Global attractor for the Navier Stokes equations with fractional deconvolution

Transkript

1 Nonlinear Differ. Equ. Appl. 5), c 4 Springer Basel -97/5/48-38 published online December 5, 4 DOI.7/s y Nonlinear Differential Equations and Applications NoDEA Global attractor for the Navier Stokes equations with fractional deconvolution Davide Catania, Alessandro Morando and Paola Trebeschi Abstract. We consider a large eddy simulation model for the 3D Navier Stokes equations obtained through fractional deconvolution of generic order. The global well-posedness of such a problem is already known. We prove the existence of the global attractor for the solution operator and find estimates for its Hausdorff and fractal dimensions both in terms of the Grashoff number and in terms of the mean dissipation length, with particular attention to the dependence on the fractional and deconvolution parameters. These results can be interpreted as bounds for the number of degrees of freedom of long-time dynamics, thus providing further information on the validity of the model for the simulation of turbulent 3D flows. Mathematics Subject Classification. 76D5, 35Q3, 76F65, 76D3. Keywords. Navier Stokes equations, Global attractor, Fractal and Hausdorff dimension, Approximate deconvolution models ADM) and methods, Fractional filter, Large eddy simulation LES).. Introduction We consider an approximate model for the Navier Stokes Equations for a homogeneous incompressible fluid, that read t u + u u) νδu + π = f, u =, u, ) =u, in [,T] D, where D is a three-space dimensional periodic domain under suitable conditions that we are going to make precise. We assume that the constant ν, called kinematic viscosity, is positive.

2 8 D. Catania, A. Morando and P. Trebeschi NoDEA It is well-known that, currently, we lack a good well-posedness theory for these equations is lacking, so that this system is not suited for numerical simulations. Moreover, the simulation of turbulent pointwise flows is prohibitively expensive and sensitive to small perturbations of the data, i.e. the pointwise flow is chaotic. Large eddy simulation may overcome these difficulties see the introductions in [, 6]). Instead of pointwise flows, averaged deterministic flows are considered, and systems satisfied by averages are studied. More precisely, some approximations are introduced in order to mitigate the nonlinearity and obtain well-posedness and the possibility to perform simulations. In particular, we are interested in fractional approximate deconvolution models ADM), introduced by Stolz Adams in [8]. We introduce the fractional filter, denoted by, setting A = I + α Δ), v = A v ) with periodic conditions in x), where α>and [, ] are fixed. Let us note that the filter is linear and commutes with differentiation. It is a generalization of the Helmholtz filter, which corresponds to the choice =, and it is known to produce excellent results in numerical simulations, at least for large Reynolds numbers and suitable geometries see [5] and the references therein). Filtering the equations for u, we obtain t u + u u) νδu + π = f, u =, u, ) =u. Set w = u and q = π, so that u = A w. We solve the so-called interior closure problem by the approximation u u D N, u D N, u = D N, w D N, w, where we have used the fractional deconvolution operator N ) D N, w := I A h w, ) h= defined for every integer N and any real [, ]. We finally deduce the fractional ADM t w + D N, w D N, w ) νδw + q = f, 3) w =, 4) w, ) =u. 5) This model has been introduced by Berselli Lewandowski [4], where global well-posedness for >3/4and convergence results are proved see Sect.. for the existence). This result has been generalized by Ali [] to the case of magnetohydrodynamics requiring only /. When the deconvolution order is, i.e. N =, and the power is nonfractional, i.e., we obtain the model introduced by Layton Lewandowski [3] and studied by Cao Lunasin Titi [5], who prove the existence of global solutions

3 Vol. 5) Global attractor for the Navier 83 and of the global attractor see Sect. 4) for the solution operator, providing also estimates of its fractal and Hausdorff dimensions. These notions are useful in order to test the validity of the model as a large eddy simulation model. Actually, the dimension of the global attractor can be interpreted as the number of degrees of freedom of the long-time dynamics of the system. In this paper, we are interested in extending such results to the fractional deconvolution model 3) 5), with particular concern to the dependence on the fractional power. First, we prove the existence of the global attractor associated to the semigroup generated by the operator St) that associates to the initial datum u the corresponding solution of 3) 4) evaluated at time t see Sect..3 for the definition of the function space H σ). Theorem.. Assume that / and let St)) t be the semigroup in H σ associated to the initial value problem 3) 5) defined in 8), with datum f H. Then a unique global attractor A exists for such a semigroup in H σ. The proof of Theorem. is given in Sect. 4. Note that the existence of solutions to 3) 5) see Sect..) is known just for /, whose assumption in our result, for this reason, is not restrictive. The subsequent step consists in finding estimates for the attractor A. We resort to the trace formula and the standard technique described for instance in [, 9]. This approach requires that the operator St) isfréchet differentiable with respect to the initial datum. After proving this result see Sect. 5), we introduce a suitable version of the Grashoff number adapted to the model we are considering: G := L3/ A / N, f ν. 6) First, we prove an estimate for the global attractor dimension in terms of this number. Theorem.. If f H for /, ], the Hausdorff dimension d H A) and the fractal dimension d F A) of the global attractor A are bounded above by d H A) d F A) C C ν +μ L +μ) 3 5+3σ )s G 6+6μ 5+3σ )s, 7) where C is a numerical constant independent of the other parameters appearing in the right-hand side, C is defined in 7), σis defined in 94), while μ and s are defined in 73). The Grashoff number takes into account the effect of the forcing term f, so that the previous theorem establishes a relation between the global attractor dimension and f. Notice that, in view of the definition 94) ofσ, it is convenient to analyze the estimate 7) by considering separately two cases corresponding to the different ranges /, /3) and [/3, ] for. i) For /3 wehaveσ )s = ; then the bound in the right-hand side reduces to C C ν +μ L +μ) 3 5 G 6+6μ 5, 8)

4 84 D. Catania, A. Morando and P. Trebeschi NoDEA ) where μ = [cf. 77)]. In particular, for =wehaveμ =, hence the exponent of G in 8) reduces to 6/5 and we recover the already known estimate proved in [5] but now it holds for all N and not only for N = ). We also observe that, when <, the exponent worsens. ii) For / <</3 the result is worse with respect to the case i), not only because of the values of, but also since the exponent σ )s. Now, in order to understand more deeply the relation between the global attractor dimension and the number of degrees of freedom of the long-time dynamics, we introduce the mean rate of energy of the model) dissipation, defined by ε := L 3 sup T lim sup ν A / N, w A T + T wt) dt. Moreover, in analogy with Kolmogorov dissipation length in the classical theory of turbulence, we define the mean dissipation length for the model as ) ν 3 /4 l d :=, ε so that T sup lim sup ν A / N, w A T + T wt) dt = L3 ν 3 l 4. 9) d Theorem.3. If f H and /, ], the Hausdorff dimension d H A) and the fractal dimension d F A) of the global attractor A are bounded above by [ ) γ ) ] 6 6γ L L d H A) d F A) C N +) γ, α l d where C is a numerical constant independent of the other parameters appearing in the right-hand side, and γ is defined in 86). Considerations similar to those for Theorem. still hold. In particular, when =, we have again that γ = and the exponent of L/l d ) reduces to /5, exactly as in [5]. The mean dissipation length l d is defined to be the smallest length scale actively participating in the dynamics of the turbulent flow. This means that, when performing simulations, the size of every mesh can not be greater than l d and L/l d points for each space dimension) are needed and the number of degrees of freedom for a 3D turbulent flow is expected to be L/l d ) 3.The previous theorem shows the dependence of the number of degrees of freedom of the fractional deconvolution model with respect to and the other parameters. The effect of filtering is the reduction of this number for =,wehave/5, which is less than 3), and a reduced regularization <) increases the number of degrees of freedom. In particular, recalling the definition of γ 86), we can easily compute that = 3 holds for =9/. 6γ

