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

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 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 Abstract. A new proof of existence of solutions for the three dimensional system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcing term. Indeed, we are able to prove that a stationary statistical solution is also an invariant probability measure under suitable assumptions. 1. Introduction. Let Ω ⊂R3be an open bounded set with regular boundary Γ, and consider the following system of 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, 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. There exist many modified versions of Navier-Stokes equations due to Leray and others with mollification (and/or cut off) of the nonlinear term as a way to approximate the original problem, see for instance the review paper of Constantin [1]. We also mention the paper [5] by Flandoli and Maslowski with a global cut off function used in a 2-dimensional stochastic context. In 2006, Caraballo, Kloeden and Real (cf. [3]) proposed a 3-dimensional model where the nonlinear term included a cut off factor FN(kuk) based on the norm of the gradient of the solution in the whole domain. Namely, for N∈(0,+∞) the Date: 17 September 2008. 2000 Mathematics Subject Classification. 35Q30, 35K90, 37L30. Key words and phrases. Modified Navier-Stokes model, statistical solutions, invariant measures. Partly supported by Ministerio de Educaci´on y Ciencia project MTM2005-01412. 1 2 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL function FN: [0,+∞)→(0,1] is defined by FN(r) := min 1,N r, r ∈[0,+∞). They called the resulting system                ∂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) the globally modified Navier-Stokes equations (GMNSE) and established the wellposedness of the model, in particular the absence of blow-up of solutions, as well as the existence of global V−attractors and certain relationships with the original NSE model. See also [7,8] for other studies and applications of the GMNSE as well as the review paper [2]. Foias et al. [6] (see also the papers cited therein) have made a very systematic study of invariant measures and statistical solutions of the NSE models in 2 and 3 dimensions. This investigation was continued in [4] by Caraballo, Kloeden and Real for the GMNSE, who proved the existence of time-averaged measures, then of invariant measures and finally that any invariant measure is a stationary statistical solution. We note that a relation between a family of time-average probability measures and the pullback attractor of a non-autonomous version of the NSE was established in [10]. Our aim in this paper is to establish the inverse of the last result in [4], i.e. that a stationary statistical solution of the GMNSE is an invariant probability measure, which we prove under suitable assumptions. The structure of the paper is as follows. We recall some preliminaries on the used functional spaces, definitions and existence results in Section 2. In Section 3we establish a new proof of