Summary of doctoral thesis in mathematics: Stability and stabilization for some evolution equations in fluid mechanics

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Purpose of thesis: Resear h thesis on the problem: The stability and stabilization of some evolution equations appear in fluid mechanics.

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Nội dung Text: Summary of doctoral thesis in mathematics: Stability and stabilization for some evolution equations in fluid mechanics

MINISTRY OF EDUCATION AND TRAINING HA NOI PEDAGOGICAL UNIVERSITY 2 NGUYEN VIET TUAN STABILITY AND STABILIZATION FOR SOME EVOLUTION EQUATIONS IN FLUID MECHANICS Spe iality: Mathemati al analysis Code: 9 46 01 02 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ha Noi - 2019
This thesis has been ompleted at the Ha Noi Pedagogi al Uni- versity 2 S ientifi Advisor: Asso .Prof. PhD. Cung The Anh Referee 1: Referee 2: Referee 3: The thesis shall be defended at the University level Thesis Assessment Coun il at Ha Noi Pedagogi al University 2 on....... The thesis an be found in the National Library and the Library of Ha Noi Pedagogi al University 2.
INTRODUCTION 1. MOTIVATION AND HISTORY OF THE PROBLEM Partial differential evolution equations appear frequently in the of physi al and biologi al pro esses, su h as heat transfer and diffusion, pro ess of wave transmission in fluid me hani s and pop- ulation models in biology. The study of this equations lass has important meaning in s ien e and te hnology. That is why it has attra ted widespread attention. After studying the well-posedness of the problem, it is im- portant to study the long-time behavior of solutions, as it allows us to understand and predi t the future dynami s, sin e we an make the appropriate adjustments to a hieve the desired results. An effe tive approa h is the study of the existen e and stability of the stationary solutions. In mathemati al, the stationary solu- tions response orresponds to the stationary state of the pro ess, and is the solution of the orresponding ellipti problem. When the stationary solutions of the pro esses is not stability, people try to stabilize it by using appropriate ontrols, or using appropriate random noise. In re ent years, stability and stabilization issues have been studied extensively for Navier-Stokes equations and some lasses of nonlinear paraboli equations. However, the orresponding re- sults for other lasses of equations in fluid me hani s and paraboli systems are still small. There are new mathemati al diffi ulties, be ause of the omplexity of the system or the intera tion be- tween nonlinear terms in the system. Therefore, this is a very urrent issue and attra ted widespread attention from domesti and international math s ientists. First, we onsider 3D Navier-Stokes-Voigt (sometimes written Voight) equations in smooth bounded domains with homogeneous 1
Diri hlet boundary onditions:   ut − ν∆u − α2 ∆ut + (u · ∇)u + ∇p = f in O × R+ ,  O × R+ ,  ∇ · u = 0 in (1)   u(x, t) = 0 on ∂O × R+ ,  u(x, 0) = u0 (x) in O.  In the last few years, mathemati al questions related to 3D Navier-Stokes-Voigt equations have attra ted the attention of a number of mathemati ians. The existen e and long-time behavior of solutions in terms of existen e of attra tors to the 3D Navier- Stokes-Voigt equations in domains that are bounded or unbounded but satisfying the Poin ar² inequality was investigated extensively in the works of C.T. Anh and P.T. Trang (2013), A.O. Celebi, V.K. Kalantarov and M. Polat (2009), J. Gar ½a-Luengo, P. Mar½n- Rubio and J. Real (2012). The de ay rate of solutions to the equations on the whole spa e was studied in the works of C.T. Anh and P.T. Trang (2016), C.J. Ni he (2016), C. Zhao and H. Zhu (2015). The main aim of this thesis to study the exponential stability and stabilization of strong stationary solutions to prob- lem (1). Next, we onsider the following 2D g-Navier-Stokes equations ∂u    − ν∆u + (u · ∇)u = ∇p + f in Ω × R+ ,    ∂t ∇ · (gu) = 0 Ω × R+ ,  in (2) +     u(x, t) = 0 on ∂Ω × R ,  u(x, 0) = u0 (x), in Ω.  In the past de ade, the existen e and long-time behavior of solutions in terms of existen e of attra tors for 2D g-Navier-Stokes equations have been studied extensively in both autonomous and non-autonomous ases (see e.g. C.T. Anh and D.T. Quyet (2012), J. Jiang, Y. Hou and X. Wang (2011), J. Jiang and X. Wang 2
