REMARKS ON GLOBAL ATTRACTORS FOR THE 3D NAVIER–STOKES EQUATIONS WITH HORIZONTAL FILTERING

. We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian ﬂuid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal ﬁlter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one deﬁned as the inverse of the Helmholtz operator. It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiﬂow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the ﬂow under periodicity in all directions and discuss the possibility to introduce a solution semiﬂow.


(Communicated by José A. Langa)
Abstract. We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.
It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.
1. 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 ) is the velocity field, π(t, x) denotes the pressure, f (t, x) = (f 1 , f 2 , f 3 ) is the external force, and ν > 0 the kinematic viscosity.
In the recent years, the so called "α-models" have been proposed to perform numerical simulations of the 3-dimensional fluid equations (1)- (2). These models are 60 LUCA BISCONTI AND DAVIDE CATANIA 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 horizontal filter given by (see [5]) As discussed in [2,14,15], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard one. Another significant advantage of this choice is that there is no need to introduce artificial boundary conditions for the Helmholtz operator. The idea behind anisotropic differential filters can be traced back to the approach used by Germano [14]. Recently, the Large Eddy Simulation (LES) community has manifested interest in models involving such a kind of filtering (e.g., [2,4,12,19]) 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 of [5] proved global existence and uniqueness of a proper class of weak solutions to the considered regularized model (see the system of equations (5)-(6) below). Again, motivated by [5], the authors of [6,7] gave 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.
In the sequel, we mainly consider the domain L > 0, with 2 π L periodicity with respect to x h (i.e. with respect to x 1 , x 2 ), and homogeneous Dirichlet boundary conditions on Observe that the filter given by (4) is acting just on the horizontal variables, so it makes sense to require the periodicity only in We consider the approximate model in terms of the filtered quantities w = u h = A −1 h u and q = π h = A −1 h π, so that u = A h w. Here, we assume homogeneous Dirichlet boundary conditions on Γ for the filtered fields as well as for the unfiltered ones, in order to prevent the introduction of artificial boundary conditions, and impose the initial datum w| t=0 = w 0 for the filtered velocity field w. This model was first introduced by Berselli in [5].
Let us note that this model represents a special case of Approximate Deconvolution LES Model (ADM), see Adams-Stolz [1], when the order of deconvolution is zero. We refer to [5,2,10,8] for some recent results in this context concerning general orders of deconvolution.
Our aim is to prove the existence of an attractor in the class of regular weak solutions (see below for details) to the horizontally filtered model (5)- (6). However, the present case does not seem to fit the classical theory of attractors (see, e.g., [3]) and a different scheme is needed to carry out our analysis. In fact, despite the smoothing created by the horizontal filter, the regularity of the considered weak solutions does not ensure the continuous dependence on their initial data, even in the fully periodic setting (the dependence on the initial data is only weakly continuous). Hence, the standard dynamical theory fails to apply to this situation since the strong continuity on the initial data is needed to get the continuity of the solution semiflow.
To overcome this problem, we follow the approach proposed by Sell [17]: in this case, the dynamics becomes the time-shift in the space of paths, and the attractor is a suitable compact set that attracts the regular weak solutions under the action of the time-shift S(t)w(·) = w(t + ·).
Let W be the space of the regular weak solutions to (5)-(6), denote by H and V the usual function spaces of fluid dynamics (see Section 2), and set We prove the following result (see Section 4). Here, for the sake of simplicity, we assume that the forcing term f is independent of time.
In the last part of the paper, we consider problem (5)-(6) in the fully spaceperiodic setting. We want to discuss the possibility to obtain an analogous result for the semigroup S(t) t≥0 on V h , where S(t) : V h → V h is given by S(t)w 0 = w(t, ·), w 0 ∈ V h , and w(t, ·) is the solution at time t to (5)-(6) corresponding to the initial datum w 0 . In such a case, the main issue is to prove that S(t) effectively defines a semiflow (the solution semiflow ) and, in particular, that S(t) is a continuous operator. In fact, the regular weak solutions to (5)-(6) depend just in a weakly continuous way on their initial data, giving no guarantees on the continuity of S(t).
In Section 5, we show that the horizontal components of the flow, i.e. w h , depend continuously on the initial datum, proving the following. Proposition 1. If w is a regular weak solution to (5)-(6) under the periodic setting, Observe that the continuity of the solution operator S(t) could be shown by proving that ∂ t ∇ h w ∈ L 2 V * ; nevertheless, we only have that ∂ t ∇ h w h ∈ L 2 V * . However, since the considered problem admits a unique regular weak solution, we may wonder whether it is possible to get more regularity for such a solution, also in the vertical component, by exploiting again the special features of the fully periodic space-domain. To this end, we give an improved regularity result (see Theorem 5.3 below) which actually shows that, although the regularization created by the filter is strong in the horizontal components (and in the derivatives with respect to the horizontal components), this smoothing is not so effectively transported to the vertical component, even in the space-periodic case. Though Theorem 5.3 is not directly useful to prove the continuity of S(t) (in fact, it might be more appropriate in order to get compactness properties of S(t)), it seems interesting by itself and we report it, at the end of Section 5, for the reader's convenience.
Because of all the above facts, different techniques seem necessary to get the continuity of the solution operator S(t) associated to (5)- (6), and to prove the existence of the global attractor in such a case. We will address these issues in a future paper, in which we will study more thoroughly the dynamics associated to problem (5)-(6), possibly supplemented with homogeneous Dirichlet boundary conditions.
Plan of the paper. In Section 2 we describe the functional setting, the horizontal filter and the notion of regular weak solution, and we recall a known result concerning the existence and uniqueness of such solutions. In Subection 2.2, we recall the notion of global attractor and describe the main results. Section 3 is devoted to the proof of some basic estimates that will be used subsequently. In Section 4, under horizontal periodicity and homogeneous Dirichlet boundary conditions on Γ , the existence of the global attractor defined through the shifting semiflow is given. Finally, in Section 5, when the domain is periodic in all directions, we prove the continuity of the horizontal components of the flow, and we also provide an improved regularity result for the solutions to (5)-(6), i.e. Theorem 5.3.

