finite dimensional asymptotic behavior of navier stokes voigt equations TT TIẾNG ANH

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI NGAN FINITE-DIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIER-STOKES-VOIGT EQUATIONS SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Speciality: Speciality Code: Differential and Integral Equations 46 01 03 HA NOI, 2021 This dissertation has been written at Hanoi National University of Education Supervisor: Prof Dr Cung The Anh Referee 1: Prof Dr Sc Nguyen Minh Tri Institute of Mathematics, Vietnam Academy of Science and Technology Referee 2: Assoc Prof Dr Le Van Hien Hanoi National University of Education Referee 3: Assoc Prof Dr Do Duc Thuan Hanoi University of Science and Technology The thesis shall be defended at the University level Thesis Assessment Council at Hanoi National University of Education on This thesis can be found in: - The National Library of Vietnam; - Library of Hanoi National University of Education INTRODUCTION Motivation and overview of the problems The Navier-Stokes-Voigt (sometimes written Voight) equations which were first introduced by Oskolkov in 1973 as a model for the motion of a linear, viscoelastic, incompressible fluid The following Navier-Stokes-Voigt equations are   ut − α2 ∆ut − ν∆u + (u · ∇)u + ∇p = f  ∇ · u = in Ω × (0, ∞), (1) in Ω × (0, ∞) posed on 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 The Navier-Stokes-Voigt equations were also proposed by Cao, Lunasin and Titi in 2006 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 for an interesting application of the Navier-Stokes-Voigt equations in image inpainting The presence of the regularizing term −α2 ∆ut in Navier-Stokes-Voigt equations has two important consequences First, it leads to the global well-posedness of Navier-Stokes-Voigt equations both forwards and backwards in time, even in the case of three dimensions Second, it changes the parabolic character of the limit Navier-Stokes equations, so the Navier-Stokes-Voigt system behaves like a damped hyperbolic system Note that when α = 0, we recover the NavierStokes equations of motion The Navier-Stokes-Voigt equations has attractive advantages over other α-models in that one does not require additional artificial boundary conditions (except the Dirichlet boundary conditions) to get the global well-posedness In the last few years, mathematical questions related to Navier-Stokes-Voigt 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 in domains that are bounded or unbounded but satisfying the Poincar´e inequality was investigated extensively by many mathematicians The time decay rates of solutions to the equations on the whole space were studied in some works As far as we know, there have been many results on the existence and long-time behavior of solutions in terms of existence of attractors for the Navier-Stokes-Voigt equations However, most of the existing results are in the case of homogeneous Dirichlet boundary conditions or periodic boundary conditions with the exception of a recent work in which the existence, uniqueness of solutions and exitence of a uniform attractor for two-dimensional Navier-Stokes-Voigt equations with non-homogeneous Dirichlet boundary conditions were studied in Lipschitz domains The conventional theory of turbulence asserts that turbulent flows, type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing so the speech of the fluid at a point is continuously undergoing changes in both magnitude and direction, are monitored by a finite number of degrees of freedom For the 3D Navier-Stokes-Voigt equations, Kalantarov and Titi in 2009 proved a result on the determining modes In 2014, Azouani and Titi proposed a new feedback control for controlling general dissipative evolution equations using any of the determining systems of parameters (modes, nodes, volume elements, etc.) without requiring the presence of separation in spatial scales, i.e