CONTINUOUS DATA ASSIMILATION ALGORITHM FOR SIMPLIFIED BARDINA MODEL

. We present a continuous data assimilation algorithm for three-dimensional viscous simpliﬁed Bardina turbulence model, based on the fact that dissipative dynamical systems possess ﬁnite degrees of freedom. We con-struct an approximating solution of simpliﬁed Barbina model through an in- terpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the ﬁnite-dimensional spatial resolution of collected measurements suﬃcient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.


1.
Introduction. Data analysis is a wide range of techniques that conciliates mathematical models and physical observations with the goal of optimizing forecasts in evolutionary phenomenons. Due to the high degree of freedom of many models, a hard obstacle is to predict suitable initialization points, which in general are deduced starting from discrete and heterogeneous grids of collected observations. Approaches to this problem have been topic of countless theoretical and numerical studies. See [15] for an extensive overview about these issues in weather prediction. In this work, we deal with a technique denominated continuous data assimilation, which consists basically in inserting a forcing term (nudging) directly into the prognostic equations, as the latter is being integrated in time. In general, nudging processes are related with the difference between the solution and the observations. The target is to drive the system towards physical considerations and there are many different ways to perform this. In [3], the authors proposed a continuous data assimilation algorithm where interpolant observables are used, introducing a feedback control term that forces the model towards the theoric solution corresponding to the observations.
Following the ideas in [3], the algorithm can be described as proceeding. Suppose that the dynamical system is governed by where initial data v(0) is unknown. Assume that is possible to obtain a linear interpolation operator I h (v(t)) through observational measurements, over the time [0, T ]. Here h > 0 is a coarse spatial resolution, related to accurency of operator. Then, we consider the system where η > 0 is a Newtonian relaxation parameter and the initial condition w 0 is taken arbitrary. The linear operator I h must be constructed via methods that fit with the sort of collected data. Some examples are determining modes ( [26] and [32]), volume elements ( [31] and [32]) and nodal values ( [25], [27] and [32]). The conjecture is to determine suitable conditions, in terms of physical parameters, for values of η and h, large and small enough respectively, to ensure w is directed to the reference solution v, when time goes to infinity. Then, we can choose an appropriated initial condition to (1), for example w(0) = v(T ), for some T > 0.
In [1], authors has proved the algorithm is valid to the three-dimensional Navier-Stokes-α equations, (also called Camassa-Holm equations, [22]) with data assimilation algorithm of the form System (3) can be seen as a Kelvin-filtered Navier-Stokes equations with the filter being the inverse of the Helmholtz operator (I − α 2 ∆) −1 and α representing the width of the filter ( [23] and [17]). Numerical results pointed the connection between Camassa-Holm equations and turbulence (see [11], [9], [10] and references therein). Similar equations are also related with second grade fluids and Euler-Poincaré models as can be found in [30], [6], [33] and [13].
Note that for appliance of continuous data assimilation algorithm (2), the diffusion relaxation ∆(I h (w) − I h (u)) was added (see (4)). From the analytical point of view, it is present due to the derivatives of higher order in the non-linearity. In [16], authors performed numerical studies with the shallow water equations comparing methods involving direct insertion, data assimilation having only Newtonian relaxation and data assimilation including diffusion relaxation. For distinct experiments, they obtained different results and concluded that further inspections are required to determine the practical efficacy of each approach (see also [39]). In the context of Navier-Stokes-α system, it seems to be unknown if the algorithm is convergent without diffusion in the nudging. In this paper, we prove that it is true for a related system called three-dimensional viscous simplified Bardina turbulence model is a given external force; ν > 0 is kinematic viscosity and α > 0 is a fixed length-scale parameter. This model was considered by Layton and Lewandowski [34] being a simpler approximation of the Reynold stress tensor proposed by Bardina et al [5], which is called Bardina model. The main difference of simplified Bardina model from other alpha subgrid scale turbulence models (see e.g., [7], [35] and [41]) is that the nonlinear term can be distinguished easily. Moreover, notice that the simplified Bardina system is consistent with other alpha models in the sense that if α = 0, we have u = v and we formally recover the classical three-dimensional Navier-Stokes equations (3D NSE).
