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1 UC Irvine UC Irvine Previously Publishe Works Title Inertial manifols for certain subgri-scale $\alpha$ moels of turbulence Permalink Journal SIAM Journal on Applie Dynamical Systems, 4(3) Authors Abu Hame, M Guo, Y Titi, ES Publication Date DOI 0.37/ Peer reviewe escholarship.org Powere by the California Digital Library University of California

2 INERTIAL MANIFOLDS FOR CERTAIN SUB-GRID SCALE α-models OF TURBULENCE arxiv: v [math.ds] 6 Sep 204 MOHAMMAD ABU HAMED, YANQIU GUO, AND EDRISS S. TITI Abstract. In this note we prove the existence of an inertial manifol, i.e., a global invariant, exponentially attracting, finite-imensional smooth manifol, for two ifferent sub-gri scale α-moels of turbulence: the simplifie Barina moel an the moifie Leray-α moel, in two-imensional space. That is, we show the existence of an exact rule that parameterizes the ynamics of small spatial scales in terms of the ynamics of the large ones. In particular, this implies that the long-time ynamics of these turbulence moels is equivalent to that of a finite-imensional system of orinary ifferential equations. MSC Classification: 35Q30, 37L30, 76BO3, 76D03, 76F20, 76F55, 76F65 Keywors: inertial manifol, turbulence moels, sub-gri scale moels, Navier-Stokes equations, moifie Leray-α moel, simplifie Barina moel.. INTRODUCTION The fielity of the Navier-Stokes equation (NSE) is in capturing the ynamics of turbulent flow. However, their ownfall is in reliable irect numerical simulation of turbulence. Therefore scientists have evelope various approximate moels which are computable an preserve some statistical properties of the physical phenomenon of turbulence, an of particular interest to us in this paper are certain sub-gri scale α-moels of turbulence. In many applications, it is enough to capture the mean features of the flow, to obtain this we nee to average the nonlinear term in the NSE an this leas to the well-known closure problem. In 980 Barina et al. [3] introuce a particular sub-gri scale moel which was later simplifie by Layton an Lewanowski (see [40]) which takes the form: v t ν v +( v ) v + p = f, v = 0, () v = v α 2 v. Here the unknowns are the flui velocity fiel v, an the filtere velocity vector v, as well as the filtere pressure scalar p. In aition, there are two given parameters: ν > 0 is the constant kinematic viscosity, an α > 0 is the length scale parameter which represents the with of the filter. The vector fiel f is a given boy forcing, assume to be time inepenent. For more etails about moel (), see [4, 5, 32, 33]. Date: September 6, 204.

3 2 M. ABU HAMED, Y. GUO, AND E. S. TITI In 2005 Cheskiov-Holm-Olson-Titi [0] introuce the Leray-α moel: w t ν w+( w )w + p = f, w = 0, w = w α 2 w. (2) Leray (934 [34]) establishe the well-poseness of the NSE in 2D an 3D, by introucing a moifie system similar to (2), for which it was easier to prove the existence an uniqueness of solutions, an then by passing with the parameter α 0 + he achieve the existence of solutions to the NSE. An upper boun of the imension of the global attractor an an analysis of the energy spectrum of the solutions of the 3D version of (2) were establishe in [0], which suggeste that the Leray-α moel has great potential to become a goo sub-gri scale large-ey simulation moel of turbulence. See also a computational stuy of this moel in [27, 36, 37]. Inspire by the remarkable performance of the Leray-α moel, Ilyin-Lunasin-Titi (2006 [29]) propose a moifie-leray-α moel: u t ν u+(u )ū+ p = f, u = 0, (3) u = ū α 2 ū. It was emonstrate in [29] that the reuce moifie-leray-α moel (3) in infinite channels an pipes is equally impressive as a closure moel to Reynols average equations as Lerayα moel (2) an other sub-gri scale α-moels, e.g. the Navier-Stokes-α (also known as the viscous Camassa-Holm equations [7, 8, 9, 20]) an the Clark-α [6]. Comparing the three turbulence moels (), (2) an (3), we see that in the simplifie Barina moel (), both arguments of the nonlinearity are regularize, while the Leray-α moel (2) regularizes only the first argument of the nonlinear term, i.e. the transport velocity, an in the moifie Leray-α moel (3), solely the secon argument of the nonlinearity is smoothe, i.e. the transporte velocity is regularize. For the moels (), (2) an (3), the global well-poseness in 3D, the existence of a finite imensional global attractor, an the analysis of their energy spectra have been establishe in [3, 0, 29, 34]. Our interest lies in the large-time behavior of the ynamics generate by turbulence moels. In particular, we aim to show existence of inertial manifols for two ifferent systems in 2D: the simplifie Barina moel () an the moifie-leray-α moel (3), subject to perioic bounary conition, with basic omain Ω = [0,2πL] 2. Long-time behavior of solutions of a large class of issipative PDEs possesses a resemblance of the behavior of finite-imensional systems. The concept of inertial manifol was introuce to capture such phenomenon. Inee, an inertial manifol of an evolution equation is a finiteimensional Lipschitz invariant manifol attracting exponentially all the trajectories of a ynamical system inuce by the unerlying evolution equation [24, 25]. The precise efinition is given in section 3.2. The existence of an inertial manifol for an infinite-imensional evolution equation represents the best analytical form of reuction of an infinite system to a finite-imensional one. This is because an inertial manifol is finite-imensional, an the restriction of the evolutionary equation to this manifol reuces to a finite system of ODEs, which calle the inertial form of the given evolutionary equation. As a result, the ynamical properties of the solution

