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

DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 00, Number 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 Abstract. We obtain regularity results for solutions of the three dimensional system of globally modified Navier-Stokes equations, and we investigate the relationship between global attractors, invariant measures, time-average measures and statistical solutions of these system in the case of temporally independent forcing. 1. Introduction. The aim of this paper is to continue with the analysis of the globally modified Navier-Stokes equations, which was initiated recently in the papers [2] and [8]. In fact, we are interested in several aspects related to the statistical analysis of these equations, since statistical solutions have proven to be very useful in the understanding of turbulence in the case of Navier-Stokes equations (see Foias et al. [5]). The main reason is that the measurements of several aspects of turbulent flows are actually measurements of time-average quantities. Although there exists an extensive literature on statistical hydrodynamics in fluid mechanics and physics (see, e.g., Kolmogorov [11, 12], Kraichnan [13], Landau and Lifshitz [14], Dubois et al. [3], ...), on the mathematical side, we would like to mention the contribution of Hopf [6], the pioneering work of Prodi [18], the book by Vishik and Fursikov [22], and the recent paper by Lukaszewicz [16]. Let us now describe our model. Let Ω ⊂R3be an open bounded set with regular boundary Γ, and consider the following system of globally modified Navier-Stokes equations (GMNSE)                ∂u ∂t −ν∆u+FN(kuk) [(u· ∇)u] + ∇p=f(t) in (0,+∞)×Ω, ∇ · u= 0 in (0,+∞)×Ω, u= 0 on (0,+∞)×Γ, u(0, x) = u0(x), x ∈Ω, (1) where N∈(0,+∞) is given and FN: [0,+∞)→(0,1] is defined by FN(r) := min 1,N r, r ∈[0,+∞). Date: 7 May 2007. 2000 Mathematics Subject Classification. 35Q30, 35K90, 37L30. Partly supported by Ministerio de Educaci´on y Ciencia project MTM2005-01412. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL The GMNSE (1) is indeed a global modification of the Navier-Stokes equations (NSE) on Ω with a homogeneous Dirichlet boundary condition                ∂u ∂t −ν∆u+ (u· ∇)u+∇p=f(t) in (0,+∞)×Ω, ∇ · u= 0 in (0,+∞)×Ω, u= 0 on (0,+∞)×Γ, u(0, x) = u0(x), x ∈Ω, (2) where ν > 0 is the kinematic viscosity, uis the velocity field of the fluid, pthe pressure, u0the initial velocity field, and f(t) a given external force field. The modifying factor FN(kuk) depends on the norm kuk=k∇uk(L2(Ω))3×3, which in turn depends on ∇uover the whole domain Ω and not just at or near the point x∈Ω under consideration. Essentially, it prevents large gradients dominating the dynamics and leading to explosions. It violates the basic laws of mechanics, but mathematically the GMNSE (1) are a well defined system of equations, just like the modified versions of the NSE of Leray and