5 Vol. 5) Global attractor for the Navier 85 Despite the use of standard techniques, the proofs are not trivial, since the presence of the deconvolution operator of generic order N and, above all, of the fractional power require a clever combination of several ad hoc estimates. When decreases, the regularization provided by the filter lessens, so that refined interpolation estimates are needed. Note that we do not compute the bound of the attractor dimension when =/; this is due to the fact that, in such a case, + =3/, and it is impossible to apply the fractional Agmon s inequality Lemma A.3) in order to estimate appropriately some L -norms by using 4). We are convinced that the theorems proved in this paper can be extended to other models, even for magnetofluids, generalizing the results concerning the global attractor in magnetohydrodynamics MHD), as in [7 9], to deconvolution and fractional deconvolution approximate MHD models, such as those in [, 3]. However, this would require several more computations, and it is postponed to future works. Outline of the paper Section is devoted to present the main notation, function spaces, the notion of regular weak solution, the properties of the filter and of the deconvolution operator. In Sect. 3 we prove some a priori estimates that will be used in the proofs of the main theorems. In Sect. 4 we recall some notions and properties concerning global attractors and prove Theorem.. Section 5 is devoted to the proof of the differentiability of the solution operator, while Sect. 6 contains all results concerning the bounds of the global attractor dimension, i.e. Theorems. and.3. Finally, the Appendix lists and proves some useful formulas and inequalities that play a crucial role in the proofs of the original results contained in this paper.. Preliminary results and basic tools.. Notation We set x =x,x,x 3 ), j = xj, Δ= + + 3, =,, 3 ). The space domain is T 3 = { x R 3 : πl<x,x,x 3 <πl}, L>, with πlperiodicity with respect to x i.e. with respect to x,x,x 3 ), i.e. a torus. We denote by L p and H s with p and s R) the standard Lebesgue and Sobolev spaces, and define C [,T]; X) and the Bochner spaces L p,t; X) in the usual way. We write H s for the Sobolev space of vector functions defined on T 3 with zero spatial mean and πlperiodicity in x, and set L = H. We denote by the subscript σ spaces of divergence-free classes of) functions, and we assume the domain T 3 when not specified. The symbol, ) is used to denote the standard scalar product in L. We will denote by C a generic positive constant, independent of the relevant parameters, which may differ at every occurrence, even in the same expression.

6 86 D. Catania, A. Morando and P. Trebeschi NoDEA.. Regular weak solutions Assume that f = fx) H 3/ is independent of time for simplicity), so that f H 3/,andu H σ, so that w := u H σ we may even require the stronger regularity u L σ, which is more natural for the unfiltered problem). We say that w :[, + ) T 3 R 3 is a regular weak solution of 3) 5) when, for all T>, the following properties are verified see also []). Regularity: w C [,T]; H σ) L,T; H + σ ), t w L,T; Hσ 3/ ), q L,T;H / ). Initial data: w, ) =w. Weak formulation: { w, t ϕ) ν w, ϕ)+d N, w D N, w, ϕ)+q ϕ } s)ds + = + f, ϕ)s)ds w), ϕ) ) for each ϕ C T 3 [,T)) ) 3. Theorem.. Assume /. Then there exists a unique regular weak solution of 3) 5). This solution depends continuously on the data the system is well-posed) and satisfies the energy of the model) identity for each t [, + ). A/ N, wt) + ν t = A/ N, w) + A / N, ws) ds t ) N, f,d/ N, ws) ds This result has been proved by Berselli Lewandowski [4] for > 3/4 and extended to / by Ali in [], where the equations are coupled with the equations for the magnetic field so the theorem above follows taking the magnetic field identically zero)..3. More on functional spaces Let L>begiven. Let T 3 be the torus defined as the compact quotient manifold, T 3 = R 3 /πlz 3. For general s R, we write H s for the Sobolev space of vector functions defined on T 3 with zero spatial mean, i.e. { } H s := w : T 3 R 3, w [H s T 3 )] 3, w dx = T 3, and set L = H.

7 Vol. 5) Global attractor for the Navier 87 We denote by the subscript σ spaces of divergence-free classes of) functions, i.e. { } H s σ := w : T 3 R 3, w [H s T 3 )] 3, w =, w dx =. T 3 Since we work with periodic boundary conditions we can describe the spaces that we use in terms of the Fourier series on the 3D torus T 3. A vector field w H s σ can be expanded in terms of Fourier series of πl-periodic functions) as wx) = ŵ k e i k x, k T 3 where the space of frequencies T3 is defined by setting T 3 := π πl Z3 = L Z3 and T3 := T 3 \{}, and the Fourier coefficients are given by ŵ k = πl) 3 wx)e i k x dx. T 3 We set k := k + k + k 3, and we introduce the H s norm by w s, = k s ŵ k, ) k T 3 for s =weset :=, ). The inner product associated with this norm is w, v) H s = ) k s ŵ k, v k, k T 3 where here v k denotes the complex conjugate of v k if v k =vk,v k,v3 k )we have v k =vk, v k, v3 k )) and, ) in the right-hand side of the above formula is the scalar product in C 3. Since we are looking for real vector valued functions, we have that the Fourier coefficients satisfy v k = v k, k T3. Therefore, the spaces H s can be rewritten as H s := wx) = ŵ k e i k x, ŵ k = ŵ k, w s, = k s ŵ k < +. k T 3 k T 3 We note that ŵ = πl) 3 T 3 wx) dx =sincew has zero mean.