4 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 3 of the evolutionary PDE, which is an infinite-imensional ynamical system can be analyze by the stuy of an inertial form which is a finite-imensional system. Inertial manifols were introuce by Foias, Sell an Temam in [24, 25]. The iea was employe to a large class of issipative equations [26] (see also [44]). A number of ynamical systems possess inertial manifols, e.g., certain nonlinear reaction-iffusion equations in 2D [8, 25, 39] an in 3D [38], the Kuramoto-Sivashinsky equation [23, 24, 26, 44], Cahn-Hilliar equation [7], as well as the von Kármán plate equations [], just to name a few. It is worth mentioning that an original purpose of eveloping the theory of inertial manifols was for treating the NSE. Unfortunately, the problem of existence of inertial manifols for the 2D NSE is still unsolve an we are unaware of any such result for a system of hyroynamics which oes not involve an artificial hyperviscosity. In particular, the question of existence of an inertial manifol is still open even for the 2D Navier-Stokes-α moel, Leray-α moel an Clark-α moel an others. Recently, the concept of etermine form was introuce in [2, 22], in which it is shown that the long-time ynamics of such moels, in particular that of the 2D NSE, is equivalent to the long-time ynamics of an ODE with continuously Lipschitz vector fiel in certain infinite-imensional space of trajectories with finite range (see also [28] for relate results). In this paper, we succee to obtain the existence of inertial manifols for the simplifie Barina moel () an the moifie Leray-α moel (3), since the nonlinear terms in these two systems are miler than that of the NSE an other α-moels of turbulence. The paper is organize as follows: section 2 is evote to the preliminaries an the functional settings. In section 3 an section 4, we stuy the simplifie Barina moel () an the moifie Leray-α moel (3), respectively, an prove the existence of absorbing balls in various Hilbert spaces, as well as the existence of an inertial manifol for both moels. In the appenix, we give a etaile justification of the strong squeezing property for these two systems. 2. PRELIMINARIES We introuce some preliminary backgroun material, which is stanar in the mathematical theory of the NSE. (i) Let F be the set of all two-imensional trigonometric vector-value polynomials with perioic omain Ω. We then set { } V = φ F : φ = 0 an φ(x) x = 0. We set H an V to be the closures of V in L 2 per an H per, respectively. (ii) We enote by P σ : L 2 per H the Helmholtz-Leray orthogonal projection operator, an by A = P σ the Stokes operator with the omain D(A) = (H 2 per (Ω))2 V. Since we work with perioic space, then it is known that Au = P σ u = u, for all u D(A). The operator A is a self-ajoint positive efinite compact operator from H into H (cf. [6, 45]). We enote by 0 < L 2 = λ λ 2... the eigenvalues of A, repeate accoring to their multiplicities. Ω

5 4 M. ABU HAMED, Y. GUO, AND E. S. TITI (iii) Weenote by an(, ) thel 2 per normanthel2 per innerprouct, respectively. Moreover, one can show that V = D(A /2 ). Therefore we enote by ((, )) = (A/2,A/2 ), an by = A /2 the inner prouct an the norm on V, respectively. We also observe that, D(A s/2 ) = (Hper s (Ω))2 V (cf. [6, 45]). In aition, we enote by V the ual space of V, an by D(A) the ual space of D(A). (iv) For r < s, we recall the following version of Poincaré inequality λ s r A r φ A s φ, (4) for every φ D(A s ). (v) For w,w 2 V, we efine the bilinear form B(w,w 2 ) = P σ ((w )w 2 ). The bilinear form B : V V V is continuous, an it satisfies B(w,w 2 ),w 3 V = B(w,w 3 ),w 2 V. (5) In particular, B(w,w 2 ),w 2 V = 0. Moreover, (B(w,w),Aw) = 0 for every w D(A) (this is only true in the 2D perioic case). See [6, 44, 45, 46] for proofs. In aition, we shall use the following estimate on the L 2 norm of B(w,w 2 ) in 2D: B(w,w 2 ) c w 2 w 2 w2 2 Aw2 2, (6) which is ue to Höler s inequality an Layzhenskaya s inequality in 2D: φ L 4 c φ 2 φ 2. Finally, we quote the following classical result (see, e.g., [44, 45]): Lemma. Let X H H X be Hilbert spaces. If u L 2 (0,T;X) with u t L 2 (0,T;X ), then u is almost everywhere equal to an absolutely continuous function from [0,T] into H an the following equality hols in the istribution sense on (0,T): t u 2 H = 2 u t,u X. (7) 3. THE SIMPLIFIED BARDINA MODEL This section is evote to prove the existence of an inertial manifol for the two-imensional simplifie Barina moel. We apply the Helmholtz-Leray orthogonal projection P σ to equation (), an obtain the following equivalent functional ifferential equation (see e.g., [6, 45]) v t +νav +B( v, v) = f, v = v +α 2 A v, (8) v(0) = v 0. Moreover, we assume that the forcing term an the initial ata have spatial zero mean, i.e., Ω f(x)x = Ω v 0(x)x = 0, an hence Ωv(x,t)x = 0, for all t 0. In [5] Cao-Lunasin-Titi prove the global well-poseness of the three-imensional viscous simplifie Barina moel (8), as well as the existence of a finite-imensional global attractor. Therefore we will not iscuss here the question of well-poseness an the attractor s imension, because the two-imensional case follows similar treatment. Notably, it was also shown in