The following continuous data assimilation is considered subject to periodic boundary condition Ω = [0, L] 3 , namely, where e 1 , e 2 and e 3 are the canonical basis of R 3 and L > 0 the fixed period. We prove that, for suitable values of the relaxation parameter η > 0 and the coarse spatial resolution h > 0, solution w of (6) converges, at exponential rate, to solution u of reference system (5), independently of the initial condition w(0). The restrictions on η and h will depend on physical constants (see Theorem 3.3 for details), such as kinematic viscosity and the size of domain.
In this paper, we consider a linear interpolant operator I h :Ḣ 2 (Ω) −→ L 2 (Ω) satisfying the approximation property  where ϕ(x) = k∈Z 3 ϕ k φ k (x). It satisfies inequality (8), with C[h] = C 2π hL. The use of Fourier modes projection, as well the concept of continuous data assimilation used along this paper, is motivated by the notion of stabilization via proportional-type feedback controllers for parabolic systems. An approach of exponential stabilization by finite-dimensional feedback controllers for the Navier-Stokes equations is found in [4].
The second example of interpolant operator is the volume element operator. Dividing Ω = [0, L] 3 in N cubes Ω k , of same edge, then I h :Ḣ 2 (Ω) →L 2 (Ω) given by with h = LN − 1 3 ,i.e., the edge of each Ω k . This interpolant operator satisfies (8) with C[h] = C 2π hL. Finally, the third example is obtained by observational measurements of velocity on discrete points x k of cube Ω, that can be divided in N cubes Ω k , as previous example, with x k ∈ Ω k . Then I h :Ḣ 2 (Ω) → L 2 (Ω) defined as See [3] and [1], for more details in two and three dimensions, respectively. For complementing the theory, we present a result involving the degree of freedom of the system obtained via interpolant operator I h (·). In addition to strengthening the theoretical approach, the goal is to provide suitable enough estimates to the size of the grid of observations, to ensure the convergence u(t) −ũ(t) → 0, when time goes to infinity, of two solutions u andũ of system (5) and their respective forces, f andf . The result is given in Theorem 3.2, where we prove that the system (5) is asymptotically determined by I h (·) if h is less than a constant that depends on some physical parameters. Furthermore, Fourier modes, nodal points and volume elements can be written in a similar abstract form where ω i are functions and s i linear functionals, as presented in (9), (10) and (11). Then, estimates to the degree of freedom given by {s i }, i = 1, ..., N h can be easily obtained (see Corollary 1). In [18], a similar result for a α-regularization model of the 3D Navier-Stokes equations was obtained. The authors applied the data assimilation algorithm for the case of 3D Leray-α model (see [12]), in periodic domain Ω = [0, L] 3 : which differs from the model proposed here by the nonlinearity, composed of the filtered (smoother) and non-filtered velocities field. Also, for data assimilation algorithm application for 3D Leray-α model, the pressure term is considered by authors, while in this work we are making use of Leray Projetor (see section 2.1). In [18], the estimates are obtained using only coarse mesh observations of any two components of the three-dimensional velocity field. However, the range we obtained for the Newtonian relaxation parameter η is greater than in [18] (see (28)). Besides, our condition to the interpolant 8 is more general. This paper is organized as follows: In Section 2, we present the functional setting concerning Navier-Stokes and simplified Bardina equations, with inequalities and notations commonly used. Then, we state our results in Section 3. In Section 4 is presented some key estimates. Finally, the main results are proved in Section 5.

