respecively. The Poincaré inequaliy implies ha v 2 λ 1 1 v 2, v V, (1.2) where λ 1 is he firs eigenvalue of he Sokes operaor A. Le V = H 1 be he dual space of V. For any f V, we denoe by f, v he acion of he funcional f V on any v V. The operaor A is an isomorphism from V o V and ((u, v)) = Au, v for all u, v V. We rewrie equaion (1.1) in a sandard shor form: v νav B(v, v) = g(x),. (1.3) Here, we denoe B(u, v) = P [(u )v] = P Recall ha for u saisfying u = we have: 3 u j xj v. (1.4) j=1 B(u, v) = P 3 ( xj u j v ) (1.5) j=1 (see [21, 11, 31]). For all w D(A) and u, v V, we have he esimae and herefore B(u, v), w C u v w L C 1 λ 1/4 1 u v w D(A) (1.6) B(u, v) D(A) C 1 λ 1/4 1 u v, (1.7) where D(A) is he dual space of D(A). Le a funcion v( ) L 2 (, M; V ) L (, M; H) be given. L 2 (, M; V ) and due o (1.7) we have Therefore, Av B(v( ), v( )) L 2 (, M; D(A) ). (1.8) Consider he space of disribuions D (, M; D(A) ) (see, e.g., [21]). Recall ha a funcion v( ) L 2 (, M; V ) L (, M; H) is said o be a weak soluion of equaion (1.3) if i saisfies his equaion in he space D (, M; D(A) ). Then i follows from (1.8) ha v( ) L 2 (, M; D(A) ) for any weak soluion v( ) of (1.3) and hence v( ) C([, M]; D(A) ). Recall ha v( ) L (, M; H). Then, by he known lemma from [33] (see also [31]), he funcion v( ) C w ([, M]; H) and, consequenly, he iniial daa v = = v (x) H (1.9) 4

5 for equaion (1.3) has a sense in he class of weak soluions from he space L 2 (, M; V ) L (, M; H). We now formulae he classical heorem on he exisence of weak soluions of he Cauchy problem for he 3D N. S. sysem in he form we need in he sequel (see, e.g., [11, 21, 32, 31]). Theorem 1.1 Le g V and v H. Then for every M >, here exiss a weak soluion v() of equaion (1.3) from he space L 2 (, M; V ) L (, M; H) such ha v() = v and v() saisfies he energy inequaliy 1 d 2 d v() 2 ν v() 2 g, v(), [, M]. (1.1) Inequaliy (1.1) means ha for any funcion ψ( ) C (], M[), ψ(), 1 2 v() 2 ψ ()d ν v() 2 ψ()d The proof of Theorem 1.1 is given, e.g., in [11, 21, 14]. g, v() ψ()d. (1.11) Remark 1.1 For he 3D Navier Sokes sysem he quesion of he uniqueness of a weak soluion of problem (1.3) and (1.9) remains open. I is also unknown, wheher every weak soluion saisfies he energy inequaliy (1.1). However, i is known ha every weak soluion resuling from he Faedo Galerkin approximaion mehod saisfies his energy inequaliy. The class of weak soluions which saisfy he energy inequaliy (1.1) or (1.11) is called Leray Hopf weak soluions. In he sequel, we define he rajecory aracor for he N. S. equaion (1.3). (For more deails, see [26, 14].) To begin wih, we define he rajecory space K of equaion (1.3). We consider a se of weak soluions v(),, belonging o he space L loc 2 (R ; V ) L loc (R ; H) ha saisfy equaion (1.3) in he space of disribuions D (, M; D(A) ) for any M >. Definiion 1.1 The rajecory space K is he se of all Leray Hopf weak soluions v( ) of equaion (1.3) in he space L loc 2 (R ; V ) L loc (R ; H) ha saisfy he energy inequaliy (1.1) for, ha is, 1 2 v() 2 ψ ()d ν for all ψ C (R ), ψ. v() 2 ψ()d g, v() ψ()d (1.12) I follows from Theorem 1.1 ha, for any v H, here is a rajecory v( ) K such ha v() = v. We need he Banach space F b = {z z( ) L b 2(R ; V ) L (R ; H), z( ) L b 2(R ; D(A) )} wih norm z F b = z L b 2 (R ;V ) z L (R ;H) z L b 2 (R ;D(A) ), (1.13) 5

6 where z 2 1 = sup L b 2 (R ;V ) z(s) 2 ds, z L (R ;H) = ess sup z(), and z 2 = sup 1 L b 2 (R ;D(A) ) z(s) 2 D(A) ds. We denoe by {T (h)} := {T (h), h } he ranslaion semigroup acing on a funcion {z(), } by he formula T (h)z() = z( h),. Clearly, he semigroup {T (h)} acs on F b. We consider he acion of he semigroup {T (h)} on he rajecory space K of equaion (1.3). I follows from he definiion of K ha if v( ) K, hen v h ( ) = T (h)v( ) = v( h) K for all h. Tha is, T (h)k K, h. (1.14) We are going o consruc he global aracor of he ranslaion semigroup {T (h)} on K. We call his aracor he rajecory aracor since he semigroup {T (h)} acs on he rajecory space K. The following key proposiion is proved in [14]. Proposiion 1.1 Le g V. Then 1. The rajecory space K F b ; 2. for any funcion v( ) K, T (h)v( ) F b C v( ) 2 C (,1;H)e νλ 1h R 2, h 1, (1.15) where he consan C depends on ν, λ 1 and R depends on ν, λ 1, g V. We need a opology in he space K. Similarly o F b, we consider he space F loc = {z z( ) L loc 2 (R ; V ) L loc (R ; H), z( ) L loc 2 (R ; D(A) )}. We define on F loc he following sequenial opology which we denoe Θ loc. By definiion, a sequence of funcions {z n } F loc converges o a funcion z F loc in he opology Θ loc as n if, for any M >, and z n ( ) z( ) (n ) weakly in L 2 (, M; V ), z n ( ) z( ) (n ) weakly- in L (, M; H), z n ( ) z( ) (n ) weakly in L 2 (, M; D(A) ). Noe ha he opology Θ loc can be described in erms of open neighborhoods. Θ loc is a Hausdorff opological space wih a counable base of is opology (however, he opology Θ loc is no merizable). Recall ha F b Θ loc. Besides, any ball B R = {z F b z F b R} is compac in Θ loc. Hence, he se B R wih opology induced by Θ loc is merizable and he corresponding meric space is complee (see he deails in [26, 14]). This propery simplifies he consrucion of he rajecory aracor (in he opology Θ loc ) of he semigroup {T (h)} acing on K. I follows from he definiion of he opology Θ loc ha he ranslaion semigroup {T (h)} is coninuous in Θ loc. The following asserion is imporan for us (see he proof in [14]). 6

