FINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONS

FINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONSFINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONSFINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONSFINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONSFINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONS

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI NGAN FINITE-DIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIER-STOKES-VOIGT EQUATIONS DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2021 MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI NGAN FINITE-DIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIER-STOKES-VOIGT EQUATIONS Speciality: Differential and Integral Equations Speciality Code: 9.46.01.03 DOCTORAL DISSERTATION OF MATHEMATICS Supervisor: PROF DR CUNG THE ANH Hanoi - 2021 COMMITTAL IN THE DISSERTATION I assure that the scientific results presented in this dissertation are new and original To my knowledge, before I published these results, there had been no such results in any scientific document I take responsibility for my research results in the dissertation The publications in common with other authors have been agreed by the co-authors when put into the dissertation February, 2021 Author Nguyen Thi Ngan ACKNOWLEDGEMENTS This dissertation was carried out at the Department of Mathematics and Informatics, Hanoi National University of Education It was completed under the supervision of Prof Cung The Anh First and foremost, I would like to express my sincere gratitude to my supervisor, Prof Cung The Anh, for the continuous support of my PhD study, for his carefulness, patience, enthusiasm and immense knowledge His guidance helped me in all the time of research to learn and grow a lot, both professionally and personally Sometimes he set me back on the road when I got lost I would like to say that I am proud to be his student Besides my supervisor, I am greatly grateful to Assoc Prof Tran Dinh Ke for his encouragement during the time I have studied at Department of Mathematics and Informatics, Hanoi National University of Education I am deeply indebted to Dr Vu Manh Toi for his help and many interesting discussions during my first one year I thank all the lecturers and PhD students at the seminar of Division of Mathematical Analysis for their stimulating consultation and valuable comments I would like to thank all my colleagues at University of Education Publisher and Foreign Language Specialized School, VNU, for support- ing me to study during the last three years I also thank my friends, who always encourage me to overcome difficulties during my period of study Last but not least, I am greatly thankful to my beloved family for respecting all my decisions and supporting me spiritually throughout my life Hanoi, 2021 Nguyen Thi Ngan CONTENTS i ii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS iii LIST OF SYMBOLS INTRODUCTION Chapter 1.1 1.2 1.3 PRELIMINARIES AND AUXILIARY RESULTS 10 Function spaces 10 1.1.1 Weak convergence in Banach spaces 10 1.1.2 The Ck-spaces 11 1.1.3 1.1.4 The Lp-spaces 12 Sobolev spaces 13 The global attractor 15 1.2.1 Existence of global attractor 15 1.2.2 Finite fractal dimension 19 Determining functionals 20 1.3.1 Determining modes 20 1.3.2 Determining nodes 21 1.3.3 Determining volume