Basic definitions and inequalities.
Let Ω = [0, L] 3 the periodic box for some period L. In order to formulate the problem, this section presents some basic definitions and functional settings. We denote by L p (L p ) and H m (Ḣ m ) the usual three-dimensional Lebesgue and Sobolev vector spaces (with mean zero), respectively. Due the fact that solutions of (5)- (7) satisfy the restriction of the analysis to functions with means zero is justified. Consider the set We define H and V the closures of V in L 2 and H 1 , respectively. They are Hilbert spaces, with V ⊂ H ≡ H ⊂ V with dense inclusions and continuous injections (see [40]), endowed with inner products respectively. The norms in H and V are denoted by |u| := u L 2 = (u, u) and u := u H 1 = ((u, u)).
We denote by P : L 2 (Ω) → H the classical Helmholtz-Leray orthogonal projection and A = −P∆ the Stokes operator under periodic boundary condition, i.e., By spectral theory, there exists a sequence of eigenfunctions (u n ) n∈N such that (u n ) n∈N is an orthonormal basis of H and (13) and where (λ n ) n∈N is the set of eigenvalues

DÉBORA A. F. ALBANEZ AND MAICON J. BENVENUTTI
Let V and D be the topological dual of V and D(A), respectively. For each α > 0, we consider the linear homeomorphism We recall some particular three-dimensional cases of the Gagliardo-Nirenberg inequality (see [28]): where C is a dimensionless constant.
We define the bilinear form B : For u, v, w ∈ D(A), the bilinear term B has the property For every u, v ∈ D(A) and w ∈ H, we have We write the incompressible three-dimensional simplified Bardina equations (5) using functional settings as with initial condition u(0) = u 0 and thereby v(0) = u 0 + α 2 Au 0 .
Next, note that, using functional setting again, the continuous data assimilation equations (6) is equivalent to on the interval [0, T ]. Moreover, Leray projector implies that (8) becomes for every g ∈ D(A), where C[h] is the same as (8).
We enunciate now the global well-posedness of data assimilation equations (21), which will be proved in section 5.1.  (20), with initial data u 0 (see Theorem 2.1). Let w 0 ∈ V and η > 0 given. Suppose that I h is linear and satisfies (22) and Then the continuous data assimilation system (21) with interpolant I h possess a unique solution with regularity Finally, there exists a continuous dependence with respect to initial data w 0 in Vnorm.

3.1.
Vanishing limits and degree of freedom to the system (20). We start stating a result concerning the vanishing limit of the length-scale parameter α. We prove that as α → 0, solutions u α of the model (20) converge to a solution u of the NSE. Hence, the system (20) can be seen as a regularized approximation to 3D NSE.
We observe that in [34], the authors have proved a similar result. There, they considered the initial data for (20) of the form u 0 = (I − α 2 ∆) −1 v 0 , with v 0 ∈ L 2 , and therefore u 0 ∈ H 2 ∩ V . Here, we consider u 0 ∈ V . In this sense, our result is different from theirs and we present a detailed proof in section (5).
Also, see [40] and [14] for definitions of Leray-weak and strong solutions to the Navier-Stokes equations.
where λ 1 is given in (15). For this result, we consider the time-dependent external force f ∈ L ∞ ([0, ∞); H). and Additionally, suppose that is valid, where C is a dimensionless positive constant, depending only on constant presented in (16). Then u(t) −ũ(t) → 0 in V -norm.
Remark 1. As mentioned in introduction, a wide range of operators are of the form (12), such as (9), (10) and (11). In these cases, {s i }, i = 1, ..., N h can be used to measure the degree of freedom of the infinity-dimensional system. Due to the fact that C[h] and N h are dependent functions (C[h] decreases when N h increases), the degree of freedom is determined by (27).
Stationary solutions are completely determined by functional related with the degree of freedom. 3.2. The convergence result. We now state the main result, which ensures that when time goes to infinity, the solution of the continuous data assimilation system (21), converges exponentially to the solution of classical simplified Bardina system (20), independently of the initial condition imposed to (21).  (20), with external force f ∈ L ∞ ([0, ∞); H). Suppose that I h is a linear interpolant operator which satisfies (22). Additionally, let w be a global solution of continuous data assimilation system (21), for η large enough such that where C 2 is a dimensionless positive constant (depending only of constant given in (16)). Suppose also that η and h are related in the sense that C[h] satisfies (23). Then u(t) − w(t) → 0, in V -norm, as t → ∞, at an exponential rate.