6 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 5 [5] that the global regularity of the three-imensional invisci simplifie Barina moel, i.e., when ν = 0. In this invisci case, moel (8) coincies with the invisci Navier-Stokes-Voigt moel, namely, Euler-Voigt moel which has been a subject of intensive recent analytical an computational stuies (cf. [30, 36, 32, 33, 35, 42]). Now we can quote the following theorem without proof (since it has been proven in the 3D case in [5]) which states the global existence an uniqueness of regular solutions of equation (8). Theorem 2. (Regular Solution) Let f V, v 0 V, an T > 0. Then there exists a unique function v C([0,T];V ) L 2 ([0,T];H) with v t L 2 ([0,T];D(A) ) an v(0) = v 0, an which satisfies (8) in the following sense: v t,w D(A) +ν Av,w D(A) +(B( v, v),w) = f,w V, (9) for every w D(A). Moreover the solution v epens continuously on the initial ata, with respect to the L ([0,T];V ) norm. Here, equation (9) is unerstoo in the following sense: for almost everywhere t 0,t [0,T] we have v(t),w V v(t 0 ),w V +ν t t 0 (v,aw) + t t 0 (B( v(s), v(s)),w)s = t t 0 f,w V s. 3.. Asymptotic estimates for the long-time ynamics. This section is evote to establishing appropriate a priori estimates for the long-time ynamics of the solution of (8). In particular, we are require to justify the existence of absorbing balls for the ynamical system inuce by equation (8), in various spaces of functions. This is neee for our proof for the existence of inertial manifols. The estimates provie here are one formally, but one can prove them rigorously, e.g., by using the Galerkin approximation scheme. Throughout the following estimates, we assume the forcing f V, an the initial ata v(0) V, thus the corresponing v(0) V H -estimate for v. We take the D(A) action of equation (8) on v an use the ientities (5) an (7) to obtain 2t ( v 2 +α 2 v 2 )+ν( v 2 +α 2 A v 2 ) = f, v. (0) By the Cauchy-Schwarz an Young s inequalities, we have Consequently, we obtain f, v = (A f,a v) A f A v A f 2 2α 2 ν + α2 ν 2 A v 2. t ( v 2 +α 2 v 2 )+ν( v 2 +α 2 A v 2 ) A f 2 α 2. ν Applying Poincaré inequality (4) we get t ( v 2 +α 2 v 2 )+ ( v 2 +α 2 v 2 ) A f 2 α 2. ν We then use Gronwall s inequality to euce v(t) 2 +α 2 v(t) 2 e (t t 0 ) ( v(t 0 ) 2 +α 2 v(t 0 ) 2 )+ e (t t 0 ) α 2 λ ν 2 A f 2,

7 6 M. ABU HAMED, Y. GUO, AND E. S. TITI for all t t 0 0. Therefore In particular, it follows that lim sup(+α 2 λ ) v(t) 2 t This immeiately implies lim sup( v(t) 2 +α 2 v(t) 2 ) t α 2 λ ν 2 A f 2 an limsup t α 2 λ ν 2 A f 2. α 2 v(t) 2 lim sup v(t) t 2 ρ 0 := [ (+α 2 λ )α 2 λ ν 2] 2 A f ; α 2 λ ν 2 A f 2. lim sup v(t) t 2 ρ := (α 4 λ ν 2 ) 2 A f. () Thanks to the above, we conclue that, the solution v(t), after long enough time, enters a ball in H, centere at the origin, with raius ρ 0. Also, v(t) enters a ball in V with raius ρ. Notice the growth of ρ 0 an ρ with respect to the shrinking of ν satisfies ρ 0 ν an ρ ν asymptotically H 2 -estimate on v (L 2 -estimate on v). We take the D(A) action of equation (8) on A v by using (7), an employ the ientity (B( v, v),a v) = 0 (which is only vali in 2D perioic case, c.f. [6, 44]). It follows that 2t ( v 2 +α 2 A v 2 )+ν( A v 2 +α 2 A 3/2 v 2 ) = f,a v. By Cauchy-Schwarz inequality an Young s inequality, we have As a result, we reach to f,a v = (A 2f,A 3 2 v) A /2 f 2 2α 2 ν + α2 ν 2 A3/2 v 2. t ( v 2 +α 2 A v 2 )+ν( A v 2 +α 2 A 3/2 v 2 ) A /2 f 2 α 2. ν Applying Poincaré inequality (4) followe by Gronwall s inequality, one has v(t) 2 +α 2 A v(t) 2 e (t t 0 ) ( v(t 0 ) 2 +α 2 A v(t 0 ) 2 )+ e (t t 0 ) for all t t 0 > 0. Thus, In particular, it follows that lim sup( v(t) 2 +α 2 A v(t) 2 ) t α 2 λ ν 2 A /2 f 2, (2) α 2 λ ν 2 A /2 f 2. (3) lim sup v(t) 2 ρ := [ (+α 2 λ )α 2 λ ν 2] 2 A 2f ; t lim sup t A v(t) 2 ρ 2 := (α 4 λ ν 2 ) 2 A 2f. The above estimate along with () shows that v(t) min{ρ, ρ } for sufficiently large time t. Also, v(t) enters a ball with raius ρ 2 in D(A) after long enough time.