2.
Functional setting and anisotropic filtering. We introduce the following function spaces: n is the outward normal to Γ ), all with L 2 norm denoted by · , and scalar product (·, ·) in L 2 . Moreover, we set h have been recalled for the reader's convenience, and denote by V * the topological dual space to V . We denote by L p and H m classical Lebesgue and Sobolev spaces. Continuous and weakly continuous functions are denoted respectively by the symbols C and C w .
In the sequel, we often use the notation [φ] h and [φ] 3 to indicate, respectively, the horizontal component, φ h , and vertical component, φ 3 of the vector field φ. Also, in the sequel, in order to keep the notation compact, we use the same symbol for scalar and vector valued functions (the same convention is used also for the related spaces), distinguishing the different cases only when it is required by the context. Given a Banach space X with norm · X and p ∈ [1, +∞[, we denote by L p loc (0, ∞; X) the usual Bochner space formed by functions φ : ]0, ∞[ → X such that, for all 0 < a ≤ b < +∞, the L p (a, b) norm of φ(·) X is finite. We will also denote by L p loc [0, ∞; X) the space of functions in L p loc (0, ∞; X) such that the L p (0, b)norm of φ(·) X is finite for every b ∈ ]0, +∞[, and analogously for H m loc [0, ∞; X) (see, e.g., [18]).

2.1.
Basic results for the filtered model. The precise notion of solution of the approximate model (5)-(6) is given by the following definition.
h ) * independent of time (for simplicity), and w(0) = w 0 ∈ V h in weak sense, when the following properties are verified.
We state the following existence theorem, proved in [5, Theorem 4.1] (and generalized, with some different estimates, in [6]). It is based on the Galerkin approximation method (see also the proof of Theorem 3.1 below) combined with a compactness argument, which uses the Aubin-Lions lemma, and suitable a priori estimates.
Then there exists a unique regular weak solution to (5)-(6), with w(0) = w 0 , depending in a weakly continuous way on the data. Moreover, the solution satisfies the energy (of the model) identity Notice that, the regularity of f implies f h ∈ L 2 0 (D).

