Tài liệu Một số bài toán điều khiển tối ưu đối với hệ phương trình navier stokes voigt

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRAN MINH NGUYET SOME OPTIMAL CONTROL PROBLEMS FOR 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 - 2019 COMMITTAL IN THE DISSERTATION I assure that my scientific results 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. December 10, 2019 Author Tran Minh Nguyet 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.Dr. Cung The Anh. First and foremost, I would like to express my deep gratefulness to Prof.Dr. Cung The Anh for his careful, patient and effective supervision. I am very lucky to have a chance to study with him. He is an excellent researcher. I would like to thank Assoc.Prof.Dr. Tran Dinh Ke for his help during the time I studied at Department of Mathematics and Informatics, Hanoi National University of Education. I would also like to thank all the lecturers and PhD students at the seminar of Division of Mathematical Analysis for their encouragement and valuable comments. A very special gratitude goes to Thang Long University for providing me the funding during the time I studied in the doctoral program. Many thanks are also due to my colleagues at Division of Mathematics, Thang Long University, who always encourage me to overcome difficulties during my period of study. Last but not least, I am grateful to my parents, my husband, my brother, and my beloved daughters for their love and support. Hanoi, December 10, 2019 Tran Minh Nguyet ii CONTENTS . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS LIST OF SYMBOLS INTRODUCTION Chapter 1. 1.1 PRELIMINARIES AND AUXILIARY RESULTS 7 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Regularities of boundaries . . . . . . . . . . . . . . . . . . . 7 1.1.2 Lp and Sobolev spaces . . . . . . . . . . . . . . . . . . . . . 7 1.1.3 Solenoidal function spaces . . . . . . . . . . . . . . . . . . . 11 1.1.4 Spaces of abstract functions . . . . . . . . . . . . . . . . . . 12 1.1.5 Some useful inequalities . . . . . . . . . . . . . . . . . . . . 13 1.2 Continuous and compact imbeddings . . . . . . . . . . . . . . . . . 14 1.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 The nonstationary 3D Navier-Stokes-Voigt equations . . . . . . . 20 1.4.1 Solvability of the 3D Navier-Stokes-Voigt equations with homogeneous boundary conditions . . . . . . . . . . . . . . 21 1.4.2 1.5 Some auxiliary results on linearized equations . . . . . . . 22 Some definitions in Convex Analysis . . . . . . . . . . . . . . . . . 25 Chapter 2. A DISTRIBUTED OPTIMAL CONTROL PROBLEM 26 2.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Existence of optimal solutions . . . . . . . . . . . . . . . . . . . . . 28 2.3 First-order necessary optimality conditions . . . . . . . . . . . . . 32 2.4 Second-order sufficient optimality conditions . . . . . . . . . . . . 41 Chapter 3. A TIME OPTIMAL CONTROL PROBLEM 47 3.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Existence of optimal solutions . . . . . . . . . . . . . . . . . . . . . 49 3.3 First-order necessary optimality conditions . . . . . . . . . . . . . 52 3.4 Second-order sufficient optimality conditions . . . . . . . . . . . . 59 iii Chapter 4. AN OPTIMAL BOUNDARY CONTROL PROBLEM 67 4.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Solvability of the 3D Navier-Stokes-Voigt equations with nonhomogeneous boundary conditions . . . . . . . . . . . . . . . . . . . . 69 4.3 Existence of optimal solutions . . . . . . . . . . . . . . . . . . . . . 75 4.4 First-order and second-order necessary optimality conditions . . . 77 4.5 4.4.1 First-order necessary optimality conditions . . . . . . . . . 77 4.4.2 Second-order necessary optimality conditions . . . . . . . . 81 Second-order sufficient optimality conditions . . . . . . . . . . . . 