others with other mollifications of the nonlinear term, see the review paper of Constantin [1]. These modifications are local in character, whereas ours is global and essentially reduces estimates of the nonlinear term to those of the two dimensional NSE when the norm of the velocity gradient exceeds a given threshhold. Moreover, unlike in other modifications, the solutions of the GMNSE coincide with those of the NSE as long as this theshold is never exceeded. (We mention in passing that Flandoli and Maslowski [4] used a global cut off function involving the D(A1/4) norm for the two dimensional stochastic NSE). The GMNSE are interesting in themselves, but, more importantly, can be used to obtain useful information about the NSE. In particular, they were recently used as an intermediate step by Kloeden and Valero [10] to prove that the attainability set of the weak solutions of the 3-dim NSE which satisfy an energy constraint is compact and connected set in the weak topology. The present paper is the first in a systematic investigation of statistical solutions of the GMNSE with the long term aim to use their properties to obtain a new understanding of the statistical solutions of the three dimensional NSE. In this paper we first prove some regularity properties of the solutions of our GMNSE. This ensures that the global attractor for the dynamical system SNgenerated by (2) (when f(t) = fdoes not depend on time t) is a bounded set of the domain of the Stokes operator (sections 3 and 4). Some properties for the invariant measures associated to SNare proved in Section 5. In particular, we show that any invariant measure is supported by the attractor. Finally, in the last sections we prove the existence of invariant measures and the relationship with the concepts of time-average solutions, statistical solutions and invariant measures. Indeed, we first prove the existence of time-average measures associated to any solution of (2) with initial value in the phase space V(see Section 2 for the definition of V). Then, the existence of invariant measures is obtained from the existence of certain time-average measures. Our analysis in this article is finalized by proving that the STATISTICAL SOLUTIONS 3 invariant probability measures are statistical solutions of our GMNSE. A proof that statistical solutions of the GMNSE are invariant probability measures will be given in [9], since it requires the development of new estimates which are too lengthy to include here. In a future paper we will investigate what information can be obtained about the statistical solutions of the three dimensional NSE on a bounded domain from the results of this paper for the GMNSE. This is not a trivial undertaking in view of the still unresolved problem of uniqueness of strong and weak solutions of the three dimensional NSE, which requires the use of set-valued dynamical systems as in [10]. 