Invariant Measures and Statistical Solutions of the Globally Modified Navier-Stokes Equations

DISCRETE AND CONTINUOUS Websi e: h p://aimSciences.o g DYNAMICAL SYSTEMS Volume 00, Numbe 0, Xxxx XXXX pp. 000–000 INVARIANT MEASURES AND STATISTICAL SOLUTIONS OF THE GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Abs ac . We ob ain egula i y esul s o solu ions o he h ee dimensional sys em o globally modi ied Na ie -S okes equa ions, and we in es iga e he ela ionship be ween global a ac o s, in a ian measu es, ime-a e age mea- su es and s a is ical solu ions o hese sys em in he case o empo ally inde- penden o cing. 1. In oduc ion. The aim o his pape is o con inue wi h he analysis o he globally modi ied Na ie -S okes equa ions, which was ini ia ed ecen ly in he pa- pe s [2] and [8]. In ac , we a e in e es ed in se e al aspec s ela ed o he s a is ical analysis o hese equa ions, since s a is ical solu ions ha e p o en o be e y use- ul in he unde s anding o u bulence in he case o Na ie -S okes equa ions (see Foias e al. [5]). The main eason is ha he measu emen s o se e al aspec s o u bulen lows a e ac ually measu emen s o ime-a e age quan i ies. Al hough he e exis s an ex ensi e li e a u e on s a is ical hyd odynamics in luid mechanics and physics (see, e.g., Kolmogo o [11, 12], K aichnan [13], Landau and Li shi z [14], Dubois e al. [3], ...), on he ma hema ical side, we would like o men ion he con ibu ion o Hop [6], he pionee ing wo k o P odi [18], he book by Vishik and Fu siko [22], and he ecen pape by Lukaszewicz [16]. Le us now desc ibe ou model. Le Ω ⊂R3be an open bounded se wi h egula bounda y Γ, and conside he ollowing sys em o globally modi ied Na ie -S okes equa ions (GMNSE)                ∂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) whe e N∈(0,+∞) is gi en and FN: [0,+∞)→(0,1] is de ined by FN( ) := min 1,N , ∈[0,+∞). Da e: 7 May 2007. 2000 Ma hema ics Subjec Classi ica ion. 35Q30, 35K90, 37L30. Pa ly suppo ed by Minis e io de Educaci´on y Ciencia p ojec MTM2005-01412. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL The GMNSE (1) is indeed a global modi ica ion o he 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, (2) 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 modi ying ac o FN(kuk) depends on he no m kuk=k∇uk(L2(Ω))3×3, which in u n depends on ∇uo e he whole domain Ω and no jus a o nea he poin x∈Ω unde conside a ion. Essen ially, i p e en s la ge g adien s domina ing he dynamics and leading o explosions. I iola es he basic laws o mechanics, bu ma hema ically he GMNSE (1) a e a well de ined sys em o equa ions, jus like he modi ied e sions o he NSE o Le ay and o he s wi h o he molli ica ions o he nonlinea e m, see he e iew