8 Furthermore, since v = v +α 2 A v, one has lim sup t INERTIAL MANIFOLDS FOR TURBULENCE MODELS 7 v(t) limsup t v(t) +α 2 limsup A v(t) (ρ 0 +α 2 ρ 2 )/2. t Thus, after sufficiently large time, v(t) enter a ball in H with the raius ρ := ρ 0 +α 2 ρ 2. Also, note that ρ ν asymptotically Existence of an inertial manifol. Denote R(v) := B( v, v), then equation (8) takes the form v +νav +R(v) = f, (4) t where we assume that f V. From the energy estimate in subsection 3..2, we see that for positive time t, one has v(t) D(A), an thus v(t) H for t > 0. Moreover, for sufficient large t, the solution v(t) enters a ball with raius ρ. Since we are concerning the large-time behavior of solutions, without loss of generality we can assume v 0 H, throughout the following iscussion. Notice that the nonlinear operator R is locally Lipschitz from H to H. Inee, let v, v 2 H, then the corresponing v, v 2 D(A). Furthermore, since v = v + α 2 A v, one has v = (I +α 2 A) v, an thus A v = A(I +α 2 A) v α2 v. (5) Then, by using (6), along with Poincaré inequality an estimate (5), we infer R(v ) R(v 2 ) = B( v, v ) B( v 2, v 2 ) = B( v, v v 2 ) + B( v v 2, v 2 ) c v 2 v 2 v v 2 2 A v A v 2 2 +c v v 2 2 v v 2 2 v2 2 A v2 2 cλ ( A v + A v 2 ) A v A v 2 cλ α 4 ( v + v 2 ) v v 2. (6) As in [6, 25, 26, 44], in orer to avoi certain technical ifficulties for large values of v, resulting from the nonlinearity, we truncate the nonlinear term by a smooth cutoff function outsie the ball of raius 2ρ in H. Inee, let θ : R + [0,] with θ(s) = for 0 s, θ(s) = 0 for s 2, an θ (s) 2 for s 0. Define θ ρ (s) = θ(s/ρ), for s 0. We consier the following prepare equation, which is a moification of (4): v t +νav +θ ρ( v )(R(v) f) = 0. (7) Notice that (4) an (7) have the same asymptotic behaviors in time, an the same ynamics in the neighborhoo of the global attractor. This is because we have shown that for t sufficiently large, v(t) enters a ball in H with raius ρ. On the other han, the avantage of (7) compare to (4) is that (7) possesses an absorbing invariant ball in H. To see this, take the scalar prouct of (7) with v, an then for v 2ρ, one has 2t v 2 +λ ν v 2 2t v 2 +ν v 2 = 0,

9 8 M. ABU HAMED, Y. GUO, AND E. S. TITI since θ ρ ( v ) = 0 for v 2ρ. It follows that, if v 0 > 2ρ, the orbit of the solution to (7) will converge exponentially to the ball of raius 2ρ in H, while if v 0 2ρ, the solution oes not leave this ball. Furthermore, since R : H H is locally Lipschitz, the truncate nonlinearity F(v) := θ ρ ( v )R(v) is globally Lipschitz from H to H. To see this, we let v, v 2 H, an calculate for three cases: (i) if v 2ρ an v 2 2ρ, then F(v ) = F(v 2 ) = 0; (ii) if v 2ρ v 2, then θ ρ ( v ) = 0, thus F(v ) F(v 2 ) = θ ρ ( v )R(v 2 ) θ ρ ( v 2 )R(v 2 ) by virtue of (6) an the property of θ. (iii) if v 2ρ an v 2 2ρ, then 2 ρ v v 2 R(v 2 ) cρλ α 4 v v 2, F(v ) F(v 2 ) θ ρ ( v )(R(v ) R(v 2 )) + R(v 2 )(θ ρ ( v ) θ ρ ( v 2 )) cρλ α 4 v v 2, ue to (6) an the property of θ. A summary of these three cases yiels F(v ) F(v 2 ) L v v 2, where L := cρλ α 4. (8) Since the nonlinearity of (7) is globally Lipschitz, we shall see that equation (7) possesses the strong squeezing property state in Proposition 3, provie certain spectral gap conition is fulfille. Inee, for γ > 0 an n N, we efine the cone { } v Γ n,γ := H H : Q v n (v v 2 ) γ P n (v v 2 ). (9) 2 The strong squeezing property asserts: if the ynamics of two trajectories starts insie the cone Γ n,γ, then the trajectories stay insie the cone forever, an the higher Fourier moes of the ifference are ominate by the lower moes (i.e. the cone invariance property); on the other han, for as long as the two trajectories are outsie the cone, then the higher Fourier moes of the ifference ecay exponentially fast (i.e. the ecay property). More precisely, we have the following result. Proposition 3. Let v an v 2 be two solutions of (7). Then (7) satisfies the following properties: (i) The cone invariance property: Assume that n( is large ) enough such that the spectral gap conition λ n+ λ n > L(γ+)2 v (t νγ hols. If 0 ) Γ v 2 (t 0 ) n,γ for some t 0 0, then v (t) Γ v 2 (t) n,γ for all t t 0 ;

