UPPER BOUNDS ON THE NUMBER OF DETERMINING MODES, NODES, AND VOLUME ELEMENTS FOR A 3D MAGENETOHYDRODYNAMIC-α MODEL

In this paper we give upper bounds on the number of determining Fourier modes, determining nodes, and determining volume elements for a 3D MHD-α model. Here the bounds are estimated explicitly in terms of flow parameters, such as viscosity, magnetic diffusivity, smoothing length, forcing and domain size.


Introduction
Let Ω = (0, L) 3 , L > 0, be a periodic box in R 3 . We consider the following 3D MHD-α model which was introduced by Linshiz and Titi in [27]  Here u = u(x, t) is the unknown velocity, B = B(x, t) is the unknown magnetic field and p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient, η > 0 is the constant magnetic diffusivity and α is a length scale parameter. When α = 0 we formally recover the 3D classical MHD equations in [30]. Notice that here we only filter the velocity field but not the magnetic field, and it contrasts with the so-called Lagrangian-averaged magnetohydrodynamic-α (LAMHD-α) model (also called hyperbolic MHD equations or MHD-Voigt model) in [15].
The MHD-α model (1.1)-(1.4) involves couping Maxwell's equations governing the magnetic field and the Navier-Stokes-α equations (sometimes called the viscous Camassa-Holm equations). In recent years, the existence and long-time behavior of solutions to this MHD-α model has attracted the attention of many mathematicians. In [27], Linshiz and Titi proved the existence, uniqueness and regularity of solutions with periodic boundary conditions, while Fan and Ozawa [9] and Liu [28] achieved the same result in the whole space R 2 for both cases (ν = 1, η = 0) and (ν = 0, η = 1). More recently, Zhou and Fan [34] also established the regularity criteria to guarantee the existence of smooth solutions for higher dimensional case. For the long-time behavior of solutions, the existence of a finite-dimensional global attractor was proved by Catania in [5] in the case of three-dimensional periodic box, and the time decay rate in L 2 (R 3 ) of solutions was proved by Jiang and Fan in [19]. The Sobolev regularity and the Gevrey regularity of the global attractor was proved recently in [2]. When B = 0, the above MHD-α model reduces to the wellknown Navier-Stokes-α equations, where the existence of a finite-dimensional global attractor was proved in [10,18] in the case of periodic boundary conditions, and the decay rate of solutions on the whole space was proved by Bjorland and Schonbek in [4] and improved recently in [3] by using the theory of decay characters. We also refer the interested reader to [6,7,16,24,26,35,36] for results related to other MHD-α models.
The conventional theory of turbulence asserts that turbulent flows are monitored by a finite number of degrees of freedom. The notions and results for the case of 2D Navier-Stokes equations on determining modes [11,12], determining nodes [13,14,20] and determining volume elements [14,21] are rigorous attempts to identify those parameters that control turbulent flows. We refer the interested reader to [8] for a general unified framework for this issue of determining parameters and [17,29,33] for some recent related results. In recent years, upper bounds on the number of determining modes and nodes were established for some α-models, which were suggested as regularization models for the 3D Navier-Stokes equations when α is a small regularization parameter. More precisely, for the 3D Navier-Stokes-Voigt equations, Kalantarov and Titi in [23] proved a result on the determining modes. The results on the determining modes and determining nodes for some regularization models of 3D Navier-Stokes equations such as 3D Navier-Stokes-α, 3D Leray-α and 3D Navier-Stokes-ω equations were proved by Korn in [25]. For MHD-α models, to the best of our knowledge, there is only a result on the determining modes for the 3D MHD-Voigt equations in [6].
In this paper, we study the number of determining modes, determining nodes and determining volume elements for the MHD-α model (1.1)-(1.4). To do this, we follow the general lines of the approach used in [22] for 2D Navier-Stokes equations. We first estimate the large time asymptotics for the solutions. Then, we establish some inequalities related to the nodal in the three-dimensional case which are extension of that in the two-dimensional case in [22], and hence we can get an upper bound on the number of determining nodes. The determining volume elements is proved in a similar manner with the help of our new inequalities related to the finite volume elements. To obtain the bound on the number of determining modes, we need some technical estimates which are similar to that used to prove the determining nodes and determining volume elements. It is worthy noticing that in the present paper the number of determining nodes, modes and finite volume elements is estimated explicitly in terms of flow parameters, such as viscosity, magnetic diffu-sivity, smoothing length, forcing and domain size, and these estimates are global as they do not depend on an individual solution. It is also noticed that our arguments in the paper and our new technical estimates in three-dimensional case given in Lemmas 3.1 and 4.1 below can be used to study the degrees of freedom for some other 3D MHD-α models in [7,16,27].
The paper is organized as follows. In Section 2, we recall the functional setting of the 3D MHD-α model. Section 3 gives an upper bound on the number of determining nodes. The number of determining volume elements is studied in Section 4. We prove an upper bound on the number of determining modes in Section 5. For clarity of the presentation, the proof of some technical results used in the proof of main results is given in the Appendix.