7 Proposiion 1.2 The rajecory space K is closed in he space Θ loc. In a sandard manner, we define an aracing se in K (see [8, 9, 1, 28]). A se P F b is called aracing for he space K in he opology Θ loc if, for any bounded (in he norm of F) b se B K, he se P aracs T (h)b in he opology Θ loc as h, ha is, for any neighborhood O(P ) (in Θ loc ), here is a number h 1 = h 1 (B, O) such ha T (h)b O(P ) for all h h 1. We now define he rajecory aracor. Definiion 1.2 A se A K is called he rajecory aracor of he semigroup {T (h)} in he opology Θ loc if 1. A is bounded in F b and compac in Θ loc ; 2. A is sricly invarian wih respec o {T (h)} : T (h)a = A, h ; 3. A is an aracing se in he opology Θ loc for {T (h)} on K. Following he erminology from [9], he se A is also called he (F, b Θ loc )-aracor of he semigroup {T (h)} K. The main inequaliy (1.15) implies ha he ball B 2R in F b is an aracing (and even absorbing) se of he semigroup {T (h)} in K. The ball B 2R is clearly compac in Θ loc and T (h)b 2R B 2R for all h. Therefore, he coninuous semigroup {T (h)} has a compac aracing se. Consequenly, he ranslaion semigroup {T (h)} has he rajecory aracor A K B 2R and moreover [ ] T (h)(k B 2R ), A = s> h s where [ ] Θ loc denoes he closure in Θ loc (see [14]). Noice ha he following embeddings are coninuous: Θ loc L loc 2 (R ; H 1 δ ), (1.16) Θ loc C loc (R ; H δ ), for < δ 1, (1.17) (see [14, 21, 34]). Hence, he rajecory aracor A saisfies he following properies: for any bounded (in F b ) se B K, Θ loc dis L2 (,M;H 1 δ )(T (h)b, A) (h ), dis C([,M];H δ )(T (h)b, A) (h ), where M is an arbirary posiive number. To describe he srucure of he rajecory aracor A we need he noion of he kernel of equaion (1.3). The kernel K is he se of all weak soluions v(), R, bounded in he space F b = {z z( ) L b 2(R; V ) L (R; H), z( ) L b 2(R; D(A) )} 7

8 ha saisfies an inequaliy similar o (1.12): for all ψ C (R), ψ, 1 2 v() 2 ψ ()d ν v() 2 ψ()d g, v() ψ()d. (1.18) (The norm in F b is defined in a similar way ha he norm in F b (see (1.13)) replacing R by R). We denoe by Π he resricion operaor ono R. I is proved in [14] ha he rajecory aracor A of he 3D Navier Sokes sysem coincides wih he resricion of he kernel K of equaion (1.3) ono R : A = Π K. (1.19) The se K is bounded in F b and compac in Θ loc. The opology Θ loc is defined similar o Θ loc where he inervals (, M) are replaced by ( M, M). 2 Navier Sokes-α model and is aracor 2.1 Some properies of he Navier Sokes-α model We consider he following sysem wih periodic boundary condiions: v ν v P (u ( v)) = g(x), (2.1) v = u α 2 u, v =, u =, x T 3. (2.2) This sysem is an approximaion of he 3D N. S. sysem (1.1) discussed in he previous secion. The unknown vecor funcion is u = u(x, ) = (u 1, u 2, u 3 ). The funcion v = v(x, ) = (v 1, v 2, v 3 ) is auxiliary. We assume ha funcions u(x, ), v(x, ), and he (known) exernal force g(x) are periodic in x T 3 and have zero spaial mean. In equaion (2.2), α is a fixed posiive parameer called he sub-grid (filer) lengh scale of he model (see he moivaions in [6] and he references herein). As in (1.1), P denoes he Leray Helmholz projecor and a b is he vecor produc in R 3. We will see shorly ha, for α =, he funcion v u and formally equaions (2.1) and (2.2) coincides wih he 3D N. S. sysem (1.1). The sysem (2.1) and (2.2) is called someimes as he 3D Camassa Holm equaions (i is also known as he Lagrange averaged Navier Sokes-α model or jus he Navier Sokes-α model). Recall ha he nonlinear erm in (2.1) saisfies he following ideniy u ( v) = 3 ( u j j v u j v j) 3 = (u )v u j v j (2.3) j=1 j=1 assuming ha u, v C 1 (see [6]). For u = v, we have v ( v) = (v )v ( v j v j ) (2.4) and hence, for α =, he sysem (2.1) and (2.2) becomes (1.1) since P projecs any gradien funcion o zero, so, P ( 3 j=1 vj v j ) = (see [11, 31]). 8 j=1

9 We now rewrie sysem (2.1) and (2.2) in he shor form v νav B(u, v) = g(x), (2.5) v = u α 2 Au. (2.6) Here as in equaion (1.3), A denoes he Sokes operaor and he bilinear operaor Recall ha B(u, v) = P (u ( v)). (2.7) B(v, v) = B(v, v), (2.8) where B(u, v) = (u )v (see (1.4)) and, for α =, sysem (2.5), (2.6) coincides wih he 3D N. S. sysem (1.3). We now formulae some properies of he bilinear operaor B ha are analogous o he properies of he operaor B. The operaor B maps V V o V and he following inequaliies holds B(u, v), w c u 1/4 u 3/4 v w 1/4 w 3/4, (2.9) B(u, v), w c u 1/2 u 1/2 v w u, v, w V. (2.1) (For he proof, see [6].) We have also he ideniy B(u, v), w = B(w, v), u, u, v, w V, (2.11) which follows from he vecor calculus formulas (a b) c = de [a, b, c] = de [c, b, a] = (c b) a, a, b, c R 3, where we se a = u, b = v, and c = w. From (2.11), we conclude ha B(u, v), u = u, v V. (2.12) We need also he following inequaliy proved in [6]: B(u, v), w c ( u 1/2 u 1/2 v Aw u v w 1/2 Aw 1/2), (2.13) u, v V and w D(A). To prove (2.13) one uses he following ideniy B(u, v), w = B(w, u) B(u, w), v (2.14) and he known properies of he operaor B (see (1.6) and [6]). The ideniy (2.14) can be verified by he direc calculaion. I follows from (2.13) ha B(u, v), w c u v Aw, u, v V, w D(A). (2.15) This means ha B maps V H ino D(A) and (compare wih (1.7)). B(u, v) D(A) c u v (2.16) 9

