Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region

We consider the 2D simplified Bardina turbulence model, with horizontal filtering, in an unbounded strip-like domain. We prove global existence and uniqueness of weak solutions in a suitable class of anisotropic weighted Sobolev spaces.


Introduction
In the present paper we give some results mainly connected with the regularity and the long-time behavior of the viscous simplified Bardina turbulence model (with horizontal filtering) in a strip-like region Ω ⊆ R 2 , aimed at proving existence and uniqueness of weak solutions in a suitable class of weighted Sobolev spaces. The Bardina closure model for turbulence was introduced in 1980 by J. Bardina, J. H. Ferziger and W. C. Reynolds in [7], and later simplified and analyzed in [18] and in [12]. Indeed, the 3D simplified Bardina turbulence system was proposed in [18] as a regularization model, for small values of the scale parameter α, of the 3D Navier-Stokes equations for the purpose of numerical simulations. Analysis of the global behavior of the pertinent solutions in a bounded domain, with periodic boundary conditions, appears in [12]. Global well-posedness for the 2D simplified Bardina model was established in [13]. Again, in space-periodic domains, the inviscid simplified Bardina model is a regularizing system for the 3D Euler equations; this because it is globally well-posed and it approximates the 3D Euler equations without adding spurious regularizing terms (see [18]).
The behavior of the solutions for the simplified Bardina model (in both 2D and 3D cases) changes considerably depending on whether the integration domain is bounded. This is a basic point in studying general properties as regularity on the long-time period and dynamics (in particular, existence of attractors). More generally, this remark applies to the solutions of a broader class of dissipative systems (see, e.g., [4,15,21,24]). In fact, unlike the case of bounded domains, for some types of solutions to PDEs in unbounded regions (such as spatially periodic patterns, travelling waves, etc...), we can not expect to have uniform control on the energy; rather the energy of these solutions may blow up to infinity. Again, due to the unboundedness of Ω, compactness for the semigroup solution operator can not be retrieved by using standard Sobolev embeddings (there are no compact inclusions). Hence, in this case, the standard choices for bounded domains of the phase space, as L p (Ω), W k,p (Ω) or H p (Ω), 1 ≤ p < +∞, k ∈ N, do not seem appropriate.
Even in the promising situation in which the solutions are bounded as |x| → +∞ in Ω, i.e. they belong to L ∞ (Ω), the study of their behavior is not necessarily simplified since this space is analytically awkward to use: on one hand, strong requirements on the initial data are needed to have solutions in such a space; on the other hand, the study of dynamics in this phase space results to be more intricate since one does not have at disposal analytical semigroups, maximal regularity properties for semigroups, etc...
A reasonable alternative is using weighted Sobolev spaces (see, e.g., [1,5,21]) that, in principle, can contain sufficiently regular, spatially bounded solutions on the long-time period. In such a situation it is possible to study the semigroup generated by the considered system and to check whether it admits a global attractor in a suitable weighted phase space. A main advantage of this approach is that weighted Sobolev spaces are rather handy to use since they enjoy regularity, interpolation and embedding properties which are similar to those of the usual Sobolev spaces W k,p (Ω) for bounded domain.
However, proving estimates in such spaces is more complicated than in the standard ones and, for our analysis, we find convenient to follow the same path as in [6] (see also [14,22]). In so doing, we consider the 2D Navier-Stokes system in terms of a stream function, v, and derive formally the 2D simplified Bardina with horizontal filtering.
We now introduce the considered 2D simplified Bardina model for the potential v connected with the vector field v = (v 1 , v 2 ) (here, v is a regularizing vector field associated with the velocity field, u, of the 2D Navier-Stokes equations (1.3) below, i.e. v ≈ u and v 1 = ∂ 2 v, v 2 = −∂ 1 v), on the strip-like region Ω ⊂ R 2 , i.e.: where B(v, v) := ∂ 2 v∂ 1 ∆v − ∂ 1 v∂ 2 ∆v, α > 0 is a scale parameter, ν > 0 is the kinematic viscosity, g is a forcing term, and the domain Ω is defined by the following inequalities (see [6,14,22]): where b 1 and b 2 are twice continuously differentiable functions bounded over the entire x 1 -axis according to In order to formally derive system (1.1), we consider the 2D Navier-Stokes equations in the space periodic setting Ω = T 2 (although it would be sufficient to consider periodicity only in the x 1 -direction), i.e.