2. Preliminaries. To set our problem in the abstract framework, we consider the following usual abstract spaces (see Lions [15] and Temam [20, 21]): V=nu∈(C∞ 0(Ω))3: div u= 0o, H= the closure of Vin (L2(Ω))3with inner product (·,·) and associate norm |·| , where for u, v ∈(L2(Ω))3, (u, v) = 3 X j=1 ZΩ uj(x)vj(x)dx, V= the closure of Vin (H1 0(Ω))3with scalar product ((·,·)) and associate norm k·k ,where for u, v ∈(H1 0(Ω))3, ((u, v)) = 3 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. It follows that V⊂H≡H0⊂V0,where the injections are dense and compact. Finally, we will use k·k∗for the norm in V0and h·,·i for the duality pairing between Vand V0. Now we define the trilinear form bon V×V×Vby b(u, v, w) = 3 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, ∀u, v, w ∈V, and we denote bN(u, v, w) = FN(kvk)b(u, v, w),∀u, v, w ∈V. The form bNis linear in uand w, but it is nonlinear in v. Evidently we have bN(u, v, v) = 0,for all u, v ∈V. Moreover, by the properties of b(see [19] or [20]), there exists a constant C1>0 only dependent on Ω such that |b(u, v, w)| ≤ C1kukkvk|w|1/4kwk3/4,∀u, v, w ∈V, (3) |b(u, v, w)| ≤ C1|u|1/4kuk3/4kvk|w|1/4kwk3/4,∀u, v, w ∈V, (4) |b(u, v, w)| ≤ C1kukkvkkwk,∀u, v, w ∈V. (5) Thus, if we denote hBN(u, v), wi=bN(u, v, w),∀u, v, w ∈V, we have for example 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL kBN(u, v)k∗≤NC1kuk,∀u, v ∈V. (6) We also consider the operator A:V→V0defined by hAu, vi= ((u, v)).Denoting D(A) = (H2(Ω))3∩V, then Au =−P∆u, ∀u∈D(A),is the Stokes operator (Pis the ortho-projector from (L2(Ω))3onto H). We recall (see [20] and [19]) that there exists a constant C2>0 depending only on Ω such that |b(u, v, w)| ≤ C2|Au|kvk|w|,∀u∈D(A), v ∈V, w ∈H, (7) |b(u, v, w)| ≤ C2|u|1/4|Au|3/4kvk|w|,∀u∈D(A), v ∈V, w ∈H, (8) |b(u, v, w)| ≤ C2kuk1/2|Au|1/2kvk|w|,∀u∈D(A), v ∈V, w ∈H, (9) Definition 1. Let u0∈Hand f∈L2(0, T;H), for all T > 0, be given. A weak solution of (1) is any u∈L2(0, T;V)for all T > 0such that (u0(t) + νAu(t) + BN(u(t), u(t)) = f(t)in D0(0,+∞;V0), u(0) = u0, or equivalently (u(t), w) + νZt 0 ((u(s), w)) ds +Zt 0 bN(u(s), u(s), w)ds = (u0, w) + Zt 0 (f(s), w)ds, for all t≥0 and all w∈V. Remark 2. Observe that if u∈L2(0, T;V)for all T > 0and satisfies the equation u0(t) + νAu(t) + BN(u(t), u(t)) = f(t)in D0(0,+∞;V0), then, as a consequence of (6), u0(t)∈L2(0, T ;V0),and consequently (see [21]) u∈C([0,+∞); H)and satisfies the energy equality |u(t)|2− |u(s)|2+ 2νZt s ku(r)k2dr = 2 Zt s (f(r), u(r)) dr for all 0≤s≤t. (10) In [2] we proved that if u0∈Vand f∈L2(0, T;H), then there exists a unique solution uof the GMNSE with u(0) = u0, and u∈L2(0, T;D(A))∩C([0, T]; V) for all T > 0.Consider the Galerkin approximations for the GMNSE, given by u0 m+νAum+PmBN(um, um) = Pmf, um(0) = Pmu0,(11) where um=Pm j=1 um,jφj,Aum=Pm j=1 λjum,jφj, with λjand φjbeing the corresponding eigenvalues and orthonormal eigenfunctions of the operator A, and Pm being the projection onto the subspace of