Attractors and main results
is a semigroup, and such that the restriction σ : ]0, ∞[×W → W is continuous.
We say that A ⊂ W is a global attractor for the semiflow if A is nonempty and compact, S(t)A = A for all t ≥ 0 (i.e. A is invariant), and for all bounded sets B ⊂ W , we have lim t→+∞ δ S(t)B, A = 0, where δ(X, Y ) := sup x∈X inf y∈Y d(x, y) is the Hausdorff semidistance between the pair of sets X, Y ⊂ W . A global attractor is necessarily unique (and it coincides with the omega-limit of an absorbing set, see Section 4).
Let us note that, for each T > 0, a regular weak solution to (5) , as shown in Theorem 3.1. Here, the role of W will be played by the set W of regular weak solutions to (5) We recall that a set B in a linear topological space Z is called bounded if for every neighborhood U of the origin in Z there exists an r > 0 such that B ⊂ { ru : u ∈ U }. In the case of the Fréchet space L 2 loc [0, ∞; V h ), this reduces to ask that sup φ L 2 (0,n;V h ) : φ ∈ B < +∞, ∀ n = 0, 1, 2, . . .
Let us observe that every set B ⊂ L 2 loc [0, ∞; V h ) has finite diameter with respect to the metric d, but this does not mean that B is bounded in the sense described by (9), which is the notion of boundedness we will always refer to in the following.
Notice that W is closed in L 2 loc [0, ∞; V h ), which is complete with respect to d, thus (W, d) is a complete metric space; this is one of the consequences of the proof of Proposition 4 below.
We use the time-shift operator to define the semigroup and hence the semiflow: S(t)w = w +t := w(· + t), for each w ∈ W. The existence of the global attractor for the time-shift S(t), i.e. Theorem 1.1, is proved in Section 4.
3. Preliminary estimates. We set λ 1 > 0 equal to the first eigenvalue of the Stokes operator with horizontal periodic conditions and homogeneous Dirichlet boundary conditions on Γ , projected on the space of divergence free functions. We recall that λ 1 can be used as the constant in the Poincaré inequality. Moreover, we set Λ h = (−∆ h ) 1/2 with domain H 2 h and the same mixed periodic-Dirichlet conditions as above (Λ h is a positive self-adjoint operator), and for each t ≥ 0.
In particular, we have w ∈ L ∞ (0, Proof. Let us consider the Galerkin approximate solutions w m (t, , where E j are smooth eigenfunctions of the Stokes operator on D, with periodicity in x. If P m denotes the projection on span { E 1 , . . . , E m }, then w m solves the Cauchy problem for i = 1, . . . , m. As shown in [5, Theorem 4.1], we have that, up to considering a subsequence, as m → +∞, where w is the regular weak solution to (5)- (6), and this convergence is enough in order to pass to the limit in the nonlinear term in (13). However, the regularity of such a solution w does not allow us to test (5) directly against A h w (other than formally). Thus, in order to conclude the proof, we still proceed through the use of the Galerkin approximate solutions w m , and following the well established path given by the use of a priori estimates and a suitable compactness criterion.
We test the first equation in (13) with A h w m and use (see [5]) We estimate the right-hand side by by Cauchy-Schwarz inequality and definition of K 1 . We deduce and, thanks to the Poincaré inequality, an application of Gronwall's lemma in the interval [t, t + r] yields and, taking the limit as m → +∞, we obtain (11) and in in particular by taking t = 0 and using the fact that k 1 (0) is finite (and independent of time), since w 0 ∈ V h . Integrating (14) over s ∈ [t, t + r], taking m → +∞ and using (11) yields (12), and consequently w ∈ L 2 loc [0, ∞; V ∩ H 2 h ).

