Exponential Attractors for the 3D Fractional-order Bardina Turbulence Model With Memory and Horizontal Filtering

We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space.


Introduction
Incompressible fluids with constant density are described by the Navier-Stokes equations supplemented with initial and boundary conditions, where u(t, x) = (u 1 , u 2 , u 3 )(t, x) is the velocity field, π(t, x) denotes the pressure, f (t, x) = ( f 1 , f 2 , f 3 ) (t, x) is the external force, and ν > 0 represents the kinematic viscosity.
In the recent years, the so called "α-models" have been proposed to perform numerical simulations of the 3-dimensional fluid Eqs. (1)- (2).
These models are based on a filtering obtained through the application of the inverse of the Helmholtz operator where α > 0 is interpreted as a spatial filtering scale.
In this paper, we are concerned with a regularized model for the 3D Navier-Stokes equations derived by the introduction of a suitable horizontal (anisotropic) differential filter and we prove the existence of a global attractor for the corresponding time-shift dynamical system in path-space. Let us consider where "h" stays for "horizontal" and, instead of choosing the filter given by (3), we take into account the following anisotropic horizontal filter given by (see [5]) As discussed in [2,23,27], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard isotropic one. The idea behind anisotropic differential filters can be traced back to the approach used by Germano [23]. Recently, the Large Eddy Simulation (LES) community has manifested interest in models involving such a kind of filtering (e.g, [1,2,4,10,18,20,33]) and the connection with the family of α-models has been highlighted and investigated by Berselli in [5]: exploiting the smoothing provided by the horizontal filtering (4), the author proved global existence and uniqueness of a proper class of weak solutions to the considered regularized model (see the system of Eqs. (6)-(7) below). Again, motivated by [5], in [8,9] a considerable mathematical support to the well-posedness of initial-boundary value problems, in suitable anisotropic Sobolev spaces, to the 3D Boussinesq equations with horizontal filter for turbulent flows is given.
In the sequel, we consider the domain L > 0, with 2 π L periodicity with respect to x = (x 1 , x 2 , x 3 ), i.e. we are considering a 3D-torus (see Remark 1.1 for motivations on such a choice).
Moreover, in what follows we assume the presence, in (1), of the fractional-order dissipative term ν 2β u, = (− ) 1/2 , 3/4 ≤ β < 1 (see (68) and what follows concerning this assumption), in place of −ν u. Although this is a situation of reduced regularity, compared to that of the standard laplacian, the presence of this fractional dissipation allows us to study long-range diffusive behaviors.
Set w = u h = A −1 h u and q = π h = A −1 h π, so that u = A h w. Then, applying the horizontal filter " · h " component by component to the various fields and tensor fields in (1)- (2), and solving the interior closure problem by the approximation we finally get the regularized model and we impose a suitable initial condition w| t=0 = w 0 . Inspired by [19] (see also [24]), we introduce in the left-hand side of (6) a memory term of the form ∞ 0 g(s)w(t − s)ds. (8) The map g : [0, ∞) → R, called memory kernel, is assumed to be convex, nonnegative, smooth on R + = (0, ∞), vanishing at infinity and satisfying the further condition ∞ 0 g(s)ds = 1.
The considered model reads as follows: ∇ · w = 0.
The presence of this additional integral term in (9) allows the system to account for memory effects in the fluid. As described in [19,21], the nature of this term owes its origins to the constitutive relations characterizing some families of polymers and geological materials and, in this case, distributed relaxation effects are described through a distributed delay. System (9)-(10) is supplemented with the initial conditions where w 0 and ϕ 0 ( · ) are assigned data. Here, w 0 and ϕ 0 ( · ) denote, respectively, the initial velocity field and the past history of the velocity, which is defined for almost every s > 0. Due to the nature of the used horizontal filter, a different scheme, with respect to the one used in [19], is needed to carry out our analysis. In fact, despite the smoothing created by the horizontal filter, the regularity gained by the system, and therefore by the weak solutions, does not guarantee neither a direct use of the standard Sobolev inequalities nor the validity of compact embeddings, which are standard tools to prove dissipation and the existence of an absorbing set for the dynamical system associated with the considered system of PDEs. In order to recover, at least partially, these properties and to be able to exploit their consequences, our estimates make explicit use of a splitting of both vector fields and differential operators into horizontal and vertical components (see, e.g., [5,13]).
Moreover, the presence of the memory term makes the situation even more intricate and this fact prevents us to obtain directly energy inequalities. Thus, before applying the just mentioned splitting technique, we rewrite the system in an equivalent but handier way [see the model given by (28)- (30] below).
In this paper we prove existence, uniqueness and regularity for this model. Also, after proving suitable energy decay for the considered problem, estimating the related decay rate, in our main result (Theorem 7.2) we show the existence of a global exponential attractor for the corresponding dynamical system.