10 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 9 (ii) The ecay property: Assume that n is large enough such that λ n+ > ν L v (t) If Γ v 2 (t) n,γ for 0 t T, then Q n (v (t) v 2 (t)) Q n (v (0)) v 2 (0) e bnt, for 0 t T, where b n := νλ n+ L γ + > 0. ( γ + ). Proof. See the appenix. Notice that, the eigenvalues of the operator A satisfies the spectral gap conition: lim sup(λ j+ λ j ) =. (20) j Inee, since the eigenvalues of A in the perioic omain are of the form L 2 (k 2 + k2 2 ), the spectral gap conition (20) is available ue to a classical result in number theory: Theorem 4. (Richars [4]) The sequence {γ k = m 2 +m2 2 : m,m 2 Z an γ k+ γ k } has a subsequence {γ kj } such that γ kj+ γ kj δlog(γ kj ) for some δ > 0. Obviously, (20) implies the require conition in Proposition 3, i.e., there exists n N such that λ n+ λ n > 4L ν an λ n+ > ν L γ +, an thus for such n large enough, the strong squeezing property hols for the prepare equation (7). Definition 5. (Inertial Manifol) [25] Consier the solution operator S(t) generate by the prepare equation (7). A subset M H is calle an initial manifol for (7) if the following properties are satisfie : (i) M is a finite-imensional Lipschitz manifol; (ii) M is invariant, i.e. S(t)M M, for all t 0; (iii) M attracts exponentially all the solutions of (7). Clearly, property (iii) implies that M contains the global attractor. Next, we state a funamental theorem concerning that the strong squeezing property implies the existence of an inertial manifol an the exponential tracking (cf. [26]) for issipative evolution equations. There are several proofs of this theorem that can be foun in [7, 23, 26, 43, 44]. Theorem 6. Consier a nonlinear evolutionary equation of the type v t +Av+N(v) = 0, where A is a linear, unboune self-ajoint positive operator, acting in a Hilbert space H, such that A is compact, an N : H H is a nonlinear operator. Assume the solution v(t) enters a ball in H with the raius ρ for sufficiently large time t. For γ > 0 an n N, we efine the cone Γ n,γ in (9). Assume there exists n N such that the prepare equation v t +Av+θ ρ ( v )N(v) = 0 satisfies the strong squeezing property, i.e., for any two solutions v an v 2 of the prepare equation, if v (t 0 ) Γ v 2 (t 0 ) n,γ for some t 0 0, then v (t) Γ v 2 (t) n,γ for all t t 0 ;

11 0 M. ABU HAMED, Y. GUO, AND E. S. TITI if v (t) v 2 (t) Γ n,γ for 0 t T, then there exists a n > 0 such that Q n (v (t) v 2 (t)) H e ant Q n (v (0)) v 2 (0) H, for 0 t T. Then the prepare equation possesses an n-imensional inertial manifol in H. In aition, the following exponential tracking property hols: for any v 0 H, there exists a time τ 0 an a solution S(t)ϕ 0 on the inertial manifol such that S(t+τ)v 0 S(t)ϕ 0 H Ce ant, where the constant C epens on S(τ)v 0 H an ϕ 0 H. Since we have shown that the strong squeezing property hols for (7) provie n is large enough, by using Theorem 6, we obtain the following result for the simplifie Barina moel. Theorem 7. The prepare equation (7) of the simplifie Barina moel possesses an n- imensional inertial manifol M in H, i.e., the solution S(t)v 0 of (7) approaches the invariant Lipschitz manifol M exponentially. Furthermore, the following exponential tracking property hols: for any v 0 H, there exists a time τ 0 an a solution S(t)ϕ 0 on the inertial manifol M such that S(t+τ)v 0 S(t)ϕ 0 Ce bnt, where b n is efine in Proposition 3, an the constant C epens on S(τ)v 0 an ϕ MODIFIED-LERAY-α MODEL This section is evote to proving the existence of an inertial manifol for the moifie- Leray-α moel (3). Applying the Helmholtz-Leray orthogonal projection P σ to (3), we obtain the following equivalent functional ifferential equation: u t +νau+b(u,ū) = f u = ū+α 2 Aū (2) u(x,0) = u 0 (x). An analytical stuy of the moifie-leray-α moel has been presente in [29]. Specifically, it was shown that (2) is globally well-pose in 3D. In aition, an upper boun for the imension of its global attractor an analysis of the energy spectrum were establishe. The proof of global well-poseness in 2D is very similar, so we just state the result an omit its proof. Theorem 8. (Regular Solution) Let f H, u 0 V, an T > 0. Then there exists a unique function u C([0,T];V ) L 2 ([0,T];H) with u t L 2 ([0,T];D(A) ) an u(0) = u 0, an which satisfies equation (2) in the following sense: u t,w +ν Au,w D(A) +(B(u,ū),w) = f,w V, (22) D(A)