pape o Cons an in [1]. These modi ica ions a e local in cha ac e , whe eas ou s is global and essen ially educes es ima es o he nonlinea e m o hose o he wo dimensional NSE when he no m o he eloci y g adien exceeds a gi en h eshhold. Mo eo e , unlike in o he modi ica ions, he solu ions o he GMNSE coincide wi h hose o he NSE as long as his heshold is ne e exceeded. (We men ion in passing ha Flandoli and Maslowski [4] used a global cu o unc ion in ol ing he D(A1/4) no m o he wo dimensional s o- chas ic NSE). The GMNSE a e in e es ing in hemsel es, bu , mo e impo an ly, can be used o ob ain use ul in o ma ion abou he NSE. In pa icula , hey we e ecen ly used as an in e media e s ep by Kloeden and Vale o [10] o p o e ha he a ainabili y se o he weak solu ions o he 3-dim NSE which sa is y an ene gy cons ain is compac and connec ed se in he weak opology. The p esen pape is he i s in a sys ema ic in es iga ion o s a is ical solu ions o he GMNSE wi h he long e m aim o use hei p ope ies o ob ain a new unde s anding o he s a is ical solu ions o he h ee dimensional NSE. In his pape we i s p o e some egula i y p ope ies o he solu ions o ou GMNSE. This ensu es ha he global a ac o o he dynamical sys em SNgen- e a ed by (2) (when ( ) = does no depend on ime ) is a bounded se o he domain o he S okes ope a o (sec ions 3 and 4). Some p ope ies o he in a ian measu es associa ed o SNa e p o ed in Sec ion 5. In pa icula , we show ha any in a ian measu e is suppo ed by he a ac o . Finally, in he las sec ions we p o e he exis ence o in a ian measu es and he ela ionship wi h he concep s o ime-a e age solu ions, s a is ical solu ions and in a ian measu es. Indeed, we i s p o e he exis ence o ime-a e age measu es associa ed o any solu ion o (2) wi h ini ial alue in he phase space V(see Sec ion 2 o he de ini ion o V). Then, he exis ence o in a ian measu es is ob ained om he exis ence o ce ain ime-a e age measu es. Ou analysis in his a icle is inalized by p o ing ha he STATISTICAL SOLUTIONS 3 in a ian p obabili y measu es a e s a is ical solu ions o ou GMNSE. A p oo ha s a is ical solu ions o he GMNSE a e in a ian p obabili y measu es will be gi en in [9], since i equi es he de elopmen o new es ima es which a e oo leng hy o include he e. In a u u e pape we will in es iga e wha in o ma ion can be ob ained abou he s a is ical solu ions o he h ee dimensional NSE on a bounded domain om he esul s o his pape o he GMNSE. This is no a i ial unde aking in iew o he s ill un esol ed p oblem o uniqueness o s ong and weak solu ions o he h ee dimensional NSE, which equi es he use o se - alued dynamical sys ems as in [10]. 