8 88 D. Catania, A. Morando and P. Trebeschi NoDEA.4. Filter properties In this paper we will use the deconvolution operator D N,, defined by ) and studied in Sect..5, which is constructed by using A defined in ), i.e. the Helmholtz operator with fractional regularization. For real p the pseudo-differential operator A p is defined in the periodic case as A p w := +α k ) p ŵk e i k x. k T 3 Lemma.. i) For all p the operator A p, as a linear operator in L, is self-adjoint and commutes with differentiation. ii) The inverse operator A is continuous from H s to H s+ with the estimate A w s+, α w s,. iii) We have A / w = w + α Δ) / w, w H. ) Proof. For the proof of i) ii) we refer to []. Concerning iii), using that A / and Δ) / are self-adjoint, we get ) A / w = A / w,a / w =w,a w) = w, I + α Δ) )w ) = w + α ) Δ) / w, Δ) / w = w + α Δ) / w. Remark.3. Since the relevant cases and applications concern small values of α, fromnowon, weassume <α ; this makes some terms less cumbersome and does not limit the generality of the results. Remark.4. Formula ) implies the following equivalence of norms A / w w, α A/ w, w H. ) By the same reason we have also that A w w, α A w, w H. 3).5. The deconvolution operator We can express the pseudo-differential deconvolution operator in terms of the Fourier series. We have: D N, w = D N, k) ŵ k e i k x k T 3 where α D N, k) =+α k k ) N+ )ρ N,k, ρ N,k = +α k.

9 Vol. 5) Global attractor for the Navier 89 The following lemma is proved in []. Lemma.5. The symbol DN, k) of the operator D N, satisfies the following properties: D, k) =, D N, k) N +, 4) D N, k) Âk) := +α k ). The following lemma is given in [4] and []). Lemma.6. For all s and N>, the operator D N, : H s H s is self-adjoint; commutes with differentiation. Moreover, for all w sufficiently smooth we have w s, D N, w s, A / N, w s,. 5) Remark.7. In [] the above inequalities are formulated in terms of a suitable constant C. In fact, one can take C =, so we do not write C in the above inequalities. Lemma.8. For all p,s and N>, and for all w sufficiently smooth we have w s, D p N, w s, N +) p w s,. 6) Proof. We use the definition of the norm ) and 4) tohave w s, = k s ŵ k k s D p N, k) ŵ k k T 3 k T 3 N +) p k s ŵ k =N +) p w s,. k T 3 In the sequel we will make use of the following estimates which come from the estimates 5) and 6) as particular cases: D N, w A / N,w ; 7) A / D N, w N +) / A / N,w ; 8) A / D N, w N +) / A / N,w ; 9) A D N, w N +) / A N,w. ) In particular estimate 7) is estimate 5) with s =, while 8) ) come from 6) taking p =/ and replacing w by A / N,w, A/ N, w and A N, w respectively.

10 8 D. Catania, A. Morando and P. Trebeschi NoDEA 3. A priori estimates In this section, we collect suitable a priori estimates for the weak solution to the problem 3) 5), that will be used from Sect. 4 on to prove the existence of the global attractor. We set λ := L first eigenvalue for the Laplace operator Δ), Λ := Δ) / with domain H and periodic boundary conditions Λ is a positive self-adjoint operator in L ), Λ := Λ,and k := A / N, w), ) K := Λ A / N, f, ) ν k := k + K. 3) νλ Theorem 3.. If w is a regular weak solution of 3) 5), for arbitrary t,r > we have A / N, wt) k e νλt + K νλ e νλt ) k, 4) t+r t t Remark 3.. ν A / N, ws) ds rk + k, 5) e νλs/c ν A / N, ws) ds CK νλ + k, C> given. 6) i) From 4) we get that t A / N, wt) L, + ). Then, in view of ), we obtain that N, w L, + ; H σ) and, in view of Lemma.8 with p =/, we also deduce ii) From 5) we find that w L, + ; H σ). t A / N, wt) L loc, + ), which yields, in view of Lemma.8 and ), w L loc, + ; Hσ + ). Proof. To be rigorous, in order to prove the estimates one should pass through a sequence of Galerkin approximating solutions of 3) 5), then pass to the limit in the resulting estimates. In order to avoid overloading calculations, here, we derive the estimates by arguing directly, at a formal level, on the solution of the problem 3) 5). We refer to [] and [4] for a detailed discussion of the Galerkin approximation technique.

11 Vol. 5) Global attractor for the Navier 8 Let us prove 4). We test the Eq. 3) against A D N, w. Leibniz s rule and differentiation under the integral sign give ) t w,a D N, w)= t A / N, w,a/ N, w = d dt A/ N, wt). Since w =, by using 93) weget DN, w D N, w ),A D N, w ) = D N, w D N, w),d N, w)=; by using that the operators A /, N, are self-adjoint, we get ν Δw,A D N, w)= ν w),a D N, w)=ν w, A D N, w) ) = ν A / N,w, A/ N, w = ν A / N, w ; by using w = and integration by parts, we get moreover q, A D N, w)= q, A D N, w)) = ; f,a D N, w)=f,d N, w)= N, f,d/ N, w). Collecting the previous estimates we get d dt A/ N, wt) + ν A / N, wt) = N, f,d/ N, wt)). We estimate the right-hand side by Λ = Δ) / ) N, f,d/ N, w) = Λ A / N, f, ΛA/ N, w) Λ A / N, f ΛA/ N, w Λ A / N, f ν = { K + ν A / N, w } + ν ΛA/ N, w by Cauchy Schwarz inequality and definition of K. We notice that in the estimate above we used that Λw = w. We deduce d dt A/ N, w + ν A / N, wt) K 7) and, thanks to the Poincaré inequality, For every w we compute by Parseval s identity w = j w = i k j ŵ k = k j ŵ k = k ŵ k = Λw. j= j= k j= k k

12 8 D. Catania, A. Morando and P. Trebeschi NoDEA d dt A/ N, wt) + νλ A / an application of Gronwall s Lemma yields N, wt) K ; A / N, wt) k e νλt + K e νλt ) νλ which gives 4) and in particular A / N, wt) k, which implies see Remark 3.) w L, ; H σ). Let us prove now 5). Integrating 7) over[t, t + r] and neglecting A / N, wt + r) in the left-hand side, we get t+r t ν A / N, ws) ds rk + A / N, wt) and using 4) yields 5). Let us prove 6). Now, let us multiply 7)bye νλt/c > and integrate in time over the interval [,t]: t e νλs/c t d ds A/ N, ws) ds + t e νλs/c ν A / N, ws) dt K e νλs/c ds = CK e νλt/c) CK. νλ νλ Integrating by parts, the first integral in the left-hand side of the above inequality becomes [ e νλs/c A / N, ws) ] t + νλ t e νλs/c A / N, C ws) ds, hence we have e νλt/c A / N, wt) + νλ C + t t e νλs/c ν A / N, ws) ds e νλs/c A / N, ws) ds CK + k ; νλ neglecting the first two terms, which are positive quantities, provides 6). 4. Existence of the global attractor 4.. Some basic definitions To make the exposition self-contained, we recall below some basic concepts and results on invariant sets and attractors in dynamical systems that will be used later on.