Hspanned by {φ1, . . . , φm}. From the proof of Theorem 7 in [2] and the uniqueness of u, it follows that if u0∈Vand f∈L2(0, T;H), then among other things,        um→ustrong 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 for all T > 0. It was also proved in [2] that if u0∈H\V, and f∈L∞(0,+∞;H),then there exists a solution uof GMNSE with u(0) = u0, but we do not know if it is unique. Nevertheless, in this last case, we know that every solution uof the GMNSE with u(0) = u0satisfies u∈L2(ε, T;D(A)) ∩C([ε, T]; V) for all 0 < ε < T. 3. Regularity of the solutions. Existence of an absorbing ball in D(A). Let f∈L∞(0,+∞;H),and denote |f|∞=kfkL∞(0,+∞;H). Suppose first that u0∈V, and let u=u(t) be the corresponding solution of the GMNSE. For the Galerkin approximations umwe easily have d dt|um(t)|2+νλ1|um(t)|2≤|f(t)|2 νλ1 , t ≥0, thus multiplying by eνλ1tand integrating, one obtains |um(t)|2≤ |u0|2e−νλ1t+|f|2 ∞ ν2λ2 1 for all t≥0.(13) If we now take the inner product of the Galerkin ODE (11) with Aum(t) we obtain for all t≥0 1 2 d dtkum(t)k2+ν|Aum(t)|2+bN(um(t), um(t), Aum(t)) = (f(t), Aum(t)).(14) Evidently, |(f(t), Aum(t))| ≤ ν 4|Aum(t)|2+|f|2 ∞ ν. Taking into account that λ1kum(t)k2≤ |Aum(t)|2and that, by (8), |bN(um(t), um(t), Aum(t))| ≤ NC2|um(t)|1/4|Aum(t)|7/4, we obtain d dtkum(t)k2+νλ1kum(t)k2≤2 ν|f|2 ∞+C(N)|um(t)|2,(15) with C(N)given by C(N)=(NC2)877 29ν7.(16) Substituting the bound (13) for |um(t)|2in the differential inequality (15) gives d dtkum(t)k2+νλ1kum(t)k2≤C(N)|u0|2e−νλ1t+|f|2 ∞ ν2 + C(N) νλ2 1. Integrating this inequality then gives the solution estimate kum(t)k2≤(ku0k2+C(N)t|u0|2)e−νλ1t+|f|2 ∞ ν2λ12 + C(N) νλ2 1,∀t≥0,(17) On the other hand, by (9) and Young’s inequality, one obtains 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL |bN(um(t), um(t), Aum(t))| ≤ NC2kum(t)k1/2|Aum(t)|3/2 ≤ν 4|Aum(t)|2+C(N)kum(t)k2, with C(N)=27(NC2)4 4ν3.(18) Thus (14) simplifies to d dtkum(t)k2+ν|Aum(t)|2≤2 ν|f|2 ∞+ 2C(N)kum(t)k2t≥0.(19) Let us fix 0 < ε ≤1.Integrating (19) between tand t+ε, we obtain in particular νZt+ε t |Aum(s)|2ds ≤2 ν|f|2 ∞+ 2C(N)Zt+ε t kum(s)k2ds +kum(t)k2∀t≥0, and then, by (17), one obtains Zt+ε t |Aum(s)|2ds (20) ≤1+2C(N) ν(ku0k2+C(N)(t+ 1)|u0|2)e−νλ1t +|f|2 ∞ ν22 + 1+2C(N) νλ12 + C(N) νλ2 1 ∀t≥0∀m≥1. Suppose now that f0, the time derivative of f, also belongs to L∞(0,+∞;H).In [8] it is proved that 1 2 d dt|u0 m(t)|2+νku0 m(t)k2(21) =−(FN(kum(t)k))0b(um(t), um(t), u0 m(t)) −bN(u0 m(t), um(t), u0 m(t)) + (f0(t), u0 m(t)) t≥0, where |(FN(kum(t)k))0| ≤ Nku0 m(t)k kum(t)k2χO(t) a.e. in (0,+∞),(22) with O={t∈(0,+∞) : kum(t)k ≥ N}. From (3), (22) and Young’s inequality, we have |2(FN(kum(t)k))0b(um(t), um(t), u0 m(t))| ≤2Nku0 m(t)kC1|u0 m(t)|1/4ku0 m(t)k3/4 = 2NC1|u0 m(t)|1/4ku0 m(t)k7/4(23) ≤νku0 m(t)k2+7 8ν7 25(NC1)8|u0 m(t)|2. STATISTICAL SOLUTIONS 7 By (4) and Young’s inequality again |2bN(u0 m(t), um(t), u0 m(t))| ≤2NC1|u0 m(t)|1/2ku0 m(t)k3/2 ≤νku0 m(t)k2+27 16ν3(NC1)4|u0 m(t)|2.