where u(x, t) = (u 1 , u 2 ) is the velocity field, π(x, t) denotes the pressure, f (x, t) = (f 1 , f 2 ) is the external force, and ν > 0 the kinematic viscosity. First, we rewrite the Navier-Stokes equations (1.3) in terms of the vorticity ξ := curl u := ∂ 1 u 2 − ∂ 2 u 1 ∈ R and then we introduce the stream function ω associated to the velocity field u, i.e. a scalar function ω ∈ R such that u = curl ω = (∂ 2 ω, −∂ 1 ω) ∈ R 2 (notice that ξ = −∆ω), to get where the bilinear operator B is as above (i.e. B(ω, ω) = ∂ 2 ω∂ 1 ∆ω − ∂ 1 ω∂ 2 ∆ω) and For a function w, we introduce the horizontal filter (related to the horizontal Helmholtz operator), given by [3,16,17], from the point of view of the numerical simulations, this filter is less memory consuming with respect to the standard one. Further, another interesting feature of this filter is that, even in the case of domains which are not periodic in the vertical direction, there is no need to introduce artificial boundary conditions for the Helmholtz operator (see, e.g., [3,8,9,10,11]). We and solve the interior closure problem by using the approximations to get the following initial value problem: By applying the operator A h = I − α 2 ∂ 2 1 to the above system, term by term, and considering the obtained equations on the channel-like domain described by (1.2) (introducing suitable boundary conditions), we get (1.1). Here and in the sequel, for simplicity, we always assume that g(x, t) = g(x).
Set the following anisotropic Sobolev spaces: and As first step in our analysis we provide an existence theorem to (1.1) in standard anisotropic Sobolev spaces. In this case we deal with a proper class of weak solutions to the considered problem (see Definition 3.1 below). This result reads as follows. In the main theorem of the paper we prove that the global weak solution v = v(t) in Theorem 1.1 is actually defined in a suitable class of anisotropic weighted Sobolev spaces (see Theorem 3.1 below).
In proving Theorem 3.1 we do not follow directly the scheme behind the standard Aubin-Lions lemma, rather we use a different compactness method (see [23,Corollary 2.34], see also Lemma 5.2 below) by which we perform our analysis on approximating open bounded subsets O of Ω. This result allows us to surmount the difficulties due to the boundary conditions and the unboundedness of the considered domain Ω.
The results obtained here open the way to the analysis of dynamics in terms of attractors; this will be the matter of a forthcoming paper.
Plan of the paper. In Section 2 we introduce the main notation and we also give some preliminary results. In Section 3 we give the precise definition of weak solution, we state our main result (Theorem 3.1) and we also present some remarks on the existence of weak solutions. Section 4 is devoted to the proof of Theorem 1.1. In Section 5, we study problem (1.1) in suitable Sobolev weighted spaces proving Theorem 3.1. Finally, the appendix is dedicated to the properties of the weight functions used to define the weighted Sobolev spaces used in Theorem 3.1.

Notation and preliminary results
In what follows, we denote by L p := L p (Ω), and W k,p := W k,p (Ω), with H k := W 2,k , k, p ∈ N, the usual Lebesgue and Sobolev spaces, respectively. Also, we denote by ( , ) and · the standard L 2 -inner product and norm in L 2 (Ω), respectively. We denote by (H k ) ′ the dual space to H k , k, p ∈ N, and this notation will be adapted in a straightforward manner, when it makes sense, to the further spaces that will be introduced in the sequel.
Given a Banach space X, for p ∈ [1, ∞), we denote the usual Bochner spaces L p (0, T ; X) with associated norm f p L p (0,T ;X) := T 0 f (s) p X ds (the lower bound of f (s) X if p = ∞), with · X the norm of X.
Hereafter, C will denote a dimensionless constant which might depend on the shape of the domain Ω and that may assume different values, even in the same line.
Again, arguing as in [6,14], we take ψ := ϕ 1/2 . Notice that we can choose ϕ so that This property will play a crucial role in the subsequent computations.
We denote by H l γ := H l γ (Ω) the space of functions equipped with the following norm: Using the above notation, we introduce the further spaces Let us recall the following results taken from [6] (see also [14]).