12 INERTIAL MANIFOLDS FOR TURBULENCE MODELS for every w D(A). Moreover the solution v epens continually on the initial ata with respect to the L ([0,T];V ) norm. Here, the equation (22) is unerstoo in the following sense: for almost everywhere t 0,t [0,T] we have u(t),w V u(t 0 ),w V +ν t t 0 (u,aw)+ t t 0 (B(u(s),ū(s)),w)s = t t 0 f,w V s. 4.. Asymptotic estimates for the long-time ynamics. In orer to prove the existence of an inertial manifol, it is require to establish appropriate a priori estimates on the long-time ynamics of the solution. In particular, we are require to fin absorbing balls for the ynamical system inuce by the equation (2) in various spaces of functions. The estimates provie here are one formally, but can be justifie rigorously, for instance, by using the stanar Galerkin approximation metho. During our estimates, u 0 V an f H H -estimate on ū. Taking the D(A) action of the equation (2) on ū by using the fact (B(u,ū),ū) = 0 an (7), we obtain 2t ( ū 2 +α 2 ū 2 )+ν( ū 2 +α 2 Aū 2 ) = (f,ū). (23) Notice that, the energy ientity (23) is almost ientical to (0) from the analysis of the simplifie Barina moel. Therefore, we can aopt the estimate in the subsection 3.. to conclue lim sup ū(t) t 2 ρ 0 := [ (+α 2 λ )α 2 λ ν 2] 2 A f ; lim sup ū(t) t 2 ρ := (α 4 λ ν 2 ) 2 A f. From this, we conclue that, the solution ū(t), after a sufficiently large time, enters a ball in H with raius ρ 0, an also enters a ball in V with raius ρ. In aition the growth of the raii ρ 0 an ρ with respect to the shrinking of the viscosity ν satisfies ρ 0 ν an ρ ν L 2 -estimate on u (H 2 -estimate on ū). By taking the D(A) action of the equation (2) on u an using (7), we have 2t u 2 +ν u 2 +(B(u,ū),u) = (f,u). Recall insubsection3..2 whenweerivel 2 -estimate onv (H 2 -estimate on v) forthesimplifie Barina moel, we use the ientity (B( v, v),a v) = 0 (in the perioic 2D case) to eliminate the nonlinearity. On the other han, for the NSE, the L 2 -estimate is fairly easy, since (B(u,u),u) = 0. However, uner the current situation, the nonlinear term (B(u, ū), u) oes not vanish, which causes the estimate to be slightly more involve. Inee, by using Höler s inequality, an the Layzhenskaya inequality u L 4 c u 2 u 2, as well as the Young s inequality, we infer (B(u,ū),u) u 2 L 4 ū c u u ū ν 4 u 2 + c ν u 2 ū 2.

13 2 M. ABU HAMED, Y. GUO, AND E. S. TITI Also, (f,u) = (A 2f,A 2u) A 2f u ν 4 u 2 + ν A 2f 2. Combining the above estimates, we obtain t u 2 +ν u 2 c ν u 2 ū ν A 2 f 2. In subsection 4.., we have shown that there exists t > 0 such that ū(t) ρ 0 an ū(t) ρ provie t t. As a result, t u 2 +ν u 2 c ν ρ2 u f 2 ν A, for all t t. (24) We attempt to erive a uniform boun for u(t). To this en, we integrate between s an t+ for t t s t+ : u(t+ ) 2 u(s) 2 + c ν ρ2 t+ Then, integrating with respect to s from t to t+ gives u(t+ ) 2 t u(s) 2 s+ 2 ν 2 λ A 2f 2. c t+ ν 2 ρ 2 λ + u(s) 2 s+ 2 t ν 3 λ 2 A 2f 2, for all t t. (25) In orer to control the right-han sie, we shoul obtain a boun on t+ t u(s) 2 s. To this en, we euce from (23) by using Cauchy-Schwarz an Young s inequalities: t ( ū 2 +α 2 ū 2 )+ν( ū 2 +α 2 Aū 2 ) A f 2 α 2. ν Integrating the above inequality from t to t+ yiels να 2 t+ t Aū(s) 2 s ū(t) 2 +α 2 ū(t) 2 + A f 2 α 2 ν 2 λ ρ 2 0 +α2 ρ 2 + A f 2 α 2 ν 2, for t t, λ where we have use the fact that ū(t) ρ 0 an ū(t) ρ for t t. ) By efinition u = ū+α 2 Aū, it follows that u 2 2( ū 2 +α 4 Aū 2 ) 2( +α 4 Aū 2 ue λ 2 to Poincaré inequality. Consequently, for t t, one has t+ t ( u(s) 2 s 2 λ 2 ( C 0 := Substituting (26) into (25), we conclue +α 4 ) t+ λ 2 u(t+ ) 2 ρ 2 3 := ( c ν ρ2 + t +α 4 ) 2 να 2 ) Aū(s) 2 s (ρ 20 +α2 ρ 2 + A f 2 α 2 ν 2 λ C ν 2 λ A 2f 2, for t t. ). (26) This inicates that, for t t +, the solution u(t) enters a ball in H with the raius ρ 3.

