Equivalence of Invariant Measures and Stationary Statistical Solutions for The Autonomous Globally Modified Navier-Stokes Equations

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume 00, Numbe 0, Xxxx XXXX pp. 000–000 EQUIVALENCE OF INVARIANT MEASURES AND STATIONARY STATISTICAL SOLUTIONS FOR THE AUTONOMOUS GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Abs ac . A new p oo o exis ence o solu ions o he h ee dimensional sys em o globally modi ied Na ie -S okes equa ions in oduced in [3] by Ca a- ballo, Kloeden and Real is ob ained using a smoo he Gale kin scheme. This is hen used o in es iga e he ela ionship be ween in a ian measu es and s a is ical solu ions o his sys em in he case o empo ally independen o c- ing e m. Indeed, we a e able o p o e ha a s a iona y s a is ical solu ion is also an in a ian p obabili y measu e unde sui able assump ions. 1. In oduc ion. Le Ω ⊂R3be an open bounded se wi h egula bounda y Γ, and conside he ollowing sys em o Na ie -S okes equa ions (NSE) on Ω wi h a homogeneous Di ichle bounda y condi ion                ∂u ∂ −ν∆u+ (u· ∇)u+∇p= ( ) in (0,+∞)×Ω, ∇ · u= 0 in (0,+∞)×Ω, u= 0 on (0,+∞)×Γ, u(0, x) = u0(x), x ∈Ω, whe e ν > 0 is he kinema ic iscosi y, uis he eloci y ield o he luid, p he p essu e, u0 he ini ial eloci y ield, and ( ) a gi en ex e nal o ce ield. The e exis many modi ied e sions o Na ie -S okes equa ions due o Le ay and o he s wi h molli ica ion (and/o cu o ) o he nonlinea e m as a way o app oxima e he o iginal p oblem, see o ins ance he e iew pape o Cons an in [1]. We also men ion he pape [5] by Flandoli and Maslowski wi h a global cu o unc ion used in a 2-dimensional s ochas ic con ex . In 2006, Ca aballo, Kloeden and Real (c . [3]) p oposed a 3-dimensional model whe e he nonlinea e m included a cu o ac o FN(kuk) based on he no m o he g adien o he solu ion in he whole domain. Namely, o N∈(0,+∞) he Da e: 17 Sep embe 2008. 2000 Ma hema ics Subjec Classi ica ion. 35Q30, 35K90, 37L30. Key wo ds and ph ases. Modi ied Na ie -S okes model, s a is ical solu ions, in a ian measu es. Pa ly suppo ed by Minis e io de Educaci´on y Ciencia p ojec MTM2005-01412. 1 2 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL unc ion FN: [0,+∞)→(0,1] is de ined by FN( ) := min 1,N , ∈[0,+∞). They called he esul ing sys em                ∂u ∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) in (0,+∞)×Ω, ∇ · u= 0 in (0,+∞)×Ω, u= 0 on (0,+∞)×Γ, u(0, x) = u0(x), x ∈Ω, (1) he globally modi ied Na ie -S okes equa ions (GMNSE) and es ablished he well- posedness o he model, in pa icula he absence o blow-up o solu ions, as well as he exis ence o global V−a ac o s and ce ain ela ionships wi h he o iginal NSE model. See also [7,8] o o he s udies and applica ions o he GMNSE as well as he e iew pape [2]. Foias e al. [6] (see