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 [15] and Temam [20, 21]): V=nu∈(C∞ 0(Ω))3: di u= 0o, H= he closu e o Vin (L2(Ω))3wi h inne p oduc (·,·) and associa e no m |·| , whe e o u, ∈(L2(Ω))3, (u, ) = 3 X j=1 ZΩ uj(x) j(x)dx, V= he closu e o Vin (H1 0(Ω))3wi h scala p oduc ((·,·)) and associa e 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 [19] o [20]), 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, (3) |b(u, , w)| ≤ C1|u|1/4kuk3/4k k|w|1/4kwk3/4,∀u, , w ∈V, (4) |b(u, , w)| ≤ C1kukk kkwk,∀u, , w ∈V. (5) Thus, i we deno e hBN(u, ), wi=bN(u, , w),∀u, , w ∈V, we ha e o example 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL kBN(u, )k∗≤NC1kuk,∀u, ∈V. (6) 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 [20] and [19]) 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, (7) |b(u, , w)| ≤ C2|u|1/4|Au|3/4k k|w|,∀u∈D(A), ∈V, w ∈H, (8) |b(u, , w)| ≤ C2kuk1/2|Au|1/2k k|w|,∀u∈D(A), ∈V, w ∈H, (9) De ini ion 1. Le u0∈Hand ∈L2(0, T;H), o all T > 0, be 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 (6), u0( )∈L2(0, T ;V0),and consequen ly (see [21]) 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≤ . (10) In [2] we p o ed ha i u0∈Vand ∈L2(0, T;H), hen he e exis s a unique solu ion uo he GMNSE wi h u(0) = u0, and u∈L2(0, T;D(A))∩C([0, T]; V) o all T > 0.Conside he Gale kin app oxima ions o he GMNSE, gi en by u0 m+νAum+PmBN(um, um) = Pm , um(0) = Pmu0,(11) whe e um=Pm j=1 um,jφj,Aum=Pm j=1 λjum,jφj, wi h λjand φjbeing he co - esponding eigen alues and o hono mal eigen unc ions o he ope a o A, and Pm being he p ojec ion on o he subspace o Hspanned by {φ1, . . . , φm}. F om he p oo o Theo em 7 in [2] and he uniqueness o u, i ollows ha i u0∈Vand ∈L2(0, T;H), hen among o he hings,        um→us ong in L2(0, T;V), um* u weak in L2(0, T ;D(A)), u0 m* u0weak in L2(0, T;H), (12) STATISTICAL SOLUTIONS 5 o all T > 0. I was also p o ed in [2] ha i u0∈H V, and ∈L∞(0,+∞;H), hen he e exis s a solu ion uo GMNSE wi h u(0) = u0, bu we do no know i i is unique. Ne e heless, in his las case, we know ha e e y solu ion uo he GMNSE wi h u(0) = u0sa is ies u∈L2(ε, T;D(A)) ∩C([ε, T]; V) o all 0 < ε < T. 3. Regula i y o he solu ions. Exis ence o an abso bing ball in D(A). Le ∈L∞(0,+∞;H),and deno e | |∞=k kL∞(0,+∞;H). Suppose i s ha u0∈V, and le u=u( ) be he co esponding solu ion o he GMNSE. Fo he Gale kin app oxima ions umwe easily ha e d d |um( )|2+νλ1|um( )|2≤| ( )|2 νλ1 , ≥0, hus mul iplying by eνλ1 and in eg a ing, one ob ains |um( )|2≤ |u0|2e−νλ1 +| |2 ∞ ν2λ2 1 o all ≥0.(13) I we now ake he inne p oduc o he Gale kin ODE (11) wi h Aum( ) we ob ain o all ≥0 1 2 d d kum( )k2+ν|Aum( )|2+bN(um( ), um( ), Aum( )) = ( ( ), Aum( )).