(24) Thus, if we denote L(N)= 1 + 7 8ν7 25(NC1)8+27 16ν3(NC1)4, from (21), (23) and (24) we easily obtain d dt|u0 m(t)|2≤L(N)|u0 m(t)|2+|f0|2 ∞∀t≥0∀m≥1.(25) If we integrate this inequality between s∈[t, t +ε] and t+ε, we have |u0 m(t+ε)|2≤ |u0 m(s)|2+L(N)Zt+ε s |u0 m(r)|2dr +ε|f0|2 ∞∀0≤t≤s≤t+ε, for all m≥1.Integrating now this last inequality for sbetween tand t+ε, we obtain |u0 m(t+ε)|2≤(ε−1+L(N))Zt+ε t |u0 m(s)|2ds +|f0|2 ∞∀t≥0,(26) for all m≥1. Now, observe that by (11), the definition of FNand (7), |u0 m(t)| ≤ ν|Aum(t)|+|BN(um(t), um(t))|+|f(t)| ≤ν|Aum(t)|+N kum(t)k|b(um(t), um(t),·)|+|f|∞ ≤(ν+NC2)|Aum(t)|+|f|∞, t ≥0, and therefore Zt+ε t |u0 m(s)|2ds ≤2|f|2 ∞+ 2(ν+NC2)2Zt+ε t |Aum(s)|2ds ∀t≥0,(27) for all m≥1. From (20), (26) and (27), it is clear that there exist two positive constants C(N) f and D(N) f, independent of ε, u0, t and m, and increasing with |f|∞and |f0|∞, such that |u0 m(t+ε)|2≤(1 + ε−1)hC(N) f+D(N) f(ku0k2+ (t+ 1)|u0|2)e−νλ1ti,(28) for all t≥0, m ≥1, ε ∈(0,1], u0∈V. Again, by (11) and (9), ν|Aum(t)| ≤ |u0 m(t)|+|BN(um(t), um(t))|+|f(t)| ≤ |u0 m(t)|+NC2kum(t)k1/2|Aum(t)|1/2+|f|∞ ≤ |u0 m(t)|+ν 2|Aum(t)|+N2C2 2 2νkum(t)k+|f|∞, t ≥0, and therefore 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL |Aum(t)|2≤12 ν2|u0 m(t)|2+3N4C4 2 ν4kum(t)k2+ 12|f|2 ∞,∀t≥0,(29) for all m≥1. From (17), (28) and (29), one finds that there exist two positive constants K(N) f and R(N) f, independent of ε,u0, t and m, and increasing with |f|∞and |f0|∞, such that |Aum(t)|2≤(1 + ε−1)hR(N) f+K(N) f(1 + t)ku0k2e−νλ1ti∀t≥ε, (30) for all m≥1, ε ∈(0,1], u0∈V. Let t≥εbe fixed. By (30) we obtain |Aum(s)|2≤(1 + ε−1)hR(N) f+K(N) f(2 + t)ku0k2e−νλ1ti∀s∈[t, t + 1],(31) for all m≥1. Now, we will make use of the following result (see [19] for a proof). Lemma 3. Let X⊂Ybe Banach spaces such that Xis reflexive and the injection of Xin Yis compact. Suppose that {un}is a bounded sequence in L∞(t0, T ;X)such that un* u weakly in Lp(t0, T ;X)for some p∈[1,+∞)and u∈C0([t0, T]; Y). Then, u(t)∈Xfor all t∈[t0, T]and ku(t)kX≤sup n≥1 kunkL∞(t0,T ;X),∀t∈[t0, T].(32) From this lemma, inequality (31) and convergences in (12), we have u(t)∈D(A),|Au(t)|2≤(1+ε−1)hR(N) f+K(N) f(2 + t)ku0k2e−νλ1ti∀t≥ε, (33) where the inequality is valid for all u0∈Vand all ε∈(0,1]. Suppose now that u0∈Hand u(t) is a solution of the GMNSE with initial datum u0.We know that u(t)∈Vfor all t > 0. Let ε∈(0,1] be fixed and let v(t) be the unique solution of the GMNSE with initial datum u(ε) and forcing term b f(t) = f(t+ε).By (33), v(t)∈D(A) and |Av(t)|2≤(1+ε−1)hR(N) f+K(N) f(2 + t)ku(ε)k2e−νλ1ti∀t≥ε. But, by uniqueness, v(t) = u(t+ε) for all t≥0,and thus, from the above inequality we have u(t)∈D(A)∀t≥2ε, (34) |Au(t)|2≤(1 + ε−1)hR(N) f+K(N) f(2 + t)ku(ε)k2e−νλ1(t−1)i∀t≥2ε. (35) Now let w(t) be the unique solution of the GMNSE with initial datum u(ε/2) and forcing term e f(t) = f(t+ε/2).By uniqueness we know that w(t) = u(t+ε/2) for all t≥0. STATISTICAL SOLUTIONS 9 From estimate (39) in Proposition 15 in [8] we have ε/2kw(1/2)k2≤KNeKN |u(ε/2)|2+Z1/2 0 |e f(s)|2+Z1/2 0 |e f0(s)|2ds!, where KN>0, is a constant depending only on C1,N,νand λ1.Consequently, ku(ε)k2≤2ε−1KNeKN|u(ε/2)|2+|f|2 ∞+|f0|2 ∞.