4.
Global attractor for the time-shift semiflow. This section is devoted to the proof of Theorem 1.1. First, we recall some notions concerning semiflows and a fundamental result that we will exploit in order to prove the existence of the global attractor. Abounded subset B ⊂ W is called an absorbing set if for every w ∈ W , there exists t 1 = t 1 (w) such that S(t)w ∈ B for all t ≥ t 1 . A semiflow is said compact if, for every bounded set B ⊂ W and for every t > 0, S(t)B lies in compact subset of W . We recall that, in the following, boundedness is always intended in the sense illustrated in Subsection 2.2.
Theorem 4.1. Let S(t) define a compact semiflow admitting an absorbing set B on a complete metric space W . Then S(t) has a global attractor A in W and it coincides with the omega-limit set of B: where the closure is taken in W .
Here, we stress the fact that the notion of regular weak solution (cf. Definition 2.1) is somehow more complicated than the classical one of weak solution (although natural in the present context), since the former involves anisotropic Sobolev spaces. Thus, the main difficulty in adapting the theory developed by Sell [17], which is well established in the case of isotropic Banach spaces as in the case of the standard spaces H and V , is related to the fact that the regular weak solutions have certain derivatives (especially the vertical one of the vertical component of the velocity field) which are less regular than the others. As a direct consequence of this fact, by Theorem 3.1, we have that the energy dissipation occurs just in the V h -norm (in particular, in the horizontal derivatives of the velocity field), but as we will prove in the sequel this is enough in order to retrieve the needed semigroup theory for the considered case.
In order to prove Theorem 1.1, we show that the time-shift operator S(t) verifies the hypotheses of Theorem 4.1; this is the content of the next results.
The proof of this proposition follows closely the one of [17,Lemma 7]; however, we give it here for the sake of completeness.
Proof of Proposition 2. Clearly, S(t) is a semigroup. We need to prove that the mapping (τ, w) → S(τ )w = w +τ is continuous for (τ, w) ∈ ]0, +∞[×L 2 loc [0, ∞; V h ). It is sufficient to show that, if τ n and w n are sequences such that τ n → τ in ]0, +∞[ and w n → w in L 2 loc [0, ∞; V h ), then d(w n +τn , w +τ ) → 0 as n → +∞, which holds provided b a w n +τn − w +τ Here and in the following, we omit "ds" in several integrals to keep the notation more compact.
Following the same steps of the proof of [17,Lemma 7], the main difference consists in the use of the norm · V h in place of · . Let us note that, since τ > 0, we can assume 0 < 1 2 τ ≤ τ n ≤ 2τ . It is sufficient to prove that d(w n +τn , w +τn ) → 0 and that d(w +τn , w +τ ) → 0 , when n → +∞.
this implies the first relation in (15). In order to prove the second relation in (15), let us fix ε > 0 and take ψ ∈ for all σ ∈ [τ /2, 2τ ] (this is possible since C 1 is dense in L 2 ).

Moreover, if K denotes an upper bound for ∂ t ψ(s)
for n ≥ N sufficiently large. Using the triangular inequality, (16) and (17), we infer that b a w +τn − w +τ for all n ≥ N , which proves that d(w +τn , w +τ ) → 0, as n → +∞, and ends the proof.

Proposition 3. There exists an absorbing set B ⊂ W that is bounded in W.
Proof. We define B as the subset of W such that the inequality is satisfied for every t ≥ 0. According to relation (9), B is bounded in W.
We need to prove that, if w ∈ W, then S(t)w ∈ B for each t sufficiently large. Actually, from (11), there exists t 1 > 0 such that the inequality (18) holds for all t ≥ t 1 . Thus S(t)w belongs to B for each t ≥ t 1 , and B is an absorbing set. Proposition 4. The semiflow defined by S(t) on the metric space W is compact, i.e for each bounded set B in W and for each t > 0, then S(t)B lies in a compact subset of W.
Proof. Let B be a bounded subset of W. Thanks to the semigroup property of S(t), if S(t)B is contained in a compact set of W for some t > 0, then S(t + s)B lies in a compact set of W too. Then, to prove the claim, it suffices to prove that S(t)B lies in a compact set of W for 0 < t ≤ 1.
Here, we have the inclusions h ⊂ V h being compact and the embedding V h ⊂ L 2 being continuous. Therefore, by using Aubin-Lions compactness theorem we have that, up to a subsequence, S(t)w n converges strongly to γ(t) in L 2 loc [0, T ; V h ) and weakly in L 2 loc 0, T ; H 2 h , as n → +∞, with γ(t) ∈ L ∞ (0, ∞; V h ) ∩ L 2 loc 0, ∞; V ∩ H 2 h . Now, since S(t)w n (τ ) = w n (τ + t), the same compactness argument also yields, up to a subsequence, that w n converges strongly to w in L 2 loc [0, T ; V h ) and weakly in L 2 loc 0, T ; H 2 h , as n → +∞, with w ∈ L ∞ (0, ∞; V h ) ∩ L 2 loc 0, ∞; V ∩ H 2 h . By the continuity of (τ, w) → S(t)w in ]0, +∞[ × L 2 loc [0, ∞; V h ) we also obtain that S(t)w n → S(t)w and the uniqueness of the limit implies that γ(t) = S(t)w ∈ W. This concludes the proof.