elements 22 1.3.4 Determining functionals .22 1.4 The Navier-Stokes-Voigt equations with periodic boundary conditions 23 1.5 The Gronwall inequalities 25 Chapter BOUNDS ON THE NUMBER OF DETERMINING NODES FOR 3D NAVIER-STOKES-VOIGT EQUATIONS 27 2.1 Problem setting .27 2.2 Preliminaries 28 2.3 Determining nodes for instationary solutions .31 2.4 Determining nodes for stationary solutions 33 2.5 Determining nodes for periodic solutions 36 Chapter FEEDBACK CONTROL OF NAVIER-STOKES-VOIGT EQUATIONS BY FINITE DETERMINING PARAMETERS 42 3.1 Problem setting .42 3.2 Preliminaries 43 3.3 Stabilization of Navier-Stokes-Voigt equations by using an interpolant operator as feedback controllers .44 3.3.1 Feedback control employing finite volume elements or projection onto Fourier modes as an interpolant operator .45 3.3.2 Chapter Feedback control employing finitely many nodal valued observables 51 ASYMPTOTIC BEHAVIOR OF THREE-DIMENSIONAL NON-HOMOGENEOUS NAVIER-STOKES-VOIGT EQUATIONS 58 4.1 Problem setting .58 4.2 Preliminaries 60 4.3 Existence and uniqueness of weak solutions 61 4.4 Existence of a global attractor .71 4.4.1 Existence of an absorbing set .71 4.4.2 The asymptotic compactness .72 4.5 Fractal dimension estimate of the global attractor .77 4.6 Existence and exponential stability of a stationary solution 81 4.7 Determining projections and functionals for weak solutions 84 CONCLUSIONS AND FUTURE WORKS 90 LIST OF PUBLICATIONS 91 REFERENCES .92 LIST OF SYMBOLS R the set of real numbers Rd A := B A¯ d-dimensional Euclidean vector space A is defined by B (., )X ∥x∥X X∗ ⟨x′, x⟩X∗,X X X ‹→ Y Lp(Ω) L∞(Ω) C0∞ (Ω) C(Ω¯ )  W  m,p (Ω), m H (Ω) ,  H m( Ω) H−m(Ω) L2(Ω) the closure of the set A scalar product in the Hilbert space X norm of x in the space X the dual space of the space X duality pairing between x′ ∈ X ∗ and x ∈ X is imbedded in Y the space ∫of Lebesgue measurable functions f such that Ω |f (x)|pdx < +∞ the space of almost everywhere bounded functions on Ω the space of infinitely differentiable functions with compact support in Ω the space of continuous functions on Ω¯ Sobolev spaces the dual space of Hm(Ω) L2(Ω) × L2(Ω) × L2(Ω) (analogously applied for all other kinds of spaces) (., ) ((., )) ((., ))1 |.| ∥.∥ ∥.∥1 the scalar product in L2(Ω) the scalar product in H10(Ω) the scalar product in H1(Ω) the norm in L2(Ω) the norm in H01(Ω) the norm in H1(Ω) x·y the scalar product between x, y ∈ Rn ∇ ∇y ∂ (∂x , ∂y1 (∂x , ∂ ∂ , · · · ∂y , ∂x ) ∂y2 ∂x n , · · · , ) ∂x ∂x n ∂ ∂ ∂ ∂x ∂x + y ∂ + · · · + n ∂y1 ∂y2 ∂yn x + ∂x + · ·y·n + ∂x ∂x1 n {y ∈ C∞0 (Ω) : div y = 0} y·∇ ∇ · y, div y V H, V the closures of V in L (Ω) and H01(Ω) Lp(0, T ; X), < p < f : [0, T ] → X such ∞ the space ∫ T of functions p that ∥f (t)∥ dt < ∞ X ∞ L (0, T ; X) the space of functions f : [0, T ] → X such that ∥f (.)∥X is almost everywhere bounded on [0, T ] 1,p W (0, T ; X) {y ∈ Lp(0, T ; X) : yt ∈ Lp(0, T ; X)} C([0, T ]; X) the space of continuous functions from [0, T ] to X {xk} sequence of vectors xk xk → x xk converges strongly to x xk ⇀ x xk converges weakly to x ∗ xk ⇀ x xk converges weakly-∗ to x i.e id est (that is) a.e almost every p page 2D two-dimensional 3D three-dimensional Q The proof is complete INTRODUCTION Motivation and overview of the problems Fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids (liquids and