84 CONCLUSION AND FUTURE WORK LIST OF PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 iv LIST OF SYMBOLS R the set of real numbers R+ the set of positive real numbers Rn n-dimensional Euclidean vector space A := B A is defined by B Ā the closure of the set A (., .)X scalar product in the Hilbert space X kxkX norm of x in the space X X′ the dual space of the space X hx′ , xiX ′ ,X duality pairing between x′ ∈ X ′ and x ∈ X X ,→ Y X is imbedded in Y Lp (Ω) the space of Lebesgue measurable functions f R such that Ω |f (x)|p dx < +∞ L20 (Ω) the space of functions f ∈ L2 (Ω) such that R Ω L∞ (Ω) f (x)dx = 0 the space of almost everywhere bounded functions on Ω C0∞ (Ω) the space of infinitely differentiable functions with compact support in Ω C(Ω̄)    W m,p (Ω),       H m (Ω),   H m (Ω), 0      H s (Ω),     H s (Γ) the space of continuous functions on Ω̄ Sobolev spaces H −m (Ω) the dual space of H0m (Ω) H −s (Γ) the dual space of H s (Γ) L2 (Ω) L2 (Ω) × L2 (Ω) × L2 (Ω) (analogously applied for all other kinds of spaces) 1 (., .) the scalar product in L2 (Ω) ((., .)) the scalar product in H10 (Ω) ((., .))1 the scalar product in H1 (Ω) |.| the norm in L2 (Ω) k.k the norm in H10 (Ω) k.k1 the norm in H1 (Ω) x·y the scalar product between x, y ∈ Rn ∇ ( ∂x∂ 1 , ∂x∂ 2 , · · · , ∂x∂ n ) ∇y ∂y ∂y ∂y ( ∂x , , · · · , ∂x ) 1 ∂x2 n y·∇ y1 ∂x∂ 1 + y2 ∂x∂ 2 + · · · + yn ∂x∂ n ∇ · y , div y ∂y1 ∂x1 V {y ∈ H, V the closures of V in L2 (Ω) and H10 (Ω) ∂y2 ∂yn ∂x2 + · · · + ∂xn C∞ 0 (Ω) : div y = + 0} Lp (0, T ; X), 1 < p < ∞ the space of functions f : [0, T ] → X such that L∞ (0, T ; X) RT 0 kf (t)kpX dt < ∞ the space of functions f : [0, T ] → X such that kf (.)kX is almost everywhere bounded on [0, T ] W 1,p (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 NU (u) the normal cone of U at the point u TU (u) the polar cone of tangents of U at u i.e. id est (that is) a.e. almost every p. 5 page 5 2D two-dimensional 3D three-dimensional 2 The proof is complete 2 INTRODUCTION 1. Literature survey and motivation The Navier-Stokes-Voigt (sometimes written Voight) equations was first introduced by Oskolkov in [57] as a model of motion of certain linear viscoelastic incompressible fluids. This system was also proposed by Cao, Lunasin and Titi in [12] as a regularization, for small values of α, of the three-dimensional Navier-Stokes equations for the sake of direct numerical simulations. In fact, the Navier-Stokes-Voigt system belongs to the so-called α-models in fluid mechanics (see e.g. [38]), but it has attractive advantages over other α-models in that one does not need to impose any additional artificial boundary conditions (except the Dirichlet boundary conditions) to get the global well-posedness. We also refer the interested reader to [21] for some interesting applications of NavierStokes-Voigt equations in image inpainting. In the past years, the existence and long-time behavior of solutions to the Navier-Stokes-Voigt equations has attracted the attention of many mathematicians. In bounded domains or unbounded domains satisfying the Poincaré inequality, there are many results on the existence and long-time behavior of solutions in terms of existence of attractors for the Navier-Stokes-Voigt equations, see e.g. [3, 18, 19, 31, 41, 42, 60, 74]. In the whole space R3 , the existence and decay rates of solutions have been studied recently in [4, 56, 75]. The optimal control theory has been developed rapidly in the past few decades and becomes an important and separate field of applied mathematics. The optimal control of ordinary differential equations is of interest for its applications in many fileds such as aviation and space technology, robotics and the control of chemical processes. However, in many situations, the processes to be optimized may not be modeled by ordinary differential equations, instead partial differential equations are used. For example, heat conduction, diffusion, electromagnetic waves, fluid flows can be modeled by partial differential equations. In particular, optimal control of partial differential equations in fluid mechanics was first studied in 1980s by Fursikov when he established several theorems about the existence of solutions to some optimal control problems governed by Navier-Stokes equations (see [25, 26, 27]). 