(14) E iden ly, |( ( ), Aum( ))| ≤ ν 4|Aum( )|2+| |2 ∞ ν. Taking in o accoun ha λ1kum( )k2≤ |Aum( )|2and ha , by (8), |bN(um( ), um( ), Aum( ))| ≤ NC2|um( )|1/4|Aum( )|7/4, we ob ain d d kum( )k2+νλ1kum( )k2≤2 ν| |2 ∞+C(N)|um( )|2,(15) wi h C(N)gi en by C(N)=(NC2)877 29ν7.(16) Subs i u ing he bound (13) o |um( )|2in he di e en ial inequali y (15) gi es d d kum( )k2+νλ1kum( )k2≤C(N)|u0|2e−νλ1 +| |2 ∞ ν2 + C(N) νλ2 1. In eg a ing his inequali y hen gi es he solu ion es ima e kum( )k2≤(ku0k2+C(N) |u0|2)e−νλ1 +| |2 ∞ ν2λ12 + C(N) νλ2 1,∀ ≥0,(17) On he o he hand, by (9) and Young’s inequali y, one ob ains 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL |bN(um( ), um( ), Aum( ))| ≤ NC2kum( )k1/2|Aum( )|3/2 ≤ν 4|Aum( )|2+C(N)kum( )k2, wi h C(N)=27(NC2)4 4ν3.(18) Thus (14) simpli ies o d d kum( )k2+ν|Aum( )|2≤2 ν| |2 ∞+ 2C(N)kum( )k2 ≥0.(19) Le us ix 0 < ε ≤1.In eg a ing (19) be ween and +ε, we ob ain in pa icula νZ +ε |Aum(s)|2ds ≤2 ν| |2 ∞+ 2C(N)Z +ε kum(s)k2ds +kum( )k2∀ ≥0, and hen, by (17), one ob ains Z +ε |Aum(s)|2ds (20) ≤1+2C(N) ν(ku0k2+C(N)( + 1)|u0|2)e−νλ1 +| |2 ∞ ν22 + 1+2C(N) νλ12 + C(N) νλ2 1 ∀ ≥0∀m≥1. Suppose now ha 0, he ime de i a i e o , also belongs o L∞(0,+∞;H).In [8] i is p o ed ha 1 2 d d |u0 m( )|2+νku0 m( )k2(21) =−(FN(kum( )k))0b(um( ), um( ), u0 m( )) −bN(u0 m( ), um( ), u0 m( )) + ( 0( ), u0 m( )) ≥0, whe e |(FN(kum( )k))0| ≤ Nku0 m( )k kum( )k2χO( ) a.e. in (0,+∞),(22) wi h O={ ∈(0,+∞) : kum( )k ≥ N}. F om (3), (22) and Young’s inequali y, we ha e |2(FN(kum( )k))0b(um( ), um( ), u0 m( ))| ≤2Nku0 m( )kC1|u0 m( )|1/4ku0 m( )k3/4 = 2NC1|u0 m( )|1/4ku0 m( )k7/4(23) ≤νku0 m( )k2+7 8ν7 25(NC1)8|u0 m( )|2. STATISTICAL SOLUTIONS 7 By (4) and Young’s inequali y again |2bN(u0 m( ), um( ), u0 m( ))| ≤2NC1|u0 m( )|1/2ku0 m( )k3/2 ≤νku0 m( )k2+27 16ν3(NC1)4|u0 m( )|2.(24) Thus, i we deno e L(N)= 1 + 7 8ν7 25(NC1)8+27 16ν3(NC1)4, om (21), (23) and (24) we easily ob ain d d |u0 m( )|2≤L(N)|u0 m( )|2+| 0|2 ∞∀ ≥0∀m≥1.(25) I we in eg a e his inequali y be ween s∈[ , +ε] and +ε, we ha e |u0 m( +ε)|2≤ |u0 m(s)|2+L(N)Z +ε s |u0 m( )|2d +ε| 0|2 ∞∀0≤ ≤s≤ +ε, o all m≥1.In eg a ing now his las inequali y o sbe ween and +ε, we ob ain |u0 m( +ε)|2≤(ε−1+L(N))Z +ε |u0 m(s)|2ds +| 0|2 ∞∀ ≥0,(26) o all m≥1. Now, obse e ha by (11), he de ini ion o FNand (7), |u0 m( )| ≤ ν|Aum( )|+|BN(um( ), um( ))|+| ( )| ≤ν|Aum( )|+N kum( )k|b(um( ), um( ),·)|+| |∞ ≤(ν+NC2)|Aum( )|+| |∞, ≥0, and he e o e Z +ε |u0 m(s)|2ds ≤2| |2 ∞+ 2(ν+NC2)2Z +ε |Aum(s)|2ds ∀ ≥0,(27) o all m≥1. F om (20), (26) and (27), i is clea ha he e exis wo posi i e cons an s C(N) and D(N) , independen o ε, u0, and m, and inc easing wi h | |∞and | 0|∞, such ha |u0 m( +ε)|2≤(1 + ε−1)hC(N) +D(N) (ku0k2+ ( + 1)|u0|2)e−νλ1 i,(28) o all ≥0, m ≥1, ε ∈(0,1], u0∈V. Again, by (11) and (9), ν|Aum( )| ≤ |u0 