without assuming the existence of an inertial manifold Therefore we have chosen the topic ”Finite-dimensional asymptotic behavior of NavierStokes-Voigt equations” We will study the following problems: (P1) The bounds on the number of determining nodes for instationary, stationary and periodic solutions to the 3D Navier-Stokes-Voigt equations with periodic boundary conditions (P2) The stabilization of stationary solutions of the Navier-Stokes-Voigt equations in both cases of dimension two and dimension three with periodic boundary conditions by finitedimensional feedback controls (P3) The long-time behavior of solutions of the 3D Navier-Stokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions in term of the existence of a finite-dimensional global attractor and the existence of the determining projections Objectives The objectives of this dissertation are mentioned the following to solve the problems (P1), (P2) and (P3), respectively: (i) to show that the solutions of the 3D Navier-Stokes-Voigt equations are entirely determined by their values on a discrete set, provided this set contains enough points and these points are sufficiently densely distributed (ii) to show that an unstable stationary solution can be stabilized in both cases of dimension two and dimension three by using finite-dimensional feedback control scheme employing finite volume elements or projection onto Fourier modes, and in case of dimension two by using a finite-dimensional feedback control employing finitely many nodal values (iii) to prove the existence and uniqueness of global weak solutions of the problem (P3), then prove that the dynamical system generated by weak solutions has a finite-dimensional global attractor with an explicit upper bound of fractal dimension; study the existence and stability of stationary solutions; show that if there exists an operator which projects the solutions onto a finite-dimensional space and it satisfies a suitable approximation inequality then the long-time behavior of the solutions is determined Methodology (i) The Faedo-Galerkin approximate method and compact methods: to study the existence and uniqueness of the solution (ii) The methods of the theory of infinite-dimensional dynamic systems: to study the existence and properties of the global attractor (iii) The methods in works of Titi et al in 1993 and 1997: to study the bounds on the number of determining parameters (iv) The method in work of Azouani and Titi in 2014: to study the stabilization of stationary solutions by finite-dimensional feedback controls The structure and results of the dissertation Chapter presents several basic concepts and recalls some auxiliary results for the NavierStokes-Voigt equations Chapter gives bounds on the number of determining nodes for instationary, stationary and periodic solutions to the Navier-Stokes-Voigt equations with periodic boundary conditions Chapter studies the stabilization of stationary solutions to the Navier-Stokes-Voigt equations with periodic boundary conditions by finite-dimensional feedback controls Chapter studies the existence and long-time behavior of weak solutions to the 3D NavierStokes-Voigt equations with non-homogeneous Dirichlet boundary conditions The results obtained in Chapters 2, and are the answers for problems (P1), (P2), (P3), respectively Chapter and Chapter are based on the papers [CT1], [CT2] in the List of Publications which were published The results of Chapter 4, which has been submitted for publication, are in the work [CT3] in the List of Publications Chapter PRELIMINARIES AND AUXILIARY RESULTS In this chapter, we recall some basic concepts and results on function spaces, imbeddings, operators, global attractor, determining modes, nodes and finite volume, the Navier-StokesVoigt equations and present some auxiliary results Function spaces ❼ C k , Lp and Sobolev spaces: C k (Ω), Lp (Ω), W m,p (Ω), H m (Ω) (m ∈ N) ❼ Spaces of abstract