3 One of the most important objectives of optimal control theory is to obtain necessary (or possibly necessary and sufficient) conditions for the control to be an extremum. Since the pioneering work [1] of Abergel and Temam in 1990, where the first optimality conditions to the optimal control problem for fluid flows can be found, this matter has been studied very intensively by many authors, and in various research directions such as distributed optimal control, time optimal control, boundary optimal control and sparse optimal control. Let us briefly review some results on optimality conditions of optimal control problems governed by Navier-Stokes equations that is one of the most important equations in fluid mechanics. For distributed control problems, this matter was studied in [23, 33, 36, 68]. These works are all in the case of absence of state constraints. In the case of the presence of state constraints, the problem was investigated by Wang [71] and Liu [52]. The time optimal control problem of Navier-Stokes equations was investigated by Barbu in [7] and Fernandez-Cara in [24]. Optimal boundary control problems of the Navier-Stokes equations have been studied by many authors, see for instance, [32, 39, 40, 61] in the stationary case, and [10, 17, 28, 29, 34, 37] in the nonstationary case. One interesting result about Pontryagin’s principle for optimal control problem governed by 3D NavierStokes equations is introduced by B.T. Kien, A. Rösch and D. Wachsmuth in [43]. We can see also the habilitation [35], the theses [69], [63] and references therein, for other works on optimal control of Navier-Stokes equations. As described above, the unique existence and long-time behavior of solutions to the Navier-Stokes-Voigt equations, as well as the optimal control problems for fluid flows, in particular for Navier-Stokes equations, have been considered by many mathematicians. However, to the best of our knowledge, the optimal control of three-dimensional Navier-Stokes-Voigt equations has not been studied before. This is our motivation to choose the topic ”Some optimal control problems for Navier-Stokes-Voigt equations”. Because of the physical and practical significance, one only considers Navier-Stokes-Voigt equations in the case of three or two dimensions. The thesis presents results on some optimal control problems for this equations in the three-dimensional space (the most physically meaningful case). However, all results of the thesis are still true in the two-dimensional one (with very similar statements of results and corresponding proofs). Namely, we will study the following problems: (P1) The distributed optimal control problem of the nonstationary three di- 4 mensional Navier-Stokes-Voigt equations, where the objective functional is of quadratic form and the distributed control belongs to a non-empty, closed, convex subset, (P2) The time optimal control problem of the nonstationary three dimensional Navier-Stokes-Voigt equations, where the set of admissible controls is an arbitrary non-empty, closed, convex subset, (P3) The boundary optimal control problem of the nonstationary three dimensional Navier-Stokes-Voigt equations, where the objective functional is of quadratic form and the boundary control variable has to satisfy some compatibility conditions. 2. Objectives The objectives of this dissertation are to prove the existence of optimal solutions and to give the necessary and sufficient optimality conditions for problems (P1), (P2), (P3), namely, (i) to show the existence of optimal solutions and to establish the first-order necessary and the second-order sufficient optimality conditions for problem (P1). (ii) to prove the existence of optimal solutions and to derive the first-order necessary and the second-order sufficient optimality conditions for problem (P2). (iii) to get the existence of optimal solutions and to give the first-order and second-order necessary, the second-order sufficient optimality conditions for problem (P3). 