also he pape s ci ed he ein) ha e made a e y sys ema ic s udy o in a ian measu es and s a is ical solu ions o he NSE models in 2 and 3 dimensions. This in es iga ion was con inued in [4] by Ca aballo, Kloeden and Real o he GMNSE, who p o ed he exis ence o ime-a e aged measu es, hen o in a ian measu es and inally ha any in a ian measu e is a s a iona y s a is ical solu ion. We no e ha a ela ion be ween a amily o ime-a e age p obabili y measu es and he pullback a ac o o a non-au onomous e sion o he NSE was es ablished in [10]. Ou aim in his pape is o es ablish he in e se o he las esul in [4], i.e. ha a s a iona y s a is ical solu ion o he GMNSE is an in a ian p obabili y measu e, which we p o e unde sui able assump ions. The s uc u e o he pape is as ollows. We ecall some p elimina ies on he used unc ional spaces, de ini ions and exis ence esul s in Sec ion 2. In Sec ion 3we es ablish a new p oo o he exis ence o s ong solu ions o he GMNSE wi h a di e en app oach om ha used in [3] since we need mo e egula i y on he Gale kin app oxima ions. Es ima es in D(A) a e ob ained o hese solu ions o he new Gale kin scheme in Sec ion 4. Then, in Sec ion 5we es ablish ou main esul , Theo em 15, p o ing unde addi ional assump ions ha a s a is ical solu ion is an in a ian p obabili y measu e. Finally, o imp o e he cla i y o he pape , p oo s o some echnical esul s used ea lie a e gi en in he Appendix. 2. P elimina ies. To se ou p oblem in he abs ac amewo k, we conside he ollowing usual abs ac spaces (see Lions [9] and Temam [12,13]): V=nu∈(C∞ 0(Ω))3: di u= 0o, H= he closu e o Vin (L2(Ω))3wi h inne p oduc (·,·) and associa ed no m |·| , whe e o u, ∈(L2(Ω))3, (u, ) = 3 X j=1 ZΩ uj(x) j(x)dx, STATISTICAL SOLUTIONS 3 V= he closu e o Vin (H1 0(Ω))3wi h inne p oduc ((·,·)) and associa ed no m k·k ,whe e o u, ∈(H1 0(Ω))3, ((u, )) = 3 X i,j=1 ZΩ ∂uj ∂xi ∂ j ∂xi dx. I ollows ha V⊂H≡H0⊂V0,whe e he injec ions a e dense and compac . Finally, we will use k·k∗ o he no m in V0and h·,·i o he duali y pai ing be ween Vand V0. Now we de ine he ilinea o m bon V×V×Vby b(u, , w) = 3 X i,j=1 ZΩ ui ∂ j ∂xi wjdx, ∀u, , w ∈V, and we deno e bN(u, , w) = FN(k k)b(u, , w),∀u, , w ∈V. The o m bNis linea in uand w, bu i is nonlinea in . E iden ly we ha e bN(u, , ) = 0, o all u, ∈V. Mo eo e , by he p ope ies o b(see [11] o [12]), he e exis s a cons an C1>0 only dependen on Ω such ha |b(u, , w)| ≤ C1kukk k|w|1/4kwk3/4,∀u, , w ∈V, (2) |b(u, , w)| ≤ C1|u|1/4kuk3/4k k|w|1/4kwk3/4,∀u, , w ∈V, (3) |b(u, , w)| ≤ C1kukk kkwk,∀u, , w ∈V. (4) Thus, i we deno e hBN(u, ), wi=bN(u, , w),∀u, , w ∈V, we ha e o example kBN(u, )k∗≤NC1kuk,∀u, ∈V. (5) We also conside he ope a o A:V→V0de ined by hAu, i= ((u, )).Deno ing D(A) = (H2(Ω))3∩V, hen Au =−P∆u, ∀u∈D(A),is he S okes ope a o (Pis he o ho-p ojec o om (L2(Ω))3on