14 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 3 Furthermore, the growth of the raius ρ 3 with respect to the shrinking of the viscosity ν satisfies ρ 3 ν H -estimate on u. We take the D(A) action of the equation (2) on Au. It follows from (7) that 2t u 2 +ν Au 2 +(B(u,ū),Au) = (f,au). By using Cauchy-Schwarz an Young s inequalities, one has t u 2 +ν Au 2 2 ν ( B(u,ū) 2 + f 2 ). Recall we have shown that ū(t) ρ for t t, as well as u(t) ρ 3 for t t +. Therefore, by employing (6) along with (5), we euce B(u,ū) c u 2 u 2 ū 2 Aū 2 c α u u 2 ū 2 c α ρ 3ρ 2 u 2, for t t +. As a result, for t t +, t u 2 c να 2ρ2 3 ρ u + 2 ν f 2. To obtain a uniform boun for u(t), we integrate between s an t+, for t + s t+, u(t+ ) 2 u(s) 2 + c t+ να 2ρ2 3 ρ u(s) s+ 2 t ν 2 f 2. λ Then, using Cauchy-Schwarz an integrating with respect to s between t an t+ u(t+ t+ ) 2 u(s) 2 s+ t c ν 5 2α 2 λ 3 2 ρ 2 3 ρ ( t+ t u(s) 2 s ) 2 t yiel + 2 ν 3 λ 2 f 2, for t t + νλ. Now we ought to fin a boun for t+ t u(s) 2 s. Inee, integrating (24) from t to t+ for t t + gives ( t+ u(s) 2 s u(t) 2 + c ) t+ t ν ν ρ2 u(s) 2 s+ 2 t ν 2 A 2f 2 λ C := ( ρ c ν ν ρ2 C ) ν 2 A 2 f 2, λ where we have use (26) an the fact that u(t) ρ 3 provie t t +. Finally, we conclue u(t) 2 ρ 2 c := C + ρ 2 2 3ρ ν 3 2α 2 λ C ν 2 f 2, for t t + 2. λ This shows the solution u(t) enters of a ball in V of raius ρ for t t + 2.

15 4 M. ABU HAMED, Y. GUO, AND E. S. TITI Also, recall ρ 0 ν, ρ ν, ρ 3 ν 3, then by (26) one has C 0 ν 3, an thus we see that C ν 7. Hence, ρ ν Existence of an inertial manifol. From energy estimates establishe in section 4., we see that for positive time t, one has u(t) V because of the parabolic nature of the equation, an for sufficiently large time t t + 2, the solution u(t) enters a ball in V of raius ρ. So, without loss of generality, as far as inertial manifol is concerne, which is a long-time behavior, we assume the initial ata u 0 V. We set R(u) := B(u,ū). Then the equation (2) takes the form u t +νau+r(u) = f. (27) Recall that the nonlinear term B( v, v) = P σ ( v ) v in the simplifie Barina moel () is locally Lipschitz from H to H, which is a conition for () possessing an inertial manifol. However, R(u) oes not have this property, since it is not a mapping from H to H. However, we will be able to show that R is locally Lipschitz continuous from V to V. To see this, we calculate R(u) = A 2 B(u,ū) B(A 2 u,ū) + B(u,A 2ū) u ū L +c u 3 2 u 2 Aū 2 A 2ū 2 c u ū 2 A 3 2ū 2 +c u 2 u 2 Aū 2 A 3 2ū 2 cλ 2 u A 3 2ū cλ 2 α 2 u 2. (28) Note that, throughout the above calculation, we have employe (6), an Agmon s inequality in 2D: φ L c φ 2 Aφ 2, as well as (5), where c is a positive constant. This shows that R is a mapping from V to V. By similar computation, we euce, for u, u 2 V: R(u ) R(u 2 ) cλ 2 α 2 ( u + u 2 ) u u 2, (29) that is, R : V V is locally Lipschitz continuous. Recall in the subsection 4..3, we have shown that u(t) ρ for sufficiently large time t t + 2. As in [25, 26], in orer to avoi certain technical ifficulties for large values of u, resulting from the nonlinearity, we truncate the nonlinear term outsie the ball of raius 2 ρ in V by a smooth cutoff function θ : R + [0,] with θ(s) = for 0 s, θ(s) = 0 for s 2, an θ (s) 2 for s 0. Define θ ρ (s) = θ(s/ ρ) for s 0. We consier the following prepare equation, which is a moification of (27): u t +νau+θ ρ ( u )(R(u) f) = 0. (30) Since R : V V is locally Lipschitz, by similar calculation as in subsection 3.2, it can be shown that the truncate nonlinearity F(u) := θ ρ ( u )R(u) is globally Lipschitz continuous with Lipschitz constant L := c ρλ 2 α 2.