(2013), H. Kwean and J. Roh (2005), D. Wu and J. Tao (2012), and referen es therein). However, there are still many open issues that need to be investigated regarding the system (2), su h as: 1) Existen e, uniqueness and exponential stability of strong stationary solutions. 2) Stabilization of strong stationary solutions. 3) Stabilization of long-time behavior of solutions. Finally, we onsider the following sto hasti 2D g-Navier-Stokes equations with finite delays     du = [ν∆u − (u · ∇)u − ∇p + f + F (u(t − ρ(t)))]dt  +G(u(t − ρ(t)))dW (t), x ∈ O, t > 0,     ∇ · (gu) = 0, x ∈ O, t > 0, (3)  u(x, t) = 0, x ∈ ∂O, t > 0,       u(x, t) = ϕ(x, t), x ∈ O, t ∈ [−τ, 0], The existen e and stability of stationary solutions to 2D Navier- Stokes equations with delays have been studied by many authors in re ent years, see for instan e, Caraballo and Han (2014, 2015), Caraballo and Real (2001, 2003), Chen (2012), Garrido-Atienza and Mar½n-Rubio (2006), Mar½n-Rubio, Real and Valero (2011), Wan and Zhou (2011). The existen e and stability of stationary solutions to the 2D g-Navier-Stokes equations without/with de- lays have been studied in re ent works (see C.T. Anh and D.T. Quyet (2012), D.T. Quyet (2014)). However, to the best of our knowledge, there is no result on the stability of solutions to prob- lem (3). 2. PURPOSE OF THESIS Resear h thesis on the problem: The stability and stabilization of some evolution equations appear in fluid me hani s. 3
3. OBJECT AND SCOPE OF THESIS • Resear h obje t: The stability and stabilization of some evo- lution equations appear in fluid me hani s, namely: three- dimensional Navier-Stokes-Voigt system, g-Navier-Stokes sys- tem, sto hasti 2D g-Navier-Stokes equations with finite de- lays. • Resear h s ope: ◦ Content 1: Three-dimensional Navier-Stokes-Voigt sys- tem. 1) Existen e, uniqueness and the exponential stability of strong stationary solutions. 2) Stabilization of strong stationary solutions by us- ing either an internal feedba k ontrol with sup- port large enough or a multipli ative noise of suf- fi ient intensity. ◦ Content 2: Two-dimensional g-Navier-Stokes system. 1) Existen e, uniqueness and the exponential stability of strong stationary solutions. 2) Stabilization of strong stationary solutions by us- ing either an internal feedba k ontrol with sup- port large enough or finite-dimensional feedba k ontrol. 3) Stabilization of long-time behavior of solutions un- der a tion of fast os illating-in-time external for es. ◦ Content 3: Sto hasti 2D g-Navier-Stokes equations with finite delays. 1) Existen e and uniqueness of weak stationary solu- tions to the deterministi system. 2) The exponential stability in mean square and al- most sure exponential stability of the weak solu- tions to the sto hasti equations. 4
4. RESEARCH METHODS • To study the existen e of solutions: Galerkin approximation, the ompa tness and energy methods. • To study the stability of stationary and ir ulating solutions: Energy ratings and Gronwall's inequality. • To study the stabilization problem: The methods of Mathe- mati al ontrol theory and Sto hasti analysis. 