FINITEDIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIERSTOKESVOIGT EQUATIONS

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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 i 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 supporting 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 ii CONTENTS i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS LIST OF SYMBOLS INTRODUCTION Chapter 1.1 PRELIMINARIES AND AUXILIARY RESULTS 10 Function spaces 10 1.1.1 Weak convergence in Banach spaces 10 1.1.2 The C k -spaces 11 1.1.3 The Lp -spaces 12 1.1.4 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 1.2 1.3 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 iii 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 3.3.1 Feedback control employing finite volume elements or projection onto Fourier modes as an interpolant operator 3.3.2 Chapter 44 45 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 90 91 REFERENCES 92 CONCLUSIONS AND FUTURE WORKS LIST OF PUBLICATIONS iv LIST OF SYMBOLS R Rd A := B A¯ (., )X ∥x∥X X∗ ⟨x′ , x⟩X ∗ ,X X →Y Lp (Ω) L∞ (Ω) C0∞ (Ω) ¯ C(Ω)  m,p   W (Ω), H m (Ω),    m H0 (Ω) H −m (Ω) L2 (Ω) (., ) ((., )) ((., ))1 |.| ∥.∥ ∥.∥1 x·y the set of real numbers d-dimensional Euclidean vector space A is defined by B 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 X is imbedded in Y the space of Lebesgue measurable functions f such that Ω |f (x)|p dx < +∞ 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 H0m (Ω) L2 (Ω) × L2 (Ω) × L2 (Ω) (analogously applied for all other kinds of spaces) 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 H10 (Ω) the norm in H1 (Ω) the scalar product between x, y ∈ Rn ∇ ∇y y·∇ ∇ · y, div y V H, V Lp (0, T ; X), < p < ∞ L∞ (0, T ; X) W 1,p (0, T ; X) C([0, T ]; X) {xk } xk → x xk ⇀ x xk ⇀∗ x i.e a.e p 2D 3D ✷ ( ∂x∂ , ∂x∂ , · · · , ∂x∂ n ) ∂y ∂y ∂y ( ∂x , , · · · , ) ∂x ∂x n y1 ∂x∂ + y2 ∂x∂ + · · · + yn ∂x∂ n ∂y1 ∂y2 ∂yn ∂x1 + ∂x2 + · · · + ∂xn {y ∈ C∞ (Ω) : div y = 0} the closures of V in L2 (Ω) and H10 (Ω) the space of functions f : [0, T ] → X such T that ∥f (t)∥pX dt < ∞ the space of functions f : [0, T ] → X such that ∥f (.)∥X is almost everywhere bounded on [0, T ] {y ∈ Lp (0, T ; X) : yt ∈ Lp (0, T ; X)} the space of continuous functions from [0, T ] to X sequence of vectors xk xk converges strongly to x xk converges weakly to x xk converges weakly-∗ to x id est (that is) almost every page two-dimensional three-dimensional 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  u − ν∆u + (u · ∇)u + ∇p = f (x, t), t ∇ · u = 0, posed on a spatial domain Ω ⊂ Rd , d ∈ {2, 3}, supplemented with 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 t (1) ∇ · u = in Ω × (0, ∞) 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 Using Lemma 4.3.2 we get ν ν∥z ∗ ∥2 ≤ |fˆ| |z ∗ | + ∥z ∗ ∥2 Hence, ∥z ∗ ∥ ≤ 1/2 3νλ1 |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  u + α2 Au + νAu + B(u∗ , u) + B(u, u∗ ) + B(u, u) = 0, t t (4.42) u(0) = u0 − u∗ Multiplying the first equation of (4.42) by u and using (4.8), we have 1d (|u|2 + α2 ∥u∥2 ) + ν∥u∥2 = −b(u, u∗ , u) dt Applying the Hăolder inequality, we have |b(u, u , u)| ≤ ∥u∥L2 (Ω) ∥u∗ ∥L6 (Ω) ∥u∥L3 (Ω) ≤ c|u|3/2 ∥u∗ ∥1 ∥u∥1/2 c ≤ 3/4 ∥u∗ ∥1 ∥u∥2 , λ1 where we have used the Sobolev embedding H1 (Ω) → L6 (Ω), the inequality ∥u∥L3 (Ω) ≤ c|u|1/2 ∥u∥1/2 and inequality (4.6) Then, we get 1d c (|u|2 + α2 ∥u∥2 ) + ν∥u∥2 ≤ 3/4 ∥u∗ ∥1 ∥u∥2 dt λ1 Using (4.39), we have 1d 4c ˆ c (|u|2 + α2 ∥u∥2 ) + ν − | f | − ∥ψ∥1 ∥u∥2 ≤ 5/4 3/4 dt 3νλ1 λ1 Hence, d 2λ1 (|u|2 + α2 ∥u∥2 ) + dt + λ1 α2 ν− 4c ˆ| − c ∥ψ∥1 × | f 5/4 3/4 3νλ1 λ1 × (|u|2 + α2 ∥u∥2 ) ≤ Applying the Gronwall lemma, we get (4.40) The proof is complete 83 4.7 Determining projections and functionals for weak solutions 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 − α2 ∆ut − ν∆u + (u · ∇)u + ∇p = f in Ω × (0, T ),     ∇ · u = in Ω × (0, T ), (4.43)  u = φ on ∂Ω,     u(0) = u ,   vt − α2 ∆vt − ν∆v + (v · ∇)v + ∇p = g in Ω × (0, T ),     ∇ · v = in Ω × (0, T ), (4.44)  v = φ on ∂Ω,     v(0) = v , where f, g are given external forces in L∞ (0, ∞; L2 (Ω)) and satisfy lim |f (t) − g(t)| = 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 lim |RN (u(t) − v(t))| = 0, t→∞ (4.45) implies that lim (|u(t) − v(t)|2 + α2 ∥u(t) − v(t)∥2 ) = t→∞ N Let {ϕi }N i=1 be a basis of VN and let {li }i=1 be a set of bounded linear functionals from L2 (Ω) We can construct a projection operator as N RN (u) := li (u)ϕi i=1 Suppose that whenever lim |li (u(t) − v(t))| = 0, i = 1, , N, t→∞ 84 it implies that (4.45) holds, then we can say that the set {li }N i=1 forms N a set of determining functionals We see that the basis {ϕi }i=1 neither be divergence-free nor span a subspace of H1 (Ω) Moreover, Definition 4.7.1 encompasses each of the concepts of determining nodes, modes and N volumes by making particular choices for the sets {ϕi }N i=1 and {li }i=1 For more details, we refer the interested reader to [27, 28] We define the generalized Grashof number Gr0 in dimension three as Gr0 = 5/4 ν λ1 lim sup|fˆ(t)|, (4.46) t→∞ 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)| = 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 inequality |u − RN u| ≤ C1 N −θ ∥u∥1 , (4.47) for some positive constant C1 and N satisfying √ ∞>N > 24 6c C1 (1 + λ1 α α2 1/2 )λ1 θ Gr02 , (4.48) where c and Gr0 are given in (4.9) and (4.46), respectively, then lim (|u(t) − v(t)|2 + α2 ∥u(t) − v(t)∥2 ) = t→∞ Proof Let ψ ∈ H2 (Ω) be a fixed function satisfying (4.11)-(4.12) We set u = u − ψ, v = v − ψ then u, v ∈ V and satisfy the following problem, respectively ut + α2 Aut + νAu + B(u, u) + B(u, ψ) + B(ψ, u) = fˆ, u(0) = u0 , 85 vt + α2 Avt + νAv + B(v, v) + B(v, ψ) + B(ψ, v) = gˆ, v(0) = v0 , where fˆ = 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 (z + α2 Az) + νAz + B(u, u) − B(v, v) + B(u, ψ) − B(v, ψ) dt + B(ψ, u) − B(ψ, v) = fˆ − gˆ, (4.49) and z(0) := z0 = u0 − v0 = u0 − v0 We take the scalar product of (4.49) with z to obtain 1d (|z|2 + α2 ∥z∥2 ) + ν∥z∥2 = −b(u, u, z) + b(v, v, z) − b(u, ψ, z) dt + b(v, ψ, z) − b(ψ, u, z) + b(ψ, v, z) + (fˆ − gˆ, z) = −b(z, v, z) − b(z, ψ, z) + (fˆ − gˆ, z) (4.50) Using Lemma 4.3.2, inequality (4.9) and Young’s inequality, we have ν b(z, ψ, z) ≤ ∥z∥2 , b(z, v, z) ≤ c∥z∥ ∥v∥ ∥z∥1/2 |z|1/2 ≤ c∥z∥3/2 ∥v∥ |z|1/2 54c4 ν ≤ ∥z∥ + ∥v∥4 |z|2 ν Applying the Cauchy inequality we get ν ˆ (fˆ − gˆ, z) ≤ ∥z∥2 + |f − gˆ|2 νλ1 Therefore, (4.50) implies that 108c4 ˆ d 2 2 (|z| + α ∥z∥ ) + ν∥z∥ − ∥ v∥ |z| ≤ |f − gˆ|2 dt ν νλ1 From the supposition (4.47), we have |z|2 ≤ 2C12 N −2θ ∥z∥2 + 2|RN z|2 86 (4.51) Therefore, (4.51) is equivalent to d 216c4 C12 −2θ 2 (|z| + α ∥z∥ ) + ν − N ∥v∥4 ∥z∥2 dt ν ˆ 216c4 ∥ v∥ |R z| + |f − gˆ|2 ≤ N ν νλ1 (4.52) From (4.24), z and fˆ are replaced by v and gˆ, respectively, and using (4.46) then 16(1 + λ1 α2 )2 ν λ1 4 ∥v(t)∥ ≤ Gr0 , α4 for t large enough Therefore, when N satisfies (4.48) we have 216c4 C12 −2θ N ∥v∥4 > ν− ν Hence, using (4.6) then (4.52) becomes (4.53) λ1 216c4 C12 −2θ d 2 N ∥v∥4 (|z|2 + α2 ∥z∥2 ) (|z| + α ∥z∥ ) + ν− dt + λ1 α ν 2 3456c (1 + λ1 α ) νλ1 4 ˆ ≤ Gr |R z| + |f − gˆ|2 , N α νλ1 for t large enough From the assumption, we have lim |fˆ(t) − gˆ(t)| = lim |f (t) − g(t)| = t→∞ t→∞ and lim |RN z(t)| = lim |RN (u(t) − v(t))| = lim |RN (u(t) − v(t))| = t→∞ t→∞ t→∞ then t+T + lim γ (τ )dτ = 0, t→∞ T t 3456c4 (1 + λ1 α2 )2 νλ1 4 ˆ where γ = Gr |f − gˆ|2 |R z| + N α νλ1 Furthermore, from (4.53) we have t+T lim inf t→∞ T λ1 where β = + λ1 α2 β(τ )dτ > 0, t 216c4 C12 −2θ ν− N ∥v∥4 ν3 lim sup t→∞ T t+T and β − (τ )dτ < ∞ t 87 Applying Lemma 1.5.2, we obtain lim (|z(t)|2 + α2 ∥z(t)∥2 ) = lim (|u(t) − v(t)|2 + α2 ∥u(t) − v(t)∥2 ) = t→∞ t→∞ Since u = u + ψ and v = v + ψ, we have lim (|u(t) − v(t)|2 + α2 ∥u(t) − v(t)∥2 ) = t→∞ This completes