3. The structure and results of the dissertation The dissertation has four chapters and a list of references. Chapter 1 collects several basic concepts and facts on Sobolev spaces and partial differential equations associated with solutions of Navier-Stokes-Voigt equations as well as some auxiliary results. Chapter 2 presents results on the distributed optimal control problem governed by Navier-Stokes-Voigt equations. Chapter 3 provides results on the time optimal control problem governed by Navier-Stokes-Voigt equations. Chapter 4 presents results on the boundary optimal control problem governed 5 by Navier-Stokes-Voigt equations. The results obtained in Chapters 2, 3 and 4 are answers for problems (P1), (P2), (P3), respectively. Chapter 2 and Chapter 3 are based on the papers [CT1], [CT2] in the List of Publications which were published in the journals Numerical Functional Analysis and Optimization and Applied Mathematics and Optimization, respectively. The results of Chapter 4 is the content of the work [CT3] in the List of Publications, which has been submitted for publication. These results have been presented at: • Mini-workshop ”Partial Differential Equations: Analysis and Numerics”, September 2019, Vietnam Institute for Advanced Studies in Mathematics, Ha Noi. • ”Vietnam-Korea Joint Conference on Selected Topics in Mathematics”, February 2017, Da Nang, Vietnam. • The 14th Workshop on Optimization and Scientific Computation, April 2016, Ba Vi, Ha Noi. • Seminar at Vietnam Institute for Advanced Studies in Mathematics. • Seminar of Division of Mathematical Analysis at Hanoi National University of Education. 6 Chapter 1 PRELIMINARIES AND AUXILIARY RESULTS In this chapter, we review some basic concepts and results on function spaces, imbeddings, operators, Navier-Stokes-Voigt equations and present some auxiliary results on linearized equations. 1.1 1.1.1 Function spaces Regularities of boundaries Let Ω be an open bounded set in Rn . Its boundary is denoted by Γ. The outward normal vector of the boundary is denoted by n. We will need some smoothness properties of Γ. In some situations, it is sufficient to assume that Γ is locally Lipschitz, i.e. each point x on the boundary Γ has a neighborhood Ux such that Γ∩Ux is the graph of a Lipschitz continuous function, see e.g. [2, Subsection 4.5]. However, this smoothness might be insufficient in some other cases. Sometimes, we will need that Ω is of class C m , where m ∈ Z+ or m = ∞. This means that the boundary Γ is a (n−1)-dimensional manifold of class C m and Ω is locally located on one side of Γ (see [66, p. 2] and [51, p. 34]). It is clear that the latter property implies the first one. 1.1.2 Lp and Sobolev spaces Let Ω be a nonempty and Lebesgue measurable subset of Rn . We denote by Lp (Ω), 1 ≤ p < ∞, the space of real-valued functions defined on Ω whose p−th power is integrable for the Lebesgue measure dx. It is a Banach space endowed with the norm Z 1/p kukLp (Ω) = |u(x)|p dx Ω 7 . For p = 2, L2 (Ω) is a Hilbert space with the scalar product Z (u, v)L2 (Ω) = u(x)v(x)dx. Ω We denote by L20 (Ω) the space of all functions in L2 (Ω) which have average of  zero on Ω L20 (Ω) :=  Z f ∈ L (Ω) : 2 f (x)dx = 0 . Ω For a multiindex α := (α1 , α2 , ..., αn ) ∈ Nn , we set |α| = Pn i=1 αi and ∂ |α| . ∂xα1 1 ∂xα2 2 · · · ∂xαnn Dα = For m ∈ Z+ , p ∈ [1, ∞), we define the Sobolev space W m,p (Ω) to be the space of all functions whose weak derivatives up to order m are functions in Lp (Ω). The norm in W m,p (Ω) is defined by  kukW m,p (Ω) :=  1/p X kDα ukpLp (Ω)  . 