the existence of strong solutions for the GMNSE with a different approach from that used in [3] since we need more regularity on the Galerkin approximations. Estimates in D(A) are obtained for these solutions of the new Galerkin scheme in Section 4. Then, in Section 5we establish our main result, Theorem 15, proving under additional assumptions that a statistical solution is an invariant probability measure. Finally, to improve the clarity of the paper, proofs of some technical results used earlier are given in the Appendix. 2. Preliminaries. To set our problem in the abstract framework, we consider the following usual abstract spaces (see Lions [9] and Temam [12,13]): V=nu∈(C∞ 0(Ω))3: div u= 0o, H= the closure of Vin (L2(Ω))3with inner product (·,·) and associated norm |·| , where for u, v ∈(L2(Ω))3, (u, v) = 3 X j=1 ZΩ uj(x)vj(x)dx, STATISTICAL SOLUTIONS 3 V= the closure of Vin (H1 0(Ω))3with inner product ((·,·)) and associated 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 [11] or [12]), 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, (2) |b(u, v, w)| ≤ C1|u|1/4kuk3/4kvk|w|1/4kwk3/4,∀u, v, w ∈V, (3) |b(u, v, w)| ≤ C1kukkvkkwk,∀u, v, w ∈V. (4) Thus, if we denote hBN(u, v), wi=bN(u, v, w),∀u, v, w ∈V, we have for example kBN(u, v)k∗≤NC1kuk,∀u, v ∈V. (5) 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 [12] and [11]) 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, (6) |b(u, v, w)| ≤ C2|u|1/4|Au|3/4kvk|w|,∀u∈D(A), v ∈V, w ∈H, (7) |b(u, v, w)| ≤ C2kuk1/2|Au|1/2kvk|w|,∀u∈D(A), v ∈V, w ∈H, (8) |b(u, v, w)| ≤ C2kukkvk1/2|Av|1/2|w|,∀u∈V, v ∈D(A), w ∈H, (9) |b(u, v, w)| ≤ C2|u|kvk1/2|Av|1/2kwk,∀u∈H, v ∈D(A), w ∈V. (10) 4 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Definition 1. Let u0∈Hand f∈L2(0, T ; (L2(Ω))3)for all T > 0be 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 (5), u0(t)∈L2(0, T;V0),and hence (see [13])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. The following result was proved in [3]. Theorem 3. [cf. [3, Th.7(a)]] Suppose that f∈L2(0, T ; (L2(Ω))3)for all T > 0, and let u0∈Vbe given. Then, there exists a unique weak solution uof (1), which is in fact a strong solution, i.e., it satisfies u∈C([0, T]; V)∩L2(0, T;D(A)) for all T > 0. In the next section, for technical reasons, we obtain a similar result on existence of solution (see Proposition 5and Remark 6below), using a different Galerkin approximation from that used in [3]. 3. A smooth Galerkin approximation for GMNSE. By regularity requirements for the manipulations in the proof of our main result (cf. Theorem 15), we need to introduce a new Galerkin approximation scheme for the GMNSE. Our first step is to consider a family of smoother functions that approximates the truncated term FNappearing in the GMNSE. This can be carried out in the classical way, by mollification. Let {ρm, m ≥1} ⊂ C∞(R) be a regularizing sequence in R, i.e., such that 0≤ρm(r)≤1, the support of ρmis included in the interval [−1/m, 1/m],and RRρm(r)dr = 1,for all m≥1. Let us consider FNprolonged as FN(r) = 1 if r≤0,and denote FN,m the convolution ρm∗FN,i.e., FN,m(r) = ZR ρm(r−s)FN(s)ds ∀r∈R.