the proof Remark 4.7.3 It is worthy noticing that when ψ ≡ 0, from Theorem 4.7.2 we can get the bounds on some determining functionals such as determining modes, nodes and finite volumes for 3D Navier-Stokes-Voigt equations with homogeneous Dirichlet boundary conditions, which are similar to those in the case of periodic boundary conditions obtained recently in [29, 36] The existence of global solutions and of a finite-dimensional global attractor to three-dimensional Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions in this chapter can be seen as an extension of previous results in the case of homogeneous boundary conditions (i.e φ ≡ 0) obtained in [17, 36] Conclusion of Chapter In this chapter, we have studied the existence and long-time behavior of solutions to the three-dimensional Navier-Stokes-Voigt equations with non-homogeneous Dirichlet boundary conditions We have achieved the following results: 1) The existence and uniqueness of global weak solutions (Theorem 4.3.3); 2) The existence of a global attractor for the semigroup generated by weak solutions (see Remark 4.1.1 and Theorem 4.4.4); 3) The finiteness of the fractal dimension of the global attractor (see Remark 4.1.1 and Theorem 4.7.2); 4) A sufficient condition for exponential stability of weak stationary solutions (Theorem 4.6.2); 88 5) The existence of determining projections for weak solutions (Theorem 4.7.2) These are the first results about long-time behavior of solutions to the three-dimensional Navier-Stokes-Voigt equations with non-homogeneous Dirichlet boundary conditions 89 CONCLUSIONS AND FUTURE WORKS Conclusion In this dissertation, we have been investigated the long-time behavior of solutions to three-dimensional Navier-Stokes-Voigt equations by some approaches: the global attractor, determining nodes and stabilization of stationary solutions by finite-dimensional feedback controls The main contributions of the dissertation are to the following: 1) Give bounds on the number of determining nodes for solutions to the three-dimensional Navier-Stokes-Voigt equations with periodic boundary conditions 2) Prove the stabilization of stationary solutions to the Navier-StokesVoigt equations with periodic boundary conditions by finite-dimensional feedback controls 3) Prove the existence and long-time behavior of solutions to the threedimensional Navier-Stokes-Voigt equations with non-homogeneous Dirichlet boundary conditions in terms of the existence of a finitedimensional global attractor and the existence of determining projections The results obtained in the dissertation are meaningful contributions to the Navier-Stokes-Voigt equations as well as the theory of infinitedimensional dissipative dynamical systems Future works Some suggestions for potential future works are proposed below: 1) Existence of approximate internal manifold for the three-dimensional Navier-Stokes-Voigt equations (see the survey article [56] for related results on Navier-Stokes equations) 2) Data assimilation for three-dimensional Navier-Stokes-Voigt equations (see [9, 19] for results on Navier-Stokes equations) 90 LIST OF PUBLICATIONS Published papers [CT1 ] V.M Toi and N.T Ngan (2020), Upper bounds on the number of determining nodes for 3D Navier-Stokes-Voigt equations, Ann Pol Math 125, no 1, 83-99 [CT2 ] N.T Ngan and V.M Toi (2020), Feedback control of Navier-StokesVoigt equations by finite determining parameters Acta Math Vietnam 45 (2020), no 4, 917-930 Submitted papers [CT3 ] C.T Anh and N.T Ngan (2020), Asymptotic behavior of threedimensional non-homogeneous Navier-Stokes-Voigt equations, submitted to Acta Applicandae Mathematicae 91 REFERENCES [1] R.A 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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 solutions 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-Stokes-Voigt equations is not smoother than initial