functions: Lp (0, T ; X), ≤ p ≤ ∞, W 1,p (0, T ; X) Some useful inequalities: Hăolder inequality, Poincares inequality, Gronwall’s inequalities, Young’s inequality with ϵ Continuous and compact imbeddings: Rellich-Kondrachov theorem, imbeddings in abstract function spaces Operators: the Stokes operator, bilinear operator, and the trilinear form The theory of the global attractor and the finite fractal dimension Some auxiliary results about finite number of degrees of freedom: determining modes, determining nodes, determining volume elements The Navier-Stokes-Voigt equations with periodic boundary conditions Chapter BOUNDS ON THE NUMBER OF DETERMINING NODES FOR 3D NAVIER-STOKES-VOIGT EQUATIONS In this chapter, we consider the 3D Navier-Stokes-Voigt equations with periodic boundary conditions We give bounds on the number of determining nodes for instationary, stationary and periodic solutions of this equations Here the number of determining nodes is estimated explicitly in terms of flow parameters such as viscosity, smoothing length, forcing and domain size The content of this chapter is based on the work [CT1] in the List of Publications 2.1 Problem setting Let Ω = (0, L)3 , L > 0, be a box in R3 We consider the 3D Navier-Stokes-Voigt equations   ut − α2 ∆ut − ν∆u + (u · ∇)u + ∇p = f in Ω × (0, ∞), (2.1)  ∇ · u = in Ω × (0, ∞), with the periodic boundary conditions u(x, t) = u(x + Lei , t) for all t > 0, (2.2) where {e1 , e2 , e3 } is the canonical basis of R3 and initial condition u(0) = u0 in Ω (2.3) We have the 3D Navier-Stokes-Voigt equations in the functional form d (u + α2 Au) + νAu + B(u, u) = f dt (2.4) If f ∈ L∞ (0, ∞; H), we define the generalized Grashof number Gr in three dimensions as follows Gr = 3/4 ν λ1 lim sup |f (t)| (2.5) t→∞ In this chapter, we give bounds on the number of determining nodes for solutions to the 3D Navier-Stokes-Voigt equations We prove that two stationary solutions coincide if they coincide on a suitable set of finite points, and if the large time behavior of the solutions to the NavierStokes-Voigt equations is known on an appropriate discrete set, then the large time behavior of the solution itself is totally determined 2.2 Preliminaries Remark 2.2.1 It is noticing that for any f and g belonging to L∞ (0, ∞; H) such that lim |f (t) − g(t)| = then the generalized Grashof number Gr defined on f is the same the one t→∞ defined on g Theorem 2.2.2 Let f ∈ L∞ (0, ∞; H) If u0 ∈ V then problem (2.3)-(2.4) has a unique global weak solution u ∈ L∞ (0, ∞; V ) that satisfies 1/2 lim sup |u(t)|2 + α2 ∥u(t)∥2 ≤ t→∞ ν λ1 Gr2 d0 (2.6) Moreover, if u0 ∈ D(A) then problem (2.3)-(2.4) has a unique global strong solution u ∈ L∞ (0, ∞; D(A)) that satisfies (2.6) and 3/2 2 lim sup ∥u(t)∥ + α |Au(t)| t→∞ 3/2 27c41 ν λ1 2ν λ1 Gr2 + ≤ Gr6 d0 2α d0 (2.7) We divide the domain Ω = (0, L)3 into N equal cubics Ωj , j = 1, , N , where Ωj is the √ j-th cubic with edge h = L/ N Furthermore, we place the point xj ∈ Ωj , j = 1, , N Lemma 2.2.3 For every w ∈ D(A), there exist some positive constants c2 , c3 , c4 , c5 such that c2 L |Aw|2 , 4/3 N c4 L ∥w∥2 ≤ c3 LN 2/3 ϑ2 (w) + 2/3 |Aw|2 , N c L ∥w∥2L∞ (Ω) ≤ c4 LN ϑ2 (w) + 1/3 |Aw|2 , N |w|2 ≤ 4L3 ϑ2 (w) + where ϑ(w) = max |w(xj )| 1≤j≤N (2.11) 2.3 Determining nodes for instationary solutions A finite set of points N := {x1 , x2 , , xN } ⊂ Ω is called a set of determining nodes if for any two strong solutions u and v of (2.3)-(2.4) with the initial data u0 , v ∈ D(A) and forcings f, g ∈ L∞ (0, ∞; H), respectively, the assumptions lim |u(xj , t) − v(xj , t)| = for xj ∈ N , j = 1, , N, t→∞ and lim |f (t) − g(t)| = 0, t→∞ imply that lim ∥u(t) − v(t)∥2 + α2 |Au(t) − Av(t)|2 = t→∞ Theorem 2.3.1 Let Ω = (0, L)3 be divided into N equal cubics with the points N := {x1 , x2 , , xN } distributed one in each cubic Then N is a set of determining nodes provided 216c41 c4 λ1 L2 Gr4 α4 d20 N> 2.4 3/2 (2.12) Determining nodes for stationary solutions