o H). We ecall (see [12] and [11]) ha he e exis s a cons an C2>0 depending only on Ω such ha |b(u, , w)| ≤ C2|Au|k k|w|,∀u∈D(A), ∈V, w ∈H, (6) |b(u, , w)| ≤ C2|u|1/4|Au|3/4k k|w|,∀u∈D(A), ∈V, w ∈H, (7) |b(u, , w)| ≤ C2kuk1/2|Au|1/2k k|w|,∀u∈D(A), ∈V, w ∈H, (8) |b(u, , w)| ≤ C2kukk k1/2|A |1/2|w|,∀u∈V, ∈D(A), w ∈H, (9) |b(u, , w)| ≤ C2|u|k k1/2|A |1/2kwk,∀u∈H, ∈D(A), w ∈V. (10) 4 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL De ini ion 1. Le u0∈Hand ∈L2(0, T ; (L2(Ω))3) o all T > 0be gi en. A weak solu ion o (1) is any u∈L2(0, T ;V) o all T > 0such ha (u0( ) + νAu( ) + BN(u( ), u( )) = ( )in D0(0,+∞;V0), u(0) = u0, o equi alen ly (u( ), w) + νZ 0 ((u(s), w)) ds +Z 0 bN(u(s), u(s), w)ds = (u0, w) + Z 0 ( (s), w)ds, o all ≥0 and all w∈V. Rema k 2. Obse e ha i u∈L2(0, T;V) o all T > 0and sa is ies he equa ion u0( ) + νAu( ) + BN(u( ), u( )) = ( )in D0(0,+∞;V0), hen, as a consequence o (5), u0( )∈L2(0, T;V0),and hence (see [13])u∈ C([0,+∞); H)and sa is ies he ene gy equali y |u( )|2− |u(s)|2+ 2νZ s ku( )k2d = 2 Z s ( ( ), u( )) d o all 0≤s≤ . The ollowing esul was p o ed in [3]. Theo em 3. [c . [3, Th.7(a)]] Suppose ha ∈L2(0, T ; (L2(Ω))3) o all T > 0, and le u0∈Vbe gi en. Then, he e exis s a unique weak solu ion uo (1), which is in ac a s ong solu ion, i.e., i sa is ies u∈C([0, T]; V)∩L2(0, T;D(A)) o all T > 0. In he nex sec ion, o echnical easons, we ob ain a simila esul on exis ence o solu ion (see P oposi ion 5and Rema k 6below), using a di e en Gale kin app oxima ion om ha used in [3]. 3. A smoo h Gale kin app oxima ion o GMNSE. By egula i y equi e- men s o he manipula ions in he p oo o ou main esul (c . Theo em 15), we need o in oduce a new Gale kin app oxima ion scheme o he GMNSE. Ou i s s ep is o conside a amily o smoo he unc ions ha app oxima es he unca ed e m FNappea ing in he GMNSE. This can be ca ied ou in he classical way, by molli ica ion. Le {ρm, m ≥1} ⊂ C∞(R) be a egula izing sequence in R, i.e., such ha 0≤ρm( )≤1, he suppo o ρmis included in he in e al [−1/m, 1/m],and RRρm( )d = 1, o all m≥1. Le us conside FNp olonged as FN( ) = 1 i ≤0,and deno e FN,m he con olu ion ρm∗FN,i.e., FN,m( ) = ZR ρm( −s)FN(s)ds ∀ ∈R.(11) I is well known ha FN,m ∈C∞(R)∩L∞(R),and also FN,m( )→FN( ), as m→+∞, uni o mly on bounded subse s o R.Mo eo e , 0≤FN,m( )≤ kFNkL∞(R)= 1 ∀ ∈R,∀m≥1. In ac , his las es ima e can be imp o ed. The ollowing esul conce ning he unc ions FN,m holds. STATISTICAL SOLUTIONS 5 Lemma 4. The unc ions FN,m de ined by (11) sa is y he ollowing inequali ies: 0≤ FN,m( )≤N+ 1 ∀ ≥0,∀m≥1.(12) |F0 N,m( )| ≤ 1/N ∀ ∈R,∀m≥1.(13) |F0 N,m( )| ≤ N+ 1 N, 2|F0 N,m( )| ≤ (N+ 1)2 N∀ ≥0,∀m≥1.