m( )|+|BN(um( ), um( ))|+| ( )| ≤ |u0 m( )|+NC2kum( )k1/2|Aum( )|1/2+| |∞ ≤ |u0 m( )|+ν 2|Aum( )|+N2C2 2 2νkum( )k+| |∞, ≥0, and he e o e 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL |Aum( )|2≤12 ν2|u0 m( )|2+3N4C4 2 ν4kum( )k2+ 12| |2 ∞,∀ ≥0,(29) o all m≥1. F om (17), (28) and (29), one inds ha he e exis wo posi i e cons an s K(N) and R(N) , independen o ε,u0, and m, and inc easing wi h | |∞and | 0|∞, such ha |Aum( )|2≤(1 + ε−1)hR(N) +K(N) (1 + )ku0k2e−νλ1 i∀ ≥ε, (30) o all m≥1, ε ∈(0,1], u0∈V. Le ≥εbe ixed. By (30) we ob ain |Aum(s)|2≤(1 + ε−1)hR(N) +K(N) (2 + )ku0k2e−νλ1 i∀s∈[ , + 1],(31) o all m≥1. Now, we will make use o he ollowing esul (see [19] o a p oo ). Lemma 3. Le X⊂Ybe Banach spaces such ha Xis e lexi e and he injec ion o Xin Yis compac . Suppose ha {un}is a bounded sequence in L∞( 0, T ;X)such ha un* u weakly in Lp( 0, T ;X) o some p∈[1,+∞)and u∈C0([ 0, T]; Y). Then, u( )∈X o all ∈[ 0, T]and ku( )kX≤sup n≥1 kunkL∞( 0,T ;X),∀ ∈[ 0, T].(32) F om his lemma, inequali y (31) and con e gences in (12), we ha e u( )∈D(A),|Au( )|2≤(1+ε−1)hR(N) +K(N) (2 + )ku0k2e−νλ1 i∀ ≥ε, (33) whe e he inequali y is alid o all u0∈Vand all ε∈(0,1]. Suppose now ha u0∈Hand u( ) is a solu ion o he GMNSE wi h ini ial da um u0.We know ha u( )∈V o all > 0. Le ε∈(0,1] be ixed and le ( ) be he unique solu ion o he GMNSE wi h ini ial da um u(ε) and o cing e m b ( ) = ( +ε).By (33), ( )∈D(A) and |A ( )|2≤(1+ε−1)hR(N) +K(N) (2 + )ku(ε)k2e−νλ1 i∀ ≥ε. Bu , by uniqueness, ( ) = u( +ε) o all ≥0,and hus, om he abo e inequali y we ha e u( )∈D(A)∀ ≥2ε, (34) |Au( )|2≤(1 + ε−1)hR(N) +K(N) (2 + )ku(ε)k2e−νλ1( −1)i∀ ≥2ε. (35) Now le w( ) be he unique solu ion o he GMNSE wi h ini ial da um u(ε/2) and o cing e m e ( ) = ( +ε/2).By uniqueness we know ha w( ) = u( +ε/2) o all ≥0. STATISTICAL SOLUTIONS 9 F om es ima e (39) in P oposi ion 15 in [8] we ha e ε/2kw(1/2)k2≤KNeKN |u(ε/2)|2+Z1/2 0 |e (s)|2+Z1/2 0 |e 0(s)|2ds!, whe e KN>0, is a cons an depending only on C1,N,νand λ1.Consequen ly, ku(ε)k2≤2ε−1KNeKN|u(ε/2)|2+| |2 ∞+| 0|2 ∞.(36) Finally, he es ima e d d |u( )|2+νku( )k2≤| ( )|2 νλ1 ≥0, is well known and, in pa icula , implies ha |u(ε/2)|2≤ |u0|2+| |2 ∞ νλ1 .(37) F om (36), (37), (33) y (35), we ob ain he ollowing esul . P oposi ion 4. Suppose ha ∈W1,∞(0,+∞;H),and le u=u( )be a solu ion o GMNSE. Then u( )∈D(A)∀ > 0,(38) and he e exis wo posi i e cons an s K(N) and M(N) , independen o ε, u0and , and inc easing wi h | |∞and | 0|∞, such ha a) i u(0) ∈V, hen |Au( )|2≤(1 + ε−1)hR(N) +M(N) (1 + )ku0k2e−νλ1 i∀ ≥ε, (39) o all ε∈(0,1]; b) in gene al, i u(0) ∈H, hen |Au( )|2≤(1 + ε−1)R(N) +ε−1(1 + ε−1)M(N) (1 + )(1 + |u0|2)e−νλ1 ,(40) o all ≥2ε, 0< ε ≤1. In pa icula , he e exis s a T0=T0(|u0|)depending only on |u0|,| |∞,| 0|∞,C1, C2, N,νand λ1such ha |Au( )|2≤2R(N) ∀ ≥T0(|u0|).