gases) Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation One of the most important equations in fluid dynamics is the Navier-Stokes equations Navier-Stokes equations, are partial differential equations that describe the flow of incompressible fluids The equations are generalization of the equations devised by Swiss mathematician Leonhard Euler in the 18th century to describe the flow of incompressible and frictionless fluids In 1821, French engineer ClaudeLouis Navier introduced the element of viscosity (friction) for the more realistic and vastly more difficult problem of viscous fluids Throughout the middle of the 19th century, British physicist and mathematician Sir George Gabriel Stokes improved on this work, though complete solutions were obtained only for the case of simple two-dimensional flows The complex vortices and turbulence, or chaos, that occur in threedimensional fluid (including gas) flows as velocities increase have proven intractable to any but approximate numerical analysis methods The Navier-Stokes equations are   ut − ν∆u + (u · ∇)u + ∇p = f (x, t), ∇ · u = posed on a spatial domain Ω ⊂ Rd, d ∈ {2, 3}, supplemented with 0, appropriate boundary conditions Here u is the unknown velocity, the parameter ν > is the kinematic viscosity, and p is the scalar pressure, which serves to enforce the divergence-free condition ∇·u = The right hand side is a ”body force”, which serves to maintain some nontrivial motion of the fluid The three central questions of every partial differential equations are about existence, uniqueness and smooth dependence on initial data can develop singularities in finite time, and what these might mean For the Navier-Stokes equations satisfactory answers to those questions are available in two dimensions, i.e two-dimensional Navier-Stokes equations with smooth initial data possess a unique solution which stays smooth forever In three dimensions, those questions are still open Therefore, in recent years, many regularized equations have been proposed for the purpose of direct numerical simulations of turbulent incompressible flows modeled by the Navier-Stokes equations [27] They were called α-models, including the Navier-Stokes-α model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model and so on, by replacing the nonlinear term (u · ∇)u in Navier-Stokes equations The Navier-Stokes-Voigt (sometimes written Voight) equations were first introduced by Oskolkov [47] as a model for the motion of a linear, viscoelastic, incompressible fluid The Navier-Stokes-Voigt equations are   u − α2 ∆u − ν∆u + (u · ∇)u + ∇p = f in Ω × (0, t ∞), ∇ · u = ∞) t in Ω × (0, (1) posed on the domain Ω which is a subset of Rd, d ∈ {2, 3} Here u = u(x, t) is the unknown velocity and p = p(x, t) is the unknown pressure, ν > is the kinematic viscosity coefficient, α is a length scale parameter characterizing the elasticity of the fluid The right hand side f is a body force The Navier-Stokes-Voigt equations are nowadays considered as a regularized model of the Navier-Stokes equations and perhaps the newest model in the so-called α-models in fluid mechanics (see e.g [27]) The Navier-Stokes-Voigt equations were also proposed by Cao, Lunasin and Titi in [43] as a regularization, for small value of α, of the three dimension Navier-Stokes equations for the sake of direct numerical simulations Furthermore, we