(36) Finally, the estimate d dt|u(t)|2+νku(t)k2≤|f(t)|2 νλ1 t≥0, is well known and, in particular, implies that |u(ε/2)|2≤ |u0|2+|f|2 ∞ νλ1 .(37) From (36), (37), (33) y (35), we obtain the following result. Proposition 4. Suppose that f∈W1,∞(0,+∞;H),and let u=u(t)be a solution of GMNSE. Then u(t)∈D(A)∀t > 0,(38) and there exist two positive constants K(N) fand M(N) f, independent of ε, u0and t, and increasing with |f|∞and |f0|∞, such that a) if u(0) ∈V, then |Au(t)|2≤(1 + ε−1)hR(N) f+M(N) f(1 + t)ku0k2e−νλ1ti∀t≥ε, (39) for all ε∈(0,1]; b) in general, if u(0) ∈H, then |Au(t)|2≤(1 + ε−1)R(N) f+ε−1(1 + ε−1)M(N) f(1 + t)(1 + |u0|2)e−νλ1t,(40) for all t≥2ε, 0< ε ≤1. In particular, there exists a T0=T0(|u0|)depending only on |u0|,|f|∞,|f0|∞,C1, C2, N,νand λ1such that |Au(t)|2≤2R(N) f∀t≥T0(|u0|).(41) Remark 5. Observe that (40) implies that if f∈W1,∞(0,+∞;H),then every solution of the GMNSE belongs to L∞(ε, +∞;D(A)) for all ε > 0. If, moreover, the initial datum u0∈D(A), then it can be proved that the corresponding solution u=u(t)of the GMNSE belongs to L∞(0,+∞;D(A)),and, more exactly, sup t≥0 |Au(t)|<+∞. 16 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL L(ϕ)−L(eϕ) = LIMT→∞ 1 TZT 0 (ϕ(u(t)) −eϕ(u(t))) dt = LIMT→∞ 1 TZT0 0 (ϕ(u(t)) −eϕ(u(t))) dt = 0. Thus L(ϕ) = L(eϕ). Let now ψ∈C(BN), where BNis considered as a metric subspace of H. As BN is a closed subset of Hand ψis continuous and bounded, we can extend ψto a continuous function ϕ∈C(H). By the considerations above, the value L(ϕ) is the same for any ϕ∈C(H)∪C(V) such that ϕ|BN=ψ. Therefore, we can define a functional lon C(BN) by l(ψ) = L(ϕ),where ϕ∈C(H)∪C(V) is any continuous extension of ψ. It is evident that lis a positive linear functional on C(BN), and because BNis compact, it follows from Kakutani-Riesz representation theorem (see [5]) that there exists a positive measure µon BNsuch that l(ψ) = ZBN ψ(v)dµ(v)∀ψ∈C(BN). The measure µcan be extended to a measure on Hby setting µ(F) = µ(F∩ BN) for all Borel measurable subset Fof H. It is clear that µ(H\ BN) = 0,and observe that if ϕ∈C(V), then ϕ|BN∈C(BN) (if vn→v0in BN, then, as BNis a compact subset of V,vn→v0in V, and therefore ϕ(vn)→ϕ(v0)). Consequently for any ϕ∈C(H)∪C(V) we have LIMT→∞ 1 TZT 0 ϕ(u(t)) dt =L(ϕ) = l(ϕ|BN) =ZBN ϕ|BN(v)dµ(v) =ZH ϕ(v)dµ(v). Finally, note that taking ϕ≡1,we deduce that µ(H) = LIMT→∞1 = 1,so that µ is a probability measure on H. Remark 15. With an almost identical proof to that of the preceding theorem, one can prove that there exists a time-average measure of any solution of the autonomous GMNSE. Now, we can obtain existence of SN-invariant measures. Proposition 16. Let u(t) = SN(t)u0be the solution of the autonomous GMNSE corresponding to u0∈V, and let µbe a time-average measure of u(t)such that C(V)⊂L1(H, µ)and (59) is satisfied for all ϕ∈C(V).Then µis an SN-invariant measure. STATISTICAL SOLUTIONS 17 Proof.