(11) It is well known that FN,m ∈C∞(R)∩L∞(R),and also FN,m(r)→FN(r), as m→+∞, uniformly on bounded subsets of R.Moreover, 0≤FN,m(r)≤ kFNkL∞(R)= 1 ∀r∈R,∀m≥1. In fact, this last estimate can be improved. The following result concerning the functions FN,m holds. STATISTICAL SOLUTIONS 5 Lemma 4. The functions FN,m defined by (11) satisfy the following inequalities: 0≤rFN,m(r)≤N+ 1 ∀r≥0,∀m≥1.(12) |F0 N,m(r)| ≤ 1/N ∀r∈R,∀m≥1.(13) r|F0 N,m(r)| ≤ N+ 1 N, r2|F0 N,m(r)| ≤ (N+ 1)2 N∀r≥0,∀m≥1.(14) F0 N,m(r) = 0 ∀r≤1/m, ∀m > 2/N. (15) Its proof is given in the Appendix to avoid distracting from the main flow of ideas here. Now, let us denote BN,m(v, v) the element of V0defined by hBN,m(v, v), wi=FN,m(kvk)b(v, v, w)∀u, w ∈V. Consider the Galerkin approximations for the GMNSE given by u0 m+νAum+PmBN,m(um, um) = Pmf, um(0) = Pmu0,(16) where um=Pm j=1 um,jφj,Aum=Pm j=1 λjum,jφj. Here the λjand φjare the corresponding eigenvalues and eigenfunctions (orthonormal in H, orthogonal in V) of the operator Aand Pmis the projection onto the subspace of Hspanned by {φ1, . . . , φm}. Now, we establish the main result of this section, which provides a new proof of the existence of solution given in Theorem 3. Proposition 5. Under the above notation, if f∈L∞(0, T;H)for all T > 0,the scheme (16) is well defined, i.e. for each u0∈V, there exists a smooth family (C1in time) of functions {um},solutions of (16), that are defined in (0,+∞).Moreover, there exists a subsequence of {um}mthat converges (in several senses) to the unique solution of (1). Remark 6. The assumption f∈L∞(0, T ;H)instead of L2(0, T; (L2(Ω))3)as in Theorem 3is not essential, but only for simplicity in the notation. Indeed, since projector operators are involved in the test-functions space, there is no restriction in considering ftaking values in Hinstead of (L2(Ω))3.On the other hand, the stronger requirement in time, L∞instead of L2,is also for the sake of convenience in the calculus. Observe that our final goal in Section 5is with an autonomous term f. Proof. [of Proposition 5] The existence and uniqueness of local solution of (16) is a consequence of the Picard Theorem, and the fact that the local solution is a global one defined in the same interval of time as fis in view of the estimates (19) and (24) below. After that, we will proceed by the compactness method: firstly proving uniform estimates for umin Hand then in V. Observe that for the estimates in Hwe only need u0∈H. Fix a value T > 0 and let us denote |f|∞=kfkL∞(0,T ;H). 6 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL It is standard that if we take the inner product of the Galerkin ODE (16) with umand use b(um, um, um) = 0, we have d dt|um|2+ 2νkumk2= 2(f, um),(17) and then, using λ1|um|2≤ kumk2,we obtain d dt|um|2+νkumk2≤|f|2 ∞ νλ1 ,(18) and consequently |um(t)|2+νZt 0 kum(s)k2ds ≤ |u0|2+|f|2 ∞ νλ1 t(19) for all t≥0 in the interval of definition of um.The well posedness of umin (0, T) follows from Gronwall’s lemma. Now we obtain uniform estimates in Vfor all umin (0, T ) using u0∈V. We take the inner product of the Galerkin ODE (16) with Aumand obtain 1 2 d dtkumk2+ν|Aum|2+bN,m(um, um, Aum) = (f, Aum).