We know that if the external force f ∈ H then there exists a stationary solution u to the 3D Navier-Stokes-Voigt equations, i.e., u satisfies νAu + B(u, u) = f in H, (2.21) and the following estimates hold ∥u∥ ≤ |f | and 1/2 |Au| ≤ νλ1 c21 |f | + |f |3 3/2 ν ν λ1 (2.22) Theorem 2.4.1 Given f ∈ H Let u and v be two stationary solutions of the Navier-StokesVoigt equations, i.e., νAu + B(u, u) = f, νAv + B(v, v) = f, such that u(xj ) = v(xj ), j = 1, , N Then u = v if N> 8c31 |f |3/2 3/4 ν λ1 c21 |f | + |f |3 3/2 ν ν λ1 3/2 3/2 c4 L (2.23) We now assume that the external force f ∈ L∞ (0, ∞; H) satisfies f (t) → f∞ in H as t → ∞, (2.27) where f∞ ∈ H We will prove the following result Theorem 2.4.2 Assume that f ∈ L∞ (0, ∞; H) satisfies (2.27) and we denote by u the solution to problem (2.3)-(2.4) with u0 ∈ D(A) We assume that 3/2 N > max 216c41 c4 λ1 L2 Gr4 , α4 d20 8c31 |f∞ |3/2 c21 |f∞ | + |f∞ |3 3/4 3/2 ν ν λ1 ν λ1 3/2 (2.28) 3/2 c4 L3 , and furthermore, for j = 1, , N , u(xj , t) → ξj as t → ∞, (2.29) where ξj being a vector in R3 Then there exists a unique solution u∞ to (2.21) with f replaced by f∞ , which satisfies u∞ (xj ) = ξj , j = 1, , N , and as t → ∞, u(t) converges to u∞ , in the D(A)-norm and the uniform convergence norm 2.5 Determining nodes for periodic solutions We fix a function γ : R → H which is periodic of period T Moreover, assume that γ ∈ L∞ (0, T ; H) The corresponding periodic solution to the 3D Navier-Stokes-Voigt equations is a function φ : R → V , which is periodic of period T , φ(t + T ) = φ(t), ∀t, (2.32) with φ|[0,T ] ∈ L∞ (0, T ; D(A)), (2.33) d φ + α2 Aφ + νAφ + B(φ, φ) = γ dt (2.34) and such that The existence of such a solution φ satisfying (2.32)-(2.34) was proved Moreover, there exists a positive constant Γ depending on ∥γ∥L∞ (0,T ;H) and some constants α, ν, λ1 such that the time-periodic solution φ satisfies ∥φ∥2 + α2 |Aφ|2 ≤ Γ (2.35) Chapter FEEDBACK CONTROL OF NAVIER-STOKES-VOIGT EQUATIONS BY FINITE DETERMINING PARAMETERS In this chapter we study the stabilization of stationary solutions to Navier-Stokes-Voigt equations by finite-dimensional feedback control scheme introduced by Azouani and Titi in 2014 The designed feedback control scheme are based on the finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes and volume elements The content of this chapter is based on the work [CT2] in the List of Publications 3.1 Problem setting Let Ω = (0, L)d , L > 0, be a box in Rd , d ∈ {2, 3} We consider the Navier-Stokes-Voigt equations   ut − α2 ∆ut − ν∆u + (u · ∇)u + ∇p = f in Ω × (0, ∞),  ∇ · u = in Ω × (0, ∞), (3.1) with the periodic boundary conditions u(x, t) = u(x + Lei , t), x ∈ Ω, t > 0, 10 (3.2) where {ei } is the canonical basis of Rd and initial condition u(x, 0) = u0 , x ∈ Ω (3.3) We have the 3D Navier-Stokes-Voigt equations in the functional form d (u + α2 Au) + νAu + B(u, u) = f dt (3.4) In this chapter, we propose a simple finite-dimensional feedback control scheme for stabilizing stationary solutions of Navier-Stokes-Voigt equations with periodic boundary conditions Here the feedback control scheme only uses finitely many of observables and controllers, such as finite number of determining Fourier modes, determining nodes, and determining finite volumes 3.2 Preliminaries We have the following estimates on the trilinear form b(u, v, w):   |u|1/2 ∥u∥1/2 ∥v∥ |w|1/2 ∥w∥1/2 if d = 2, |b(u, v, w)| ≤ c0 ∀u, v, w ∈ V,  ∥u∥∥v∥|w|1/2 ∥w∥1/2 if d = 3, |b(u, v, w)|   |u|1/2 ∥u∥1/2 ∥v∥1/2 |Av|1/2 |w| if d = 2, ≤ c0 ∀u ∈ V, v ∈ D(A), w ∈ H  ∥u∥∥v∥1/2 |Av|1/2 |w| if d = 3, (3.5) (3.6) We also have that |b(u, v, w)| ≤ c0 ∥u∥∥v∥|w|1/2 ∥w∥1/2 where  c0    1/4 c0 = λ   c ∀u, v, w ∈ V, (3.7) if d = 2, (3.8) if d = 3, with c0 is given in (3.5) Theorem 3.2.1 For any f ∈ H, there exists at least one stationary solution u∗ of problem (3.1)-(3.2)-(3.3) satisfying |f |2 , ν λ1 (3.9) 2 54c40 |f |6 |f | + 10 , ν2 ν λ1 (3.10) ∥u∗ ∥2 ≤ and |Au∗ |2 ≤ 11 where