16 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 5 Now, for γ > 0 an N N, we efine the cone in the prouct space V V: { } u Γ N,γ := V V : Q u N (u u 2 ) γ P N (u u 2 ). 2 The following result states the equation (30) possesses the strong squeezing property: Proposition 9. Let u an u 2 be two solutions of (30). Then (30) satisfies the following properties: (i) The cone invariance property: Assume that N( is large ) enough such that the spectral gap conition λ N+ λ N > L(γ+)2 u (t νγ hols. If 0 ) u 2 (t 0 ) Γ N,γ for some t 0 0, then u (t) Γ u 2 (t) N,γ for all t t 0 ; (ii) The ecayproperty: Assume that N is sufficiently large suchthat λ N+ > ν L u (t) If u 2 (t) Γ N,γ for 0 t T, then Q N (u (t) u 2 (t)) Q N (u (0)) u 2 (0) e β Nt, for 0 t T, where β N := νλ N+ L( γ + ) > 0. ( γ + ). Proof. See the appenix. Note that the spectral gap conition is satisfie for sufficiently large N, by virtue of Theorem 4. Consequently the strong squeezing property hols for the equation (30). Then, accoring to Theorem 6 which concerns the existence of an inertial manifol, we have the following result. Theorem 0. The prepare equation (30) of the moifie-leray-α moel possesses an N- imensional inertial manifol M in V, i.e., the solution S(t)u 0 of (30) approaches the invariant Lipschitz manifol M exponentially in V. Furthermore, the following exponential tracking property hols: for any u 0 V, there exists a time τ 0 an a solution S(t)ϕ 0 on the inertial manifol M such that S(t+τ)u 0 S(t)ϕ 0 Ce β Nt, where β N is efine in Proposition 9, an the constant C epens on S(τ)u 0 an ϕ 0. Remark. Concerning the Leray-α moel (2), the nonlinearity is ( w )w an clearly there is a loss of erivative. It can be shown that the operator R(v) := B( v,v) = P σ ( v )v is Lipschitz continuous from V to H in 2D. As far as inertial manifol is concerne, this prouces the similar ifficulty as what we face for the 2D NSE. Inee, uner such scenario, using the classical theory, theexistence of an inertial manifolrequiresastronger gap conition: λ 2 j+ λ 2 j must be sufficiently big, which only hols for very large viscosity ν (see, e.g. [43]). But our main interest lies in flui flow with small viscosity, which is the situation when turbulence occurs, so a result vali for only large ν is of no account.

17 6 M. ABU HAMED, Y. GUO, AND E. S. TITI 5. APPENDIX We present the proof of Propositions 3 an 9 for the sake of completion. Since the proof of these two propositions are similar, we only show Proposition 9. Proof. The metho of the proof is stanar (see, e.g. [26]). Assume u an( u 2 are ) two solutions u (t) of (30). To show the cone invariance property (i), it is sufficient to show can not pass u 2 (t) through the bounary of the cone if the ynamics starts insie the cone. More ( precisely, ) we shall show t ( Q u (t) N(u (t) u 2 (t)) γ P N (u (t) u 2 (t)) ) < 0 whenever Γ u 2 (t) N,γ, where Γ N,γ stans for the bounary of the cone Γ N,γ. Recall F(u) = θ ρ ( u )R(u). Then by the equation (30), t (u u 2 )+νa(u u 2 )+F(u ) F(u 2 ) = 0. By setting p = P N (u u 2 ) an q = Q N (u u 2 ), we obtain We take the scalar prouct of (3) with Ap, p t +νap+p N (F(u ) F(u 2 )) = 0 (3) q t +νaq +Q N (F(u ) F(u 2 )) = 0. (32) 2t p 2 +ν Ap 2 +(P N (F(u ) F(u 2 )),Ap) = 0. Thus by the global Lipschitz continuity of F, we have 2t p 2 νλ N p 2 F(u ) F(u 2 ) p νλ N p 2 L u u 2 p. (33) Without loss of generality, we can assume p(t) > 0. (Otherwise, if p(t ) = 0 for some t, then since we consier the bounary of the cone, we can assume q(t ) = γ p(t ) = 0, an thus u (t ) = u 2 (t ). By the uniqueness of solutions, we obtain u (t) = u 2 (t) for all t t, an the cone invariance property follows.) Now we can ivie both sies of (33) by p(t), so t p νλ N p L u u 2. (34) Analogously, by taking the scalar prouct of (32) with Aq, we can euce t q νλ N+ q +L u u 2. (35) Multiplying (34) with γ an subtracting the result from (35), we infer, by using the fact p+q = u u 2, t ( q γ p ) ν(λ Nγ p λ N+ q )+L(γ +)( p + q ). u (t) So whenever q(t) = γ p(t), i.e. Γ u 2 (t) N,γ, we have ) (ν(λ t ( q γ p ) (γ +)2 N λ N+ )+L q < 0, γ

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20 INERTIAL MANIFOLDS FOR TURBULENCE MODELS 9 [42] F. Ramos an E. S. Titi, Invariant measures for the 3D Navier-Stokes-Voigt equations an their Navier- Stokes limit, Discrete an Continuous Dynamical Systems 28 (200), [43] J. C. Robinson, Infinite-imensional ynamical systems. An introuction to issipative parabolic PDEs an the theory of global attractors, Cambrige Texts in Applie Mathematics, Cambrige University Press, Cambrige, 200. [44] R. Temam, Infinite-imensional Dynamical Systems in Mechanics an Physics, Secon eition, Applie Mathematical Sciences 68, Springer-Verlag, New York, 997. [45] R. Temam, Navier-Stokes Equations, Theory an Numerical Analysis, 3r revise eition, North-Hollan, 200. [46] R. Temam, Navier-Stokes Equations an Nonlinear Functional Analysis, Secon eition, CBMS-NSF Regional Conference Series in Applie Mathematics 66, SIAM, Philaelphia, PA, 995. (M. Abu Hame) Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. ALSO, Department of Mathematics, The College of Sakhnin - Acaemic College for Teacher Eucation, Sakhnin 3080, Israel. aress: mohamma@tx.technion.ac.il (Y. Guo) Department of Computer Science an Applie Mathematics, Weizmann Institute of Science, Rehovot 7600, Israel. aress: yanqiu.guo@weizmann.ac.il (E. S. Titi) Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX , USA. ALSO, Department of Computer Science an Applie Mathematics, Weizmann Institute of Science, Rehovot 7600, Israel. aress: titi@math.tamu.eu an eriss.titi@weizmann.ac.il