On the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0024 Natural Science, 2018, Volume 63, Issue 6, pp. 17-22 This paper is available online at ON THE NUMBER OF DETERMINING FINITE VOLUME ELEMENTS FOR 3D NAVIER-STOKES-VOIGT EQUATIONS Nguyen Thi Ngan1, Nguyen Thi Minh Toai1 and Tran Quoc Tuan2 1University of Education Publishing House, Hanoi National University of Education 1Faculty of Mathematics, Hanoi National University of Education Abstract. We give an upper bound on the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations in bounded domains with periodic boundary conditions, which is estimated explicitly in terms of flow parameters, such as viscosity, smoothing length, forcing and domain size. Keywords: Navier-Stokes-Voigt equations, weak solutions, determining finite volume elements. 1. Introduction Let Ω = (0, L)3, L > 0 be a periodic box in R3. We consider the following 3D Navier-Stokes-Voigt equations{ ut − α2∆ut − ν∆u+ (u · ∇)u+∇p = f in Ω× (0,∞), ∇ · u = 0 in Ω× (0,∞), (1.1) subject to the periodic boundary conditions u(x, t) = u(x+ L, t), x ∈ Ω, t > 0, (1.2) and the initial condition u(x, 0) = u0(x), x ∈ Ω. (1.3) Here u = u(x, t) is the unknown velocity and p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient. When α = 0, we formally recover the 3D classical Navier-Stokes equations. The system (1.1) was first introduced by Oskolkov in [1] as a model of motion of certain linear viscoelastic fluids. It was also proposed by Cao, Lunasin and Titi in [2] as a regularization, for small value of α, of the 3D Navier-Stokes equations for the sake of direct numerical simulations. The presence of the regularizing term−α2∆ut in (1.1) has many important consequences. On one hand, it leads to the global well-posedness of (1.1) both forward and Received July 12, 2018. Revised August 13, 2018. Accepted August 20, 2018. Contact Nguyen Thi Ngan, e-mail: ngannt.nxb@hnue.edu.vn 17 Nguyen Thi Ngan and Nguyen Thi Minh Toai backward in time, even in the case of three dimensions. On the other hand, it changes the parabolic character of the limit Navier-Stokes equations, so the Navier-Stokes-Voigt system behaves like a damped hyperbolic system. In fact, the Navier-Stokes-Voigt system belongs to the so-called α-models in fluid mechanics (see e.g. [3]). We also refer the reader to [4] for an interesting application of the Navier-Stokes-Voigt equations in image inpainting. 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, determining nodes and determining finite volume elements (see e.g. [5, 6, 7]) are rigorous attempts to identify those parameters that control turbulent flows. In the last decade, some results on the number of determining modes and determining nodes were proved for 3D Navier-Stokes-Voigt equations [8] and some other regularization models of 3D Navier-Stokes equations such as 3D Navier-Stokes-α, 3D Leray-α and 3D Navier-Stokes-ω equations [9]. However, to the best of our knowledge, there is no existing result on the number of determining finite volume elements for α-models in fluid mechanics. This is the main motivation of the present paper. In this paper, following the general lines of the approach in [7], we give an upper bound on the number of determining finite volume elements for the 3D Navier-Stokes-Voigt equations. Here the number of determining finite volume elements is estimated explicitly in terms of flow parameters, such as viscosity, smoothing length, forcing and domain size, and the obtained estimate is global as it does not depend on an individual solution. 