c0 is given in (3.8) Moreover, if f satisfies c0 |f | ν> , (3.11) 3/4 νλ1 then the solution u∗ is unique and globally exponentially stable, that is, for any solution u of problem (3.1)-(3.2)-(3.3), we have |u(t) − u∗ |2 + α2 ∥u(t) − u∗ ∥2 ≤ |u0 − u∗ | + α2 ∥u0 − u∗ ∥2 e−λt , for any t > 0, where λ ∈ 3.3 0, 2d−1 ν− c0 |f | 3/4 νλ1 Stabilization of Navier-Stokes-Voigt equations by using an interpolant operator as feedback controllers By Theorem 3.2.1, we see that the stationary solution u∗ may be unstable when the condition (3.11) does not hold, that is, when c0 |f | ν≤ 3/4 νλ1 The aim of this section is to use the notions of finite number of determining modes, nodes and volume elements to design feedback controllers which stabilize the stationary solution u∗ in this case We consider the system with interpolant operator Ih as the controller:    ∂t (u − α2 ∆u) − ν∆u + (u · ∇)u + ∇p = f − µIh (u − u∗ ), x ∈ Ω, t > 0,       ∇ · u = 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0,    u(x, t) = u(x + Lei , t),      u(x, 0) = u0 , x ∈ Ω By applying the projector P then the system is equivalent to    d (u + α2 Au) + νAu + B(u, u) = P f − µP (Ih (u − u∗ )), dt  u(0) = u0 (3.12) We will consider two cases of Ih 3.3.1 Feedback control employing finite volume elements or projection onto Fourier modes as an interpolant operator Let us consider the linear map Ih : V → H which is an interpolant operator that approximates identity with error of order h satisfying the estimate |φ − Ih (φ)|2 ≤ c21 h2 ∥φ∥2 , 12 for all φ ∈ V, (3.13) for some positive constant c1 > We now give two examples of Ih satisfying (3.13) ❼ Finite volume elements We divide the domain Ω into Ωk , k = 1, , N , where Ωk is k-th √ cubic with edge L/ d N , and so the volume of Ωk is |Ωk | = Ld /N We denote by 1Ωk the characteristic function of Ωk The interpolant operator Ih has the following form N φk 1Ωk (x), Ih (φ) = k=1 where φk is the local average of φ in Ωk defined by |Ωk | φk = φdx Ωk ❼ Projection onto Fourier modes as an interpolant operator Let us consider the operator Ih given by the projection onto the low Fourier modes, up to wave number |k| such that |k| ≤ L 2πh defined as Ih (φ) = φk wk (x), L |k|≤⌊ 2πh ⌋ for every φ = φk wk (x), where k∈Zd \{0} wk (x) = exp Ld 2πi k · x , ∀k ∈ Zd \ {0} and φk = (φ, wk ) d L Theorem 3.3.1 Let d ∈ {2, 3} Given f ∈ H and u∗ as in Theorem 3.2.1 Let µ and h be positive parameters satisfying µc21 h2 ≤ ν and µ ≥ 54c40 |f |4 , ν λ21 and let Ih satisfy (3.13) Then for any u0 ∈ V , there exist a unique solution u to problem (3.12) such that for any T > 0, u ∈ C([0, T ]; V ), du ∈ L2 (0, T ; V ), dt (3.14) and for all t > 0, |u(t) − u∗ |2 + α2 ∥u(t) − u∗ ∥2 ≤ |u0 − u∗ |2 + α2 ∥u0 − u∗ ∥2 e− 3.3.2 νd0 t (3.15) Feedback control employing finitely many nodal valued observables In this subsection we consider system (3.12) with the interpolant operator Ih : D(A) → H satisfying |φ − Ih (φ)|2 ≤ c22 h2 ∥φ∥2 + c22 h4 |Aφ|2 for all φ ∈ D(A) 13 (3.30) One example of Ih is the following form N Ih (φ) = φ(xk )1Ωk (x), k=1 for the points xk ∈ Ωk , k = 1, , N are arbitrary, with Ωk and 1Ωk are the same as in Subsection 3.3.1 Theorem 3.3.2 Let d = Given f ∈ H and u∗ as in Theorem 3.2.1 Let µ and h be positive parameters satisfying 2µ max{c2 , c22 }h2 108c40 |f |4 4c20 |f |2 + ≤ ν and µ ≥ ν λ31 ν λ1 54c40 |f |4 2+ , ν λ31 (3.31) and let Ih satisfy (3.30) Then for any u0 ∈ D(A), the unique corresponding solution u to system (3.12) satisfies for any T > 0, u ∈ C([0, T ]; D(A)) and du ∈ L2 (0, T ; D(A)), dt (3.32) and for all t > 0, ∥u(t) − u∗ ∥2 + α2 |A(u(t) − u∗ )|2 ≤ ∥u0 − u∗ ∥2 + α2 |A(u0 − u∗ )|2 e− νd0 t (3.33) Remark 3.3.3 In the case d = 3, we not have the property b(z, z, Az) = as in the case d = 2, so the above proof of Theorem 3.3.2 is no longer valid The stabilization question is still open in this case 14 Chapter ASYMPTOTIC BEHAVIOR OF THREE-DIMENSIONAL NON-HOMOGENEOUS NAVIER-STOKES-VOIGT EQUATIONS