5. RESULTS OF THESIS The thesis a hieves the following main results: • Proving the uniqueness and exponential stability of strong stationary solutions; proving the onditions for stabilization of strong stationary solutions by feedba k ontrol with sup- port in the domain and by random noise for 3D Navier- Stokes-Voigt equations in bounded domains. These are the basi ontents of Chapter 2. • Proving the existen e, uniqueness and exponential stabil- ity of strong stationary solutions; proving the onditions for stabilization of strong stationary solutions by using an in- ternal feedba k ontrols with support in domain and finite dimensional feedba k ontrols; proving the onditions for stabilization of the long-time behavior under a tion of fast os illating-in-time external for es for 2D g-Navier-Stokes equa- tions in bounded domains. These are the basi ontents of Chapter 3. • Proving the existen e and uniqueness of weak stationary so- lutions to the deterministi system; the exponential mean square stability and almost sure exponential stability of the weak solution to the sto hasti 2D g-Navier-Stokes equa- tions with finite delays in bounded domains. These are the basi ontents of Chapter 4. 5
6. STRUCTURES OF THESIS Beside Introdu tion, Con lusion, Author's works related to the thesis and Referen es, the thesis in ludes 4 hapters: • Chapter 1. Preliminaries. • Chapter 2. Stabilization of 3D Navier-Stokes-Voigt equa- tions. • Chapter 3. Stabilization of 2D g-Navier-Stokes equations. • Chapter 4. The stability of solutions to sto hasti 2D g- Navier-Stokes equations with finite delays. 6
Chapter 1 PRELIMINARIES In this hapter, we re all some general on epts and results about the fun tion spa es, operators, sto hasti analysis, inequal- ities for the nonlinear term and some additional results (the usual inequalities, the ompa tness methods) to prove the main results of the thesis in the following hapters. 1.1. THE FUNCTION SPACES In this se tion, we repeat some of the results about the fun tion spa es that will be used in the thesis: Sobolev spa e p m m p (spa e L (O), spa e H (O), spa e H0 (O)), spa e L (0, T ; Y ) and C([0, T ]; Y ). In addition, we also present fun tion spa es H and V related to Navier-Stokes-Voigt equations; fun tion spa es Hg and Vg related to g-Navier-Stokes equations. 1.2. THE OPERATORS 1.2.1. Operators A, B We define the Stokes operator A:V →V′ by (Au, v) = ((u, v)), for all u, v ∈ V. We also define the operator B :V ×V →V′ by (B(u, v), w) = b(u, v, w), for all u, v, w ∈ V, 3 ∂vj Z P where b(u, v, w) = ui wj dx. i,j=1 ∂xi O Lemma 1.1. We have  1/4 3/4 1/4 3/4 c|u| kuk kvk|w| kwk , ∀u, v, w ∈ V,   |b(u, v, w)| ≤ cλ−1/4 kukkvkkwk, ∀u, v, w ∈ V,  ckukkvk1/2 |Av|1/2 |w|, ∀u ∈ V, v ∈ D(A), w ∈ H,  7
where c are appropriate onstants. 1.2.2. Operators Ag , Bg and Cg We define the operator Ag : Vg → Vg′ by hAg u, vig = ((u, v))g , ∀u, v ∈ Vg . We denote by η1 the first eigenvalue of the operator Ag . We also define the operator Bg : Vg × Vg → Vg′ by hBg (u, v), wig = bg (u, v, w), ∀u, v, w ∈ Vg , where 2 Z X ∂vj bg (u, v, w) = ui wj gdx. i,j=1 O ∂xi We onsider the operator Cg : V g → H g defined by ∇g ∇g (Cg u, v)g = (( · ∇)u, v)g = bg ( , u, v), ∀v ∈ Vg . g g Lemma 1.2. We have 1/2 1/2 1/2 1/2   c1 |u|g kukg kvkg |w|g kwkg ,  c |u|1/2 kuk1/2 kvk1/2 |A v|1/2 |w| ,  2 g g g g g g |bg (u, v, w)| ≤ 1/2 1/2   c3 |u|g |Ag u|g kvkg |w|g , 1/2 1/2  c4 |u|g kvkg |w|g |Ag w|g ,  where ci , i = 1, . . . , 4, are appropriate onstants. Lemma 1.3. Let u ∈ L2 (0, T ; Vg ), then the fun tion Cg u defined by ∇g ∇g (Cg u(t), v)g = (( · ∇)u, v)g = bg ( , u, v), ∀v ∈ Vg , g g 8
belongs to L2 (0, T ; Hg ), and hen e also belongs to L2 (0, T ; Vg′ ). Moreover, |∇g|∞ |Cg u(t)|g ≤ · ku(t)kg , for a.e. t ∈ (0, T ), m0 and |∇g|∞ kCg u(t)k∗ ≤ 1/2 · ku(t)kg , for a.e. t ∈ (0, T ). m0 η1 1.3. RESULTS OF STOCHASTIC ANALYSIS In this se tion, we repeat some of the results about the probability theory, Brownian motions or Wiener pro esses and sto hasti integrals that will be used in the thesis. 1.4. RESULTS OF NORMAL USED In this se tion, we re all some of the primary but important inequalities that are frequently used in the thesis. We also present a number of important propositions and theorems often used to prove the results of the thesis: Aubin-Lions ompa t lemma, the onsequen e of the Brouwer fixed point theorem, the Ty honoff fixed point theorem. 9