10 2.2 Cauchy problem and aracor for he N. S.-α model Le now a funcion u( ) L (, M; V ) L 2 (, M; D(A)) be given. Then he funcion v( ) = u( ) α 2 Au( ) L (, M; V ) L 2 (, M; H) and Av( ) L 2 (, M; D(A) ). Consequenly, from inequaliy (2.16) we conclude ha he corresponding funcion B(u( ), v( )) L 2 (, M; D(A) ). Therefore, all he erms of equaion (2.5) (excep he ime derivaive) belongs o he space L 2 (, M; D(A) ) and he equaion iself is meaningful in he disribuion space D (, M; D(A) ). We supplemen sysem (2.5) and (2.6) wih iniial daa u = = u V. (2.17) (Compare wih (1.9), where v H.) Definiion 2.1 Le g H, u V, and M >. A funcion u( ) L (, M; V ) L 2 (, M; D(A)) is called a soluion of sysem (2.5), (2.6), and (2.17) if (i) u() saisfies he equaion in he space of disribuions D (, M; D(A) ), i.e., for every ω D(A) d u α 2 Au, ω A(u α 2 Au), ω d B(u, u α 2 Au), ω = g, ω, (2.18) where (2.18) is undersood in he scalar disribuion sense of he space D (, M), ha is, for every ϕ C (], M[), v(), ω ϕ ()d B(u(), v()), ω ϕ()d = Av(), ω ϕ()d g, ω ϕ()d, (2.19) where v() = u() α 2 Au(). (ii) u() = u. Since u() is a soluion of (2.5) and (2.6), dv/d L 2 (, M; D(A) ). We noe ha u = (1 α 2 A) 1 v, so du/d L 2 (, M; H), herefore, u C([, M]; H), and he iniial condiion (2.17) is meaningful. Remark 2.1 Someimes, he funcion v() = (1α 2 A)u() is also called he soluion of sysem (2.5) and (2.6). This erminology forms a correspondence beween he soluions of (2.5), (2.6) and he soluions of he exac Navier Sokes sysem (1.3). In he work [6], he following heorem was proved. Theorem 2.1 Le g H and u V. Then, for every M >, he Cauchy problem (2.5), (2.6), and (2.17) has a unique soluion u() ha belongs o he space C([, M]; V ) L (, M; D(A)). Here, we formulae and prove some corollaries from his heorem we need in he sequel. Firs of all, we are ineresed in soluion esimaes ha are independen of α as α. 1

11 Corollary 2.1 (The energy equaliy) Le u() be a soluion of (2.5), (2.6), and (2.17) hen he following ideniy holds: 1 d { u() 2 α 2 u() 2} 2 d ν { u() 2 α 2 Au() 2} = g, u(), [, M], (2.2) he funcion u() 2 α 2 u() 2 is absoluely coninuous, and is ime derivaive saisfies (2.2) in he usual sense for a.e. (, M). Proof. We ake he scalar produc in H of equaion (2.5) wih u() and use he facs ha u L 2 (, M; D(A)) and u L 2 (, M; H). Then due o he known heorem from [31] d d u() 2 = 2(u, u), d d u() 2 = 2(A 1/2 u, A 1/2 u) = 2(Au, u) = 2(u, Au), (recall ha A 1/2 u L 2 (, M; V ) and A 1/2 u L 2 (, M; V )). Besides, ( B(u, v), u) = (see (2.12)). To complee he proof we noe ha (Av, u) = u() 2 α 2 Au() 2. (Analogous approach is used o prove he uniqueness of a soluion of (2.5), (2.6), and (2.17), see [6].) Corollary 2.2 (A priori esimaes) If u() is a soluion of (2.5), (2.6), and (2.17), hen he following inequaliies hold: ν 1 Proof. follows: Hence, u() 2 α 2 u() 2 ( u() 2 α 2 u() 2) e νλ1 g 2 λ 2 1ν,(2.21) 2 { u(s) 2 α 2 Au(s) 2} ds ( u() 2 α 2 u() 2) e νλ 1 (2.22) g 2 λ 2 1ν g 2,. 2 λ 1 ν We use he energy equaliy (2.2) and esimae he righ-hand side as (g, u) ν 2 u 2 1 2ν g 2 V ν 2 u 2 1 2νλ 1 g 2 ν 2 { u 2 α 2 Au 2} 1 2νλ 1 g 2. d { u() 2 α 2 u() 2} ν { u() 2 α 2 Au() 2} 1 g 2. (2.23) d νλ 1 11

12 I follows from he Poincaré inequaliy ha { u() 2 α 2 u() 2 λ 1 1 u() 2 α 2 Au() 2} (since λ 1 u 2 u 2 and λ 1 u 2 Au 2 ). Consequenly, from (2.23) we have d { u() 2 α 2 u() 2} { νλ 1 u() 2 α 2 u() 2} 1 g 2. d νλ 1 Using now he known asserion d d ϕ νλ 1ϕ 1 g 2 = ϕ() ϕ()e νλ1 1 g 2, νλ 1 ν 2 λ 2 1 where ϕ() = u() 2 α 2 u() 2, we obain (2.21). Inegraing (2.23) over [, 1] we find ha u( 1) 2 α 2 u( 1) 2 ν 1 u() 2 α 2 u() 2 1 g 2 νλ 1 ( u() 2 α 2 u() 2) e νλ1 g 2 λ 2 1ν 1 g 2, 2 νλ 1 where we have applied (2.21). Thus, (2.22) is also proved. { u(s) 2 α 2 Au(s) 2} ds Remark 2.2 (i) Esimaes (2.21) and (2.22) imply ha, for α >, u L (R ; V ) L loc 2 (R ; D(A)). This inclusion is essenially used in he proof of Theorem 2.1 (see [6]). (ii) I follows also ha v L b 2(R ; H). Remark 2.3 We noe ha he consans in he righ-hand sides of esimaes (2.21) and (2.22) are independen of α (for < α 1). This fac plays he key role in he proof of he convergence of soluions of he Navier Sokes-α model o he soluions of he real Navier Sokes sysem as α. Remark 2.4 In he work [6], he following smoohing propery for soluions of (2.5), (2.6), and (2.17) is esablished: Au() 2 ν s A 3/2 u(s) 2 ds C(α,, u(), g ), (2.24) where C(α, z, r 1, r 2 ) is a monoone increasing funcion in each variable z, r 1, r 2 and C(α, z, r 1, r 2 ) as α. We now consider he semigroup {S α ()} = {S()}, α >, acing in he space V by he formula S()u = u(), where u() is a soluion of problem (2.5), (2.6), and (2.17). I follows from (2.21) ha he semigroup {S()} has bounded (in V ) absorbing se P = { u u 2 g αλ 1 ν }. The se P 1 = S(1)P is also absorbing and inequaliy (2.24) implies ha P 1 is precompac in V. I can be verified ha he semigroup {S()} is coninuous in V. These facs are sufficien o sae ha he semigroup {S()} corresponding o he 12