0≤|α|≤m The following theorem gives a basic property of this space. Theorem 1.1.1. [2, Theorem 3.2 and Theorem 3.5] W m,p (Ω) is a separable Banach space if 1 ≤ p < ∞. In the case p = 2, the space H m (Ω) := W m,2 (Ω) is a separable Hilbert space with scalar product (u, v)H m (Ω) = X (Dα u, Dα v)L2 (Ω) . 0≤|α|≤m We now recall an extension theorem valid for Sobolev spaces on bounded domains. Lemma 1.1.2. [62, Theorem 5.20] If Ω is a bounded domain of class C k in Rn , then for each open set Ω′ ⊃ Ω̄ there exists a bounded linear extension operator E such that if u ∈ H k (Ω) then Eu ∈ H0k (Ω′ ) and kEukH k (Ω′ ) ≤ Ck,Ω′ kukH k (Ω) . (1.1) In fact, we have (1.1) for each H j with 0 ≤ j ≤ k. Let C0∞ (Ω) be the space of all C ∞ -functions with compact support in Ω. The closure of C0∞ (Ω) in H m (Ω) is denoted by H0m (Ω), and we use the notation H −m (Ω) for the dual space of H0m (Ω). 8 Next, we introduce the space H s (Ω) when s is a positive real number (see [2, Section 7.42 - 7.48]). Assume that Ω is the whole space Rn or an open bounded subset of class C 1 . Let s be a positive real number that is not an integer. Then we can write s = m + σ where m ∈ N and 0 < σ < 1. We denote by H s (Ω) the space consisting of functions in H m (Ω) for which the norm  1/2   X Z Z |Dα u(x) − Dα u(y)|2 kukH s (Ω) = kuk2H m (Ω) + dxdy , |x − y|n+2σ   Ω Ω |α|=m is finite. With this norm, H s (Ω) is a Hilbert space. Now, we recall the definition of integral of a function defined on the boundary (see [20, p. 143-146]). We assume that Ω is an open bounded set with C ∞ boundary Γ. Then there exists a finite family of bounded open sets {Uj }rj=1 covering Γ and a corresponding family of diffeomorphisms {φj }rj=1 mapping Uj onto the set B = {y ∈ Rn : y = (y ′ , yn ), |y ′ | < 1, −1 < yn < 1} with φj (Uj ∩ Ω) = B+ := {y = (y ′ , yn ) ∈ B, yn > 0}, and so φj (Uj ∩ Γ) = B0 := {y = (y ′ , yn ) ∈ B, yn = 0}. If f is a function having the support in Uj , we define Z Z Z f (x)ds = ′ ′ ′ f ◦ φ−1 j (y , 0)Jj (y )dy . f (x)ds =: Uj ∩Γ Γ B0 Here, Jj (y ′ ) = n X Jjk (y ′ )
2 !1/2 , k=1 with  Jjk (y ′ ) = det ∂(x1 , ..., xk−1 , xk+1 , ..., xn )   ∂(y1 , ..., yn−1 ) yn =0 ! , and x = φ−1 j (y). For an arbitrary function f defined on Γ, we set Z f (x)ds = Γ r Z X j=1 f (x)ωj (x)ds, Γ where {ωj } is a partition of unity for Γ subordinate to {Uj }. This definition depends neither on the mapping φj nor the partition of unity ωj considered. In 9 fact, this definition is valid in the case that Γ is locally Lipschitz (see [55, p. 120]). For s ∈ R, s ≥ 0, we are going to define the Sobolev spaces H s (Γ) (see [20, p. 143-146]). Assume that Γ is of class C m (m ≥ s). Let u be a function on Γ, we define θj u on Rn−1 by ′ θj u(y ) =  (ωj u)(φ−1 (y ′ , 0)) if |y ′ | < 1, 0 otherwise. j We denote by H s (Γ) the space of functions defined on Γ such that θj u ∈ H s (Rn−1 ), j = 1, . . . , r. Endowed with the norm kukH s (Γ) = ( r X )1/2 kθj uk2H s (Rn−1 ) , (1.2) j=1 the space H s (Γ) is a Hilbert space. One can prove that the definition of H s (Γ) is independent of the choice of Uj , φj , ωj . The norm (1.2) depends on system {Uj , φj , ωj }, however, one can prove that these different norms are equivalent. We denote by H −s (Γ) the dual spaces of H s (Γ). In the sequel, we will be concerned with vector functions in three-dimensional space. For convenience, we will make use of the notation L2 (Ω) := L2 (Ω)3 = L2 (Ω) × L2 (Ω) × L2 (Ω), which is analogously applied for all other kinds of spaces. These product spaces are equipped with the usual product norm, except the case C∞ 0 (Ω) that is not a normed space at all. We will use the notations (., .), ((., .)), ((., .))1 and |.