also refer the interested reader to [18] for an interesting application of the Navier-Stokes-Voigt equations in image inpainting The presence of the regularizing term −α2∆ut in (1) has two important consequences First, it leads to the global well-posedness of NavierStokes-Voigt equations both forwards and backwards in time, even in the Now, we prove that Aˆα has finite fractal dimension in H, with the same bound (4.31) Consider an initial orthogonal set of infinitesimal displacements w1,0, , wn,0 for some n ≥ The volume of the parallelepiped they span is given by Vn(0) = |w1,0 ∧ ∧ wn,0| It follows that the evolution of such displacements obeys the evolution equation w˙ i = L(zˆ; t)wi , wi(0) = wi,0, for all i = 1, , n Then (see for instance, [16, 51]) the volume elements Vn(t) = |w1(t) ∧ ∧ wn(t)| satisfy Vn(t) = Vn(0) exp Σ ∫ Tr(Pn (s)L(zˆ; s))dsΣ, t where the orthogonal projection Pn(s) is onto the linear span of {w1(s), , wn(s)} in H, and n Σ Tr(Pn (s)L(zˆ; s)) = ⟨L(zˆ; s)ϕj (s), ϕj (s)⟩, j=1 with n ≥ and {ϕ1(s), , ϕn(s)} an orthonormal set spanning Pn(s)H Letting 1∫ T ⟨⟨Pn L(zˆ)⟩⟩ := lim sup Tr(Pn (t)L(zˆ; t))dt, T T we obtain →∞ Vn (t) ≤ Vn (0) exp Σt sup sup ⟨⟨Pn L(zˆ)⟩⟩Σ zˆ∈Aˆα Pn (0) for all t ≥ 0, where supremum over Pn(0) is a supremum over all choices of initial n orthogonal set of infinitesimal displacements that we take around zˆ We then need to show that Vn (t) decays exponentially in time whenever n ≥ N , with N > to be determined later To achieve this, we are going to estimate ⟨L(zˆ; t)w, w⟩ for w ∈ H One first sees that |G−1w|2 = |w|2 α α by w, we have Taking the inner product of (4.35) that ν |2 ν ⟨L(zˆ; t)w, w⟩ w + |G−1w|2 − = | α2 α2 − b(G−1 w, G−1zˆ, G−1w) ∥G−1w∥2 + (4.36) − b(G−1w, ψ, G−1w) From (4.9) and noting that G−1 zˆ = z, we obtain |b(G−1 w, G−1zˆ, G−1w)| ≤ c|G−1 w|1/2 ∥G−1 w∥3/2 ∥z∥ So, by using (4.36) and the Young inequality, we have c |b(G−1 w, G−1zˆ, G−1w)| ≤ |G−1 w|1/2 |w|3/2 ∥z∥ αν3/ |w|2 + c |G−1w|2∥z∥4 ≤2 ν From inequality (4.32) and 4α2 ∥z∥4 ∥z∥ = we get estimate ∥z∥ ∥z + α ∥ )∥z∥2, 2≤ z| α2 |b(G2 −1 w, G−1 zˆ, G−1 w)| ≤ (| ν Applying Lemma 4.3.2 we obtain cM12 |w + z G −1w 4α | α ν∥ ∥ | | |b(G−1w, ψ, G−1w)| ν G−1w∥2 ≤ ν ∥ Hence, Σ ν cM12 ν (4.37) ∥z∥ |G−1w| ν + ⟨L(zˆ; t)w, w⟩ ≤ 2α |w| + α 2 Using−(4.37), Lemma 6.2 of [55] and noting that the eigenvalues of the Stokes operator in three dimensions satisfy λj ≥ cλ1j2/3, ∀j ≥ 1, we obtain 1∫ = T T Tr(Pn (t)L(zˆ; t))dt ∫ T Σn ⟨L(zˆ; t)ϕj (t),0 ϕj (t)⟩dt j=1 ∫ ≤ T n Σ T ∫ ν − 2α |ϕ (t)| dt j 0 j=1 Σ cM T Σν + ∥z∥ν + T n |G−1ϕj (t)|2 dt j = α2 ν ≤ ∫ 1 ≤ − Σ Σ ν+ α2 ∥z∥ ν3 n n T ν c M ∫ T 1/3 α4 T 2α2 n+c dt + 2/3 j α j = λ1 cM Σ dt, ν+ ∥z∥ ν λ where we have used the following estimate Σ n n 1+ for some constant c ≥ λ1 α2j ≤ /3 j=1 2/3 So, from the bounds on the global attractor given by (4.33), we get ν cM 1/3 K1 Σ ⟨⟨Pn L(zˆ)⟩⟩ ≤ − n+ ν+ n 2α2 λ α4 ν3 In view of M1 and K1 , in order to get ⟨⟨Pn L(zˆ)⟩⟩ ≤ 0, we need the requirement c Σ n : = 1+ λ1 α2 λ + | f ˆ | ty of a stationary solu- tion 3/2 , ν8 λ where c > is a dimensionless scale invariant constant independent of α, ν, λ1, fˆ Hence, the global attractor Aˆα has finite fractal dimension in H, with dim1 | ≤1 +f λˆ This α | comp letes λ the proof + ν8 λ 4.6 3/2 Existe nce and expon ential stabili Definition 4.6.1 Suppose f ∈ L2(Ω) A function u ∗ is called a weak