13 Navier Sokes-α model has he global aracor A α, ha is A α compac in V, sricly invarian wih respec o {S()} : S()A α = A α, for all, and, dis V (S()B, A α ) as for any bounded (in V ) se of iniial daa B = {u } (see [8, 9, 1, 14, 28]). Moreover, A α is bounded in D(A) H 3 (T) 2 for every fixed α > bu no uniformly wih respec o α (see [6]). In he nex secion, we sudy he behaviour of he Navier Sokes-α model as α. We esablish is relaion wih soluions of he 3D Navier Sokes sysem. 3 On he convergence of soluions of N. S.-α model Firs of all, we need an esimae for he derivaive v in which consans are independen of α similar o ha proved for u in Corollaries 2.1 and 2.2. Proposiion 3.1 Le g H. Then any soluion u() of (2.5), (2.6), and (2.17) saisfies he inequaliy ( 1 ) 1/2 v(s) 2 D(A) ds C ( u() 2 α 2 u() 2) e νλ1 R 2, (3.1) where C depends on λ 1, ν; R depends on λ 1, ν, g and he values C and R are independen of α. Proof. We use inequaliy (2.16): B(u, v) D(A) c u v, u V, v H. (3.2) Replacing here a soluion u() of (2.5), (2.6), (2.17) and v = u α 2 Au, we obain B(u(), v()) D(A) c u() { u() α 2 Au() } = c { u() u() α u() α Au() } c ( u() 2 α 2 u() 2) 1/2 ( u() 2 α 2 Au() 2) 1/2, (3.3) where we have used he simples Cauchy inequaliy. Applying inequaliy (2.21), we have ha { } B(u(), v()) 2 D(A) c2 ϕ()e νλ1 g 2 ( u() 2 α 2 Au() 2), λ 2 1ν 2 where ϕ() = u() 2 α 2 u() 2. Inegraing his inequaliy over [, 1], we find 1 B(u(s), v(s)) 2 D(A) ds { } 1 c 2 ϕ()e νλ1 g 2 ( u(s) 2 α 2 Au(s) 2) ds λ 2 1ν 2 (The funcion in braces was jus majorized on [, 1] by is values a.) We now use (2.22) and obain 1 { c 2 B(u(s), v(s)) 2 D(A) ds } ϕ()e νλ1 g 2 1 λ 2 1ν 2 ν 13 {ϕ()e νλ 1 g 2 λ 2 1ν 2 g 2 λ 1 ν }

14 Hence, ( 1 where C 1 = cν 1/2 and R 2 1 = g 2 λ 2 1 ν2 g 2 λ 1 ν. From he esimae we conclude ha 1 B(u(s), v(s)) 2 D(A) ds ) 1/2 C 1 ϕ()e νλ 1 R 2 1, (3.4) Av D(A) = v H = v u α 2 Au ( 1 1 Av(s) 2 D(A) ds 2 u(s) 2 ds α { u(s) 2 α 2 Au(s) 2} ds ) α 2 Au(s) 2 ds (Recall ha α 1.) Using once more inequaliy (2.22), we obain ha 1 for an appropriae C 2 and R 2 independen of α. The funcions u and v saisfy equaion (2.5), i.e., Av(s) 2 D(A) ds C 2ϕ()e νλ 1 R 2 2 (3.5) v = νav B(u, v) g (3.6) We apply o (3.6) he riangle inequaliy aking ino accoun inequaliies (3.4) and (3.5): ( 1 ( 1 v(s) 2 D(A) ds ) 1/2 ν ( 1 B(u(s), v(s)) 2 D(A) ds ) 1/2 g D(A) Av(s) 2 D(A) ds ) 1/2 ν ( ) C 2 ϕ()e νλ1 R2 2 1/2 C1 ϕ()e νλ1 R1 2 λ 1 1 g ν ( C 2 ϕ()e νλ1 R2 2 1 ) C 1 ϕ()e νλ1 R1 2 λ 1 1 g Cϕ()e νλ1 R 2 = C ( u() 2 α 2 u() 2) e νλ1 R 2, where C = νc 2 C 1 and R = ν(r 2 2 1) R 2 1 λ 1 1 g. The proof is compleed. The following inequaliy holds: f 2 D(A) f α2 Af 2 D(A), f D(A). Indeed, he operaor A is self-adjoin and posiive. Therefore, f 2 D(A) = j=1 f j 2 λ 1 j j=1 f j 2 (1 α 2 λ j )λ 1 j = (1 α 2 A)f 2 D(A) = f α2 Af 2 D(A), 14

15 where f = j=1 f je j, Ae j = λ j e j, j = 1, 2,..., {e j } are he eigenvecors of he operaor A and {λ j } are he corresponding eigenvalues. Thus we conclude ha 1 where v = u α 2 Au. u(s) 2 D(A) ds 1 v(s) 2 D(A) ds,, (3.7) Corollary 3.1 The inequaliy (3.1) also holds for he funcion u : ( 1 ) 1/2 u(s) 2 D(A) ds C ( u() 2 α 2 u() 2) e νλ1 R 2, (3.8) wih he same consans C and R. To consruc he rajecory aracor for sysem (2.5) and (2.6) we have o pass o a new funcion variable w ha occupies an inermediae posiion beween he funcion u and v. We consider he funcion Then we clearly have I is easy o verify he following ideniies: w = (1 α 2 A) 1/2 u. (3.9) v = (1 α 2 A)u = (1 α 2 A) 1/2 w. (3.1) w 2 = u 2 α 2 u 2, (3.11) w 2 = (1 α 2 A) 1/2 u 2 = A 1/2 (1 α 2 A) 1/2 u 2 = (A(1 α 2 A)u, u) = ((1 α 2 A)u, Au) = u 2 α 2 Au 2. (3.12) We noe ha he funcion w = w(x, ) saisfies he following equaion: w νaw (1 α 2 A) 1/2 B((1 α 2 A) 1/2 w, (1 α 2 A) 1/2 w) = (1 α 2 A) 1/2 g (3.13) ha is a consequence of (2.5), (3.9), and (3.1). Using he funcion w, we rewrie inequaliies (2.21), (2.22), and (3.1). Corollary 3.2 The following inequaliies hold: ( 1 ν 1 w() 2 w() 2 e νλ 1 g 2 λ 2 1ν 2, (3.14) w(s) 2 ds w() 2 e νλ 1 g 2 λ 2 1ν 2 g 2 λ 1 ν, (3.15) w(s) 2 D(A) ds ) 1/2 C w() 2 e νλ 1 R 2, >. (3.16) We noe ha (3.16) follows from (3.1) if one akes ino accoun similarly o (3.7) ha 1 w(s) 2 D(A) ds 1 15 v(s) 2 D(A) ds,. (3.17)

16 We now consider he Banach space F b defined in Sec. 1. Recall ha F b = {z z( ) L b 2(R ; V ) L (R ; H), z( ) L b 2(R ; D(A) )} Inequaliies (3.14) (3.16) provide he following Proposiion 3.2 If g H, hen, for any soluion u() of problem (2.5), (2.6), and (2.17), he corresponding funcion w() = (1 α 2 A) 1/2 u() being a soluion of (3.13) saisfies he inequaliy T (h)w( ) F b C 3 w() 2 e νλ 1h R 2 3, h, (3.18) where he consan C 3 depends on ν, λ 1 and R 3 depends on ν, λ 1, g. (We sress ha C 3 and R 3 are independen of α.) We consider he rajecory space K α of sysem (2.5) and (2.6). By definiion, he space K α is he union of all funcions w() = (1 α 2 A) 1/2 u(), where u() is a soluion of (2.5), (2.6), and (2.17) wih an arbirary u V. Proposiion 3.2 implies ha K α F b for α >. We rewrie he energy equaliy (2.2) in he inegral form we need in he sequel. Proposiion 3.3 For every w K α, 1 2 for all ψ C (R ). w() 2 ψ ()d ν w() 2 ψ()d = To prove (3.19) we rewrie he ideniy (2.2) in he form 1 d 2 d w() 2 ν w() 2 = g, u(),, g, u() ψ()d (3.19) muliply by an arbirary es funcion ψ C (R ), and inegrae in from o. Then, inegraing by par in he firs inegral erm (ha is legiimae since he funcion w() 2 is absoluely coninuous), we obain he needed resul (3.19). We also consider he opological space Θ loc inroduced in Sec. 1 in connecion wih he iniial Navier Sokes sysem. Recall ha F b Θ loc Lemma 3.1 Le wo sequences {u n ()} F b and {α n } ], 1] be given such ha α n as n. We denoe w n = (1 α n A) 1/2 u n for n N. We assume ha he sequence {w n ()} is bounded in F b and w n () w() in Θ loc as n. Then he sequence {u n ()} is bounded in F b and u n () w() in Θ loc as n. Proof. The firs asserion follows from he apparen inequaliies. u n 2 u n 2 α 2 u n 2 = w n 2, (3.2) u n 2 u n 2 α 2 Au n 2 = w n 2 (3.21) 16