|, k.k, k.k1 to denote the scalar products and corresponding norms in the spaces L2 (Ω), H10 (Ω), H1 (Ω) respectively, namely Z X 3 (u, v) := Ω ((u, v)) := j=1 Z X 3 Ω uj vj dx, u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ L2 (Ω), ∇uj · ∇vj dx, u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ H10 (Ω), j=1 ((u, v))1 = (u, v) + ((u, v)), u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ) ∈ H1 (Ω), 10 |u| := p (u, u), kuk := p ((u, u)) and kuk1 := p ((u, u))1 . By Poincaré’s inequality, we see that two norms k.k and k.k1 are equivalent in the space H10 (Ω). We have the following theorem, which is called a trace theorem. Theorem 1.1.3. [30, Theorem 1.5.1.2] Let Ω be a bounded open subset with locally Lipschitz boundary Γ and s ∈ (1/2, 1] be a real number. Then the mapping T : u → u|Γ which is defined for every Lipschitz continuous function u on Ω̄, has a unique continuous extension as an operator from H s (Ω) onto H s−1/2 (Γ), which is still denoted by T. This operator has a right continuous inverse. The right inverse operator in the theorem above is called a lifting operator. If Ω is sufficiently smooth, of class C 1 , for example, then the unit outer normal n to Γ is well defined and continuous. However, we sometimes need less regularity on Ω but still would like to have n well-defined. In this regard, we have the following result. Lemma 1.1.4. [55, Chapter II, Lemma 4.2] If Γ is locally Lipschitz then the unit outer normal n exists almost everywhere on Γ. We have the following Green’s formula on the relationship between integrals on the domain Ω and integrals on its boundary Γ. Theorem 1.1.5. [30, Theorem 1.5.3.1] Let Ω be a bounded open subset in Rn with locally Lipschitz boundary Γ. Then, for every u, v ∈ H 1 (Ω) we have Z Z Z ∂u ∂v vdx + u dx = T(u)T(v)ni ds, Ω ∂xi Ω ∂xi Γ where ni denotes the ith complement of the unit outward normal vector n to Γ and T is the trace operator. 1.1.3 Solenoidal function spaces We will usually work with functions that satisfy the constraint div y = 0. Let V be the space V = {y ∈ C∞ 0 (Ω) : div y = 0}. The closure of V in the space L2 (Ω) and H10 (Ω) are denoted by H and V , which are Hilbert spaces with scalar products (., .) and ((., .)), respectively. The space V has the following useful characterization. 11 Theorem 1.1.6. [66, Theorem I.1.6] Let Ω be an open Lipschitz domain in R3 . Then it holds V = {u ∈ H10 (Ω) : div u = 0}. We will use notation V ′ for the dual space of V and h., .iV ′ ,V for the duality pairing between V ′ and V . 1.1.4 Spaces of abstract functions Let T < ∞ be a given final time. We denote by Q the time-space cylinder Ω × (0, T ). We present here some spaces of Banach-space valued functions (see [67, p. 143] and [51, p. 6-7]). Let X be a real Banach space. We denote by Lp (0, T ; X), 1 ≤ p ≤ ∞, the standard Banach space of all functions from (0, T ) to X , endowed with the norm !1/p Z T kykLp (0,T ;X) := 0 ky(t)kpX dt , 1 ≤ p < ∞, kykL∞ (0,T ;X) := esssup ky(t)kX . t∈(0,T ) ′ The dual space of Lp (0, T ; X), 1 < p < ∞, is Lp (0, T ; X ′ ) with p′ := p/(p − 1) and X ′ is the dual space of X . In particular, if X is a Hilbert space then L2 (0, T ; X) is a Hilbert space with the following inner product Z T hu, viL2 (0,T ;X) = hu(t), v(t)iX dt. 0 In the sequel, we will identify the spaces L2 (0, T ; L2 (Ω)) and L2 (Q). We denote by C([0, T ]; X) the space of all continuous functions from [0, T ] to X . To deal with time derivatives in state equations, we need the following lemma. Lemma 1.1.7. [66, Lemma 1.1] Let X be a given Banach space with dual space X ′ and let u and g be two functions belonging to L1 (0, T ; X). Then, the following three conditions are equivalent. (i). The function u is a.e. equal to a primitive function of a function g , i.e. Z t u(t) = ξ + g(s)ds, ξ ∈ X, a.e. t ∈ [0, T ]. 0 (ii). For each test function φ ∈ C0∞ ((0, T )), Z T Z u(t)φ′ (t)dt = − 0 g(t)φ(t)dt. 0 12 T (iii). For each η ∈ X ′ , d hη, uiX ′ ,X = hη, giX ′ ,X , dt in the scalar distribution sense on (0, T ). Next, we introduce the common spaces of functions y whose time derivatives yt exist as abstract functions W 1,p (0, T ; X) := {y ∈ Lp (0, T ; X) : yt ∈ Lp (0, T ; X)}. Endowed with the norm  kykW 1,p (0,T ;X) := kyk2Lp (0,T ;X) + kyt k2Lp (0,T ;X) 1/2 , it is a Banach space. 