(20) Evidently, |(f, Aum)| ≤ ν 4|Aum|2+|f|2 ∞ ν In addition, by (8), (12) and Young’s inequality, one obtains |bN,m(um, um, Aum)| ≤ C2(N+ 1)kumk1/2|Aum|3/2 ≤ν 4|Aum|2+C0 Nkumk2, with C0 N=27(N+ 1)4C4 2 4ν3. Thus (20) simplifies to d dtkumk2+ν|Aum|2≤2 ν|f|2 ∞+ 2C0 Nkumk2.(21) From (21) and the fact that kum(0)k=kPmu0k ≤ ku0k by the choice of the basis {φj}of H, one easily concludes that the sequence {um} is bounded in C([0, T]; V) and in L2(0, T;D(A)) for all T > 0. Then, observe that |bN,m(um, um, w)| ≤ (N+ 1)C2kumk1/2|Aum|1/2|w|,for any w∈H, and thus, the sequence {PmBN,m(um, um)}is bounded in L2(0, T;H) for all T > 0. Therefore, from the equation u0 m=−νAum−PmBN,m(um, um)+Pmf, one sees that the sequence {u0 m}is also bounded in L2(0, T;H). Consequently, as D(A)⊂V⊂Hwith compact injection, by Theorem 5.1 in Chapter 1 of [9] there exists an element u∈L∞(0, T;V)∩L2(0, T;D(A)) for all STATISTICAL SOLUTIONS 7 T > 0, and a subsequence {uµ}of {um}, such that                    uµ→ustrong in L2(0, T;V), uµ→ua.e. in (0, T)×Ω, uµ* u weak in L2(0, T;D(A)), uµ ∗ * u weak-star in L∞(0, T;V), u0 µ* u0weak in L2(0, T;H), (22) for all T > 0. Also, as uµconverges to vin L2(0, T ;V) for all T > 0, we can assume, possibly extracting a subsequence, that kuµ(t)k → ku(t)ka.e. in (0,+∞). Then, by the boundedness of {uµ}in C([0, T]; V) and the fact that FNis continuous and FN,m converges to FNuniformly on bounded subsets of Ras m→+∞, one obtains FN,m(kuµ(t)k)→FN(ku(t)k) a.e. in (0,+∞).(23) From (22) and (23) we can take limits in (16) exactly as in [3], and we obtain that uis a solution of (1). Remark 7. If we assume that fis also defined for negative time, for instance, f∈L∞(−T, T;H),the above solutions umare also well defined for all (−T, T). This will be used below -in the proof of Theorem 15- to ensure that we can go backwards in time on each solution umof the Galerkin scheme. This is due to (17), whence we have d dt|um|2+ 2νλm|um|2≥ −|f|2 ∞− |um|2, where |f|∞denotes now kfkL∞(−T,T ;H).Therefore, integrating between t∈(−T, 0) and 0we deduce |um(t)|2≤ |u0|2+|t||f|2 ∞+(1+2νλm)Z0 t |um(s)|2ds (24) for all t < 0in the interval of definition of um. Thus, by (19), (24) and again Gronwall’s lemma, umis defined in all (−T, T ). Remark 8. Observe that, since the sequence {u0 m}is bounded in L2(0, T ;H)for any T > 0,the sequence umas a sequence of functions from [0, T ]into His equicontinuous. Then, since {um}is bounded in C([0, T ]; V)and V⊂Hwith compact injection, we can assert that from any subsequence of {um}we can extract a subsequence that converges in C([0, T]; H), and taking into account the previous arguments for the existence of solution uof (1), and the uniqueness of such a solution (cf. Theorem 3), one has that um→uin C([0, T ]; H),as m→+∞,for all T > 0. 4. Estimates in D(A)for the Galerkin approximations. In this section we go further in developing new estimates for the solutions of (16), but now with initial data u0∈D(A),and assuming that f∈W1,∞(R;H).