5.
The fully space-periodic case. In this section, in order to improve the regularity of the solutions, we consider a torus as a space domain, i.e. a domain periodic in all directions: D = { x ∈ R 3 : − π L < x 1 , x 2 , x 3 < π L }, L > 0, with 2 π L periodicity with respect to x. This setting enables us to perfom some computations (more precisely, the usage of some test functions) that are not allowed in the presence of Dirichlet boundary conditions; in such a way we retrive some additional information on the horizontal components of velocity field, w h , and on the pressure, q. Function spaces are defined accordingly to our periodic setting, in particular with zero mean with respect to x } . We can still consider the horizontal filtering and thus problem (5)- (6), and prove the existence of a unique regular weak solution (defined like in Section 2) essentially as done in the presence of Dirichlet boundary conditions on the top and bottom boundary Γ .
Also in this case, we use the notations · := · L 2 (D) and (·, ·) := (·, ·) L 2 (D) for the L 2 (D) norm and scalar product respectively. 5.1. Continuity of the horizontal flow. To prove the claimed continuity of w h , we need the following two lemmas.
Proof. Taking the divergence operator in (5) and using (6), we obtain and, by applying the operator A h , we deduce by elliptic estimates used along with w ⊗ w H −1 (D) ≤ C w ∇w . This last control follows from the Hölder and the Gagliardo-Nirenberg inequalities: taking ϕ ∈ H 1 (D), we have that , which proves the bound. Thus, from relation (20) we get and hence A h q ∈ L 2 H −1 , so that ∇ h q ∈ L 2 L 2 , where L 2 L 2 denotes L 2 (0, T ; L 2 (D)).
Proof. From (5), we have Considering the last term in the right-hand side of the above equation, and recalling that the filter improves the regularity by two horizontal derivatives, we have by the Gagliardo-Nirenberg inequality. Using Lemma 5.1, the previous estimate, the regularity of f and of the regular weak solution w, we deduce that all terms in the right-hand side of (21) have the same regularity L 2 H −1 , therefore ∂ t ∇ h w h ∈ L 2 V * . Now, we are ready to prove Proposition 1 (continuity of w h ).
Proof of Proposition 1. Since w ∈ L 2 (V ∩ H 2 h ), thus ∇ h w ∈ L 2 V , and ∂ t ∇ h w h ∈ L 2 V * , we obtain by interpolation (see [16,18] This implies the continuity of the map w 0 → w h (t), with w 0 ∈ V h .

5.2.
A higher order estimate. We refer to the beginning of Section 3 for the definitions of Λ h , k 1 (t), and set λ 1 = L −2 (first eigenvalue of the Laplace operator −∆ on D fully periodic, and Poincaré constant) and k 2 (t) = ∇w(t) 2 + α 2 ∇∇ h w(t) 2 , We state and prove the following result.
for each t, r > 0.
Proof. In the space periodic setting we can use −∆A h w as test function for Equation (5), because the periodic boundary conditions are preserved by the operator ∆A h , differently from the Dirichlet conditions, and hence integration by parts can be easily performed. Note that we proceed formally (since we lack the needed regularity to test directly), but the procedure actually goes through the Galerkin approximation, as done in Theorem 3.1.
First, observe that Thus, testing the equation (5) against −∆A h w, we get Take into account the nonlinear term in the right-hand side of the above estimate. We have that (w · ∇)w, ∆w = ∇ (w · ∇)w , ∇w where in the last step we used the following relations (w · ∇)∇ h w, ∇ h w = 0 and (w · ∇)∂ 3 w, ∂ 3 w = 0 .
Therefore, exploiting components, we obtain where [ · ] h and [ · ] 3 indicate, respectively, the horizontal and vertical components of the considered vector fields. Next, we estimate the nonlinear terms A 1 , A 2 and B defined above. First, we estimate A 1 and we get For the term A 2 we have the following control where we used the Gagliardo-Nirenberg inequality D j w L p ≤ C D m w α L r w 1−α L q , with j = 1, p = 4, m = r = q = 2 and α = 7/8, and Young's inequality.
Let us now consider the term B in (25). We have that Now, arguing as in the case of A 2 , the term B can be controlled as follows.