If v ∈ H 2 0 , then it holds true that Next, we have a weighted version of the classical Poincaré inequality.
Let ǫ in the definition of ϕ be sufficiently small. Let v ∈ H 2 ∩ H 1 0 . Then We also have the following controls in the L 4 -norm.

Weak solutions and existence results
Consider the simplified-Bardina model (1.1). Observe that the bilinear form where the second line is obtained integrating by parts and exploiting the boundary conditions. Here and in the sequel, unless stated otherwise, we drop the dx in the space-integrals to keep the notation as compact as possible.
We now give the following definition.
, for a.e. t ∈ R (and the initial datum is assumed in weak sense).
In the next section we give a proof of Theorem 1.1 that guarantees existence and uniqueness of a weak solution to problem (1.1).
The anisotropic weighted Sobolev spaces introduced in (2.2) provide the appropriate functional framework for studying the existence of weak solutions to (1.1) enjoying extra regularity properties. Then, in Section 5 we prove our main result, that reads as follows.

Existence in anisotropic Sobolev spaces
This section is devoted to the proof of Theorem 1.1, which provides the existence of a unique weak solution of the problem (1.1). Since the proof follows standard methods, we proceed formally in order to find appropriate a priori estimates. A rigorous proof can be easily obtained by introducing a Galerkin approximation and finding similar estimates.
We are now ready to proceed with the proof of Theorem 1.1.
Proof of Theorem 1.1. Testing formally (1.1) against v, we get which implies . Multiplying (1.1) against v t and integrating over Ω, we get and, thanks to the Hölder, the Gagliardo-Nirenberg and the Young inequalities, we have also for a suitable constant C = C(λ 1 , α, ν) > 0. Plugging this estimate in (4.2), we obtain Since we have already proved that v ∈ L 2 loc (H 2 ), an application of the Grönwall lemma gives the claimed regularity of v (here we use the full regularity of v 0 ), and consequently by the previous inequality, the regularity of v t .
Lastly, notice that the proof of the uniqueness of weak solutions is quite standard and very similar to the proof of uniqueness for the case of a bounded domain (mainly because of the validity of the Poincaré inequality) and this last part of the proof is left to the reader.
Lemma 5.1. Under the assumption γ ≤ 2/3, setting ψ = ϕ 1/2 , then it holds true that The precise construction of the weight function ϕ and the proof of this lemma are postponed to Appendix A. For the remainder of the paper we always assume that γ ≤ 2/3.
In the proof of existence in weighted spaces, we will use the following result (see also [20,Theorem 2.2] and [2]) to overcome the difficulties arising because of the unboundedness of the strip-like region Ω.
We are now ready to prove Theorem 3.1.
for every j = 1, . . . , m. The existence of solutions is guaranteed by the Peano theorem.
We split the proof in a number of steps.
(1) We establish a priori estimates for {v m } in the space L ∞ loc (0, ∞;

STEP 1: Establishing a priori estimates in
. Here, we proceed again formally by dealing with v and the equation satisfied by it. However, the a priori estimates that we are about to derive can be rigorously justified. Indeed, a rigorous proof uses v m instead of v and w j as test functions, as it will be made in the second and third steps below.
We multiply equation (1.1) by vψ 2 in L 2 (Ω) and use integration by parts to get In particular, we have used that and noticed that all boundary terms are zero.
Using (5.2) along with the estimates (5.3)-(5.5) we get 1 2 d dt ψ∇v 2 + α 2 ψ∂ 1 ∇v 2 + ν ψ∆v 2 + να 2 ψ∂ 1 ∆v 2 Using the control on v t and ∂ 1 v t provided by Theorem 1.1 together with the Grönwall inequality, we get the claimed regularity on v, i.e. v ∈ L ∞ loc (0, ∞; H 2,h γ ) ∩ L 2 loc (0, ∞; H 3,h γ ). This concludes STEP 1. Before proceeding with the next steps, we open a parenthesis to outline the scheme behind the remaining part of the proof. Until now, we have used v in place of v m for a matter of convenience; however, in view of extracting a convergent subsequence of {v m }, here below we will employ this latter notation. From the above estimates, we can extract a subsequence of {v m }, still denoted by {v m }, such that v m ⇀ṽ weak-star in L ∞ (τ, T ; H 2,h γ (Ω)), v m ⇀ṽ weak in L 2 (τ, T ; H 3,h γ (Ω)).