[The more restrictive hypothesis of f∈W1,∞(R;H),which we will use from here on instead of W1,∞((−T, T); H) for all T, is only for the sake of brevity in the statements.] 8 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Proposition 9. With the notation of Section 3, if f∈W1,∞(R;H),there exists a positive constant ˜ M(N) f, independent of u0, t and m, such that |Aum(t)| ≤ ˜ M(N) f(1 + |Au0|)∀t≥0,for all m > 2/N, u0∈D(A).(25) Proof. Now let us denote |·|∞=k·kL∞(R;H).From (18) we have d dt eνλ1t|um(t)|2≤|f|2 ∞ νλ1 eνλ1t, t ≥0, and integrating one obtains |um(t)|2≤ |u0|2e−νλ1t+|f|2 ∞ ν2λ2 1 for all t≥0.(26) Now, if we use (20) and take into account that λ1kum(t)k2≤ |Aum(t)|2and that, by (7) and (12), |bN,m(um(t), um(t), Aum(t))| ≤ (N+ 1)C2|um(t)|1/4|Aum(t)|7/4, then we can obtain the inequality d dtkum(t)k2+νλ1kum(t)k2≤2 ν|f|2 ∞+C(N) 1|um(t)|2,(27) with C(N) 1=(N+ 1)8C8 277 29ν7. Substituting the bound (26) for |um(t)|2in the differential inequality (27) gives d dtkum(t)k2+νλ1kum(t)k2≤C(N) 1|u0|2e−νλ1t+|f|2 ∞ ν 2 + C(N) 1 νλ2 1!. Integrating this inequality we deduce that kum(t)k2≤(ku0k2+C(N) 1t|u0|2)e−νλ1t+|f|2 ∞ ν2λ1 2 + C(N) 1 νλ2 1!,∀t≥0.(28) On the other hand, integrating (21) between tand t+ 1, we obtain in particular νZt+1 t |Aum(s)|2ds ≤2 ν|f|2 ∞+ 2C0 NZt+1 t kum(s)k2ds +kum(t)k2∀t≥0, and then, by (28), one obtains Zt+1 t |Aum(s)|2ds ≤1+2C0 N ν(ku0k2+C(N) 1(t+ 1)|u0|2)e−νλ1t(29) +|f|2 ∞ ν2"2 + 1+2C0 N νλ1 2 + C(N) 1 νλ2 1!# ∀t≥0∀m≥1. Suppose now that m > 2/N. From (16) and (15), we obtain for the second derivative of um, u00 m(t) = −νAu0 m(t)−F0 N,m(kum(t)k)((u0 m(t), um(t))) kum(t)kχOm(t)PmB(um(t), um(t)) −FN,m(kum(t)k)PmB(u0 m(t), um(t)) −FN,m(kum(t)k)PmB(um(t), u0 m(t)) + Pmf0(t),(30) where Om={t∈[0,+∞) : kum(t)k>1/m} and χOm(t) is the indicator function of the set Om. STATISTICAL SOLUTIONS 9 Multiplying in (30) by u0 m,we obtain 1 2 d dt|u0 m(t)|2+νku0 m(t)k2 =−F0 N,m(kum(t)k)((u0 m(t), um(t))) kum(t)kχOm(t)b(um(t), um(t), u0 m(t)) −FN,m(kum(t)k)b(u0 m(t), um(t), u0 m(t)) + (f0(t), u0 m(t)) t≥0.(31) From (2), (14) and Young’s inequality, we have |F0 N,m(kum(t)k)((u0 m(t), um(t))) kum(t)kχOm(t)b(um(t), um(t), u0 m(t))| ≤2C1(N+ 1)2 Nku0 m(t)k|u0 m(t)|1/4ku0 m(t)k3/4 =2C1(N+ 1)2 NC1|u0 m(t)|1/4ku0 m(t)k7/4 ≤ν 2ku0 m(t)k2+7 4ν7 25C2 1(N+ 1)2 N8 |u0 m(t)|2.(32) By (3), (12) and Young’s inequality again |2FN,m(kum(t)k)b(u0 m(t), um(t), u0 m(t))| ≤2(N+ 1)C1|u0 m(t)|1/2ku0 m(t)k3/2 ≤ν 2ku0 m(t)k2+27 2ν3(N+ 1)4C4 1|u0 m(t)|2.(33) Thus, if we denote L(N) 1= 1 + 1 27 2ν7C2 1(N+ 1)2 N8 +27 ν3(N+ 1)4C4 1, from (31), (32) and (33) we easily obtain d dt|u0 m(t)|2≤L(N) 1|u0 m(t)|2+|f0|2 ∞∀t≥0∀m > 2/N. (34) If we integrate this inequality between s∈[t, t + 1] and t+ 1,we have |u0 m(t+ 1)|2≤ |u0 m(s)|2 +L(N) 1Zt+1 s |u0 m(r)|2dr +|f0|2 ∞∀0≤t≤s≤t+ 1,∀m > 2/N. Integrating now this last inequality for sbetween tand t+ 1,we obtain |u0 m(t+ 1)|2≤(1 + L(N) 1)Zt+1 t |u0 m(s)|2ds +|f0|2 ∞∀t≥0,∀m > 2/N. (35) Now, observe that by (16), (6) and (12), |u0 m(t)| ≤ ν|Aum(t)|+|FN,m(kum(t)k)B(um(t), um(t))|+|f(t)| ≤[ν+ (N+ 1)C2]|Aum(t)|+|f|∞, t ≥0, and therefore Zt+1 t |u0 m(s)|2ds ≤2|f|2 ∞+ 2[ν+ (N+ 1)C2]2 ×Zt+1 t |Aum(s)|2ds ∀t≥0,∀m≥2/N. (36) 16 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Then, since Pm∈ L(H) is self-adjoint, from (57) and (58) we obtain ZH Φ(S(m) N(T)v)dµ(v) =ZH Φ(Pmv)dµ(v) (59) +ZT 0ZH (PmGN,m(Pmv)−PmGN(v), Dv(Φm)(t, v)) dµ(v)dt ∀T > 0. Now, observe that PmGN,m(Pmv)−PmGN(v) = PmBN(v, v)−PmBN,m(Pmv, Pmv), and then, by (58) and since P∗ m=Pm,we obtain from (59) ZH Φ(S(m) N(T)v)dµ(v) (60) =ZH Φ(Pmv)dµ(v) +ZT 0ZH (PmBN(v, v)−PmBN,m(Pmv, Pmv), Dv(Φm)(t, v)) dµ(v)dt =ZH Φ(Pmv)dµ(v) +ZT 0ZH (BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v)) dµ(v)dt ∀T > 0. Now, we prove that as m→+∞, the last integral term goes to zero for all T > 0. Observe that |(BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v))| =|FN(kvk)b(v, v, Dv(Φm)(t, v)) −FN,m(kPmvk)b(Pmv, Pmv, Dv(Φm)(t, v))| ≤ |FN(kvk)b(v−Pm, v, Dv(Φm)(t, v))| +|FN(kvk)−FN,m(kvk)||b(Pmv, v, Dv(Φm)(t, v))| +|FN,m(kvk)b(Pmv, v −Pm, Dv(Φm)(t, v))| +|FN,m(kvk)−FN,m(kPmvk)||b(Pmv, Pmv, Dv(Φm)(t, v))|, and therefore, by (4), (12) and (13), |(BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v))| ≤NC1kv−PmvkkDv(Φm)(t, v))k +|FN(kvk)−FN,m(kvk)|C1kvk2kDv(Φm)(t, v))k +(N+ 1)C1kv−PmvkkDv(Φm)(t, v))k +C1 Nkv−Pmvkkvk2kDv(Φm)(t, v))k.(61) On the other hand, reasoning exactly as on page 232 in [6], one obtains kDv(Φm)(t, v))k =kΦ0(S(m) N(t)vksup w∈H, |A−1/2w| ≤ 1 |A−1/2(DvS(m) N(t)v)w| ∀ t≥0,∀v, w ∈H, STATISTICAL SOLUTIONS 17 for any m≥1,and then, taking into account that Φ ∈ T has derivative Φ0(·) bounded on V, there exists a constant CΦ>0 such that kDv(Φm)(t, v))k ≤ CΦsup w∈H, |A−1/2w| ≤ 1 |A−1/2(DvS(m) N(t)v)w|(62) ∀t≥0,∀v, w ∈H, for any m≥1. Since BNis a bounded subset of D(A), from (50), Proposition 9and (62) we deduce that there exists a constant Cf,BNsuch that kDv(Φm)(t, v))k ≤ CΦeT Cf,BN∀0≤t≤T, ∀v∈ BN,(63) for any m > 2/N. From (61) and (63) we deduce that for each T > 0 there exists a constant CT>0, such that |(BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v))| ≤CT(kv−Pmvk+|FN(kvk)−FN,m(kvk)|) for all (t, v)∈[0, T ]× BN,for any m > 2/N. Then, as µ(H\ BN) = 0,we obtain that for each T > 0, ZT 0ZH |(BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v))|dµ(v)dt (64) ≤TCTZBN (kv−Pmvk+|FN(kvk)−FN,m(kvk)|)dµ(v)∀m > 2/N. As kv−Pmvk → 0 as m→+∞and kv−Pmvk ≤ 2kvkfor any v∈ BN,and FN,m converges to FNas m→+∞uniformly on bounded subsets of R,it is immediate form (64) that ZT 0ZH |(BN(v, v)−BN,m(Pmv, Pmv), Dv(Φm)(t, v))|dµ(v)dt →0 as m→+∞, and then, from (60) we conclude that ZH Φ(S(m) N(T)v)dµ(v)−ZH Φ(Pmv)dµ(v)→0 as m→+∞, (65) for all T > 0. But, taking again into account that µ(H\BN) = 0,that BNis a bounded subset of D(A),and that Φ is bounded on bounded subsets of H, we deduce from Remark 8and (52) that ZH Φ(S(m) N(T)v)dµ(v)→ZV Φ(SN(T)v)dµ(v) as m→+∞, for all T > 0, and ZH Φ(Pmv)dµ(v)→ZV Φ(v)dµ(v) as m→+∞, and then, by (65) we have (45), as desired. 