Functional setting and preliminaries
Let V be the set of all vector valued trigonometric polynomials u defined in Ω such that ∇ · u = 0 and Ω u(x)dx = 0. Denote by H and V the closures of V in L 2 (Ω) 3 and in H 1 (Ω) 3 , respectively. We denote by (·, ·) and | · | the inner product and the norm in H, and by ((·, ·)) = (∇·, ∇·) and ∥ · ∥ = |∇ · | the inner product and norm in V .
Let P be the Helmholtz-Leray orthogonal projection in L 2 (Ω) 3 onto the space H. Following the notations for the MHD-α equations, we denote Using the identity one can easily show that (2.1) Here, for a Banach space X, we have used the notation ⟨., .⟩ X ′ ,X to denote the dual pairing between X and its dual space X ′ . We denote by A = −P ∆ the Stokes operator with domain D(A) = H 2 (Ω) 3 ∩ V . Notice the fact that in the case of periodic boundary conditions, A = −∆ is a self-adjoint positive operator with compact inverse. Hence there exists a complete set of eigenfunctions {w j } ∞ j=1 which is orthonormal in H, and orthogonal in both V and D(A) such that Aw j = λ j w j with We have the following Poincaré type inequalities Notice that |u + α 2 Au| 2 = |u| 2 + 2α 2 ∥u∥ 2 + α 4 |Au| 2 , so |u + α 2 Au| 2 ≥ 2 3/2 α 3 ∥u∥ |Au|, (2.4) and From the definitions of B and B, we have and in particular, Also, since (2.1) we have We have the following estimates (see e.g. [31]): for some positive constants c i , i = 1, . . . , 4.
We apply the projection P to (1.1)-(1.5) to obtain the equivalent system of equations and for every w ∈ D(A), φ ∈ V and for almost every t ∈ [0, T ]. When (u 0 , B 0 ) ∈ D(A) × V , we call a strong solution of (2.14)-(2.17) in the interval [0, T ] the solution that satisfies We define the generalized three-dimensional Grashof number Gr as follows where µ = min{ν, η}. To prove our main results, we will use the following well-posedness and largetime asymptotic result, whose proof will be postponed in the Appendix.
We also need the following generalized Gronwall inequality.

Lemma 2.1 ( [11, 20]). Suppose that ϕ(t) is an absolutely continuous non-negative function on [0, ∞) that satisfies the following inequality
where β and γ are locally integrable real-valued functions on [0, ∞) that satisfy the following conditions for some T > 0 lim inf

Determining nodes
We divide the domain Ω into N equal squares Ω j , j = 1, . . . , N , where Ω j is the j-th cubic with edge h = L/ 3 √ N . Furthermore, we place the point x j ∈ Ω j , j = 1, . . . , N . To estimate the number of determining nodes, we need the following lemma whose proof is given in the Appendix.
The main result in this section is the following theorem.
Multiplying (3.4) by A v and (3.5) by A B, we get We now estimate the terms on the right-hand side. First, we have Using (2.10), the Cauchy inequality and the Poincaré inequality (2.2), we have where we have used (2.4). Analogously, we derive (3.8) Now, using (2.9) and the Cauchy inequality, we obtain (3.9) By (2.9), the Cauchy inequality and the Poincaré inequality (2.3), we have (3.10) Using (2.9) and the Cauchy inequality, we deduce that where we have used (2.4). Analogously with using the Poincaré inequality (2.2), we arrive at (3.12) Using (2.9), the Cauchy inequality and the Poincaré inequality (2.3) yields where we have used (2.5). Using (2.9) and the Young inequality, we have (3.14) From (3.6)-(3.14), we get where µ = min{ν, η}. We rewrite the last inequality in the form , and Hence, to complete the proof, we will show that β(t) and γ(t) fulfill the requirements of Lemma 2.1. First, since the assumption of f 1 , f 2 and ϑ one sees that Now, from the definitions of K 1 (t) and K 2 (t) we have By using the facts that So, using the large-time asymptotic estimates (2.18)-(2.22) together Remark 2.1, we deduce with the choice T = (µλ 1 ) −1 that This completes the proof.