17 (see (3.11 and (3.12)). Besides similar o (3.17), we prove ha Consequenly, 1 u n (s) 2 D(A) ds 1 w n (s) 2 D(A) ds. (3.22) u n F b w n F b, n N. (3.23) From (3.23), we conclude ha {u n ()} is bounded in F. b Since a ball in F b is a weakly compac se in Θ loc, we can exrac from {u n ()} a convergen subsequence and we denoe he limi of his subsequence by u(). For simpliciy, we denoe his subsequence by {u n ()}. We also keep he corresponding subsequence of {w n ()}. Then we have u n () u(), w n () w() in Θ loc as n. We sae ha u w. Consider an arbirary inerval [, M]. By our assumpion, w n () w() (n ) weakly in L 2 (, M; V ) and w n () w() (n ) weakly in L 2 (, M; D(A) ). Then, by he Aubin heorem (see [34, 21, 35]), we obain ha w n () w() (n ) srongly in L 2 (, M; H). Arguing similarly, we have ha u n () u() (n ) srongly in L 2 (, M; H). We noe ha (1 α n A) 1/2 L(H,H) < 1 and herefore (1 α n A) 1/2 w n (1 α n A) 1/2 w L2 (,M;H) w n w L2 (,M;H) (n ). (3.24) I follows from Lemma 3.2 (see below) ha (1 α n A) 1/2 w w L2 (,M;H) (n ). (3.25) Combining (3.24) and (3.25), we observe ha ha is, u n w L2 (,M;H) = (1 α n A) 1/2 w n w L2 (1 α n A) 1/2 w n (1 α n A) 1/2 w L2 (1 α n A) 1/2 w w L2 (n ), u n () w() srongly in L 2 (, M; H), consequenly, u() w() and Lemma 3.1 is compleely proved. Lemma 3.2 Le f() L 2 (, M; H) and le α n (n ). Then (1 α n A) 1/2 f() f() srongly in L 2 (, M; H). The proof is given in Appendix. We now formulae and prove he main heorem of his secion. Theorem 3.1 Le a sequence {w n } K α n be given such ha {w n } is bounded in F, b α n (n ), and w n () w() in Θ loc as n. Then w() is a weak soluion of he 3D Navier Sokes sysem such ha w saisfies he energy inequaliy (1.12), i.e., w K, where K is he rajecory space of he 3D Navier Sokes sysem. 17

18 Proof. By he assumpion, we have w n F b C, n N (3.26) and since w n () w() in Θ loc as n we conclude ha w F b C. (3.27) We se u n = (1 αna) 2 1/2 w n. I is clear ha u n is a soluion of he original sysem (2.5) and (2.6). Inequaliy (3.26) implies ha { ess sup un () 2 αn u 2 n () 2} C, (3.28) sup 1 sup 1 u n (s) 2 D(A) ds sup { un (s) 2 α 2 n Au n (s) 2} ds C, (3.29) 1 w n (s) 2 D(A) ds C. (3.3) We now prove ha w() is a weak soluion of he 3D Navier Sokes sysem on any inerval (, M). The funcion w n () saisfies he equaion w n νaw n (1 α 2 na) 1/2 B(un, v n ) = (1 α 2 na) 1/2 g (3.31) in he space D (, M; D(A) ). Here v n = u n α 2 nau n. By he assumpion of he heorem, weakly in L 2 (, M; V ), weak- in L (, M; H), and w n () w() (n ) (3.32) w n () w() (n ) (3.33) weakly in L 2 (, M; D(A) ). Then, hese convergencies ake place in he space of disribuions D (, M; D(A) ). Moreover, i follows from (3.32) ha Aw n () Aw() (n ) (3.34) weakly in L 2 (, M; V ) and, hence, in he opology of D (, M; D(A) ) as well. Applying Lemma 3.2 in a paricular case, where he funcion f() g is ime independen, we find ha (1 α 2 na) 1/2 g g (n ) (3.35) srongly in L 2 (, M; H) and, clearly, in D (, M; D(A) ) as well. Thus having (3.33) (3.35), o prove ha w saisfy he equaion we mus esablish ha w νaw B(w, w) = g (3.36) (1 α 2 na) 1/2 B(un, v n ) B(w, w) as n (3.37) 18

19 in he space D (, M; D(A) ). Firsly, we prove ha B(u n, v n ) B(w, w) (n ) (3.38) weakly in he space L q (, M; D(A) ) for some q, 1 < q < 2. I follows from Lemma 3.1 ha We noe ha u n () w() (n ) in Θ loc. (3.39) B(u n, v n ) = B(u n, u n α 2 nau n ) = B(u n, u n ) α 2 n B(u n, Au n ) = B(u n, u n ) α 2 n B(u n, Au n ) (3.4) (Here, we have used he ideniy (2.8).) Consider boh erms of (3.4) separaely. We sar wih he second. By (2.16), we have α 2 n B(u n, Au n ) D(A) cα 2 n u n Au n. (3.41) Fixing an arbirary β, 1 < β < 2, we obain he following chain of inequaliies c β α 2β n c β α 2β n αn 2 B(u n (), Au n ()) β D(A) d c β α 2β n ( ( ) sup u n () γ [,M] ) [ sup u n γ u n q(β γ) d [,M] u n () β γ Au n () β d ] 1 q [ u n () β Au n () β d ] 1 p Au n pβ d, (3.42) where γ is an arbirary number such ha < γ < β, and, in (3.42), we have applied he Hölder inequaliy wih 1/p 1/q = 1 (hese numbers will be deermined laer on). Coninuing he chain of inequaliies afer (3.42), we have c β α 2β n α 2 n B(u n, Au n ) β D(A) d ( ) γ 2 [ sup u n 2 [,M] ] 1 [ q M ] 1 p u n q(β γ) d Au n pβ d. (3.43) We now se p = 2/β, q = 2/(2 β), and find he number γ from he equaion q(β γ) = 2, ha is, 2 (β γ) = 2 γ = 2(β 1). 2 β We see ha such γ saisfies he inequaliy < γ < β, since γ = 2(β 1) < β β < 2. 19