18 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL Appendix: Estimates on the truncated function. This section is devoted to prove some technical estimates concerning the functions FN,m,defined by (11), namely those of Lemma 4. To start, observe that FN,m(r) = Zr+1/m r−1/m ρm(r−τ)FN(τ)dτ. Consequently, if N < r −1/m, then FN,m(r) = Zr+1/m r−1/m ρm(r−τ)N τdτ ≤N r−1/m. Thus, •if N≥r−1/m, then rFN,m(r)≤r≤N+ 1/m, and •if N < r −1/m, then rFN,m(r)≤rN r−1/m =N1 + 1 m(r−1/m) ≤N1 + 1 mN =N+ 1/m. Hence, (12) is proved. Moreover, it is straightforward to check that F0 N,m(r)=(ρm∗F0 N)(r),where F0 N(r) = −N r2χ(N,+∞)(r),and therefore (13) also holds. For (14), related to r|F0 N,m(r)|and r2|F0 N,m(r)|,we have to proceed more carefully. Observe that |F0 N,m(r)| ≤ Zr+1/m r−1/m ρm(r−τ)|F0 N(τ)|dτ ≤Zr+1/m r−1/m ρm(r−τ)N τ2dτ, and consequently, if r > N + 1,then |F0 N,m(r)| ≤ N (r−1/m)2 ≤N (r−1)2. Thus, STATISTICAL SOLUTIONS 19 •if r > N + 1, r|F0 N,m(r)| ≤ Nr (r−1)2 =N r−11 + 1 r−1 <1 + 1 N =N+ 1 N, and r2|F0 N,m(r)| ≤ Nr2 (r−1)2 =N1 + 1 r−12 < N 1 + 1 N2 =(N+ 1)2 N. •if 0 ≤r≤N+ 1,then, by (13), one also has r|F0 N,m(r)| ≤ N+ 1 N, r2|F0 N,m(r)| ≤ (N+ 1)2 N. Hence, combining all situations one recovers the inequality in (14). Finally, in order to prove (15) observe that if m > 2/N and r≤1/m, then FN,m(r) = Zr+1/m r−1/m ρm(r−τ)FN(τ)dτ = 1, whence the last result follows. REFERENCES [1] P. Constantin, Near identity transformations for the Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, Vol. II, 117–141, North-Holland, Amsterdam, 2003. [2] T. Caraballo, P.E. Kloeden, J.A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations, in “Legacy of the Legend, Professor V. Lakshmikantham” (eds. J. Vasundhara Devi, S. Sivasundaram, Zahia Drici and Farzana Mcrae), Cambridge University Press. To appear. [3] T. Caraballo, P.E. Kloeden and J. Real, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Studies 6 (2006), 411–436. [4] T. Caraballo, P.E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B,10 (2008), 761–781. [5] F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes Equation under random perturbations, Commun. Math. Phys. 171 (1995), 119–141. [6] C. Foias, O. Manley, R. Rosa and R. Temam Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. [7] P.E. Kloeden, J.A. Langa and J. Real, Pullback V-attractors of the three dimensional globally modified Navier-Stokes equations: existence and finite fractal dimension, Commun. Pure Appl. Anal. 6(2007), 937–955. 20 PETER E. KLOEDEN, PEDRO MAR´ IN-RUBIO, AND JOS´ E REAL [8] P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), 1491–1508. [9] J.L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Paris, 1969. [10] G. Lukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B,9(2008), 643–659. [11] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. [12] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977. [13] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, SIAM, Philadelphia, 1995. E-mail address, Peter E. Kloeden: [email protected] E-mail address, Pedro Mar´ın-Rubio: [email protected] E-mail address, Jos´e Real: [email protected] (Peter E. Kloeden) Institut f¨ ur Mathematik, Johann Wolfgang Goethe-Universit¨ at, D-60054 Frankfurt am Main, Germany (Pedro Mar´ın-Rubio and Jos´e Real) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla (Spain)