Remark 1.1 A natural domain for the horizontal filter A h would be
L > 0, with 2 π L periodicity with respect to x h := (x 1 , x 2 ), and homogeneous Dirichlet boundary conditions on Actually, another significant advantage in choosing A h instead of the isotropic operator A is that there is no need to introduce artificial boundary conditions for the Helmholtz operator.
However, such a choice makes problematic to properly define the fractional-order operator 2β so that it might enjoy all the properties satisfied when defined in a fully periodic setting. A possible way to circumvent this problem is to consider in the space domain D h , with the aforementioned boundary conditions, the fractional anisotropic filter [5,6,12,14]) coupled with the stronger dissipation term −ν w, i.e. a full Laplacian appears instead of ν 2β w in (6) (and also in (9)). The case β = 1 is more standard and it will not be considered here. Proceeding as before, this yields the equations It is possible to verify that the results that we obtain in this paper can be achieved also for the above model, following the same approach. Actually, some parts of the proof can be even simplified and become quite standard. In order to give a more precise idea about this point, apply A β h term by term to (13), to get ∇ · w = 0. (15) Assuming to have at disposal sufficient regularity to test and manipulate Eq. (14) against w in L 2 (D h ), after an integration by parts we reach with · L 2 and ( · , · ) L 2 , respectively, the norm and the scalar product in L 2 (D h ). Essentially, the presence of the term να 2β β h ∇w 2 L 2 in the left-hand side of the above relation, allows us to recover existence, uniqueness, well-posedness and regularity results adapting, to this case, the calculations in Sections 3-to-5 performed in the fully periodic setting. Thus, we judge more interesting the mathematical tools utilized for the proofs when the weaker dissipation term ν 2β w in (9), i.e. ν 2β u in (1), is assumed (and produces να 2 β ∇ h w 2 L 2 , after applying A h to (9) and performing the L 2 -test); for this reason, in what follows, we will consider the Eqs. (9)-(10) in the periodic context.
The proposed scheme for our study is as follows.
-Part 2 Existence of suitable class of weak solutions is established in Sect. 3. In Sect. 3.1, we provide a priori estimates for the considered system in suitable lower order norms. Higher order energy estimates, which are needed to prove the existence of an exponential attractor, are provided in subsequent sections. Section 4 is devoted to the study of continuous dependence and uniqueness of weak solutions.
-Part 3 Then, in Sects. 5 and 6 , we use the dissipative properties of the system to analyze energy decay and dynamics. In so doing we prove the existence of a bounded absorbing set. Finally, in Sect. 7 we show the existence of the global exponential attractor for the dynamical system associated to Eqs. (9)-(10).

Preliminaries on the Regularized Model, Basic Facts and Notation
We introduce the following function spaces: all with L 2 norm denoted by · , and scalar product (·, ·) in L 2 . Moreover, we set V = {φ ∈ H: ∇φ ∈ (L 2 (D)) 9 }, The space V h is endowed with the inner product where u 2 V h = u 2 + α 2 ∇ h u 2 . Further, for any 0 < β < 1, we define In the sequel, in order to keep the notation compact, we omit the superscript indicating the dimension of the considered spaces, reintroducing it only when it is strictly required by the context.
In the sequel (especially to get estimates in H s , where s is not integer) we shall also use some commutator-type estimates as the one in the following lemma concerning the operator s , s ∈ R + (see, e.g., [25,26,34], see also [7,36]), with = (− ) 1/2 .
We will make use of the following result about product-laws in Sobolev spaces (see [17,32]) holds, provided that If in (18) the equality sign holds, inequalities (19)-(21) must be strict.