Moreover, as a consequence of the estimates in the proof of Theorem 1.1 we also have that (5.6) v m →ṽ strong in L 2 (τ, T ; H 2,h (Ω)).
To conclude our argument, obtaining that {v m } is relatively compact in L 2 (τ, T ; H 2,h γ (Ω)), we would need some control on dv m /dt. When it is possible to choose a special basis w j ∈ C ∞ 0 (Ω) to generate the Galerkin elements v m (x, t) = m j=1 a m j (t)w j (x), m ∈ N, such that a uniform control on dv m /dt L 2 (τ,T ;H 1,h γ ) (Ω) holds true, this is enough to use a compactness resultà la Aubin-Lions to get the existence of a subsequence such that v m →ṽ in L 2 (τ, T ; H 2,h γ (Ω)), and even more. Here, using Lemma 5.
). Since we also have that {v m } is weakly convergent toṽ in L 2 (τ, T ; H 3,h γ ), due to the uniqueness of the limit it follows that (ṽ)| O = v| O for every bounded subset O ⊂ Ω. This fact along with (5.7) will be enough to prove thatṽ is a weak solution to (1.1) . Indeed, to conclude our analysis on Ω× (τ, T ), and to prove that the weak formulation for v m is stable when m → +∞, we consider a proper family of test functions with separate variables and bounded supports (see, e.g., [2]). Let {w j } j=1,...,m be the basis of the space H m approximating H 3,h γ ∩ H 2 0 , for m ∈ N. Let σ = σ(t) be a continuously differentiable function on [τ, T ] with σ(T ) = 0. Then, we set the following weak formulation (where w j (x)σ(t) are the tests) on Ω × (τ, T ): for all j = 1, . . . , m. Using Lemma 5.2 (the intersection supp w j ∩ Ω is bounded) we will prove that the above relation passes to the limit as m → +∞. To proceed to the next steps, and prove that Lemma 5.2 applies to our case, we Also here, to keep the notation as compact as possible, we write v in place of v m . Consider k > 0 arbitrarily small and set V k (t) := (v(t + k) − v(t)). We take the product of (1.1) against −ψ 2 w j , integrate in time over (t, t + k) ⊂ (τ, T ); subsequently, we multiply it by and by summing over j, we reach (here below, we reintroduce the dx in the spacedepending integrals) For the terms in the right-hand side of the above equality, exploiting the Fubini's theorem along with the properties of the following functions which are used to change the order of integration, we get With similar computations, we also obtain that Now, by exploiting (2.9), we have that Now, we estimate the terms J i , i = 1, . . . , 10. Let us start with J 1 to get (5.10) For the terms J 2 and J 3 we have that and that (5.12) Next, for the terms J 4 and J 5 we have where we used again (5.1), and T τ ψ∆v(s) ds J 6 is estimated as follows: . In a very similar way we also get Whence, for i = 7, 8, 9, 10, we have that To conclude we reabsorb the terms (5.10), (5.11) and (5.12) in the left-hand side of (5.8). Then, using standard manipulations along with the above estimates we get this concludes STEP 3.
) for a suitable v. By STEP 1 we have that {v m } is also bounded in L 2 (τ, T ; H 3,h γ ) (let us recall that v m | O is the restriction of v m to O), and hence v m ⇀ṽ weakly in L 2 (τ, T ; H 3,h γ (Ω)). Thus, due to the uniqueness of the limit, one has that (ṽ)| O = v on every ball O ⊂ Ω. Therefore v is defined on Ω and v ∈ L 2 (τ, T ; H 3,h γ (Ω)). STEP 5: The limiting function v is a weak solution.
We now show that v is a weak solution of problem (1.1). Hence, we have to check that, for any w j ∈ H 3,h γ ∩ H 2 0 (the elements of the basis for the considered test functions), the weak formulation for v m passes to the limit as m → +∞. It is enough to verify that the nonlinear term passes to the limit, i.e., setting Ω ′ = supp w j ∩ Ω, we take into account the difference We estimate singularly the above terms, so that Again, for the second term Analogously, we have that Therefore, all the above four terms go to 0 as m goes to +∞.
We are ready to show Lemma 5.1.