- Let ψ∈C(H) and τ > 0. The function ψ◦SN(τ) : v7→ ψ(SN(τ)v) is also continuous in V, and by (59) with ϕreplaced by ψ◦SN(τ), we have ZH ψ(SN(τ)v)dµ(v) = LIMT→∞ 1 TZT 0 ψ(SN(t+τ)u0)dt = LIMT→∞ 1 TZT+τ τ ψ(SN(t)u0)dt = LIMT→∞ "1 TZT 0 ψ(SN(t)u0)dt +1 TZT+τ T ψ(SN(t)u0)dt −1 TZτ 0 ψ(SN(t)u0)dt#. But, observe that SN(t)u0belongs to a compact set of V, and hence also of H, for all t≥0. Therefore ψ(SN(t)u0) remains bounded for all t≥0,so LIMT→∞ "1 TZT+τ T ψ(SN(t)u0)dt −1 TZτ 0 ψ(SN(t)u0)dt#= 0. Thus, ZH ψ(SN(τ)v)dµ(v) = LIMT→∞ 1 TZT 0 ψ(SN(t)u0)dt =ZH ψ(v)dµ(v), for all τ > 0 and any ψ∈C(H). By density, we then obtain ZH φ(SN(τ)v)dµ(v) = ZH φ(v)dµ(v)∀φ∈L1(H, µ). In particular, taking the characteristic function of any measurable subset Eof V, we then have µ(E) = µ(SN(τ)−1E)∀τ > 0, and the SN-invariance of µfollows. 8. Stationary Statistical Solutions of the GMNSE in the autonomous case. Definition 17. We define Tas the set of real valued functionals Φ = Φ(v)on H such that (i) cr:= sup |v|≤r |Φ(v)|<+∞for all r > 0; (ii) for any v∈Vthere exists Φ0(v)∈Vsuch that |Φ(v+w)−Φ(v)−(Φ0(v), w)| |w|→0 as |w| → 0 with w∈V; (60) (iii) the mapping v7→ Φ0(v)is continuous and bounded as function from Vinto V. 18 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL Let us denote kvk= +∞if v∈H\V. With this convention, if µis a probability measure on Hand RHkvk2dµ(v)<+∞,then µ(H\V) = 0. We define GN(v) = −νAv −BN(v, v) + f∀v∈V. (61) Taking into account (5) and that r|FN(r)−FN(s)| ≤ |r−s| ∀ r, s ≥0, it is easy to obtain that kBN(v, v)−BN(u, u)k∗≤NC1(2kvk+kuk)kv−uk ∀ u, v ∈V, and therefore the mapping GN:V→V0is continuous. Also, by (6), kGN(v)k∗≤(ν+NC1)kvk+λ−1/2 1|f| ∀ v∈V. (62) Thus, if Φ ∈ T , |hGN(v),Φ0(v)i| ≤ [(ν+NC1)kvk+λ1−1/2|f|] sup w∈V kΦ0(w)k ∀ v∈V, and consequently, if µis a probability measure on Hwith RHkvkdµ(v)<+∞, then the integral RHhGN(v),Φ0(v)idµ(v) is finite. Definition 18. A stationary statistical solution of the GMNSE is a probability measure µon Hsuch that (i) ZH kvk2dµ(v)<+∞; (ii) ZH hGN(v),Φ0(v)idµ(v) = 0 for any Φ∈ T ; (iii) Z{a≤|v|2<b} {νkvk2−(f, v)}dµ(v)≤0for any 0≤a < b ≤+∞. We have the following result Theorem 19. Any SN-invariant probability measure on His a stationary statistical solution of the GMNSE. Proof.- Let µbe a SN-invariant probability measure on H. We know by Proposition 4 and Lemma 9 that µ(H\ BN) = 0.The set BNis a compact subset of V and hence the function kvkis bounded on BN. Thus, for any β > 0, ZH kvkβdµ(v) = ZBN kvkβdµ(v)<+∞, and in particular condition (i) in Definition 18 holds. Let us fix 0 ≤a < b ≤+∞,an let us denote E={v∈V:a≤ |v|2< b}, F ={v∈H:a≤ |v|2< b}. Since µis SN-invariant, and by (i) the function v7→ νkvk2−(f, v) is µ-integrable, we have