In this chapter we study the existence and long-time behavior of solutions to 3D NavierStokes-Voigt equations with non-homogeneous Dirichlet boundary conditions Firstly, we prove the existence and uniqueness of global weak solutions to this system Secondly, we prove that the dynamical system generated by weak solutions has a finite-dimensional global attractor with an explicit upper bound of fractal dimension Next, we study the existence and stability of stationary solutions Finally, we show that if there exists an operator which projects the solutions onto a finite-dimensional space and it satisfies a suitable approximation inequality, then the long-time behavior of the solutions is determined The content of this chapter is based on the work [CT3] in the List of Publications, which has been submitted for publication 4.1 Problem setting Let Ω be a bounded domain in R3 with boundary ∂Ω of class C We consider the 3D Navier-Stokes-Voigt equations   ut − α2 ∆ut − ν∆u + (u · ∇)u + ∇p = f  ∇ · u = in Ω × (0, ∞), (4.1) in Ω × (0, ∞), 15 with non-homogeneous Dirichlet boundary conditions u(x, t) = φ(x), x ∈ ∂Ω, t > 0, (4.2) u(x, 0) = u0 (x), x ∈ Ω (4.3) and initial condition Here we assume that ❼ f = f (x) is a time-independent density force per unit volume ❼ φ = φ(x) is a vector function defined on ∂Ω and satisfies some certain conditions More precisely, we assume that φ is given in the following slightly restrictive form φ = curl ζ, where ζ = (ζ1 , ζ2 , ζ3 ) ∈ H3 (Ω), and curl denotes the usual operator, i.e., curl ζ = (∂x2 ζ3 − ∂x3 ζ2 , ∂x3 ζ1 − ∂x1 ζ3 , ∂x1 ζ2 − ∂x2 ζ1 ), for ζ = (ζ1 , ζ2 , ζ3 ) ❼ The parameter α ∈ (0, 1] is a length scale parameter characterizing the elasticity of the fluid Remark 4.1.1 In this problem, we consider the function φ which has the above slightly restrictive form to use the results of Temam These results show that we can extend the given boundary data φ to a function ψ in the whole domain Ω satisfying some necessary conditions (see Lemma 4.3.2 below) It plays an essential role of our investigation In this chapter, we first use the Faedo-Galerkin approximation method to prove the existence and uniqueness of weak solutions to system (4.1)-(4.2)-(4.3) We then prove the existence of a global attractor for the semigroup generated by weak solutions of problem (4.1)-(4.2)-(4.3) Next, we show that the global attractor has finite fractal dimension After that, a sufficient condition for exponential stability of the weak stationary solution to the non-homogeneous Navier-Stokes-Voigt equations is given The last, we prove that if there exists a projection operator which satisfies some certain approximation inequalities, then the long-time behavior of solutions to 3D non-homogeneous Navier-Stokes-Voigt equations is determined 16 4.2 Preliminaries We consider the following function spaces H = u ∈ L2 (Ω) : ∇ · u = , V = {u ∈ H10 (Ω) : ∇ · u = 0}, which are endowed with their usual scalar products and norms which we denote (·, ·) and | · | for H; ((·, ·)) and ∥ · ∥ for V Denote by P the Helmholtz-Leray orthogonal projection in L2 (Ω) onto H The Stokes operator A is defined as A = −P ∆, D(A) = H2 (Ω) ∩ V We set B(u, v) = P ((u · ∇)v) From the bilinear operator B(·, ·), we can define the trilinear form b(u, v, w) = ⟨B(u, v), w⟩ satisfying some estimates 4.3 Existence and uniqueness a weak solution Definition 4.3.1 Let f ∈ L2 (Ω) A function u is called a weak solution of problem (4.1)(4.2)-(4.3) on the interval (0, T ) if the following conditions are satisfied    u ∈ C([0, T ]; H1 (Ω)), du/dt ∈ L2 (0, T ; H1 (Ω)),      d    (u(t) + α2 Au(t)) + νAu(t) + B(u(t), u(t)) = f in H−1 (Ω),   dt  ∇ · u = in Ω × (0, T ),       u(t) = φ on ∂Ω,       u(0) = u0 Lemma 4.3.2 Let Ω be a bounded domain in R3 with boundary ∂Ω of class C and let φ = curl ζ, where ζ = (ζ1 , ζ2 , ζ3 ) ∈ H3 (Ω), is a vector function defined on ∂Ω Then there exists some ψ ∈ H2 (Ω) such that ∇ · ψ = 0, (4.11) ψ = φ on ∂Ω, and |b(v, ψ, v)| ≤ ν ∥v∥2 , ∀v ∈ V (4.12) Theorem 4.3.3 Let f ∈ L2 (Ω), u0 ∈ H1 (Ω), ∇ · u0 = in Ω, and u0 = φ on ∂Ω Then there exists a unique weak solution u to problem (4.1)-(4.2)-(4.3) on every interval (0, T ) Moreover, for all t > 0, the mapping u0 → u(t) is continuous on H1 (Ω) 17 Remark 4.3.4 Let ψ ∈ H2 (Ω) be a fixed function satisfying conditions (4.11)-(4.12) From the proof of Theorem 4.3.3 we see that if z ∈ C([0, T ]; V ) is a weak solution of problem   zt + α2 Azt + νAz + B(z, z) + B(z, ψ) + B(ψ, z) = fˆ, (4.21)  z(0) = z0 , where z0 = u0 − ψ ∈ V and fˆ = f + ν∆ψ − (ψ · ∇)ψ, then u := ψ + z ∈ C([0, T ]; H1 (Ω)) is a weak solution of problem (4.1)-(4.2)-(4.3) with initial datum u0 Therefore, if there exists a global attractor Aα in V for the continuous semigroup generated by weak solutions z of problem (4.21), then ψ + Aα := {ψ + z : z ∈ Aα } is a global attractor in H1 (Ω) for the continuous semigroup generated by weak solutions u of problem (4.1)-(4.2)-(4.3) Because of the above reason, from now on we only need to study the existence, finiteness of fractal dimension of global attractor Aα of the semigroup Sα (t) : V → V z0 → z(t) generated by weak solutions z of problem (4.21) Then we will get the corresponding results for the global attractor ψ + Aα of the continuous semigroup generated by weak solutions u of problem (4.1)-(4.2)-(4.3) 4.4 Existence of a global attractor In this section, we will prove the existence and regularity of the global attractor Aα for the semigroup Sα (t) generated by weak solutions z of problem (4.21) 4.4.1 Existence of an absorbing set We first show that the trajectories originating from any given bounded set eventually fall (uniformly in time) into a bounded absorbing set B1α , where B1α = 4(1 + λ1 α2 )|fˆ|2 , z ∈ V : |z| + α ∥z∥ ≤ ν λ21 2 with fˆ = f + ν∆ψ − (ψ · ∇)ψ Proposition 4.4.1 The set B1α is a bounded absorbing set for Sα (t) 18 4.4.2 The asymptotic compactness We now prove the asymptotic compactness of the semigroup Sα (t) Step 1: (Splitting the solution) We have sup sup |Sα (t)z0 |2 + α2 ∥Sα (t)z0 ∥2 ≤ M12 , (4.25) t≥0 z0 ∈B1α where M1 = 4(1 + λ1 α2 )|fˆ|2 ν λ21 1/2 (4.26) For z0 ∈ B1α , we split the solution as Sα (t)z0 = Lα (t)z0 + Kα (t)z0 , where Lα (t) is the semigroup generated by the problem   vt + α2 Avt + νAv + B(z, v) + B(ψ, v) = 0, (4.27)  v(0) = z0 , and w(t) = Kα (t)z0 is the solution to the problem   wt + α2 Awt + νAw + B(z, w) + B(z, ψ) + B(ψ, w) = fˆ, (4.28)  w(0) = 0, where fˆ = f + ν∆ψ − (ψ · ∇)ψ Step 2: (The exponential decay of the solution operator Lα (t)) Multiplying the first equation of system (4.27) by v, we have 1d (|v|2 + α2 ∥v∥2 ) + ν∥v∥2 + b(z, v, v) + b(ψ, v, v) = dt Hence, d 2νλ1 (|v|2 + α2 ∥v∥2 ) + (|v|2 + α2 ∥v∥2 ) ≤ dt + λ1 α2 Applying the Gronwall lemma we obtain − |v|2 + α2 ∥v∥2 ≤ (|z0 |2 + α2 ∥z0 ∥2 )e 2νλ1 t 1+λ1 α2 , for any t ≥ (4.29) Step 3: (The component related to Kα (t)u0 belongs to a compact subset of V ) Next we show that, for every fixed time, the component related to Kα (t)z0 belongs to a compact subset of V , uniformly as the initial data z0 belongs to the absorbing set B1α Lemma 4.4.2 For every α ∈ (0, 1] and z0 ∈ B1α , we have the estimate sup (∥w∥2 + α2 |Aw|2 ) ≤ rα , t≥0 19 where rα = + λ1 α2 νλ1 ˆ 2cM14 M12 4c2 (M12 + M12 ) |f | + + ∥ψ∥H2 (Ω) ∥ψ∥1 , ν α6 ν να2 with M1 is defined in (4.26) and 2(1 + λ1 α2 )|fˆ|2 c2 M12 ∥ψ∥21 (1 + λ1 α2 ) + M1 = 3/2 ν λ21 λ1 ν α2 1/2 We now define B2α (rα ) = z ∈ H2 (Ω) : ∥z∥2 + α2 |Az|2 ≤ rα ∩ B1α From (4.29) and Lemma 4.4.2, we obtain the following result Theorem 4.4.3 The set B2α (rα ) is a compact exponentially attracting set for the semigroup Sα (t), namely distV (Sα (t)B1α , B2α (rα )) ≤ M e − νλ1 t 2(1+λ1 α2 ) , where distV denotes the Hausdorff semi-distance in V ; M = sup ∥z0 ∥ z0 ∈B1α Theorem 4.4.4 Let α ∈ (0, 1] Then there exists a global attractor Aα in V for the semigroup Sα (t) generated by weak solutions of problem (4.21) Moreover, Aα ⊂ B2α (rα ), and it is therefore bounded in H2 (Ω) 4.5 Fractal dimension estimate of the global attractor We now establish an estimate for the fractal dimension in V for the global attractor Aα Theorem 4.5.1 For every α ∈ (0, 1], the global attractor Aα has a finite fractal dimension in V More precisely, dimV Aα ≤ c α3 1 + λ1 α ˆ + |f | λ1 ν λ41 3/2 , (4.31) where c > is a dimensionless scale invariant constant and fˆ = f + ν∆ψ − (ψ · ∇)ψ 4.6 Existence and exponential stability of a stationary solution 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 (Ω) 20 (4.38) Theorem 4.6.2 Let f ∈ L2 (Ω) Then, there exists at least a solution u∗ of problem (4.38) satisfying ∥u∗ ∥1 ≤ 1/2 3νλ1 |fˆ| + ∥ψ∥1 , (4.39) 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 4c c 3νλ1 λ1 ν> |fˆ| + 5/4 3/4 ∥ψ∥1 , then for any solution u(t) of problem (4.1)-(4.2)-(4.3), the following inequality holds for all t>0 |u(t) − u∗ |2 + α2 ∥u(t) − u∗ ∥2 ≤ (|u0 − u∗ |2 + α2 ∥u0 − u∗ ∥2 )× × exp − 2λ1 + λ1 α ν− 4c c 3νλ1 λ1 |fˆ| − 5/4 3/4 ∥ψ∥1 t , (4.40) that is, the stationary solution u∗ is globally exponentially stable 4.7 Determining projections and functionals for weak solutions Definition 4.7.1 Let u, v ∈ L∞ (0, ∞; H1 (Ω)) be weak solutions to the two following NavierStokes-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 ),    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→∞ 21 (4.44) The projection operator RN : H1 (Ω) → VN ⊂ L2 (Ω), N = dim(VN ) < ∞, is called a determining projection for weak solutions of 3D Navier-Stokes-Voigt equations if lim |RN (u(t) − v(t))| = 0, (4.45) t→∞ 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→∞ it implies that (4.45) holds, then we can say that the set {li }N i=1 forms a set of determining functionals We see that the basis {ϕi }N 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, N modes and volumes by making particular choices for the sets {ϕi }N i=1 and {li }i=1 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)) 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 > √ 1/2 24 6c2 C1 (1 + λ1 α2 )λ1 Gr02 α2 θ then lim (|u(t) − v(t)|2 + α2 ∥u(t) − v(t)∥2 ) = t→∞ 22 , (4.48) CONCLUSIONS AND FUTURE WORKS Conclusions In this dissertation, we have been investigated the long-time behavior of solutions to threedimensional 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: Give bounds on the number of determining nodes for solutions to the three-dimensional Navier-Stokes-Voigt equations with periodic boundary conditions Prove the stabilization of stationary solutions to the Navier-Stokes-Voigt equations with periodic boundary conditions by finite-dimensional feedback controls Prove the existence and long-time behavior of solutions to the three-dimensional NavierStokes-Voigt equations with non-homogeneous Dirichlet boundary conditions in terms of the existence of a finite-dimensional global attractor and the existence of determining projections Future works Approximate internal manifold for the three-dimensional Navier-Stokes-Voigt equations Data assimilation for three-dimensional Navier-Stokes-Voigt equations 23 LIST OF PUBLICATIONS [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-Stokes-Voigt equations by finite determining parameters Acta Math Vietnam 45 (2020), no 4, 917-930 [CT3] C.T Anh and N.T Ngan (2020), Asymptotic behavior of three-dimensional non-homogeneous Navier-Stokes-Voigt equations, submitted to Acta Applicandae Mathematicae ... to Navier- Stokes- Voigt 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. .. CONTROL OF NAVIER- STOKES- VOIGT EQUATIONS BY FINITE DETERMINING PARAMETERS In this chapter we study the stabilization of stationary solutions to Navier- Stokes- Voigt equations by finite- dimensional. .. so the Navier- Stokes- Voigt system behaves like a damped hyperbolic system Note that when α = 0, we recover the NavierStokes equations of motion The Navier- Stokes- Voigt equations has attractive