(14) F0 N,m( ) = 0 ∀ ≤1/m, ∀m > 2/N. (15) I s p oo is gi en in he Appendix o a oid dis ac ing om he main low o ideas he e. Now, le us deno e BN,m( , ) he elemen o V0de ined by hBN,m( , ), wi=FN,m(k k)b( , , w)∀u, w ∈V. Conside he Gale kin app oxima ions o he GMNSE gi en by u0 m+νAum+PmBN,m(um, um) = Pm , um(0) = Pmu0,(16) whe e um=Pm j=1 um,jφj,Aum=Pm j=1 λjum,jφj. He e he λjand φja e he co esponding eigen alues and eigen unc ions (o hono mal in H, o hogonal in V) o he ope a o Aand Pmis he p ojec ion on o he subspace o Hspanned by {φ1, . . . , φm}. Now, we es ablish he main esul o his sec ion, which p o ides a new p oo o he exis ence o solu ion gi en in Theo em 3. P oposi ion 5. Unde he abo e no a ion, i ∈L∞(0, T;H) o all T > 0, he scheme (16) is well de ined, i.e. o each u0∈V, he e exis s a smoo h amily (C1in ime) o unc ions {um},solu ions o (16), ha a e de ined in (0,+∞).Mo eo e , he e exis s a subsequence o {um}m ha con e ges (in se e al senses) o he unique solu ion o (1). Rema k 6. The assump ion ∈L∞(0, T ;H)ins ead o L2(0, T; (L2(Ω))3)as in Theo em 3is no essen ial, bu only o simplici y in he no a ion. Indeed, since p ojec o ope a o s a e in ol ed in he es - unc ions space, he e is no es ic ion in conside ing aking alues in Hins ead o (L2(Ω))3.On he o he hand, he s onge equi emen in ime, L∞ins ead o L2,is also o he sake o con enience in he calculus. Obse e ha ou inal goal in Sec ion 5is wi h an au onomous e m . P oo . [o P oposi ion 5] The exis ence and uniqueness o local solu ion o (16) is a consequence o he Pica d Theo em, and he ac ha he local solu ion is a global one de ined in he same in e al o ime as is in iew o he es ima es (19) and (24) below. A e ha , we will p oceed by he compac ness me hod: i s ly p o ing uni o m es ima es o umin Hand hen in V. Obse e ha o he es ima es in Hwe only need u0∈H. Fix a alue T > 0 and le us deno e | |∞=k kL∞(0,T ;H). 6 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL I is s anda d ha i we ake he inne p oduc o he Gale kin ODE (16) wi h umand use b(um, um, um) = 0, we ha e d d |um|2+ 2νkumk2= 2( , um),(17) and hen, using λ1|um|2≤ kumk2,we ob ain d d |um|2+νkumk2≤| |2 ∞ νλ1 ,(18) and consequen ly |um( )|2+νZ 0 kum(s)k2ds ≤ |u0|2+| |2 ∞ νλ1 (19) o all ≥0 in he in e al o de ini ion o um.The well posedness o umin (0, T) ollows om G onwall’s lemma. Now we ob ain uni o m es ima es in V o all umin (0, T ) using u0∈V. We ake he inne p oduc o he Gale kin ODE (16) wi h Aumand ob ain 1 2 d d kumk2+ν|Aum|2+bN,m(um, um, Aum) = ( , Aum).(20) E iden ly, |( , Aum)| ≤ ν 4|Aum|2+| |2 ∞ ν In addi ion, by (8), (12) and Young’s inequali y, one ob ains |bN,m(um, um, Aum)| ≤ C2(N+ 1)kumk1/2|Aum|3/2 ≤ν 4|Aum|2+C0 Nkumk2, wi h C0 N=27(N+ 1)4C4 2 4ν3. Thus (20) simpli ies o d d kumk2+ν|Aum|2≤2 ν| |2 ∞+ 2C0 Nkumk2.(21) F om (21) and he ac ha kum(0)k=kPmu0k ≤ ku0k by he choice o he basis {φj}o H, one easily concludes ha he sequence {um} is bounded in C([0, T]; V) and in L2(0, T;D(A)) o all T > 0. Then, obse e ha |bN,m(um, um, w)| ≤ (N+ 1)C2kumk1/2|Aum|1/2|w|, o any w∈H, and hus, he sequence {PmBN,m(um, um)}is bounded in L2(0, T;H) o all T > 0. The e o e, om he equa ion u0 m=−νAum−PmBN,m(um, um)+Pm , one sees ha he sequence {u0 m}is also bounded in L2(0, T;H). Consequen ly, as D(A)⊂V⊂Hwi h compac injec ion, by Theo em 5.1 in Chap e 1 o [9] he e exis s an elemen u∈L∞(0, T;V)∩L2(0, T;D(A)) o all STATISTICAL SOLUTIONS 7 T > 0, and a subsequence {uµ}o {um}, such ha                    uµ→us ong in L2(0, T;V), uµ→ua.e. in (0, T)×Ω, uµ* u weak in L2(0, T;D(A)), uµ ∗ * u weak-s a in L∞(0, T;V), u0 µ* u0weak in L2(0, T;H), (22) o all T > 0. Also, as uµcon e ges o in L2(0, T ;V) o all T > 0, we can assume, possibly ex ac ing a subsequence, ha kuµ( )k → ku( )ka.e. in (0,+∞). Then, by he boundedness o {uµ}in C([0, T]; V) and he ac ha FNis con in- uous and FN,m con e ges o FNuni o mly on bounded subse s o Ras m→+∞, one ob ains FN,m(kuµ( )k)→FN(ku( )k) a.e. in (0,+∞).(23) F om (22) and (23) we can ake limi s in (16) exac ly as in [3], and we ob ain ha uis a solu ion o (1). Rema k 7. I we assume ha is also de ined o nega i e ime, o ins ance, ∈L∞(−T, T;H), he abo e solu ions uma e also well de ined o all (−T, T). This will be used below -in he p oo o Theo em 15- o ensu e ha we can go backwa ds in ime on each solu ion umo he Gale kin scheme. This is due o (17), whence we ha e d d |um|2+ 2νλm|um|2≥ −| |2 ∞− |um|2, whe e | |∞deno es now k kL∞(−T,T ;H).The e o e, in eg a ing be ween ∈(−T, 0) and 0we deduce |um( )|2≤ |u0|2+| || |2 ∞+(1+2νλm)Z0 |um(s)|2ds (24) o all < 0in he in e al o de ini ion o um. Thus, by (19), (24) and again G onwall’s lemma, umis de ined in all (−T, T ). Rema k 8. Obse e ha , since he sequence {u0 m}is bounded in L2(0, T ;H) o any T > 0, he sequence umas a sequence o unc ions om [0, T ]in o His equicon inuous. Then, since {um}is bounded in C([0, T ]; V)and V⊂Hwi h compac injec ion, we can asse ha om any subsequence o {um}we can ex ac a subsequence ha con e ges in C([0, T]; H), and aking in o accoun he p e ious a gumen s o he exis ence o solu ion uo (1), and he uniqueness o such a so- lu ion (c . Theo em 3), one has ha um→uin C([0, T ]; H),as m→+∞, o all T > 0. 4. Es ima es in D(A) o he Gale kin app oxima ions. In his sec ion we go u he in de eloping new es ima es o he solu ions o (16), bu now wi h ini ial da a u0∈D(A),and assuming ha ∈W1,∞(R;H).[The mo e es ic- i e hypo hesis o ∈W1,∞(R;H),which we will use om he e on ins ead o W1,∞((−T, T); H) o all T, is only o he sake o b e i y in he s a emen s.] 8 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL P oposi ion 9. Wi h he no a ion o Sec ion 3, i ∈W1,∞(R;H), he e exis s a posi i e cons an ˜ M(N) , independen o u0, and m, such ha |Aum( )| ≤ ˜ M(N) (1 + |Au0|)∀ ≥0, o all m > 2/N, u0∈D(A).