(41) Rema k 5. Obse e ha (40) implies ha i ∈W1,∞(0,+∞;H), hen e e y solu ion o he GMNSE belongs o L∞(ε, +∞;D(A)) o all ε > 0. I , mo eo e , he ini ial da um u0∈D(A), hen i can be p o ed ha he co esponding solu ion u=u( )o he GMNSE belongs o L∞(0,+∞;D(A)),and, mo e exac ly, sup ≥0 |Au( )|<+∞. 16 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL L(ϕ)−L(eϕ) = LIMT→∞ 1 TZT 0 (ϕ(u( )) −eϕ(u( ))) d = LIMT→∞ 1 TZT0 0 (ϕ(u( )) −eϕ(u( ))) d = 0. Thus L(ϕ) = L(eϕ). Le now ψ∈C(BN), whe e BNis conside ed as a me ic subspace o H. As BN is a closed subse o Hand ψis con inuous and bounded, we can ex end ψ o a con inuous unc ion ϕ∈C(H). By he conside a ions abo e, he alue L(ϕ) is he same o any ϕ∈C(H)∪C(V) such ha ϕ|BN=ψ. The e o e, we can de ine a unc ional lon C(BN) by l(ψ) = L(ϕ),whe e ϕ∈C(H)∪C(V) is any con inuous ex ension o ψ. I is e iden ha lis a posi i e linea unc ional on C(BN), and because BNis compac , i ollows om Kaku ani-Riesz ep esen a ion heo em (see [5]) ha he e exis s a posi i e measu e µon BNsuch ha l(ψ) = ZBN ψ( )dµ( )∀ψ∈C(BN). The measu e µcan be ex ended o a measu e on Hby se ing µ(F) = µ(F∩ BN) o all Bo el measu able subse Fo H. I is clea ha µ(H BN) = 0,and obse e ha i ϕ∈C(V), hen ϕ|BN∈C(BN) (i n→ 0in BN, hen, as BNis a compac subse o V, n→ 0in V, and he e o e ϕ( n)→ϕ( 0)). Consequen ly o any ϕ∈C(H)∪C(V) we ha e LIMT→∞ 1 TZT 0 ϕ(u( )) d =L(ϕ) = l(ϕ|BN) =ZBN ϕ|BN( )dµ( ) =ZH ϕ( )dµ( ). Finally, no e ha aking ϕ≡1,we deduce ha µ(H) = LIMT→∞1 = 1,so ha µ is a p obabili y measu e on H. Rema k 15. Wi h an almos iden ical p oo o ha o he p eceding heo em, one can p o e ha he e exis s a ime-a e age measu e o any solu ion o he au- onomous GMNSE. Now, we can ob ain exis ence o SN-in a ian measu es. P oposi ion 16. Le u( ) = SN( )u0be he solu ion o he au onomous GMNSE co esponding o u0∈V, and le µbe a ime-a e age measu e o u( )such ha C(V)⊂L1(H, µ)and (59) is sa is ied o all ϕ∈C(V).Then µis an SN-in a ian measu e. STATISTICAL SOLUTIONS 17 P oo .- Le ψ∈C(H) and τ > 0. The unc ion ψ◦SN(τ) : 7→ ψ(SN(τ) ) is also con inuous in V, and by (59) wi h ϕ eplaced by ψ◦SN(τ), we ha e ZH ψ(SN(τ) )dµ( ) = LIMT→∞ 1 TZT 0 ψ(SN( +τ)u0)d = LIMT→∞ 1 TZT+τ τ ψ(SN( )u0)d = LIMT→∞ "1 TZT 0 ψ(SN( )u0)d +1 TZT+τ T ψ(SN( )u0)d −1 TZτ 0 ψ(SN( )u0)d #. Bu , obse e ha SN( )u0belongs o a compac se o V, and hence also o H, o all ≥0. The e o e ψ(SN( )u0) emains bounded o all ≥0,so LIMT→∞ "1 TZT+τ T ψ(SN( )u0)d −1 TZτ 0 ψ(SN( )u0)d #= 0. Thus, ZH ψ(SN(τ) )dµ( ) = LIMT→∞ 1 TZT 0 ψ(SN( )u0)d =ZH ψ( )dµ( ), o all τ > 0 and any ψ∈C(H). By densi y, we hen ob ain ZH φ(SN(τ) )dµ( ) = ZH φ( )dµ( )∀φ∈L1(H, µ). In pa icula , aking he cha ac e is ic unc ion o any measu able subse Eo V, we hen ha e µ(E) = µ(SN(τ)−1E)∀τ > 0, and he SN-in a iance o µ ollows. 