ZF [νkvk2−(f, v)] dµ(v) (63) =ZE [νkvk2−(f, v)] dµ(v) =ZE [νkSN(t)vk2−(f, SN(t)v)] dµ(v)∀t≥0. STATISTICAL SOLUTIONS 19 Now, observe that reasoning as in the proof of Lemma 6 one can obtain that kSN(t)vk2≤2 ν|f|2+kvk2e2C(N)t∀t≥0 (64) for any v∈V. Consequently, taking into account condition (i), we can integrate in (63) and apply Fubini’s theorem, to obtain ZF [νkvk2−(f, v)] dµ(v) (65) =1 TZT 0ZE [νkSN(t)vk2−(f, SN(t)v)] dµ(v)dt =1 TZEZT 0 [νkSN(t)vk2−(f, SN(t)v)] dt dµ(v) for all T > 0. But we know that for all v∈Vand all T > 0, |SN(T)v|2− |v|2+ 2νZT 0 kSN(t)vk2dt = 2 ZT 0 (f, SN(t)v)dt, and hence, by (65), ZF [νkvk2−(f, v)] dµ(v) = 1 2TZE (|v|2− |SN(T)v|2)dµ(v)∀T > 0.(66) Now observe that |SN(t)v|2≤ |v|2e−νλ1t+|f|2 ν2λ2 1 ∀t≥0∀v∈V. (67) Suppose first that b < +∞. Then, from (66) and (67), and making T→+∞,we find that ZF [νkvk2−(f, v)] dµ(v) = 0. If b= +∞,it is enough to consider a sequence bn%+∞.Thus we have proved that µsatisfies condition (iii) in Definition 18. Finally, we must prove that µsatisfies condition (ii) in Definition 18. Let Φ ∈ T be given. For each integer m≥1,denote Φm(v) = Φ(Pmv)∀v∈H. It is easy to see that Φm∈C1(H), with Φ0 m(v) = PmΦ0(Pmv) for all v∈H. Evidently, sup |v|≤r |Φm(v)| ≤ sup |w|≤r |Φ(w)|=cr<+∞, and the mapping v7→ Φ0 m(v) is continuous and bounded as a function from Vinto V. Thus (see for example [17] Theorem 4.2, page 65) for any m≥1 and all v∈V we have Φm(SN(T)v)−Φm(v) = ZT 0 hGN(SN(t)v),Φ0 m(SN(t)v)idt ∀T > 0.(68) Since µis SN-invariant, ZH hGN(v),Φ0 m(v)idµ(v) = ZV hGN(SN(t)v),Φ0 m(SN(t)v)idµ(v) 20 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND JOS´ E REAL for all t≥0.Now we integrate, and taking into account (62), (64) and condition (i), we can apply Fubini theorem, and we obtain ZH hGN(v),Φ0 m(v)idµ(v) = 1 TZT 0ZV hGN(SN(t)v),Φ0 m(SN(t)v)idµ(v)dt =1 TZVZT 0 hGN(SN(t)v),Φ0 m(SN(t)v)idt dµ(v), and thus, by (68), ZH hGN(v),Φ0 m(v)idµ(v) = 1 TZV [Φm(SN(T)v)−Φm(v)] dµ(v)∀T > 0. Taking T→+∞in the last equality, and using the mean value theorem, the boundedness of Φ0 mon V, and the inequality (67), we obtain ZH hGN(v),Φ0 m(v)idµ(v) = 0.(69) Now, observe that kΦ0 m(v)−Φ0(v)k=kPmΦ0(Pmv)−Φ0(v)k ≤ kPmΦ0(Pmv)−PmΦ0(v)k+kPmΦ0(v)−Φ0(v)k ≤ kΦ0(Pmv)−Φ0(v)k+kPmΦ0(v)−Φ0(v)k. Therefore, by the continuity of Φ0on V, we obtain kΦ0 m(v)−Φ0(v)k → 0 as m→+∞for all v∈V. 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[21] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, SIAM, Philadelphia, 1995. [22] M.I. Vishik and A.V. Fursikov, Mathematical Problems of Statistical Hydrodynamics, Kluwer, Dordrecht, 1988. E-mail address, Tom´as Caraballo: [email protected] E-mail address, Peter E. Kloeden: [email protected] E-mail address, Jos´e Real: [email protected] (Tom´as Caraballo and Jos´e Real) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain) (Peter E. Kloeden) Institut f¨ ur Mathematik, Johann Wolfgang Goethe-Universit¨ at, D60054 Frankfurt am Main, Germany