(25) P oo . Now le us deno e |·|∞=k·kL∞(R;H).F om (18) we ha e d d eνλ1 |um( )|2≤| |2 ∞ νλ1 eνλ1 , ≥0, and in eg a ing one ob ains |um( )|2≤ |u0|2e−νλ1 +| |2 ∞ ν2λ2 1 o all ≥0.(26) Now, i we use (20) and ake in o accoun ha λ1kum( )k2≤ |Aum( )|2and ha , by (7) and (12), |bN,m(um( ), um( ), Aum( ))| ≤ (N+ 1)C2|um( )|1/4|Aum( )|7/4, hen we can ob ain he inequali y d d kum( )k2+νλ1kum( )k2≤2 ν| |2 ∞+C(N) 1|um( )|2,(27) wi h C(N) 1=(N+ 1)8C8 277 29ν7. Subs i u ing he bound (26) o |um( )|2in he di e en ial inequali y (27) gi es d d kum( )k2+νλ1kum( )k2≤C(N) 1|u0|2e−νλ1 +| |2 ∞ ν 2 + C(N) 1 νλ2 1!. In eg a ing his inequali y we deduce ha kum( )k2≤(ku0k2+C(N) 1 |u0|2)e−νλ1 +| |2 ∞ ν2λ1 2 + C(N) 1 νλ2 1!,∀ ≥0.(28) On he o he hand, in eg a ing (21) be ween and + 1, we ob ain in pa icula νZ +1 |Aum(s)|2ds ≤2 ν| |2 ∞+ 2C0 NZ +1 kum(s)k2ds +kum( )k2∀ ≥0, and hen, by (28), one ob ains Z +1 |Aum(s)|2ds ≤1+2C0 N ν(ku0k2+C(N) 1( + 1)|u0|2)e−νλ1 (29) +| |2 ∞ ν2"2 + 1+2C0 N νλ1 2 + C(N) 1 νλ2 1!# ∀ ≥0∀m≥1. Suppose now ha m > 2/N. F om (16) and (15), we ob ain o he second de i a i e o um, u00 m( ) = −νAu0 m( )−F0 N,m(kum( )k)((u0 m( ), um( ))) kum( )kχOm( )PmB(um( ), um( )) −FN,m(kum( )k)PmB(u0 m( ), um( )) −FN,m(kum( )k)PmB(um( ), u0 m( )) + Pm 0( ),(30) whe e Om={ ∈[0,+∞) : kum( )k>1/m} and χOm( ) is he indica o unc ion o he se Om. STATISTICAL SOLUTIONS 9 Mul iplying in (30) by u0 m,we ob ain 1 2 d d |u0 m( )|2+νku0 m( )k2 =−F0 N,m(kum( )k)((u0 m( ), um( ))) kum( )kχOm( )b(um( ), um( ), u0 m( )) −FN,m(kum( )k)b(u0 m( ), um( ), u0 m( )) + ( 0( ), u0 m( )) ≥0.(31) F om (2), (14) and Young’s inequali y, we ha e |F0 N,m(kum( )k)((u0 m( ), um( ))) kum( )kχOm( )b(um( ), um( ), u0 m( ))| ≤2C1(N+ 1)2 Nku0 m( )k|u0 m( )|1/4ku0 m( )k3/4 =2C1(N+ 1)2 NC1|u0 m( )|1/4ku0 m( )k7/4 ≤ν 2ku0 m( )k2+7 4ν7 25C2 1(N+ 1)2 N8 |u0 m( )|2.(32) By (3), (12) and Young’s inequali y again |2FN,m(kum( )k)b(u0 m( ), um( ), u0 m( ))| ≤2(N+ 1)C1|u0 m( )|1/2ku0 m( )k3/2 ≤ν 2ku0 m( )k2+27 2ν3(N+ 1)4C4 1|u0 m( )|2.(33) Thus, i we deno e L(N) 1= 1 + 1 27 2ν7C2 1(N+ 1)2 N8 +27 ν3(N+ 1)4C4 1, om (31), (32) and (33) we easily ob ain d d |u0 m( )|2≤L(N) 1|u0 m( )|2+| 0|2 ∞∀ ≥0∀m > 2/N. (34) I we in eg a e his inequali y be ween s∈[ , + 1] and + 1,we ha e |u0 m( + 1)|2≤ |u0 m(s)|2 +L(N) 1Z +1 s |u0 m( )|2d +| 0|2 ∞∀0≤ ≤s≤ + 1,∀m > 2/N. In eg a ing now his las inequali y o sbe ween and + 1,we ob ain |u0 m( + 1)|2≤(1 + L(N) 1)Z +1 |u0 m(s)|2ds +| 0|2 ∞∀ ≥0,∀m > 2/N. (35) Now, obse e ha by (16), (6) and (12), |u0 m( )| ≤ ν|Aum( )|+|FN,m(kum( )k)B(um( ), um( ))|+| ( )| ≤[ν+ (N+ 1)C2]|Aum( )|+| |∞, ≥0, and he e o e Z +1 |u0 m(s)|2ds ≤2| |2 ∞+ 2[ν+ (N+ 1)C2]2 ×Z +1 |Aum(s)|2ds ∀ ≥0,∀m≥2/N. (36) 16 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Then, since Pm∈ L(H) is sel -adjoin , om (57) and (58) we ob ain ZH Φ(S(m) N(T) )dµ( ) =ZH Φ(Pm )dµ( ) (59) +ZT 0ZH (PmGN,m(Pm )−PmGN( ), D (Φm)( , )) dµ( )d ∀T > 0. Now, obse e ha PmGN,m(Pm )−PmGN( ) = PmBN( , )−PmBN,m(Pm , Pm ), and hen, by (58) and since P∗ m=Pm,we ob ain om (59) ZH Φ(S(m) N(T) )dµ( ) (60) =ZH Φ(Pm )dµ( ) +ZT 0ZH (PmBN( , )−PmBN,m(Pm , Pm ), D (Φm)( , )) dµ( )d =ZH Φ(Pm )dµ( ) +ZT 0ZH (BN( , )−BN,m(Pm , Pm ), D (Φm)( , )) dµ( )d ∀T > 0. Now, we p o e ha as m→+∞, he las in eg al e m goes o ze o o all T > 0. Obse e ha |(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))| =|FN(k k)b( , , D (Φm)( , )) −FN,m(kPm k)b(Pm , Pm , D (Φm)( , ))| ≤ |FN(k k)b( −Pm, , D (Φm)( , ))| +|FN(k k)−FN,m(k k)||b(Pm , , D (Φm)( , ))| +|FN,m(k k)b(Pm , −Pm, D (Φm)( , ))| +|FN,m(k k)−FN,m(kPm k)||b(Pm , Pm , D (Φm)( , ))|, and he e o e, by (4), (12) and (13), |(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))| ≤NC1k −Pm kkD (Φm)( , ))k +|FN(k k)−FN,m(k k)|C1k k2kD (Φm)( , ))k +(N+ 1)C1k −Pm kkD (Φm)( , ))k +C1 Nk −Pm kk k2kD (Φm)( , ))k.(61) On he o he hand, easoning exac ly as on page 232 in [6], one ob ains kD (Φm)( , ))k =kΦ0(S(m) N( ) ksup w∈H, |A−1/2w| ≤ 1 |A−1/2(D S(m) N( ) )w| ∀ ≥0,∀ , w ∈H, STATISTICAL SOLUTIONS 17 o any m≥1,and hen, aking in o accoun ha Φ ∈ T has de i a i e Φ0(·) bounded on V, he e exis s a cons an CΦ>0 such ha kD (Φm)( , ))k ≤ CΦsup w∈H, |A−1/2w| ≤ 1 |A−1/2(D S(m) N( ) )w|(62) ∀ ≥0,∀ , w ∈H, o any m≥1. Since BNis a bounded subse o D(A), om (50), P oposi ion 9and (62) we deduce ha he e exis s a cons an C ,BNsuch ha kD (Φm)( , ))k ≤ CΦeT C ,BN∀0≤ ≤T, ∀ ∈ BN,(63) o any m > 2/N. F om (61) and (63) we deduce ha o each T > 0 he e exis s a cons an CT>0, such ha |(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))| ≤CT(k −Pm k+|FN(k k)−FN,m(k k)|) o all ( , )∈[0, T ]× BN, o any m > 2/N. Then, as µ(H BN) = 0,we ob ain ha o each T > 0, ZT 0ZH |(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|dµ( )d (64) ≤TCTZBN (k −Pm k+|FN(k k)−FN,m(k k)|)dµ( )∀m > 2/N. As k −Pm k → 0 as m→+∞and k −Pm k ≤ 2k k o any ∈ BN,and FN,m con e ges o FNas m→+∞uni o mly on bounded subse s o R,i is immedia e o m (64) ha ZT 0ZH |(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|dµ( )d →0 as m→+∞, and hen, om (60) we conclude ha ZH Φ(S(m) N(T) )dµ( )−ZH Φ(Pm )dµ( )→0 as m→+∞, (65) o all T > 0. Bu , aking again in o accoun ha µ(H BN) = 0, ha BNis a bounded subse o D(A),and ha Φ is bounded on bounded subse s o H, we deduce om Rema k 8and (52) ha ZH Φ(S(m) N(T) )dµ( )→ZV Φ(SN(T) )dµ( ) as m→+∞, o all T > 0, and ZH Φ(Pm )dµ( )→ZV Φ( )dµ( ) as m→+∞, and hen, by (65) we ha e (45), as desi ed. 18 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Appendix: Es ima es on he unca ed unc ion. This sec ion is de o ed o p o e some echnical es ima es conce ning he unc ions FN,m,de ined by (11), namely