Regularized Model With Memory
To get the considered model, we apply the operator A h defined in (4), term by term to (9)-(10) and, following the scheme proposed in [19], we finally reach where g has been introduced in (8). This is the simplified Bardina model with horizontal filter, memory and fractional viscosity and we supply this problem with periodic boundary conditions. Thus, system (23)- (24) can be thought on the 3D-torus T 3 := R 3 /2π LZ 3 . Before starting our analysis, we give further notation and hypotheses to plug system (23)-(24) in a suitable framework. We introduce the following L 2 -weighted Hilbert space on R + , i.e.
with ( · , · ) the L 2 -inner product and Here, we assume μ nonnegative, absolutely continuous, decreasing (hence μ ≤ 0 a. e. on R + ), and summable on R + with total mass Further conditions on the kernel μ will be provided at the end of this subsection. The infinitesimal generator of the right-translation semigroup on M 1 h is given by Here, ∂ s η is the distributional derivative of η(s) with respect to the internal variable s. Finally, we define the extended memory space We also use more regular spaces: To this end we define with inner product analogous to that of M h , and norm given by The corresponding higher order extended memory space H Again, in the sequel we will also consider M ) and the extended Following [19] (see also [24]), we introduce the past history variable which satisfies the following differential identity Now, consider system (23)-(24) coupled with the previous equation. Using the operator T = −∂ s , an integration by parts in ds leads to an equivalent differential problem in the variables w = w(t) and η = η t (·), i.e.
For the system (28)- (30), but also for the case corresponding to (23)-(24), the initial conditions (11) can be rewritten as follows: We will provide a precise definition of weak solution for system (23)-(24) at the beginning of Sect. 3.

Existence
Let us introduce the definition of "regular weak solution".
and almost every t > 0 we have that where η is such that Also, as we will see later (in relation (58) below) that this solution satisfies a suitable energy-like equality instead of only an energy inequality as in the case of the standard Navier-Stokes equations.
Here we proceed formally to derive an energy estimate (also in this case the calculations can be made more rigorous by using a proper Galerkin scheme, see the following section) starting from initial data (w 0 , η 0 ) ∈ H 1 h .
Observe that, starting from initial data (w 0 , η 0 ) ∈ H 1 h , following the classical approach for the Navier-Stokes equations, and using suitable controls on w, ∇ h w and ∂ t w (see relations (46) and (52), below), one can prove the existence of weak solutions analogous to Leray-Hopf solutions (see, e.g., [22]).
Therefore, the existence of weak solutions is carried out by exploiting the energy estimates provided in the next parts of this section, and then passing to the limit in the usual way. We refer to [16,19] for more details on the Galerkin scheme in connection with equations with memory.
In Sect. 5, in order to study the existence of an exponential attractor for the dynamical system associated with (28)-(30), we will be interested in more regular solutions and show, beyond the control at the end of this section, higher order estimates for ∂ t w.

A Priori Estimates
In the sequel we proceed formally in order to find appropriate energy estimates. As already mentioned, a rigorous proof can be easily obtained by introducing a suitable Galerkin approximation scheme {(w k , η k )} (see, e.g., [22]).
Testing (28) and (29) against w and η, respectively, we get the above second equation can be rewritten as and summing up (33) and (35) we get To prove that T η, η M 1 h ≤ 0, we can use directly the argument in the proof of [24, Theorem 3.1], and we report it here-adapted to our context-for the sake of completeness. For any η ∈ D(T ) [see (26)], we have that Now, observing that But the left-hand side of (37) is bounded, and the remaining two terms of the right-hand side are negative. Then, we can conclude that both the integral and the limit exist and are finite.
In particular, this implies that the limit equals zero. As a direct consequence, we have that Moreover, we also have that (see, e.g., [19] for isotropic flows) with δ > 0 suitable constant. Relation (36) along with (38)-(39) produces a first energy estimate for the system (28)-(30) in M 1 h . However, to get further a priori estimates, needed in order to prove existence, some additional regularity is required in the considered model. (36), given an initial datum (w 0 , η 0 ) ∈ H 1 h , the corresponding energy at time t ≥ 0, reads

Remark 3.2 From
As a further step in order to prove the existence, we now consider relation (36), and using along with (39), we get and, in particular, we infer

Remark 3.3 Using
Hölder's and Gagliardo-Nirenberg's inequalities we have that: As a consequence of the above control, and assuming 3/4 ≤ β < 1 [see relation (68) below, and its consequences; this provides a lower bound for β], we have that Then, exploiting (42) along with (44), we reach , from the above differential inequality we obtain that is, the following global estimate (here f ∈ L 2 (D)): In particular,

Remark 3.6
Hereafter, Q(t) will denote any positive increasing function, that can change every time.

Energy Equality, Continuous Dependence and Uniqueness
We start by recalling the following result. Let w ε , η ε denote the standard regularization (convolution in time) of w, η, with 0 < t 0 < t 1 < T fixed, and 0 < ε < t 0 , ε < T − t 1 , ε < t 1 − t 0 (see [5,22]). For each t ∈ [t 0 , t 1 ], we have where the smooth function j ε is even, positive, supported in ] − ε, ε[, and such that ε −ε j ε (s) ds = 1. Under these assumptions, we have, for any w ∈ L q (t 0 , t 1 ; X ), with 1 ≤ q < +∞ and X Hilbert space, the following properties (see [22]): and the same identical properties, in proper spaces, hold true for η ε . Observe that the εregularization commutes with the filter A h and more generally with space derivatives.