13 Vol. 5) Global attractor for the Navier 83 Definition 4.. Let W, d) be a metric space. A semigroup on W, d) is a family of operators St)) t, St) :W W, such that i) S)w = w, for all w W ; ii) Ss)St)w = St + s)w for all w W and for every s, t. A semigroup St)) t is said to be continuous or a semiflow) if St) :W W is a continuous operator from W to itself, for every t. Definition 4.. Let St)) t be a semigroup on W, d). We say that the operators St) areuniformly compact for t large if for every bounded set B there exists t, which may depend on B, such that St)B t t is relatively compact in W. Definition 4.3. We say that A W is a global attractor for the semiflow if ) A is nonempty and compact; ) St)A = A for all t i.e. A is invariant); 3) for all bounded sets B W,wehave lim δ St)B,A) =, t + where δx, Y ):= supinf dx, y) is the Hausdorff semidistance between x X y Y the pair of sets X, Y W. In order to establish the existence of attractors, a useful concept is the related concept of absorbing set. Definition 4.4. Let B be a subset of W and U be an open set containing B. We say that B is absorbing in U if B U, B bounded, t B ) such that St)B B, t t. When U = W in the preceding definition, then we simply say that B is an absorbing set in W. Remark 4.5. Let us observe that a global attractor for a semigroup St)) t, if it exists, is necessarily unique and coincides with the omega-limit of an absorbing set. We remind that the omega-limit of a set B W is defined by ωb) := St)B. s t s The following theorem on the existence of the global attractor is proven in [9] seealso[7]). Theorem 4.6. Let W, d) be a metric space and St)) t a continuous semigroup on W, such that St) is uniformly compact for t large. Assume moreover that there exists a bounded absorbing set B W. Then there exists the unique)

14 84 D. Catania, A. Morando and P. Trebeschi NoDEA global attractor A for the semigroup St)) t, which is given by the omegalimit of the set B, i.e. A = ωb) = St)B. s t s 4.. Existence proof of Theorem.) In this section, the theoretical results collected above will be applied to the continuous semigroup St)) t on H σ defined by t, St) :H σ H σ w St)w := wt, ), 8) where w = wt, ) is the regular weak solution of the problem 3) 5) with initial data w. We notice that the continuity of the operators St) in8) directly follows from the definition of regular weak solution, see Sect... In order to prove the existence of the global attractor for the semigroup 8), as an application of Theorem 4.6, we need the following lemmas. Observe that, in this section, we provide the spaces H σ and H σ with the norms w H := A / N, w, w H := A N, w, which are respectively equivalent to the norms w, and w, [see Remark.4 and 6)]. Lemma 4.7. If / and B is a bounded subset of H σ, then there exists t = t B ) > such that, if w is a regular weak solution of 3) 5) with arbitrary initial datum w B, then A / N, wt) K, t t, 9) νλ where K is defined in ) and λ is the first eigenvalue of the Laplace operator Δ. In particular, we have lim sup A / N, wt) K =: r t + νλ. Notice that the constant r is independent of the initial datum. Proof. The estimate 9) follows at once from the estimate 4) by noticing that the right-hand side of the latter estimate gt) :=k e νλt + K e νλt ) R e νλt + K e νλt ) νλ νλ tends to K νλ as t +, where R =sup w B A / N, w ; thus there exists t > such that gt) K νλ for t t. Remark 4.8. As a consequence of Lemma 4.7, see also 5), we obtain that the ball B = B ; r ):= { } w H σ : A / N, w r 3)

15 Vol. 5) Global attractor for the Navier 85 is a bounded absorbing set in H σ; indeed by Lemma 4.7 we get that for every bounded set B H σ we have St)B B, t t, being t > the time given in Lemma 4.7. Lemma 4.9. Assume that f H, / and let B be a bounded subset of H σ.thereexistr = r ν, N,, K,λ ) > andatimet = t B ) >t such that, if w is the regular weak solution of 3) 5) with arbitrary initial datum w B, then A N, wt) r, t t. 3) In particular, we have lim sup A N, wt) r. t + Proof. We use A D N,w as test function for the Eq. 3) now formally, but the procedure actually goes through the Galerkin approximation). Thus, testing the Eq. 3) against A D N,w, and arguing as in the proof of Theorem 3. we get t w,a D N, w ) = ) t A N, w,a N, w = d dt A N, w ; DN, w D N, w ),A D N, w ) = D N, w D N, w),a D N, w); using that Δ =Λ and w = ν Δw,A D N, w ) = ν Λ w,a D N, w ) ) = ν ΛA N, w, ΛA N, w = ν ΛA N, w = ν A N, w ; q, A D N, w ) = q, A D N, w) ) =. Finally, we observe that f,a D N, w ) =f,a D N, w) ) ) = N, f,a N, w = Λ N, f, ΛA N, w Λ N, f ΛA N, w = Λ N, f A N, w Λ N, f + ν ν 4 A N, w. Collecting the previous inequalities we get d dt A N, w + ν A N, w ) = N, f,a N, w D N, w D N, w),a D N, w) 3) Λ N, f + ν ν 4 A N, w + D N, w D N, w),a D N, w).

16 86 D. Catania, A. Morando and P. Trebeschi NoDEA Now we are going to make an estimate of the nonlinear term D N, w D N, w),a D N, w) in the right-hand side of the above inequality. We firstly make use of the formula 9) recall that w = ) and Hölder s inequality to get D N, w D N, w),a D N, w) = D N, w ) D N, w,a D N, w) D N, w L 3 D N, w L 3 A D N, w L 3. Now we estimate each of the L 3 -factors appearing in the right-hand side of the inequality above as follows. The estimate of D N, w L 3. We apply 95) to the function u = D N, w to get D N, w L 3 C A / D N, w a D N, w a. 33) The estimate of D N, w L 3. We apply 96) to the function u = D N, w to get D N, w L 3 C A D N, w b A / D N, w b. 34) The estimate of A D N, w L 3. We apply 97) to the function u = D N, w to get A D N, w L 3 C A D N, w c A / D N, w c. 35) The exponents a, b, c involved in the estimates 33) 35) above are those calculated in 99). Using 33) 35), we get D N, w D N, w),a D N, w) C A / D N, w a D N, w a A D N, w b A / D N, w b A D N, w c A / D N, w c = C D N, w a A / D N, w b c+a A D N, w b+c. 36). We estimate the factor involving D N, w by using 7) toget D N, w a A / N, w a. 37). We estimate the factor involving A / D N, w by using 8) toget A / D N, w b c+a N +) b c+a A / N, w b c+a. 38) 3. Finally, we estimate the factor involving A D N, w by using ) toget From 36) 39) we get A D N, w b+c N +) b+c A N, w b+c. 39)