20 Replacing such p, q, and γ ino (3.43), we obain he following esimae: c β α 2 β n α 2 n B(u n, Au n ) β D(A) d ( ) β 1 [ sup αn u 2 n 2 u n 2 d [,M] ] 2 β 2 [ ] β αn Au 2 n 2 2 d. (3.44) We now use esimaes (3.28) and (3.29) and find ha he righ-hand side of (3.44) is less or equal han C 1 α 2 β n : Therefore, he erm α 2 n B(u n, Au n ) β D(A) d C 1 α 2 β n, 1 < β < 2. (3.45) α 2 n B(u n, Au n ) (n ) (3.46) srongly in L β (, M; D(A) ) for any β, 1 < β < 2. We now sudy he behavior of he erm B(u n, u n ) from (3.4). I follows from (3.39) ha u n () w() (n ) weakly in L 2 (, M; V ) and {u n ()} is bounded in his space. Besides, u n () w() (n ) weakly in L 2 (, M; D(A) ) and, hereby, { u n ()} is bounded in his space. applying he Aubin compacness heorem (see [34, 21, 35]), we obain ha Thus, u n () w() (n ) (3.47) srongly in L 2 (, M; H). Recall ha L 2 (, M; H) L 2 (T 3 [, M]) 3 and herefore we may assume ha u n (x, ) w(x, ) (n ) for a.e. (x, ) T 3 [, M]. (3.48) The ideniy (1.5) implies ha B(u n, u n ) = P 3 ( ) xj u j n u n. (3.49) j=1 I follows from (3.48) ha u j n(x, )u n (x, ) w j (x, )w(x, ) (n ) for a.e. (x, ) T 3 [, M]. (3.5) Recall ha {u n } is bounded in L 2 (, M; V ) and in L (, M; H). Hence, he well-known inequaliy B(u, u) V c u 1/2 u 3/2, u V, implies ha {u j nu n } is bounded in L 4/3 (, M; H) (3.51) 2

21 and in L 4/3 (T 3 [, M]) 3. Applying he known lemma on weak convergence from [21], we conclude from (3.5) and (3.51) ha u j n()u n () w j ()w() (n ) weakly in L 4/3 (T 3 [, M]) 3 and weakly in L 4/3 (, M; H). Then, due o (3.49), B(u n (), u n ()) B(w(), w()) (n ) (3.52) weakly in L 4/3 (, M; V ). Combining (3.46) and (3.52), we find ha weakly in L 4/3 (, M; D(A) ). We now sae ha B(u n, v n ) B(w, w) (n ) (1 α 2 na) 1/2 B(un, v n ) B(w, w) (n ) (3.53) weakly in L 4/3 (, M; D(A) ). Here, we need he following lemma ha is analogous o Lemma 3.1. Lemma 3.3 Le f n () L q (, M; D(A) ) and f n f (n ) weakly in he space L q (, M; D(A) ), q > 1. Le also α n (n ). Then (1 α n A) 1/2 f n () f() weakly in L q (, M; D(A) ). Proof. By he assumpion, for all ϕ L p (, M; D(A)) (1/p 1/q = 1) We have f n (), ϕ() d f(), ϕ() d (n ). (3.54) (1 αn A) 1/2 f n (), ϕ() d = fn, (1 α n A) 1/2 ϕ d = fn, (1 α n A) 1/2 ϕ ϕ d f n, ϕ d. (3.55) By Lemma 3.4 (see below), (1 α n A) 1/2 ϕ ϕ (n ) srongly in L p (, M; D(A)). Then fn, (1 α n A) 1/2 ϕ ϕ d f n Lq(,M;D(A) ) (1 α n A) 1/2 ϕ ϕ Lp(,M;D(A)) C (1 α n A) 1/2 ϕ ϕ Lp (n ) and he righ-hand side of (3.55) ends o f(), ϕ() d as n (see (3.54)). Lemma 3.3 is proved. Lemma 3.4 Le ϕ() L p (, M; D(A)) and α n (n ). Then (1 α n A) 1/2 ϕ() ϕ() srongly in L p (, M; D(A)). 21

22 The proof is given in Appendix. We now coninue he proof of Theorem 3.1. To his momen, we have esablished relaion (3.53) implying ha he funcion w() saisfies equaion (3.36). I is lef o prove ha w() saisfies he energy inequaliy (1.11) on every inerval (, M). Indeed he funcions w n () saisfies he energy equaliy (see (3.19)) 1 2 w n () 2 ψ ()d ν w n () 2 ψ()d = g, u n () ψ()d (3.56) for all ψ C (, M). Le now ψ for ], M[. We have already proved ha u n () w() (n ) srongly in L 2 (, M; H) (see (3.47)). Similarly we prove ha w n () w() (n ) srongly in L 2 (, M; H). (3.57) Then he real funcions w n () converge o w() as n srongly in L 2 (, M). In paricular, passing o a subsequence, we may assume ha w n () 2 w() 2 (n ) for a.e. [, M]. (3.58) Consider a sequence of funcions { w n () 2 ψ ()} in he space L 1 (, M). I follows from he assumpion of Theorem 3.1 ha his sequence is essenially bounded and, hence, i has an inegrable majoran. Then, by he Lebesque dominan convergence heorem, we obain from (3.58) ha w n () 2 ψ ()d w() 2 ψ ()d (n ). (3.59) We noe ha w n () ψ() w() ψ() (n ) weakly in L 2 (, M; V ) (he assumpion of Theorem 3.1). Consequenly, w() 2 ψ()d lim inf n w n () 2 ψ()d. (3.6) We have already noice ha u n () w() (n ) srongly in L 2 (, M; H). Therefore, g, u n () ψ()d g, w() ψ()d (n ). (3.61) Using (3.59) (3.61) and passing o he limi in (3.56), we obain ha 1 2 w() 2 ψ ()d ν w() 2 ψ()d g, w() ψ()d (3.62) for all ψ C (, M), ψ. Thus, we have proved ha he limi funcion w() in Theorem 3.1 is a weak soluion of he 3D Navier Sokes sysem and saisfies he energy inequaliy, ha is, w K. We use Theorem 3.1 in he nex secion, where we sudy he convergence of he rajecory aracors of he Navier Sokes-α model o he rajecory aracor of he 3D N. S. sysem. 22