Energy Equality
Starting from initial data in (w 0 , η 0 ) ∈ H 1 h , we prove that the energy equality holds true for the model (28)- (30).
Consider a sequence {(w k , η k )} of Galerkin approximating functions such that then test (28) against w k,ε in L 2 , and integrate on the interval [t 0 , t 1 ] in dτ , to get: with A h = I − α 2 h . Testing (29) against η k,ε in M 1 h , and then integrating on the interval [t 0 , t 1 ] in dτ , we get (after using Fubini's theorem and an integration by parts): Adding relations (54) and (55), we reach Now, we use the same argument as in [9] (see also [22]): By taking k → +∞, and using the converges types (53), we get To pass to the limit as ε → 0, we use the facts listed in the remark below Remark 4.1 By using the regularity of w, we have where the equality to zero is obtained in a standard way by approximating w through smooth functions and using the fact that ∇ · w = 0. Since j ε is supported in ]−ε, ε[ and even, so that its derivative j ε := ( j ε ) is odd. Recalling the definition of w ε , we infer where, in the first integral in the second line, the integrand is the same as in the second one but omitted for brevity, and Indeed, note that E 1 is symmetric to E 2 with respect to τ = r , and j ε (τ − r ) is odd with respect to τ − r , hence E 2 = − E 1 .
Similarly, after using Fubini's theorem and the previous mentioned symmetries, we have that The remaining terms in (56) can be handled in a similar way.
Then, in light of Remark 4.1, passing to the limit as ε → 0 in (56), we find which is the appropriate version of the energy equality for the considered model.

Continuous Dependence and Uniqueness
To study the continuous dependence on initial data, let us consider two solutions (w 1 , η 1 ) and (w 2 , η 2 ) to (28)- (29), and set w = w 1 −w 2 and η = η 1 −η 2 . Then, we test the equations for w against (A h w) k,ε (note that A h w and A h η are not directly allowed as test functions). Proceeding as in the previous subsection, we can pass to the limit k → +∞, to get To pass to the limit as ε → 0, we observe that, by proceeding as in (57) and following, we have that and, in particular, we have that Hence, using (59) along with the above relations, we obtain Now, using the fact that η satisfies the representation formula (32) (see [16,19]) it follows that Also, the nonlinear term in the right-hand side of (60) can be controlled as follows: where we have used duality, Lemma 2.2 with 3/4 ≤ β < 1, , and the Young inequality. Hence Then, using (60) together with (61) and (62), we reach , and the conclusion follows by an application of Gronwall's lemma.

Remark 4.2
Thanks to an argument very close to the one just used for the continuous dependence, we can also conclude about uniqueness (see [9] for details).
Considering more regular initial data, for instance (w 0 , η 0 ) ∈ H 1+β h or (w 0 , η 0 ) ∈ H 1+2β h , we can reproduce the previous scheme -with improved properties for the considered solutions -still proving continuous dependence on initial data. When we take into account the case of (w 0 , η 0 ) ∈ H 1+β 0 , we consider again two solutions (w 1 , η 1 ) and (w 2 , η 2 ) to (28)-(29), with w = w 1 − w 2 and η = η 1 − η 2 , and we test the equation for w against (A h 2β w) k,ε . The argument follows by performing the same calculations previously presented in this subsection, where the only significant difference is the estimate for the nonlinear term (w · ∇)w, 2β w . The way to handle this term (and even higher order versions) is shown in Sect. 5; see (68), (86) and the subsequent computations for the details.
Proof Let B 0 be another bounded set in H 1 h . There exists L 0 such that for all U ∈ B 0 . Take U ∈ B 0 , t 0 ∈ [0, 1], t > 0, then thanks to (50) we have that Now integrating in t 0 over (0, 1), we obtain that (up to take a larger constant than L 0 , we also have that )ds ≤ L 0 , for all U ∈ B 0 ) the following relation holds true: We can take t 0 such that As a consequence S(t)(U ) ∈ B 0 , for any t ≥ t 0 .