17 Vol. 5) Global attractor for the Navier 87 D N, w D N, w),a D N, w) CN +) + a / A N, w 3 b c A N, w b+c ν 4 A N, w + C ν N a b+c +)+ )p A / p N, w 3 b c)p = ν 4 A N, w + C N +) + ν 4 ) + / A N, w 3+, 4) where Young s inequality has been used with p = b+c = + and p = b c = + is the conjugate exponent of p [note that p>, as b + c< thanks to )] and where we have computed [see 99)] b + c p = +, a ) p = + ) +, 3 b c)p =3+ 4. Coming back to 3) and using 4) to estimate the nonlinear term in the right-hand side, we obtain d dt A N, w + ν A N, w Λ N, f + ν ν A N, w + C N +) + ν 4 ) + / A N, w 3+. Absorbing on the left the term ν A N, w appearing in the right-hand side of the estimate above, estimating A / N, w 3+/ by 9) we then get d dt A N, w + ν A N, w Λ N, f + C ) N +) + ν ν 4 ) + K 3+ ) =: K, t t. νλ 4) Finally, using Poincaré s inequality λ A N, w A N, w we get d dt A N, w + νλ A N, w K, t t. 4) Then setting yt) := A N, wt),4) becomes y t)+νλ yt) K, t t. 43) Multiplying 43) bye νλt then gives d dt e νλ t yt) ) =e νλt y t)+νλ e νλt yt) K e νλt, t t, hence integrating in time over t,t) for an arbitrary t>t ) yt) e νλt t ) yt )+ K νλ e νλt t ), t t. 44)

18 88 D. Catania, A. Morando and P. Trebeschi NoDEA Let us observe that the value K introduced in 4) does not depend on the initial data in 5). Since the right-hand side of 44) tends to K νλ as t +, one can find a time t >t such that A N, wt) = yt) K νλ, t t. inde- This just provides an estimate of type 3), where we take r := K νλ pendent of the initial data). This ends the proof. Remark 4.. The result of Lemma 4.9 [see estimate 3)] tells that, as long as /, a regular weak solution w to problem 3) 5) with arbitrary initial datum w H σ gains some additional regularity, definitely in time, with respect to the one originally required at the beginning of Sect..: from the time t > on, the solution w belongs to H σ pointwise in time. Notice that, as a direct consequence of the definition of regular weak solution, we only deduce that w L loc, + ; H σ ) observing that + as long as ). Let us denote by B the ball in H B = B ; r ):= σ : { } w H σ : A N, w r. As a consequence of Lemma 4.9, see also the norm equivalence in 3) and 6) with p =/), we deduce that for every bounded set B H σ t t St)B B, being t = t B) > the time given in Lemma 4.9. Since H σ is compactly imbedded in H σ, the ball B is a relatively compact subset of H σ. Hence for every bounded set B H σ St)B t t is a relatively compact subset of H σ, which means, according to Definition 4., that the operators St) are uniformly compact for large t. In view of the results collected in the previous Lemmas 4.7, 4.9, and the Remarks 4.8, 4., we may conclude that the semigroup St)) t of operators defined in 8) enjoys all the assumptions required in Theorem 4.6. Hence applying the result of that theorem, we deduce that a unique global attractor A exists and is given by the omega-limit of the absorbing ball B in 3), namely We have thus proved Theorem.. A = ωb ).

19 Vol. 5) Global attractor for the Navier Differentiability In this section we study the differentiability of the semigroup St) :H σ H σ, given by St)w = wt, ) [see8)], with respect to the initial datum w.the reason is that, in the subsequent section, we will investigate the Hausdorff and fractal dimensions of the global attractor A: the differentiability of the semigroup St) is a sufficient condition in order to prove that the Hausdorff dimension of the attractor A is finite and to find an appropriate estimate of its value see [9, Theorem 3.]). Consider the following systems of equations: t w νδw + q = f D N, w D N, w, w =, w) = w ; t v νδv + q = f D N, v D N, v, v =, v) = v ; 45) t W νδw + Q = D N, W ) D N, w D N, w ) D N, W W =, W ) = w v, and set η := w v W. In order to prove the Fréchet differentiability with respect to the initial datum of St) :H σ H σ,itissufficienttoprovethat for each t [,T). A / N, ηt) C A/ N, w v ) 46) Lemma 5.. Assume /. Given T > and defined V := w v, there exists C = CT,N,,K ) > such that A / N, V t) + ν for each t [,T). t Proof. The equations satisfied by V are A / N, V s) ds C A / N, V ) t V νδv + q q )=D N, v ) D N, v D N, w ) D N, w = D N, V ) D N, V D N, V ) D N, w D N, w ) D N, V 47) V =, V ) = w v. Testing 47) bya D N, V and, recalling that u )V, V ) = provided that u =, we obtain [see formula 9) in Lemma A.] d dt A/ N, V + ν A / N, V = D N, V ) D N, w,d N, V ) =D N, V ) D N, V,D N, w). 48) We estimate D N, V ) D N, V,D N, w) C D N, V L 3 D N, V L 3 D N, w L 3. 49)

20 83 D. Catania, A. Morando and P. Trebeschi NoDEA We estimate the L 3 -norms in 49) by using the inequality 95) toget,for a =, D N, V ) D N, V,D N, w) D N, V a A / D N, V a D N, V a A / D N, V a D N, w a A / D N, w a. To bound the factors involving the norms D N, w, D N, V and D N, V in the right-hand side above, we apply the estimate 7) respectively to w, V and V ; to bound the factors involving the norms A / D N, w, A / D N, V and A / D N, V, we apply the estimates 8) and 9) to w and V. Thus, from Young s inequality, we obtain D N, V ) D N, V,D N, w) CN +) 3 a A / N, V A/ N, V A/ N, w ν A/ N, V + C ν N +)3a A / N, w A / N, V. 5) Coming back to 48) and using 5) weget d dt A/ N, V + ν A / N, V ν A/ N, V + C ν N +)3a A / N, w A / N, V. Absorbing in the left-hand side the term ν A/ N, V we find d dt A/ N, V + ν A/ N, V C ν N +)3a A / N, w A / N, V C ν N +)3a A / N, V, 5) where we have used Theorem 3., see 4) in order to bound A / N, w with a suitable constant C. We neglect in the left-hand side the term ν A/ N, V andweapply Gronwall s Lemma to obtain, for all t [,T), A / N, V t) A / N, V ) e t C ν N+)3a ds A / N, V ) e CT. 5)

21 Vol. 5) Global attractor for the Navier 83 Coming back to 5) and integrating in time over,t) and using again 5), we get A / N, V t) + ν A / t A / N, V s) ds N, V ) + C ν N +)3a t A / N, V s) ds A / N, V ) + C ν N +)3a e CT A / N, V ) T = C A / N, V ), where C =+ CT ν N +)3a e CT. We are going to prove the announced estimate 46). Lemma 5.. Assume /. There exists a constant C = C T,N,,K ) > such that, for each t [,T) A / N, ηt) C A / N, w v ). Proof. The equations satisfied by η are t η νδη + Q = D N, V ) D N, V D N, η ) D N, w D N, w ) D N, η 53) η =, η) =, for a suitable Q depending on q, q and Q in the Eq. 45). Testing 53) bya D N, η and recalling that D N, w ) D N, η,d N, η) = since w =, we obtain d dt A/ N, η + ν A / N, η =D N, V ) D N, V,D N, η) D N, η ) D N, w,d N, η) = I + II. 54) Let us firstly estimate II. Observing that from Lemma A. II =D N, η ) D N, η,d N, w), we apply the same arguments used to provide the estimate 5), to get II ν A/ N, η + C ν N +)3a A / N, w A / N, η, 55) where a is the same value involved by the Gagliardo Nirenberg inequality 95).