stationary solution of problem (4.1)(4.2)-(4.3) if u∗ ∈ H1(Ω), ∇· u∗ = 0, u∗ = φ on ∂Ω and satisfies νAu∗ + B(u∗, u∗) = f in H−1 1 (Ω) (4.38) Σ c The main result in this section is the following theorem Theorem 4.6.2 Let f ∈ L2(Ω) Then, there exists at least a solution u∗ of problem (4.38) satisfying ∥u∗∥ ≤ |fˆ| + , ∥ψ∥ (4.39) 1/2 3νλ1 where fˆ = f +ν∆ψ−(ψ·∇)ψ and ψ ∈ H2(Ω) is a fixed function satisfying conditions (4.11)-(4.12) Moreover, if the following condition is satisfied c 4c ˆ ν> |f | + ∥ψ∥ , 5/4 3νλ 1 3/4 λ1 where c is the best constant in (4.9), then for any solution u(t) of problem (4.1)-(4.2)-(4.3), the following inequality holds for all t > |u(t) − u∗|2 + α2∥u(t) − u∗∥2 ≤ (|u0 − u∗|2 + α2∥u0 − u∗∥2)× Σ tΣ , 4c ˆ × exp − ν − |f | − ∥ψ∥ c 2λ1 + λ1α2 3νλ 5/4 3/ 1 (4.40) that is, the stationary solution u∗ is globally exponentially stable Proof Noticing that a stationary solution of problem (4.1)-(4.2)-(4.3) is exactly the corresponding stationary solution of the limit NavierStokes equations with the same non-homogeneous Dirichlet boundary conditions Therefore, the existence of a stationary solution u∗ was proved in [53, Theorem 1.5] We now give the estimate (4.39) Let ψ satisfies (4.11)-(4.12), setting z∗ = u∗ − ψ, then z∗ ∈ V and satisfies νAz∗ + B(z∗, z∗) + B(z∗, ψ) + B(ψ, z∗) = fˆ, (4.41) where fˆ = f + ν∆ψ − (ψ · ∇)ψ Multiplying the first equation of (4.41) by z∗ and noting that b(z∗, z∗, z∗) = and b(ψ, z∗, z∗) = 0, we have ν∥z∗∥2 = (fˆ, z∗) − b(z∗, ψ, z∗) Using Lemma 4.3.2 we get ν∥z∗∥2 ≤ |fˆ| |z∗| + Hence, ν ∥z∗∥2 ∥z∗∥ ≤ |fˆ| ∗ ∗ Since u = z + ψ, we get (4.39) ∗ We now prove the stability of solution u∗ For any solution u of problem (4.1)-(4.2)-(4.3), let us set u˜ = u − u , then u˜ ∈ V and satisfies ut + α2Aut + νAu + B(u∗, u) + B(u, u∗) + B(u, u)  = 0, ˜ ˜ ˜ ˜ ˜ ˜ ˜ (4.42) u˜(0) = u0 − u∗ Multiplying the first equation ∥uof (4.42) by u˜ and using (4.8), we 1d +α )+ = −b(u˜, , u˜) have ˜∥ ν∥u˜∥ u 2 ∗ (|u| dt Applying the Hoălder inequality, we have |b(u, u2 ∗ , u)| ≤ ∥u∥L2(Ω)∥u∗∥L6(Ω)∥u∥L3(Ω) 3/2 1/2 ˜ ˜ ≤ c|˜u˜| ∥u∗ ∥1 ∥u˜∥ ˜ ≤ , ∥1∥u˜∥ c λ∥u where we have used the Sobolev embedding H1(Ω) ‹→ L6(Ω), the in1/ and inequality (4.6) ˜ L3 ≤ c|˜ 1/ equality ∥u∥ ∥u 3/4∗ (Ω) u| Then, we get ˜∥ 1d ∥u + α ˜∥ ) + ν∥u˜∥ ≤ (| u˜| dt Using (4.39), we have 1d ∥u˜∥2 ) 2+ ν − dt d 4c 2 2λ1 1+ d (|u˜| + α ∥u˜∥ ) + λ 1α c ∥u ∗ λ1 ∥u˜∥2 ≤ (|u˜|2 + α2 c 5/4 |fˆ| − 3νλ 2 ∥1 ∥u˜∥ 3/4 ν − c ∥ψ∥ 3/ λ4 Σ 5/4 3νλ 4c 3/ λ4 |fˆ| − 1 ∥ψ∥ Σ × × (|u˜| + α ) ≤ ∥u˜∥ Applying the Gronwall lemma, we get (4.40) The proof is complete Determining projections and functionals for weak 4.7 solu- tions Following [28], we give the following definition Definition 4.7.1 Let u, v ∈ L∞(0, ∞; H1(Ω)) be weak solutions to the two following Navier-Stokes-Voigt equations, respectively ut − α ∆ut − ν∆u + (u · ∇)u + ∇p = f in Ω ×  (0,∇T ·),u = in Ω × (0, T ), u = φ on ∂Ω,  u(0) = u0,  v − α2∆v − ν∆v + (v · ∇)v + ∇p = g in Ω × (0, t  t  T ), ∇ · v = in Ω × (0, T ), v = φ on ∂Ω,  v(0) = v0 , (4.43) (4.44) where f, g are given external forces in L∞(0, ∞; L2(Ω)) and satisfy lim |f (t)− g(t) = | The projection operator RN : H1(Ω) → VN ⊂ L2(Ω), N = dim(VN ) < ∞, is called a determining projection for weak solutions of 3D NavierStokes-Voigt equations if t→∞ lim |RN (u(t)− v(t)) = 0, | t→∞ implies that (4.45) lim (|u(t) − v(t)|2 + α2∥u(t) − v(t)∥2) = t→ N ∞ Let {ϕi }i= be a basis functionals1from L2(Ω) N of VN and let {li }i= be a set of bounded linear We can construct 1a projection operator as N Σ RN (u) := li(u)ϕi i=1 Suppose that whenever lim |li(u(t) − v(t)) = 0, i = 1, , N, | t→∞ it implies that (4.45) holds, then we can say that the set {li}N forms a set of determi ning function als We see that the basis {ϕi }N neither be divergen ce-free nor span a subspace of H1(Ω) Moreove r, Definitio n 4.7.1 enc om i= pas ses eac h of the cep ts of det erm inin g nod es, mo des and volu b m mes ya i= i= i= king particular choices for the sets {ϕi }N more details, we refer the interested reader to and [27, 28] {li}N For We define the generalized Grashof number Gr0 in dimension three as Gr = lim sup|fˆ(t)|, (4.46) 5/4 t→∞ ν2λ where fˆ = f + ν∆ψ − (ψ · ∇)ψ, fˆ ∈ L∞(0, ∞; L2(Ω)) The following theorem is the main result in this section Theorem 4.7.2 Let u, v ∈ L∞(0, ∞; H1(Ω)) be the weak solutions to problems (4.43) and (4.44), respectively, and f (t), g(t) are given forces in L∞(0, ∞; L2(Ω)) satisfying lim |f (t)− g(t) = | If there exists a projection operator RN : H1(Ω) → VN , N = dim(VN ), satisfying lim |RN (u(t)− v(t)) = 0, t→∞ | and satisfying for some θ > the approximation t→∞ inequality |u − RN u| ≤ C1N −θ ∥u∥1, (4.47) for some positive constant C1 and N satisfying Σ1 √ 1/2 24 6c2C1(1 + where c and Gr0 are given in θ (4.9) and (4.46), respectively, then lim (|u(t) − v(t)|2 + α2∥u(t) − v(t)∥2) = t→ ∞ H2(Ω) ∈ Proof Let ψ be a fixed function satisfying (4.11)(4.12) We set u = u − ψ, v = v − ψ then u, v ∈ V and ˜ the following ˜ satisfy problem, ˜respectively ˜ u˜t Au˜t + νAu˜ + B(u˜, u˜) + B(u˜, ψ) u˜(0) =ˆ u˜0, + α + B(ψ, u˜) = f , v+t ˜αˆ Av˜t + νAv˜ + B(v˜, v˜) + B(v˜, ψ) + B(ψ, v˜(0) = v˜0, ˜ v˜) = gˆ, where f = f + ν∆ψ − (ψ · ˜ ∇)ψ, gˆ = g + ν∆ψ − (ψ · ∇)ψ, u0 = u0 − ψ, v0 = v0 − ψ Setting z = u˜ − v˜ then z ∈ V is a solution of the following problem d Az) + νAz + B(u˜, + u˜) − α B(v˜, v˜) d + B(u˜, ψ) − B(v˜, ψ) + ˆ a n d B( We take the scalar product of (4.49) with z to obtain ψ, u˜ ) − B( ψ, v˜ ) = f − gˆ , z(0) : = z0 = u˜0 − v˜0 = u0 − v0 (4.49) z) ≤ 1∥ +z) = ∥ + −b(u, α u, z) + ν b(v, v, ∥ z) − z b(u, ψ, ∥ z) b(v, z) z) | ˜ ˜ ˜ ˜ ˜ ˜ ˜ ∥z∥ 1/2 ν 54c 2 ˆ ≤ +) | | zdt + 3νz f (4 αν ∥| 51) ν ∥ v − ∥z ˜ ∥ λ 1g z∥ ∥− ˆ | From the supposition (4.47), we have |z|2 ≤ 2C2N −2θ ∥z∥ + 2| RN z| ∥ z) − u, z ∥v˜ ψ, b(ψ, v˜, ∥ 1/21/ | c∥z ∥z∥2, b(z, ≤ + ν 2 ˆ c ∥ + b(ψ, v˜, z) + ≤ (f ∥z∥ + ∥ ∥ z ∥v ν˜∥ |z| − (ν ˆ ∥z∥ + |f − gˆ|2 ˆ − g= − ˜b(z, v, z) − b(z, ψ, z) + (f ,− gˆ, z) z ≤ νλ1 gˆ, z) ˆ (4.50) Using Lemma 4.3.2, inequality (4.9) and Young’s inequality, we have Therefore, (4.50) implies that d 108c4 b 2 Therefore, (4.51) is equivalent to d dt (|z| ... Navier-StokesVoigt equations have attracted the attention of a number of mathematicians The existence and long-time behavior of solutions in terms of existence of attractors to the Navier-Stokes-Voigt equations. .. arguments indicate that the asymptotic behavior of the solu- tions of certain dissipative evolution equations can be described by only a finite number of degrees of freedom Such equations include, but... easy to study the existence and the asymptotic behavior of the solution by studying the existence of global attractor because the solution of Navier-StokesVoigt equations is not smoother than initial