23 4 Convergence of rajecories of he N. S.-α model o he rajecory aracor of he 3D N. S. sysem We denoe by A he rajecory aracor of he 3D Navier Sokes sysem v νav B(v, v) = g(x), (4.1) (A A, see Sec. 1). Recall ha he se A is bounded in F b, compac in Θ loc, and A K. We denoe by B α = {w α (x, ), }, < α 1, a family of funcions w α () = (1 α 2 A) 1/2 u α (), where u α () is a soluion of sysem (2.5), (2.6), and he norms of w α () in F b are uniformly bounded w α F b = w α L b 2 (R ;V ) w α L b (R ;H) w α L b 2 (R ;D(A) ) R, w α B α, where R is an arbirary number. Recall ha every w α () saisfies he equaion w α νaw α (1 α 2 A) 1/2 B(uα, v α ) = (1 α 2 A) 1/2 g, (4.2) where v α = (1 α 2 A) 1/2 w α () and u α = (1 α 2 A) 1/2 w α () We also se ˆB α = (1 α 2 A) 1/2 B α = { u α F b (1 α 2 A) 1/2 u α = w α B α }. where u α saisfies equaions (2.5) and (2.6). We denoe by K he kernel of equaion (4.1). Recall ha K is he union of all bounded (in he nom F b ) complee weak soluions {v(), R} of he Navier Sokes sysem (4.1) ha saisfy he energy inequaliy (1.18). We saw in Sec. 1 ha A = Π K. We now formulae he main heorem of his secion. Theorem 4.1 Le B α = {w α (x, ), }, < α 1, be bounded ses of soluions of equaion (4.2) ha saisfy he inequaliy w α F b R, α, < α 1. (4.3) Then he ses of shifed soluions {T (h)b α } (recall ha T (h)w() = w( h)) converge o he rajecory aracor A = Π K of he 3D N. S. sysem (4.1) in he opology Θ loc as h and α : T (h)b α A in Θ loc as h and α. (4.4) Moreover, he same convergence holds for he corresponding ses ˆB α = (1α 2 A) 1 B α : T (h) ˆB α A in Θ loc as h and α. (4.5) 23

24 Proof. Assume ha relaion (4.4) does no hold, i.e., here exis a neighborhood O(A ) in Θ loc and wo sequences α n, h n (n ) such ha T (h n )B αn O(A ). (4.6) So, here are soluions w αn ( ) B αn such ha he funcions W αn () = T (h n )w αn () = w αn ( h n ) do no belong o O(A ) : W αn ( ) / O(A ). (4.7) Noice ha he funcion W αn () is a soluion of equaion (4.2) on he inerval [ h n, ) wih α = α n, since W αn () is a backward ime shif of w αn () on h n. Recall ha he equaion (4.2) is auonomous. Moreover, i follows from (4.3) ha ( sup W αn () sup h n h n 1 1/2 ( 1 ) 1/2 W αn (s) ds) 2 sup W αn (s) 2 D(A) ds R. h n (4.8) This inequaliy implies ha he sequence {W αn ( )} is weakly compac in he space Θ M,M = L 2 ( M, M; V ) L ( M, M; H) {v v L 2 ( M, M; D(A) )} for every M, if we consider α n wih indices n such ha h n M. Therefore, for every fixed M >, we can choose a subsequence {α n } {α n } such ha {W αn ( )} converges weakly in Θ M,M. Then, using he sandard Canor diagonal procedure, we can consruc a funcion W (), R, and a subsequence {α n } {α n } such ha W αn W weakly in Θ M,M as n for any M >. (4.9) From (4.1), we obain he inequaliy for he limi funcion W (), R ( sup W () R sup R 1 1/2 ( 1 ) 1/2 W (s) ds) 2 sup W (s) 2 D(A) ds R. (4.1) R In paricular, we have ha W F b = L b 2(R; V ) L (R; H) {u u L b 2(R; D(A) )}. We now apply Theorem 3.1, where we can assume ha all he funcions are defined on he semiaxis [ M, ) insead of [, ) (equaions are auonomous). Then, from (4.9) and (4.1), we conclude ha W (x, ) is a weak soluion of he 3D N. S. sysem for all R and W (x, ) saisfies he energy inequaliy, ha is W K, where K is he kernel of equaion (4.1). Bu Π K = A and we have Π W A. A he same ime, we have esablished ha Π W αn Π W in Θ loc as n (4.11) (see (4.9)). In paricular for a large n Π W αn O(Π W ) O(A ). (4.12) This conradics (4.6). Therefore, (4.4) is rue. To prove (4.5), we combine (4.4) and Lemma 3.1. The proof is compleed. We now use Theorem 3.1 in order o sudy he behaviour of rajecory aracors of he Navier Sokes-α model as α. 24

25 As before, we consider he rajecory space K α for α >, of he Navier Sokesα model (2.5) and (2.6) ha was consruced in Sec. 3. Recall ha K α consiss of all he funcions of he form w α () = (1 α 2 A) 1/2 u α (),, where u α () is a soluion of (2.5) and (2.6), or equivalenly w α () is a soluion of (3.13). The space K α F b for all α >. We consider he opology Θ loc on K α. I is easy prove ha he space K α is closed in Θ loc. The ranslaion semigroup {T (h)} acs on K α by he formula T (h)w α () = w α ( h), h. From he definiion of K α, i follows ha T (h)k α K α for all h. Finally, Proposiion 3.2 implies (see (3.18)) ha here exiss an absorbing se of he semigroup {T (h)} in K α, bounded in F b and compac in Θ loc. (We noe, ha his absorbing se does no depend on α, since he consans C 3 and R 3 in (3.18) are independen of α.) Then, similar o Sec. 1, we prove he exisence of he rajecory aracor A α K α such ha A α is bounded in F, b compac in Θ loc, A α F b R, α, < α 1 (4.13) for some R > (independen of α). Recall ha T (h)a α = A α for all h ; and T (h)b α A α in Θ loc as h for any bounded se B α K α. Moreover, A α = Π K α, where K α is he kernel of equaion (3.13). Since he rajecory aracors A α saisfy (4.13) Theorem 4.1 is applicable o hese ses and we obain Corollary 4.1 The following limi relaions hold: A α A in Θ loc as α, (4.14) (1 α 2 A) 1/2 A α A in Θ loc as α. (4.15) Indeed, he family {A α, < α 1} is uniformly bounded wih respec o α ], 1]. Then, in (4.4), we se B α = A α and obain (4.14) because T (h)a α = A α for all h. The relaion (4.15) is sraighforward. We noice ha he following embeddings are coninuous (see [14]): Θ M,M L 2 ( M, M; H 1 δ ), (4.16) Θ M,M C([ M, M]; H δ ), < δ 1. (4.17) We recall ha he following quaniy is called he Hausdorff (non-symmeric) semidisance from a se X o a se Y in a Banach space E dis E (X, Y ) := sup x X From (4.14) and (4.15), we deduce dis E (x, Y ) = sup inf x y E. (4.18) x X y Y Corollary 4.2 For any fixed M >, he following limi relaions hold: dis L2 (,M;H 1 δ ) (A α, A ), dis C([,M];H δ ) (A α, A ) as α. In conclusion of his secion, we esablish he relaion beween he rajecory aracor A α and he global aracor A α of he Navier Sokes-α model for a fixed α > (see [6] and Sec. 2). 25