8. S a iona y S a is ical Solu ions o he GMNSE in he au onomous case. De ini ion 17. We de ine Tas he se o eal alued unc ionals Φ = Φ( )on H such ha (i) c := sup | |≤ |Φ( )|<+∞ o all > 0; (ii) o any ∈V he e exis s Φ0( )∈Vsuch ha |Φ( +w)−Φ( )−(Φ0( ), w)| |w|→0 as |w| → 0 wi h w∈V; (60) (iii) he mapping 7→ Φ0( )is con inuous and bounded as unc ion om Vin o V. 18 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Le us deno e k k= +∞i ∈H V. Wi h his con en ion, i µis a p obabili y measu e on Hand RHk k2dµ( )<+∞, hen µ(H V) = 0. We de ine GN( ) = −νA −BN( , ) + ∀ ∈V. (61) Taking in o accoun (5) and ha |FN( )−FN(s)| ≤ | −s| ∀ , s ≥0, i is easy o ob ain ha kBN( , )−BN(u, u)k∗≤NC1(2k k+kuk)k −uk ∀ u, ∈V, and he e o e he mapping GN:V→V0is con inuous. Also, by (6), kGN( )k∗≤(ν+NC1)k k+λ−1/2 1| | ∀ ∈V. (62) Thus, i Φ ∈ T , |hGN( ),Φ0( )i| ≤ [(ν+NC1)k k+λ1−1/2| |] sup w∈V kΦ0(w)k ∀ ∈V, and consequen ly, i µis a p obabili y measu e on Hwi h RHk kdµ( )<+∞, hen he in eg al RHhGN( ),Φ0( )idµ( ) is ini e. De ini ion 18. A s a iona y s a is ical solu ion o he GMNSE is a p obabili y measu e µon Hsuch ha (i) ZH k k2dµ( )<+∞; (ii) ZH hGN( ),Φ0( )idµ( ) = 0 o any Φ∈ T ; (iii) Z{a≤| |2<b} {νk k2−( , )}dµ( )≤0 o any 0≤a < b ≤+∞. We ha e he ollowing esul Theo em 19. Any SN-in a ian p obabili y measu e on His a s a iona y s a is- ical solu ion o he GMNSE. P oo .- Le µbe a SN-in a ian p obabili y measu e on H. We know by P opo- si ion 4 and Lemma 9 ha µ(H BN) = 0.The se BNis a compac subse o V and hence he unc ion k kis bounded on BN. Thus, o any β > 0, ZH k kβdµ( ) = ZBN k kβdµ( )<+∞, and in pa icula condi ion (i) in De ini ion 18 holds. Le us ix 0 ≤a < b ≤+∞,an le us deno e E={ ∈V:a≤ | |2< b}, F ={ ∈H:a≤ | |2< b}. Since µis SN-in a ian , and by (i) he unc ion 7→ νk k2−( , ) is µ-in eg able, we ha e ZF [νk k2−( , )] dµ( ) (63) =ZE [νk k2−( , )] dµ( ) =ZE [νkSN( ) k2−( , SN( ) )] dµ( )∀ ≥0. STATISTICAL SOLUTIONS 19 Now, obse e ha easoning as in he p oo o Lemma 6 one can ob ain ha kSN( ) k2≤2 ν| |2+k k2e2C(N) ∀ ≥0 (64) o any ∈V. Consequen ly, aking in o accoun condi ion (i), we can in eg a e in (63) and apply Fubini’s heo em, o ob ain ZF [νk k2−( , )] dµ( ) (65) =1 TZT 0ZE [νkSN( ) k2−( , SN( ) )] dµ( )d =1 TZEZT 0 [νkSN( ) k2−( , SN( ) )] d dµ( ) o all T > 0. Bu we know ha o all ∈Vand all T > 0, |SN(T) |2− | |2+ 2νZT 0 kSN( ) k2d = 2 ZT 0 ( , SN( ) )d , and hence, by (65), ZF [νk k2−( , )] dµ( ) = 1 2TZE (| |2− |SN(T) |2)dµ( )∀T > 0.