Dissipation in H 1+ȟ
Here we prove the existence of an absorbing ball in H satisfies the estimate whereν is a constant depending only on the parameters involved in the system and on the domain, and To prove this result we first introduce some preliminary calculations and lemmas. Testing (28) against 2β w in L 2 and (29) against 2β η in M 1 h , and proceeding as before, we reach Now, observe that, up to lower order terms, there exists t e = t e (R) such that Taking an arbitrary U 0 ∈ B H 1+β h (R), we consider the higher-order energy functional introduced in (65), i.e.
The Proof of Theorem 5.2 follows by Lemma 5.3.
In the next section, we will use the following results concerning the existence of a higher order absorbing ball and a higher order uniform control. Using the same argument as in Lemma 5.1, we can prove that

Lemma 5.4 There exists a bounded absorbing setB
wherec > 0 is a suitable constant.
Let us now consider higher order estimates for the system (28)-(30), taking initial data in H 1+2β h .

Let us assume
Then testing (28) against w t , in L 2 , using Hölder's and Young's inequalities and properly reabsorbing the so obtained terms on the left-hand side, we get and it holds that where, up to lower order terms, we used Lemma 2.2 with s 0 = 0, s 1 = 1+β, and s 2 = −1+β (and β ≥ 3/4). In particular, from (71), we have that β w , β ∇ h w ∈ L ∞ loc (0, +∞). As a direct consequence of the above estimates, we obtain and so it holds that w t , α ∇ h w t ∈ L 2 loc (0, +∞). In the same spirit as before, if we assume (w 0 , η 0 ) ∈ H 1+2β h and f ∈ H β , then we can prove that β w t , α ∇ h β w t ∈ L 2 loc (0, +∞). Indeed, testing (28) against 2β w t , in L 2 , using Hölder's and Young's inequalities and properly reabsorbing the so obtained terms on the left-hand side, we get It is enough to control the nonlinear term. Hence, with the same identical calculations as done in Lemma 5.5, we have Using (91) together with the above control, we obtain the claimed regularity for w t .

Exponentially Attracting Sets
It is known that dynamical systems generated by equation with memory do not regularize in finite time, due to the nature of the memory term (see, e.g., [19]). This behavior still remain the same even in the case under consideration. In particular, this prevents the existence of absorbing sets having higher regularity than the initial data.
where Q is a generic positive increasing function, and

Exponential Attractor
In this section we establish the existence of a regular exponential attractor for the semigroup associated with system (28)- (30) in the phase space H 1+β h . For the reader's convenience, we recall the definition of exponential attractor (see also [3,19,29]). In the next definition we consider a general H that, however, in our case, plays the role of H Let us also recall that the fractal dimension of E in H is defined as where N (ε) is the smallest number of ε-balls of H needed to cover E.
The main result of the paper reads as follows. As a consequence of the existence of a compact attracting set, S(t) possesses the global attractor A, which is the smallest among the compact attracting sets (hence contained in the exponential attractor). 1, see also Lemma ), that we report here below as a lemma, in a specific version prepared to fit our particular case. To this end, we will make use of the projections P 1 and P 2 of H

COROLLARIO The dynamical system S(t) on H
(iv) For every fixed R ≥ 0, the semigroup S(t) admits a decomposition of the form S(t) = Here, both Q and the nonnegative function ψ vanishing at infinity depend on R. Moreover, the functionζ satisfies the Cauchy problem

Remark 7.4
The above result is a consequence of [19,Lemma 7.4] (see also [35]) which has been adapted to the present framework.

Proof of Theorem 7.2
Essentially, we have to prove the four points in Lemma 7.3. We actually have that: point (i) is the content of Proposition 6.2, while (ii) is an immediate consequence of relations (91)-(92). Accordingly, we are left to show the validity of (iii) and (iv). In what follows, the generic positive constant C may depend on f (or on f H β ) and on the radiusR 0 of the absorbing setB 0 defined in (82).  (Q(R)). Therefore, on account of (50), there exists t e = t e (R) such that Taking an arbitrary U 0 ∈ B H 1+2β h (R), we define the higher-order energy functional Now, using (99) and Lemma 5.5, and proceeding as in Lemma 5.3, we obtain point (iii) of Lemma 7.3.
-Point (iv): Now, we refer to the splitting performed in (94) and (95). In this case we have already seen that L(t) is a strongly continuous linear semigroup on H 1+β h . Besides, L(t) is exponentially stable. This is a direct consequence of the results proved in the previous section.
Let us now take R ≥ 0 be fixed, and let U 0,1 , U 0,2 ∈ B H 1+2β h (R). For the remainder of this proof, the generic positive constant C is allowed to depend on R. Then, we decompose the difference Then, we decompose the difference into the sum where Observe that, thanks to relation (iii), the following relation holds true Moreover, the exponential stability of L(t) implies the existence of a universal constant > 0 (see [19,35]) such that Essentially, we have to prove the desired estimate for the difference (ū,ζ ), solution to the system (ū(0),ζ 0 ) = (0, 0).
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