Key estimates.
4.1. Estimates to the system (20). Next, we state some lemmas which contain useful estimates.
Lemma 4.1. Let u be a weak solution of (20) in (0, T ), with initial data u(0) = u 0 ∈ V . Then 1 2 for every 0 ≤ t < T. Furthermore, there are dimensionless positive constants C 3 , C 4 and C 5 (depending only of constants given in (16)) such that for every g ∈ D(A). Also, for every g ∈ V . Furthermore, Proof. The equality (29) is immediately obtained by taking the duality ., . D of the first expression of (20) with u and using (17). Similarly, for all g ∈ D(A), (f, g) .

4.2.
Estimates to the data assimilation system (21). In this subsection, we present estimates to the data assimilation system (21), which are similar to those previously presented. 3. Let f ∈ H, w(0) = w 0 ∈ V , and w the weak solution of data assimilation system (21), with initial data w(0) = w 0 . We have that for all 0 ≤ t < T . Furthermore, there is a dimensionless positive constant C 9 (depending only of constants given in (16))such that for all g ∈ D(A), where Proof. To obtain inequality (39), we take the duality ., . D in (21) with w, use (22) and Young's Inequality for the following estimate: Using Poincaré's Inequality, estimate (40) is obtained similarly as (30), only including the following estimate  (16)) such that d dt Proof. Subtracting data assimilation equations (21) for w andw yields Since B(w, w) − B(w,w) = B(W, W ) + B(w, W ) + B(W,w), taking the duality ., . D in (42) with W and using (17), we have Finally, we have Therefore by (43) and inequalities proves (41).
Finally, we recall the generalized Gronwall inequality, which proof can be found in [31] and [24].

Proof of main results.
Proof of Theorem 2.2. Existence of solutions can be obtained via standard Galerkin procedure, using a basis of eigenfunctions with properties (13) and (14). Denoting we have that with z m = w m + α 2 Aw m , where w m (x, t) = m j=1 g j,m (t)u j (x). Existence and uniqueness of short time ([0, T m )) solutions to the above system is obtained by classical ODE theory. Subsequently, we prove now uniform bounds for w m independently of m, which guarantees existence in time [0, T ) of each w m . It is easy to check that the inequality (39) remains true to the system (44). Therefore, we have the estimates Since by hypothesis C[h] is small enough such that C[h] ≤ να 2 η , we obtain We get for all t ∈ [0, T m ), Since the estimate above is uniform in m and t, we have existence on the interval [0, T ), for all m. Moreover, we have the following estimates Now, in order to apply Aubin-Lion Theorem (see,e.g., [36]), we estabilish uniform estimates in m for dwm dt . We can proceed as in (40) Since . By Aubin-Lions compactness theorem (see e.g., [2], [36]) and Banach-Alaoglu Theorem, we obtain a subsequence of approximated solutions, which we also denote by {w m } m∈N , such that and, for non-filtered velocity, Now, it is straightforward to pass the weak limit in (44) to obtain that w is a solution of (21).

DÉBORA A. F. ALBANEZ AND MAICON J. BENVENUTTI
Let us prove the continuous dependence on initial data of the solutions and as a result, the uniqueness of solutions. Let w andw be two solutions and denote W = w −w. By (41) and C[h] ≤ να 2 η , we obtain Using Gronwall inequality, we obtain It implies continuous dependence of the regular solution.
Proof of Theorem 3.1. To obtain the convergence of the solutions u αj of the system (20), as α j → 0 + , and relate the limit to the Navier-Stokes equations, we use estimates (29) and (31) and get We can apply the limit in the weak formulation: for all g ∈ D(A), and φ ∈

Aubin-Lions compactness theorem and the
and check that u is a weak solution of Navier-Stokes equations. Applying lim inf as j → ∞ in the inequality below we obtain that u satisfies the energy inequality, and therefore u is a Leray-weak solution.