1.1.5 Some useful inequalities ′ Lemma 1.1.8 (Hölder inequality). Assume that u ∈ Lp (Ω), v ∈ Lp (Ω) where p, p′ ∈ (1, ∞), 1/p + 1/p′ = 1. Then, uv ∈ L1 (Ω) and Z Z 1/p Z u(x)v(x)dx ≤ |u(x)| dx Ω p′ 1/p′ |v(x)| dx p Ω . Ω More general, we have the following lemma. Lemma 1.1.9 (General Hölder inequality). Let p1 , . . . . , pk ∈ (1, ∞) such that Pk −1 Qk pi 1 i=1 pi = 1. If ui ∈ L (Ω) for all i ∈ {1, 2, . . . , k} then i=1 ui ∈ L (Ω). Moreover, one has k Y ui i=1 L1 (Ω) ≤ k Y kui kLpi (Ω) . i=1 Lemma 1.1.10 (Poincaré’s inequality [62, Proposition 5.8]). If Ω is bounded in some direction then there exists a constant C depending only on Ω such that !1/2 n X ∂u 2 , ∀u ∈ H01 (Ω). kukL2 (Ω) ≤ C ∂xi 2 L (Ω) i=1 Lemma 1.1.11. [66, Chapter III, Lemma 3.5] For any open set Ω in R3 we have 1/4 3/4 kvkL4 (Ω) ≤ 21/2 kvkL2 (Ω) k∇vkL2 (Ω) , ∀v ∈ H01 (Ω). Lemma 1.1.12 (Gronwall’s inequality). Let η(.) be a nonnegative, absolutely continuous function on [0, T ], which satisfies for a.e. t ∈ [0, T ] the differential inequality η ′ (t) ≤ φ(t)η(t) + ψ(t), 13 where φ(t) and ψ(t) are nonnegative, integrable functions on [0, T ]. Then   Z t ∫ η(t) ≤ e t 0 ϕ(s)ds η(0) + ψ(s)ds 0 for all 0 ≤ t ≤ T . Lemma 1.1.13 (Young’s inequality with ). Let 1 < p, q < ∞, p1 + 1q = 1. Then, for every a, b,  > 0 we have ab ≤ ap + C()bq , where C() = (p)−q/p q −1 . 1.2 Continuous and compact imbeddings Theorem 1.2.1 (Rellich-Kondrachov theorem [2, Theorem 6.2]). Let Ω ⊂ Rn be a bounded Lipschitz domain and 1 ≤ p ≤ ∞. • If 1 ≤ p < n then W 1,p (Ω) is imbedded in Lq (Ω) ∀ 1 ≤ q ≤ imbedding is compact for 1 ≤ q < np n−p , and this np n−p . • If p = n then the imbedding W 1,p (Ω) ,→ Lq (Ω), q ∈ [1, ∞), is compact. • If p > n then the imbedding W 1,p (Ω) ,→ C(Ω) is compact. Let Ω be a bounded domain in R3 with locally Lipschitz boundary Γ. It follows from the above theorem that the imbedding H 1 (Ω) ,→ L2 (Ω) is compact. As a consequence, we get the compactness of the following imbeddings: H1 (Ω) ,→ L2 (Ω), (1.3) V ,→ L2 (Ω), (1.4) V ,→ H. Theorem 1.2.2 (Schauder’s theorem [73, p. 282]). Let X, Y be Banach spaces. A linear continuous operator T is compact iff its dual operator T ′ is compact. From (1.4) and Schauder’s theorem we deduce that the imbedding L2 (Ω) ,→ V ′ is compact. 14 (1.5) Theorem 1.2.3. [51, Theorem 16.1 and Remark 16.1] Assume that Ω is a bounded domain with locally Lipschitz boundary. Let s ∈ R+ . Then, for every  > 0, the injection H s (Ω) ,→ H s−ϵ (Ω) is compact. From this theorem and the continuity property of trace operators as well as lifting operators, we can deduce that the imbedding H 1/2 (Γ) ,→ L2 (Γ) is also compact, and so is the imbedding H1/2 (Γ) ,→ L2 (Γ). Let X be a Banach space. By the following proposition, every function in the space W 1,p (0, T ; X) is, up to changes on sets of zero measure, equivalent to a function of C([0, T ]; X), and the imbedding W 1,p (0, T ; X) ,→ C([0, T ]; X) is continuous. Proposition 1.2.4. [62, Proposition 7.1] Suppose that u ∈ W 1,p (0, T ; X), 1 ≤ p ≤ ∞. Then Z u(t) = u(s) + s t du (τ )dτ dt for every 0 ≤ s ≤ t ≤ T, and u ∈ C([0, T ]; X) (with the usual caveat). Furthermore, we have the estimate sup ku(t)kX ≤ CkukW 1,p (0,T ;X) . 0≤t≤T We denote by D([0, T ]; X) the space of all functions f : [0, T ] → X , which are infinitely differentiable and have compact support in (0, T ). We have the following theorem. Theorem 1.2.5. [51, Theorem 2.1] If X is a separable Hilbert space then D([0, T ]; X) is dense in W 1,2 (0, T ; X). We know that H 1 (Ω) is separable. Hence, it follows from the continuous and surjective property of the trace operator T : H 1 (Ω) → H 1/2 (Γ) that H 1/2 (Γ) is separable, and so is the space H1/2 (Γ). By the above theorem we imply that D([0, T ]; H1/2 (Γ)) is dense in W 1,2 (0, T ; H1/2 (Γ)). The following theorem is very useful later. 15 (1.6)