26 Proposiion 4.1 The following relaion holds: A α = {w() = (1 α 2 A) 1/2 u() = (1 α 2 A) 1/2 S α ()u, u A α }, (4.19) where {S α ()} is he semigroup corresponding o he α-model (2.5), (2.6) and acing in he space V. To prove (4.19) we recall ha he rajecory aracor A α is described using he kernel K α of sysem (3.13), while he global aracor A α has he similar presenaion in erms of he kernel of sysem (2.5) and (2.6). These kernels can be ransformed o each oher by mean of he operaor (1 α 2 A) 1/2. Finally, we formulae wo more proposiions ha follow from he resuls of [6] on well-posedness of he N. S.-α model. Proposiion 4.2 For every α > he rajecory aracor A α opological space Θ loc. is conneced in he Proposiion 4.3 The family of ses {A α, < α 1} is upper semiconinuous in Θ loc, i.e., for every α, < α 1, and for any neighborhood O(A α ) here is a δ = δ(α, O) > such ha A α O(A α ), α >, α α < δ. (4.2) We omi he proofs of Proposiions 4.2 and 4.3 because hey use he sandard reasoning known for well-posed problems (see, e.g., [8, 9]). Remark 4.1 I follows from inequaliy (2.24) ha he rajecory aracor A α is a se of more regular funcions, i.e. i is bounded in he space F b,s = L b 2(R ; D(A)) L (R ; V ) { w( ) L b 2(R ; H)} and, moreover, A α aracs bounded se of rajecories from K α in he srong local opology of he space Θ loc,s = L loc 2 (R ; D(A)) L loc (R ; V ) { w( ) L loc 2 (R ; H)}. However, hese properies are no uniform in α and hey do no persis passing o he limi as α. 5 Minimal limi of rajecory aracors A α as α Le A α be he rajecory aracor of he N. S.-α model, < α 1. As i has been proved above A α B, where B is he ball in F b (see (4.13)) wih radius R independen of α: A α F b B F b = R, α, < α 1. (5.1) I is clear ha he rajecory aracor A of he exac N. S. sysem also belongs o B (see Sec. 1). Recall ha he ball B is compac in he opology Θ loc. I follows from he Uryson compacness heorem ha he subspace B Θ loc equipped wih opology Θ loc is merizable (see [14] for more deails). We denoe he corresponding meric in B Θ loc by ρ(, ). The meric space iself, we denoe by B ρ. This meric space is compac and complee. Using new noaion, he resul of he previous secion can be wrien in he form dis Bρ (A α, A ), (5.2) 26

27 where dis Bρ (, ) denoes he Hausdorff disance from one se o anoher in B ρ (see (4.18)). We noe ha, in fac, he limi relaion (5.2) is sronger han he resuls of Corollary 4.2. Recall ha he se A B ρ is closed in B ρ. Le A min be he minimal closed subse of A ha saisfies he aracing propery (5.2), i.e. lim α dis Bρ (A α, A min ) = and A min belongs o every closed subse A A for which lim α dis Bρ (A α, A ) =. We call he se A min he minimal limi of he rajecory aracors A α as α. To prove ha such a se A min exiss we jus show ha A min = [ A α. (5.3) ]Bρ <δ 1 <α δ I is easy o prove ha a poin w belongs o he righ hand side of (5.3) if and only if here exis w αn A αn, n = 1, 2,... such ha ρ(w αn, w) and α n as n. Due o (5.2), such a limi poin w always belongs o A and, moreover, o every closed aracing se A. We sae ha he se (5.3) is aracing for A α as α. Assuming he converse, we have ha here is a sequence w αn A αn, such ha α n and dis Bρ (w αn, A min ) ε (5.4) for some fixed ε >. Recall ha w αn B ρ and B ρ is a compac meric space. Then, passing o a subsequence {w αn } {w αn }, we may assume ha ρ(w αn, w ) as α n for some w B ρ. Therefore by definiion, w A min, ha conradics o (5.4). We have proved ha he se A min is a minimal closed aracing subse of A. Proposiion 5.1 The minimal limi A min of rajecory aracors A α as α is a conneced subse of A in B ρ. Proof. Assume he converse. Then he se A min is union of wo closed noninersecing subses A 1 min and A 2 min, i.e., A min = A 1 min A 2 min and A 1 min A 2 min =. Since he space B ρ is compac, here are wo open ses O 1 and O 2 in B ρ such ha A 1 min O 1, A 2 min O 2, and O 1 O 2 =. Clearly, A min O 1 O 2. Therefore, by (5.2), here is a number α > such ha A α O 1 O 2, α, < α α. (5.5) We noe ha every se A α is conneced (see Proposiion 4.2), ha is, A α O 1 or A α O 2 for all α < α. A he same ime, since A min is he minimal limi of A α, we can find α 1 and α 2 such ha A α1 O 1 and A α2 O 2. (5.6) (oherwise, we can diminish A min ). Le, for definieness, < α 2 < α 1 < α. We se δ = sup{δ : A α O 2, α 2 α < α 2 δ}. (5.7) 27

28 We noe ha α 2 δ α 1 < α, (see (5.6)) and A α2 δ O 1 O 2 since α 2 δ < α (see (5.5)). We now sae ha A α2 δ can no belong o O 2. Indeed, if A α2 δ O 2 hen, by Proposiion 4.3, here is a small δ 2 > such ha A α2 δ δ 2 O 2, which conradics o he definiion of δ in (5.7). A he same ime, A α2 δ can no belong o O 1 neiher. Indeed, if A α2 δ O 1 hen again, by Proposiion 4.3, here is a small δ 1 > such ha A α2 δ δ 1 O 1, which conradics o he definiion of δ as well. However, all hese conradic o he inclusion A α2 δ O 1 O 2. The proof is compleed. Recall ha he se A min is compac. Finally we prove Proposiion 5.2 The minimal limi A min of rajecory aracors A α as α is sricly invarian wih respec o he ranslaion semigroup {T (h)}, ha is T (h)a min = A min, h. (5.8) Proof. Consider an arbirary w A min. By definiion, here is a sequence w αn A αn such ha ρ(w αn, w) as α n. The ranslaion semigroup {T (h)} is coninuous in Θ loc and, herefore, ρ(t (h)w αn, T (h)w) as α n. Since every A αn is sricly invarian, T (h)w αn A αn. Thus, T (h)w A min and we have proved ha T (h)a min A min, h. Le us prove he inverse inclusion. For any h and an arbirary w A min wih corresponding w αn A αn, ρ(w αn, w) (α n ), we have o find W A min such ha T (h)w = w. Since A αn is sricly invarian, here is an elemen W αn A αn such ha T (h)w αn = w αn. The sequence {W αn } belongs o he compac se B ρ. Passing o a subsequence {α n }, we have ha W αn W (n ) for some W B ρ. Then W A min. Since {T (h)} is coninuous T (h)w αn T (h)w (n ). However T (h)w αn = w αn, so, w αn T (h)w (n ) bu w αn w (n ). Hence, T (h)w = w and we have proved ha Consequenly, we obain (5.8). A min T (h)a min, h. A Appendix Proof of Lemma 3.2. Le {e n } be eigenvecors of he operaor A, i.e. Ae k = λ k e k, λ k > and λ k (k ). Then f() = k=1 f k()e k, where f k () = (f(), e k ), and f 2 L 2 (,M;H) = f k () 2 d. (A.1) We have (1 α 2 na) 1/2 f() = k=1 k=1 f k () (1 α 2 nλ k ) 1/2 e k 28