22 83 D. Catania, A. Morando and P. Trebeschi NoDEA As regards to I, we use once again Hölder s inequality, the estimates 95), 7), 8) and Young s inequality to get I D N, V L 3 D N, V L 3 D N, η L 3 D N, V a A / D N, V a D N, V a A / D N, V a D N, η a A / D N, η a CN +) 3 a A / N, V A/ N, V A/ N, η Cν A / N, V A / N, V + C ν N +)3a A / N, η. 56) Coming back to 54), using 55) and 56), we get d dt A/ N, η + ν A / N, η ν A/ N, η + C ν N +)3a A / N, w A / N, η +Cν A / N, V A / N, V + C ν N +)3a A / = ν A/ N, η + C ν N +)3a A / N, w + +Cν A / N, V A / N, V ν A/ N, η + C ν N +)3a A / N, η ) N, η A / N, η / + Cν A N, V ) A / N, V, 57) where we have used Theorem 3.; see 4) in order to bound A / N, w with a suitable constant C and Lemma 5. to bound A / N, V with C A / N, V ). Absorbing ν A/ N, η in the left-hand side of 57) weget d dt A/ N, η + ν A/ N, η C ν N +)3a A / N, η / + Cν A N, V ) A / N, V. 58) Neglecting ν A/ N, η in the left-hand side of 58) and integrating in time over,t) recall that η) = ) weget A / N, ηt) C ν N +)3a t / + C A N, V ) ν C N t t A / N, ηs) ds A / N, V s) ds A / N, ηs) ds + C A / N, V ) 4, with C N := C ν N +)3a and in view of Lemma 5..

23 Vol. 5) Global attractor for the Navier 833 Applying then Gronwall s lemma to the last inequality we get for t [,T) A / N, ηt) C A / N, V ) 4 e CN t C A / N, V ) 4, where C := C e CN T. This ends the proof. 6. Estimate for the global attractor dimension First, we linearize Eq. 3), projected onto the space of divergence free functions, about a solution w. The linear equation satisfied by a perturbation δw is ) ) t δw + D N, w D N, δw + D N, δw D N, w νδδw =, that is to say where d δw + T t)δw =, 59) dt ) ) T t) = νδ+ D N, w ) + ) D N, w. 6) If δw j,, j =,...,M,areM linearly independent functions in H σ, we denote by δw j t) the corresponding solutions to 59), computed at time t, with δw j ) = δw j,.wesete = { δw t),...,δw M t) } and denote by P M t) the orthogonal projection of H σ onto the span of E, denoted hereafter by E. In this section we provide H σ with the inner product [u, v] := A / N, u,a/ N, v) 6) and introduce the operator T M t) defined by T M t) =TraceP M t) T t) P M t)). Let {ϕ j t)} M j= be an orthonormal with respect to [, ] defined in 6)) basis of E = P M t)h σ, so that [ϕ i t), ϕ j t)] = δ ij Kronecker symbol), i,j =,...,M. 6) Using 6) and 6) weget T M t) = = [T t)ϕ i t), ϕ i t)] i= [ ) νδϕ i t)+ D N, w D N, ϕ i t) i= ) ] + D N, ϕ i t) D N, w, ϕ i t). 63)

24 834 D. Catania, A. Morando and P. Trebeschi NoDEA Accordingto6) wehavethat ) [ νδϕ i t), ϕ i t)] = νδa / N, ϕ it),a / N, ϕ it) ) = ν A / N, ϕ it), A / N, ϕ it) = ν A / N, ϕ it), 64) [ ) ] D N, w D N, ϕ i t), ϕ i t) ) ) = A / N, D N, w D N, ϕ i t),a / N, ϕ it) ) ) = A D N, w D N, ϕ i t),d N, ϕ i t) = D N, w D N, ϕ i t) ),D N, ϕ i t) ) =D N, ϕ i t) )D N, w,d N, ϕ i t)), 65) [ ) ] D N, ϕ i t) D N, w, ϕ i t) ) ) = A / N, D N, ϕ i t) D N, w,a / N, ϕ it) ) ) = A D N, ϕ i t) D N, w,d N, ϕ i t) = D N, ϕ i t) D N, w ),D N, ϕ i t) ) =, 66) where integration by parts and formulas 9), 93) have been used in the calculations above remember that w = ϕ i = ). Substituting 64) 66) into63) we get T M t) = i= ν A / N, ϕ it) + D N, ϕ i t) )D N, w,d N, ϕ i t)) i= = Q M t)+r M t). 67) The next goal is to prove that T X M := lim inf T M t) dt >, 68) T + T provided that M is taken sufficiently large, and find an estimate for M see the next Lemma 6.5). In fact see [9, Sect. 3.4] and []) if M is sufficiently large such that 68) is satisfied, then M is an upper bound for the Hausdorff dimension d H A) and the fractal dimension d F A) of the global attractor A: d H A) d F A) M. To get this result, we need to provide useful estimates for Q M t) andr M t) in the representation 67) oft M t). Lemma 6.. Assume that /. There exists a numerical constant C> such that

25 Vol. 5) Global attractor for the Navier 835 Q M t) νλ M 5/3, 69) C where λ = L is the first eigenvalue of Δ. Proof. We use the Lieb Thirring inequality in Lemma A.5 for j x) =A / N, ϕ jt, x). By 6) it follows that { j } M j= defined above is an orthonormal set in L σt 3 ). For the functions j above, the right-hand side of Lieb Thirring s inequality 5) becomes A / N, ϕ jt, x) : A / N, ϕ jt, x) dx T 3 j= = j= T 3 A / N, ϕ jt, x) dx = j= A / N, ϕ jt) = Q Mt). ν As regards to the left-hand side of 5) we get by the use of Jensen s inequality 5/3 A / N, ϕ jt, x) dx T 3 = j= T 3 /3 T 3 /3 T 3 j= j= A / A / N, ϕ jt) N, ϕ jt, x) dx 5/3 = 5/3 T 3 /3 M 5/3, 7) where T 3 =πl) 3 is the Lebesgue measure of the torus T 3, and where we used that A / N, ϕ jt) = because of the orthonormality). The estimate 69) comes from combining Lieb Thirring s inequality with the estimate in 7). Remark 6.. We notice that the same estimate 69) could be obtained by the following alternative argument. The asymptotic behavior of the eigenvalues of the operator Δ is such that see [,9]) λ i λ i /3 C, i =,,... and Q M t) ν λ i. Hence Q M t) νλ M 5/3 5 3 i /3 M 5/3. i= C i= since, by induction, one can easily prove that