2. 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 L2(Ω)3 and in H1(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 Leray orthogonal projection in L2(Ω)3 onto the space H . We denote by A = −P∆ the Stokes operator with domain D(A) = H2(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 {wj}∞j=1 which is orthonormal in H , and orthogonal in both V and D(A) such that Awj = λjwj with (2pi/L)2 = λ1 ≤ λ2 ≤ · · · ≤ λj ∼ j2/3L−2 ≤ · · · We have the following Poincaré type inequalities: ‖u‖2 ≥ λ1|u|2 for all u ∈ V, |Au|2 ≥ λ1‖u‖2 for all u ∈ D(A). Hence we have the following estimates ‖w‖2 ≥ d0 (|w|2 + α2‖w‖2) , |Aw|2 ≥ d0 (‖w‖2 + α2|Aw|2) , with d0 = λ1 1 + λ1α2 . (2.1) 18 On the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations We define B(u, v) = P (u · ∇)v, ∀u, v ∈ V. From the definition of B we have 〈B(u, v), v〉V ′,V = 0 for all u, v ∈ V. (2.2) We have the following estimate (see e.g. [10, 11]): |(B(u, v), w)| ≤ c1|u|1/2‖u‖1/2‖v‖ ‖w‖, ∀u, v, w ∈ V, (2.3) for the positive constant c1 depending only on Ω. We apply the Leray projection P to (1.1) to obtain the equivalent system of equations d dt (u+ α2Au) + νAu+B(u, u) = Pf, (2.4) with the initial datum u(0) = u0. (2.5) We first recall the following well-posedness result. Theorem 2.1. [8, 12] If u0 ∈ V and f ∈ L∞(0,∞;V ′), then problem (2.4)-(2.5) has a unique global weak solution u ∈ L∞(0,∞;V ) satisfying lim sup t→∞ (|u(t)|2 + α2‖u(t)‖2) ≤ ν2λ3/21 d0 Gr2. (2.6) where d0 is given in (2.1) and Gr is the generalized Grashoff number defined by Gr = 1 ν2λ 3/4 1 lim sup t→∞ ‖f(t)‖V ′ . We also need the following generalized Gronwall inequality. Lemma 2.1. [7] Suppose that φ(t) is an absolutely continuous non-negative function on [0,∞) that satisfies the following inequality dφ dt + βφ ≤ γ, a.e. on [0,∞), where β and γ are locally integrable real-valued functions on [0,∞) that satisfy the following conditions for some T > 0 lim inf t→∞ 1 T ∫ t+T t β(τ)dτ > 0, lim sup t→∞ 1 T ∫ t+T t β−(τ)dτ <∞, and lim sup t→∞ 1 T ∫ t+T t γ+(τ)dτ = 0, with β− := max{−β, 0}, γ+ := max{γ, 0}. Then it follows that lim t→∞ φ(t) = 0. 19 Nguyen Thi Ngan and Nguyen Thi Minh Toai 3. Determining finite volume elements We divide the domain Ω into N equal squares Ωj, j = 1, . . . , N , where Ωj is the j-th cubic with edge h = L/ 3 √ N , and so the volume of Ωj is |Ωj | = L3/N . The local average of ϕ in Ωj is defined by 〈ϕ〉 Ωj = 1 |Ωj| ∫ Ωj ϕ(x)dx. To establish the result on the number of determining finite volume elements, we need the following lemma: Lemma 3.1. [5] For every w ∈ V , we have |w|2 ≤ L3χ2(w) + L 2 6N2/3 ‖w‖2, (3.1) where χ(w) = max 1≤j≤N ∣∣∣〈w〉Ωj ∣∣∣ . Let f and g belong to L∞(0,∞;V ′) satisfying lim t→∞ ‖f(t)− g(t)‖V ′ = 0. Then one can check that the generalized Grashoff numbers Gr corresponding to the external forces f and g are the same. A set of volume elements is said to be determining if for any two solutions u and v of (2.4) corresponding to the above external forces f and g, from the condition lim t→∞ ( 〈u〉 Ωj − 〈v〉 Ωj ) = 0, it implies that lim t→∞ (|u(t)− v(t)|2 + α2‖u(t) − v(t)‖2) = 0. The following theorem is the main result of the paper. Theorem 3.1. Suppose that lim t→∞ ( 〈u〉 Ωj − 〈v〉 Ωj ) = 0, for j = 1, . . . , N . Then the volume elements are determining provided that N > ( 9c41λ 3 1L 3Gr4 4α4d20 )3/2 . Proof. Let u and v be two solutions of (2.4). Then w = u− v satisfies