Determining volume elements
We divide the domain Ω into N equal squares Ω j , j = 1, . . . , N , where Ω j is the j-th cubic with edge h = L/ 3 √ N , and so the volume of Ω j is |Ω j | = L 3 /N . The local average of φ in Ω j defined by A set of volume elements is said to be determining if for any two solutions (u 1 , B 1 ) and (u 2 , B 2 ) corresponding to external forces f 1 and f 2 satisfying To establish the result on the number of determining volume elements, we need the following lemma, which will be proved in the Appendix.
where ρ M HD−α is defined in Theorem 3.1.
Similarly to the proof of inequality (3.15) in Theorem 3.1, we have the following estimate where K 1 (t) and K 2 (t) are as in (3.16). Using (4.3) we have Hence, to complete the proof, we rewrite (4.5) in the form , and Firstly, since the assumptions of f 1 , f 2 and χ one has This completes the proof.

Determining modes
Let {w 1 , . . . , w m } be the first m eigenfunctions of the Stokes operator A. We denote by P m the orthogonal projection onto span{w 1 , . . . , w m }, and Q m = I − P m . Let (u 1 , B 1 ) and (u 2 , B 2 ) be two solutions of (2.14)-(2.17) with the forcings f 1 and f 2 given in L ∞ (0, ∞; H), respectively. A set modes {w j } m j=1 is called determining if we have Then the number of determining modes is not larger than m.
We now estimate the terms on the right-hand side. First, we have Similarly to (3.7)-(3.15) with note that v = P m v + Q m v and B = P m B + Q m B, we have the following estimates where K 1 (t) and K 2 (t) are as in (3.16).
Hence from the facts that where and This completes the proof.

Appendix
We now give the proofs of some technical results that have been used in the previous sections.
Summing up these equalities and using the Cauchy inequality we get By using the Poincaré inequality, we have where µ = min{ν, η}. By using the Poincaré inequality once again, we deduce from (6.1) that Applying the Gronwall inequality, we infer that Hence, Using the definition of Gr, we get So we get (2.18) and (2.19) from (6.2) for some T > 0. From now on, we choose T = (λ 1 µ) −1 . Integrating (6.1) over (t, t + T ) we have Hence lim sup Substituting (6.2) into (6.3) and using the definition of Gr we deduce (2.20).
Taking the inner product of (2.14) with Au, the inner product of (2.16) with AB and summing up, we have By the Cauchy inequality, we have Using (2.12) and the Cauchy inequality, we have (6.6) Now, by using (2.13) and the Young inequality, we get (6.7) Using (2.9), (2.11) and the Young inequality yields | (B(B, u), AB)| + | (B(u, B), AB)| ≤ (c 1 + c 2 )∥B∥ 1/2 ∥u∥ |AB| 3/2 (6.8) Substituting (6.5), (6.6), (6.7) and (6.8) into (6.4) we deduce that Integrating the inequality (6.9) from t to t + T we get Moreover we get from (6.10) that Now, we consider For any 0 < s < t, integrating the inequality (6.9) from s to t we get Integrating the last inequality with respect to s over the interval (6.14) Using ( Using the fact that We also have the fact that

Proof of Lemma 3.1
We have where 1 Ωj is the characteristic function of Ω j . Consider Ω j for j fixed, but arbitrary. Choose z ∈ Ω j such that z is in the line of the intersection of two planes: the plane which contains the point x and is parallel to xy-plane and the plane which contains the point x j and is parallel to the xz-plane in three dimensions. In other words, if x and x j are such that x = (ξ 1 , ξ 2 , ξ 3 ) and x j = (η 1 , η 2 , η 3 ), then z = (τ 1 , η 2 , ξ 3 ). Therefore, Following the results which were proved in [1,Lemma 4], then for every w ∈ H 2 (Ω j ), we have . Hence . This implies that .

Proof of Lemma 4.1
We first prove the estimates for the domain (0, ℓ) 3 := (0, ℓ) × (0, ℓ) × (0, ℓ) for any ℓ > 0. Following the proof in the one-dimensional and two-dimensional cases in [20,Appendix], we have the following estimates: • In the case of one dimension: for all w ∈ C ∞ 0 (R), Applying the two-dimensional estimate (6.25) to w(x 1 , x 2 , x 3 ) holding x 3 fixed, we have Integrating this inequality with respect to x 3 from 0 to ℓ to obtain Hence, in any box Ω j , we have Summing in j from 1 to N we get This is the inequality (4.1).