26 836 D. Catania, A. Morando and P. Trebeschi NoDEA Lemma 6.3. Assume that / < and let σ = σ) be defined as in 94). Then the following estimate holds R M t) Q M t) where + CC M σ)s A / N, w A / N, w μ, 7) N +) σ+/ L 3/q ) s C :=, 7) α σ ν σ3/ ) for a suitable numerical constant C>, and μ := +3σ σ 3σ +σ, q := 6 + σ), s := 3σ +σ. 73) Proof. For a given, such that / <, let σ = σ) befixedasinthe statement above. The Cauchy Schwarz and the Hölder inequalities together with 7) give R M t) DN, ϕ i t) ) D N, w D N, ϕ i t) dx T 3 i= D N, ϕ i t) D N, w dx T 3 i= T 3 M ) σ M ) σ D N, ϕ i t) D N, ϕ i t) D N, w dx i= i= M ) σ M ) σ D N, ϕ i t) L D N, ϕ i t) i= D N, w L p T 3 /q σ C D N, ϕ i t) i= σ C D N, ϕ i t) i= L L i= L σ M ) σ D N, ϕ i t) D N, w L pl 3/q i= M i= A / N, ϕ it) ) σ D N, w L pl 3/q σ = C D N, ϕ i t) M σ D N, w L pl 3/q, 74) i= L where p is the exponent defined by ), through σ and, while q is given in 73) and makes satisfied the needed identity p + q = σ.

27 Vol. 5) Global attractor for the Navier 837 It is understood that the splitting of M D N, ϕ i t) into the σ and σ)- i= powers becomes trivial as long as σ = [when /3, cf. 94)]. In this case L and L σ -norms of those two powers collapse into the L -norm of the whole M D N, ϕ i t) thus M σ in the last expression above reduces to i= ), while p and q become the conjugate exponents p = 6 7 6, q = 6 6. Now, we use Lemma A.4 in the Appendix to estimate the L -norm in the right-hand side of 74) by D N, ϕ i t) i= L C ) 3/ QM t) N +), 75) α ν for a suitable numerical constant C. It is just the application of the above inequality see Lemma A.3) that requires to exclude the border value = / from our subsequent considerations. By the use of 98) in Lemma A. in the Appendix, we estimate the L p - norm of D N, w as D N, w L p C A / D N, w β A / D N, w β, with β given in ). Then we apply estimates 8), 9) toget D N, w L p CN +) / A / N, w β A / N, w β. 76) From 74), 75) and 76) we find R M t) CN +)σ+/ L 3/q M σ α σ A / N, w β. QM t) ν ) σ3/ ) A / N, w β Now, we apply to the last expression above the Young inequality ab ar r + bs s, where the conjugate exponents r = σ3/ ) = σ3 ) and s = σ3/ ) = ar QM t) 3σ+σ arechoseninsuchawaythat r = ; then we get R M t) Q Mt) N +) σ+/ L 3/q M σ + C α σ ν σ3/ ) A / N, w s β), ) s A / N, w sβ which gives the desired estimate in view of 7) and since sβ =,s β) = μ.

28 838 D. Catania, A. Morando and P. Trebeschi NoDEA Remark 6.4. According to 94) one explicitly derives the following values of the exponent μ involved in the estimate 7) [see73)] μ = {, if / </3, ), if /3. 77) In particular it follows that μ< as / </3 and μ as /3. In view of the subsequent analysis, it is also worth noticing that because of 94) and 73) one has < σ)s as / </3 and σ)s = as/3. Let us prove now that 68) holds. To this end, we are going to prove that the following lemma holds true. Lemma 6.5. Assume that / <. Then T X M = lim inf T M t) dt νλ M 5/3 T + T C where C is defined in 7). M σ)s C C K+μ ν +μ λ μ, 78) Proof. Let us start by the representation formula 67). In view of 7), we have the following estimate T M t) =Q M t)+r M t) Q M t) R M t) Q Mt) CC M σ)s A / N, w A / N, w μ, where the exponents σ, μ and s are given in 94) and 73). We recall that the use of estimate 7) requires >/. We calculate T X M = lim inf T M t) dt T + T lim inf T + T T Q M t) CC M σ)s lim sup T + νλ M 5/3 C dt T T CC M σ)s lim sup T + A / N, w A / N, w μ dt T A / N, T w A / N, w μ dt 79) in view of 69). It remains to produce an estimate for the limit superior in the previous inequality. We will show see the next Lemma 6.7) that the following estimate lim sup T + T A / N, T w A / N, w μ dt C K+μ ν +μ λ μ

29 Vol. 5) Global attractor for the Navier 839 holds true. We use now the estimate above in 79) to find T X M = lim inf T M t) dt T + T νλ M 5/3 C M σ)s C C K+μ ν +μ λ μ, which is just the estimate 78) of the lemma. Remark 6.6. The estimate 78) implies X M to be positive for M>large enough, since the exponent σ)s of the power of M in the second term of the right-hand side above lies in the interval, ] as long as / <, see Remark 6.4. Lemma 6.7. Assume that /. There exists a suitable positive constant C such that T lim sup A / N, T + T w A / N, w μ dt C K+μ ν +μ λ μ, 8) where K is defined in ). Proof. From estimate 4) we find in particular A / N, wt) k e νλt + K νλ from which { ) μ } A / N, wt) μ C k μ K e νμλt +. νλ We note that the constant C in the inequality above can be chosen to be independent of μ, since μ as/, see Remark 6.4. Hence lim sup T + T T Ck μ lim sup T T + K A / N, w A / N, w μ dt T e νμλt A / N, w dt 8) ) μ T +C lim sup A / N, νλ T + T w dt = I + I. To estimate I we use 6) with the constant C in 6) equal to μ )tofind T e νλμt A / N, ws) ds K μν + k λ ν, k, K being defined in ), ), hence I = Ck μ lim sup T + T T To estimate I,weuse5) with t =,r = T to find T e νμλt A / N, wt) dt =. 8) A / N, wt) dt TK + k, ν

30 84 D. Catania, A. Morando and P. Trebeschi NoDEA where k is defined in 3), from which ) μ K I = C lim sup νλ T + T ) μ K C lim sup νλ T + T Then 8) follows from 8), 8) and 83). TK + k T ν A / N, w dt = C K+μ ν +μ λ μ. 83) 6.. Estimate by the Grashoff number proof of Theorem.) Using 78), let us firstly derive an estimate of M in terms of the Grashoff number G defined in 6). Comparing G with K defined in ), through the Poincaré inequality where we use that λ = L ), we get K = Λ A / N, f λ ν = ν3 G L. Using 84) in78) then gives A / N, f ν = L A / N, f ν 84) X M νλ M 5/3 C M σ)s C C ν+μ G +μ L +μ λ μ = νl M 5/3 M σ)s C Cν +μ G +μ L μ, C from which X M > holds true provided that M>C C ν +μ L +μ) 3 5+3σ )s G 6+6μ 5+3σ )s. 6.. Estimate by mean dissipation length proof of Theorem.3) In order to get the desired estimate, we are going to produce an estimate from below of X M, defined in 68), different from the one provided by Lemma 6.5. We will prove the following result. Lemma 6.8. Assume that / <. Then the following estimate X M νm5/3 CL holds true, where it is set γ = 3 CN +)γ M γ) L 3+6/q ν α 4γ l 4 d and 85) + = γ. 86) q Proof. In view of 67) we firstly get an estimate of R M t), by arguing exactly as we did to obtain the estimate 74) in the proof of Lemma 6.3, but making a different choice of the exponents; namely, by the Hölder inequality and 7) we get for each / <