hose o Lemma 4. To s a , obse e ha FN,m( ) = Z +1/m −1/m ρm( −τ)FN(τ)dτ. Consequen ly, i N < −1/m, hen FN,m( ) = Z +1/m −1/m ρm( −τ)N τdτ ≤N −1/m. Thus, •i N≥ −1/m, hen FN,m( )≤ ≤N+ 1/m, and •i N < −1/m, hen FN,m( )≤ N −1/m =N1 + 1 m( −1/m) ≤N1 + 1 mN =N+ 1/m. Hence, (12) is p o ed. Mo eo e , i is s aigh o wa d o check ha F0 N,m( )=(ρm∗F0 N)( ),whe e F0 N( ) = −N 2χ(N,+∞)( ),and he e o e (13) also holds. Fo (14), ela ed o |F0 N,m( )|and 2|F0 N,m( )|,we ha e o p oceed mo e ca e- ully. Obse e ha |F0 N,m( )| ≤ Z +1/m −1/m ρm( −τ)|F0 N(τ)|dτ ≤Z +1/m −1/m ρm( −τ)N τ2dτ, and consequen ly, i > N + 1, hen |F0 N,m( )| ≤ N ( −1/m)2 ≤N ( −1)2. Thus, STATISTICAL SOLUTIONS 19 •i > N + 1, |F0 N,m( )| ≤ N ( −1)2 =N −11 + 1 −1 <1 + 1 N =N+ 1 N, and 2|F0 N,m( )| ≤ N 2 ( −1)2 =N1 + 1 −12 < N 1 + 1 N2 =(N+ 1)2 N. •i 0 ≤ ≤N+ 1, hen, by (13), one also has |F0 N,m( )| ≤ N+ 1 N, 2|F0 N,m( )| ≤ (N+ 1)2 N. Hence, combining all si ua ions one eco e s he inequali y in (14). Finally, in o de o p o e (15) obse e ha i m > 2/N and ≤1/m, hen FN,m( ) = Z +1/m −1/m ρm( −τ)FN(τ)dτ = 1, whence he las esul ollows. REFERENCES [1] P. Cons an in, Nea iden i y ans o ma ions o he Na ie -S okes equa ions, in Handbook o Ma hema ical Fluid Dynamics, Vol. II, 117–141, No h-Holland, Ams e dam, 2003. [2] T. Ca aballo, P.E. Kloeden, J.A. Langa, J. Real and J. 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Real, Pullback V-a ac o s o he h ee dimensional globally modi ied Na ie -S okes equa ions: exis ence and ini e ac al dimension, Commun. Pu e Appl. Anal. 6(2007), 937–955. 20 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL [8] P.E. Kloeden and J. Vale o, The weak connec edness o he a ainabili y se o weak solu ions o he h ee-dimensional Na ie -S okes equa ions, P oc. R. Soc. Lond. Se . A Ma h. Phys. Eng. Sci. 463 (2007), 1491–1508. [9] J.L. Lions, Quelques M´e hodes de R´esolu ion des P obl`emes aux Limi es Non Lin´eai es, Dunod, Pa is, 1969. [10] G. Lukaszewicz, Pullback a ac o s and s a is ical solu ions o 2-D Na ie -S okes equa ions, Disc e e Con in. Dyn. Sys . Se ies B,9(2008), 643–659. [11] J.C. Robinson, In ini e-Dimensional Dynamical Sys ems, Camb idge Uni e si y P ess, Cam- b idge, 2001. [12] R. Temam, Na ie -S okes Equa ions, No h-Holland, Ams e dam, 1977. [13] R. Temam, Na ie -S okes Equa ions and Nonlinea Func ional Analysis, Second Edi ion, SIAM, Philadelphia, 1995. E-mail add ess, Pe e E. Kloeden: [email p o ec ed] E-mail add ess, Ped o Ma ´ın-Rubio: [email p o ec ed] E-mail add ess, Jos´e Real: [email p o ec ed] (Pe e E. Kloeden) Ins i u ¨ u Ma hema ik, Johann Wol gang Goe he-Uni e si ¨ a , D-60054 F ank u am Main, Ge many (Ped o Ma ´ın-Rubio and Jos´e Real) Dp o. Ecuaciones Di e enciales y An´ alisis Num´ e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa (Spain)