(66) Now obse e ha |SN( ) |2≤ | |2e−νλ1 +| |2 ν2λ2 1 ∀ ≥0∀ ∈V. (67) Suppose i s ha b < +∞. Then, om (66) and (67), and making T→+∞,we ind ha ZF [νk k2−( , )] dµ( ) = 0. I b= +∞,i is enough o conside a sequence bn%+∞.Thus we ha e p o ed ha µsa is ies condi ion (iii) in De ini ion 18. Finally, we mus p o e ha µsa is ies condi ion (ii) in De ini ion 18. Le Φ ∈ T be gi en. Fo each in ege m≥1,deno e Φm( ) = Φ(Pm )∀ ∈H. I is easy o see ha Φm∈C1(H), wi h Φ0 m( ) = PmΦ0(Pm ) o all ∈H. E iden ly, sup | |≤ |Φm( )| ≤ sup |w|≤ |Φ(w)|=c <+∞, and he mapping 7→ Φ0 m( ) is con inuous and bounded as a unc ion om Vin o V. Thus (see o example [17] Theo em 4.2, page 65) o any m≥1 and all ∈V we ha e Φm(SN(T) )−Φm( ) = ZT 0 hGN(SN( ) ),Φ0 m(SN( ) )id ∀T > 0.(68) Since µis SN-in a ian , ZH hGN( ),Φ0 m( )idµ( ) = ZV hGN(SN( ) ),Φ0 m(SN( ) )idµ( ) 20 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL o all ≥0.Now we in eg a e, and aking in o accoun (62), (64) and condi ion (i), we can apply Fubini heo em, and we ob ain ZH hGN( ),Φ0 m( )idµ( ) = 1 TZT 0ZV hGN(SN( ) ),Φ0 m(SN( ) )idµ( )d =1 TZVZT 0 hGN(SN( ) ),Φ0 m(SN( ) )id dµ( ), and hus, by (68), ZH hGN( ),Φ0 m( )idµ( ) = 1 TZV [Φm(SN(T) )−Φm( )] dµ( )∀T > 0. Taking T→+∞in he las equali y, and using he mean alue heo em, he boundedness o Φ0 mon V, and he inequali y (67), we ob ain ZH hGN( ),Φ0 m( )idµ( ) = 0.(69) Now, obse e ha kΦ0 m( )−Φ0( )k=kPmΦ0(Pm )−Φ0( )k ≤ kPmΦ0(Pm )−PmΦ0( )k+kPmΦ0( )−Φ0( )k ≤ kΦ0(Pm )−Φ0( )k+kPmΦ0( )−Φ0( )k. The e o e, by he con inui y o Φ0on V, we ob ain kΦ0 m( )−Φ0( )k → 0 as m→+∞ o all ∈V. (70) Finally, by (62), he boundedness o Φ0and Φ0 mon V, and (70), om (69) we ha e ha ZH hGN( ),Φ0( )idµ( ) = 0. As a di ec consequence o P oposi ion 16 and Theo em 19, we ha e Co olla y 20. Le µbe a ime-a e age measu e o a solu ion u( )o he GMNSE such ha C(V)⊂L1(H, µ)and (59) is sa is ied o all ϕ∈C(V).Then µis a s a iona y s a is ical solu ion o he GMNSE. Rema k 21. Le u0∈V. Le µNbe o each N > 0a ime-a e age measu e o he solu ion SN( )u0o GMNSE. We know ha he e exis s a subsequence o solu ions SN0( )u0 ha con e ges in an adequa e sense o a solu ion u( )o NSE (see page 432 in [2]). To ou knowledge, he ques ion emains open i he µN0con e ge in some sense o a measu e ela ed o u( ). 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 and J. Real, Unique s ong solu ions and V-a ac o s o a h ee dimensional sys em o globally modi ied Na ie -S okes equa ions, Ad . Nonlinea S udies 6 (2006), 411-436. [3] T. Dubois, F. Jaube eau, and R. 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E-mail add ess, Tom´as Ca aballo: [email p o ec ed] E-mail add ess, Pe e E. Kloeden: [email p o ec ed] E-mail add ess, Jos´e Real: [email p o ec ed] (Tom´as Ca aballo 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) (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