31 Vol. 5) Global attractor for the Navier 84 R M t) T 3 M ) γ M ) γ D N, ϕ i t) D N, ϕ i t) D N, w dx i= i= M ) γ M ) γ D N, ϕ i t) L D N, ϕ i t) i= D N, w T 3 /q γ C D N, ϕ i t) i= γ C D N, ϕ i t) i= L L i= L γ M ) γ D N, ϕ i t) D N, w L 3/q i= M i= A / N, ϕ it) ) γ D N, w L 3/q γ = C D N, ϕ i t) M γ D N, w L 3/q, 87) i= L where the exponents γ and q are defined by means of 86). Notice that the second relation in 86) is just the needed constraint in order that the Hölder inequality applies with the exponents γ and q above. Notice also that γ = 3 is an increasing function of, hence / <γ, as long as / <. 88) Since in particular γ =, as long as =, the same observation about the estimate 75) applies to the estimate above when =. Inthiscasethe splitting of M i= D N, ϕ it) into the γ and γ)-powers becomes trivial: the L and L γ -norms of those two powers reduce to the L -norm of the whole M i= D N, ϕ it) hence M γ =)andq =. An explicit calculation gives that 3 ) q =, for arbitrary / <. Using 7) and 75) to estimate respectively the L -norm of D N, w and the L -norm of M D N, ϕ i t) in the right-hand side of 87), we obtain i= CN +)γ R M t) α γ ν Q Mt)) γ3/ ) M γ A / γ3/ ) N, w L3/q. Again, as done in the proof of Lemma 6.3 we apply to the last expression above the Young inequality ab ar r + bs s, where the conjugate exponents r = γ3/ ) = γ3 ) and s = γ3/ ) = 3γ+γ arechoseninsucha way that ar QM t) r = ; then we get

32 84 D. Catania, A. Morando and P. Trebeschi NoDEA R M t) Q Mt) + C N +)sγ L 3s/q M s γ) α sγ ν sγ3/ ) A / N, w s. 89) Now the crucial point is to observe that, with the choice of γ made in 86), we get γ3 ) =, thus s = r =. Hence, the estimate 89) becomes R M t) Q M t) + C N +)γ L 6/q M γ) α 4γ A / N, ν w. For the rest, the proof follows the same lines of Lemma 6.5. Firstly, from 67) we find T M t) =Q M t)+r M t) Q M t) R M t) Q Mt) Then we calculate T X M = lim inf T + T lim inf T + T T C N +)γ L 6/q M γ) α 4γ A / N, ν w. T M t) dt Q M t) dt C N +)γ L 6/q M γ) α 4γ ν T Q M t) = lim inf dt T + T C N +)γ L 6/q M γ) α 4γ ν lim sup T + lim sup T + T We use 69) to estimate from below lim inf T + T L )by lim inf T + T and 9) to estimate lim sup T + lim sup T + T T T T T Q M t) T A / N, T w dt T ν A / N, T w dt. 9) dt νm5/3 CL ν A/ N, w dt by ν A / N, wt) dt L3 ν 3. Q M t) dt recall that λ = We use the last two inequalities above to bound the right-hand side of 9) and to obtain X M νm5/3 CL C N +)γ L 6/q M γ) α 4γ ν L 3 ν 3 C N +)γ L 3+6/q M γ) ν α 4γ l 4, d that is just the estimate 85). l 4 d l 4 d = νm5/3 CL

33 Vol. 5) Global attractor for the Navier 843 Let us observe that in view of 88), the exponent γ) in the power of M appearing in the second term of the right-hand side of 85) is smaller than. Then, in view 85), we see that X M > holds true provided that M 5/3 γ) > which gives CN +)γ α 4γ [ N +) γ M>C α 4γ L 5+6/q l 4 d = ] 3 L +6/q [ ) γ L L = C N +) γ α where we have used the identity + 6 q =4γ. CN +)γ α 4γ 5 6 γ) L l d l d ) ] 6 6γ, ) 4 L L +6/q, l d ) 5 6 γ) Appendix: Some useful formulas and inequalities Lemma A.. Let a, b, c be sufficiently smooth vector fields such that a =. Then, b a) =a ) b; 9) a ) b, c) = a ) c, b) ; 9) b a), b) =. 93) The following estimates are particular cases of the well known Gagliardo Nirenberg inequality see [6]), where we make use of the equivalence of norms given in Remark.4. Lemma A.. Let us assume that / and define σ) by setting σ) := { + 3, if / </3,, if /3. 94) There exists a positive constant C such that for every u = ux) sufficiently smooth vector or matrix-valued function we have u L 3 C A / u a u a, 95) u L 3 C A u b A / u b, 96) A u L 3 C A u c A / u c, 97) u L p C A / u β A / u β, 98) where a =, b = 3 + p = ), c = +/ 3 +, 99) 6, β = 3σ)+σ). ) +6σ) + σ))

34 844 D. Catania, A. Morando and P. Trebeschi NoDEA Note that the exponents a, b, c involved in the estimates 95) 97) and computed above satisfy the constraints required by the Gagliardo Nirenberg inequality, in particular / a, + b, + c for every /. Moreover, we have /4 b /3, /3 c 3/4. ) As regards to the last inequality 98), due to the formula 94), it is worth considering separately the two cases below: st case: for / </3 the expressions in ) become p =, β =. ) nd case: for /3 the expressions in ) become p = 6, β =. 3) 7 6 Notice that p defined in 3) is an increasing function of and we have p 6, as /3. Notice also that in both the previous cases the exponents p and β in ) 3) satisfy the constraints required by the Gagliardo Nirenberg inequality, in particular β for every /. The fractional version of Agmon s inequality can be found in [] p. 37, Lemma 4.9; the proof is for R n, but can be easily adapted to the periodic case by resorting to the Fourier series instead of the Fourier transform). Lemma A.3. Fractional Agmon s inequality) Let us assume that <s <n/ <s, where n is the space dimension of the periodic box domain D, and let t, ) such that n/ = t)s + ts. Then there exists a constant C = Cn, s,s ) such that u L C u t H s u t H s, where the norms refer to the space D. In particular, if D = T 3 and / <, we have u L C α A/ u / A / u 3/. 4) The final estimate can be easily obtained observing that u, α A/ u, u +, α A/ u, as it is easily seen using Fourier expansions. The following inequality generalizes Lemma 5, Estimate 8), in Foias Holm Titi [] and is used in Sect. 6.