the following system d dt ( w + α2Aw ) + νAw +B(w, u) +B(v,w) = f − g. Multiplying both sides by w we get 1 2 d dt (|w|2 + α2‖w‖2)+ ν‖w‖2 + (B(w, u), w) = 〈f − g,w〉, (3.2) 20 On the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations where we have used property (2.2). Applying the Cauchy inequality, we have 〈f − g,w〉 ≤ 1 ν ‖f − g‖2V ′ + ν 4 ‖w‖2. (3.3) Now, using (2.3) and the Young inequality, we have (B(w, u), w) ≤ c1|w|1/2‖w‖1/2‖u‖ ‖w‖ = c1|w|1/2‖u‖ ‖w‖3/2 ≤ 27c 4 1 4ν3 ‖u‖4|w|2 + ν 4 ‖w‖2. (3.4) Combining (3.3) and (3.4) then (3.2) becomes d dt (|w|2 + α2‖w‖2)+ ν‖w‖2 ≤ 27c41 2ν3 ‖u‖4|w|2 + 2 ν ‖f − g‖2V ′ . Hence, using (3.1), we obtain that d dt (|w|2 + α2‖w‖2)+ (ν − 9c41L3 4ν3N2/3 ‖u‖4 ) ‖w‖2 ≤ 27 2ν3 L3χ2(w) + 2 ν ‖f − g‖2V ′ . (3.5) Using the estimate (2.6), there exists a time t1 > 0 such that ‖u(t)‖4 ≤ ν 4λ31Gr 4 α4d20 , ∀t ≥ t1. So, we get from (3.5) that d dt (|w|2 + α2‖w‖2)+ (ν − 9c41νλ31L3Gr4 4α4d20N 2/3 ) ‖w‖2 ≤ 27 2ν3 L3χ2(w) + 2 ν ‖f − g‖2V ′ , ∀t ≥ t1. (3.6) Now, for N is chosen large enough such that N ≥ ( 9c41λ 3 1L 3Gr4 4α4d20 )3/2 , we deduce from (3.6) that d dt (|w|2 + α2‖w‖2)+ β(t) (|w|2 + α2‖w‖2) ≤ γ(t), ∀t ≥ t1 where β(t) = d0 ( ν − 9c 4 1νλ 3 1L 3Gr4 4α4d20N 2/3 ) , and γ(t) = 27 2ν3 L3χ2(w) + 2 ν ‖f(t)− g(t)‖2V ′ . 21 Nguyen Thi Ngan and Nguyen Thi Minh Toai 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, g and χ one can check that for any fixed T > 0, lim sup t→∞ 1 T ∫ t+T t γ+(τ)dτ = 0. (3.7) Moreover, lim sup t→∞ 1 T ∫ t+T t β−(τ)dτ <∞ and lim inf t→∞ 1 T ∫ t+T t β(τ)dτ > 0 (3.8) hold provided N > ( 9c41λ 3 1L 3Gr4 4α4d20 )3/2 . From eqs. (3.7) and (3.8), applying Lemma 2.2 we arrive at lim t→∞ (|w(t)|2 + α2‖w(t)‖2) = 0. This completes the proof. REFERENCES [1] A.P. Oskolkov, 1973. The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Nauchn. Sem. Leningrad. Otdel. Math. Inst. Steklov. (LOMI) 38, pp. 98-136. [2] E.M. Lunasin, Y. Cao and E.S. Titi, 2006. Global well-posedness of the three-dimensional viscous and inviscid simplied Bardina turbulence models. Commun. Math. Sci. 4, pp. 823-848. [3] E. Lunasin, M. Holst and G. Tsogtgerel, 2010. Analysis of a general family of regularized Navier-Stokes and MHD models. J. Nonlinear Sci. 20, pp. 523-567. [4] M. Holst, M.A. Ebrahimi and E. Lunasin, 2013. The Navier-Stokes-Voight model for image inpainting. IMA J. Appl. Math. 78, pp. 869-894. [5] B. Cockburn, D. Jones and E.S. Titi, 1997. Estimating the asymptotic degrees of freedom for nonlinear dissipative systems. Math. Comp. 66, pp. 1073-1087. [6] D. Jones and E.S. Titi, 1992. Determining finite volume elements for the 2D Navier-Stokes equations. Phys. D 60, pp. 165-174. [7] D. Jones and E.S. Titi, 1993. Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations. Indiana Univ. Math. J. 42, pp. 875-887. [8] V.K. Kalantarov and E.S. Titi, 2009. Global attractor and determining modes for the 3D Navier-Stokes-Voight equations. Chin. Ann. Math. Ser. B 30, pp. 697-714. [9] P. Korn, 2011. On degrees of freedom of certain conservative turbulence models for the Navier-Stokes equations. J. Math. Anal. Appl. 378, pp. 49-63. [10] P. Constantin and C. Foias, 1988. Navier-Stokes Equations, University of Chicago Press, Chicago. [11] R. Temam, 1995. Navier-Stokes Equations and Nonlinear Functional Analysis. Second edition, SIAM, Philadelphia. [12] L.C. Berselli and L. Bisconti, 2012. On the structural stability of the Euler-Voigt and Navier-Stokes-Voigt models. Nonlinear Anal. 75, pp. 117-130. 22