Chapter 2 STABILIZATION OF 3D NAVIER-STOKES-VOIGT EQUATIONS In this hapter, we onsider 3D Navier-Stokes-Voigt equations in smooth bounded domains with homogeneous Diri hlet bound- ary onditions. First, we study the existen e and exponential sta- bility of strong stationary solutions to the problem. Then we show that any unstable strong stationary solution an be exponentially stabilized by using either an internal feedba k ontrol with sup- port large enough or a multipli ative Ito noise of suffi ient inten- sity. This hapter is written based on the paper 3. 2.1. SETTING OF THE PROBLEM Let O be a bounded domain in R3 with smooth boundary ∂O. We onsider the following 3D Navier-Stokes-Voigt equations:    ut − ν∆u − α2 ∆ut + (u · ∇)u + ∇p = f in O × R+ ,  O × R+ ,  ∇ · u = 0 in (2.1)    u(x, t) = 0 on ∂O × R+ ,  u(x, 0) = u0 (x) in O,  where u = u(x, t) = (u1 , u2 , u3 ) is the unknown velo ity ve tor, p = p(x, t) is the unknown pressure, ν > 0 is the kinemati vis- osity oeffi ient, α is a length s ale parameter hara terizing the elasti ity of the fluid, f = f (x) is a given for e field and u0 is the initial velo ity. 2.2. UNIQUENESS AND EXPONENTIAL STABILITY OF STA- TIONARY SOLUTIONS Definition 2.1. Let f ∈ (L2 (O))3 be given. A fun tion u∗ ∈ D(A) is said to be a strong stationary solution to problem (2.1) if νAu∗ + B(u∗ , u∗ ) = f in (L2 (Ω))3 . (2.2) 10
The following theorem is the main result in this se tion. Theorem 2.1. Let f ∈ (L2 (O))3 . Then a) There exists at least one strong stationary solution u∗ of problem (2.1) satisfying 1 ku∗ k ≤ 1/2 |f |. (2.3) λ1 ν b) Moreover, if the following ondition holds c0 |f | ν2 > 3/4 , (2.4) λ1 where c0 is the best onstant in Lemma 1.1, then the strong stationary solution to problem (2.1) is unique and globally exponentially stable. 2.3. STABILIZATION OF STATIONARY SOLUTIONS BY US- ING AN INTERNAL FEEDBACK CONTROL We onsider the following ontrolled 3D Navier-Stokes-Voigt equations     ut − ν∆u − α2 ∆ut + (u · ∇)u + ∇p O × R+ ,  = 1ω h + f in     ∇·u=0 in O × R+ , (2.5)  u(x, t) = 0 ∂O × R+ ,  on      u(x, 0) = u (x) 0 in O, where 1ω is the hara teristi fun tion of the subdomain ω ⊂ O 2 3 with smooth boundary ∂ω , f ∈ (L (O)) and u0 ∈ V are given, h = h(x, t) is the ontrol. Let us define Oω = O\ω, Vω = u ∈ (C0∞ (Oω ))3 : ∇ · u = 0 . 11
Denote by Aω be the Stokes operator defined on Oω . We denote ∗ by λ1 (ω) the first eigenvalue of the operator Aω . Consider the feedba k ontroller h = −k(u − u∗ ), k ∈ R+ , and the orresponding losed loop system     ut − ν∆u − α2 ∆ut + (u · ∇)u +∇p + 1ω k(u − u∗ ) = f O × R+ ,  in     ∇·u=0 in O × R+ , (2.6)  u(x, t) = 0 ∂O × R+ ,  on      u(x, 0) = u (x) 0 in O. We set γ ∗ (u∗ ) := sup {|b(u, u∗ , u)| : |u| = 1} ≤ γ ku∗ kH α . We are now in position to state the main result of this se tion. Theorem 2.2. Let u∗ ∈ V ∩ (H β (O))3 , β > 5/2, be any strong stationary solution to (2.1) su h that νλ∗1 (ω) > γ ∗ (u∗ ). Then for ea h u0 ∈ V and k ≥ k0 suffi iently large but indepen- dent of u0 , there is a weak solution u ∈ C([0, ∞); V ) to (2.6) su h that ku(t) − u∗ kα ≤ e−ηt ku0 − u∗ kα , ∀t ≥ 0, for some η > 0. Here kuk2α := |u|2 + α2 kuk2 . Remark 2.1. By the Poin ar² inequality, we have −2 λ∗1 (ω) ≥C sup dist(x, ∂O) . x∈Oω Hen e λ∗1 (ω) an be made arbitrarily large by making the annular domain Oω = O \ ω ¯ is thin enough. Therefore, it follows from ∗ Theorem 2.2 that the steady state u is exponentially stabilizable if Oω is suffi iently thin . 12
2.4. STABILIZATION OF STATIONARY SOLUTIONS BY A MULTIPLICATIVE ITO NOISE We onsider the following sto hasti 3D Navier-Stokes-Voigt equations     d(u − α2 ∆u) + [−ν∆u + (u · ∇)u+ ∇p]dt 2 ∗ O × R+ ,   = f dt + σ(I − α ∆)(u − u )dWt in    ∇·u=0 in O × R+ , (2.7)  u(x, t) = 0 ∂O × R+ ,  on      u(x, 0) = u (x) 0 in O, where σ > 0, Wt : Ω → R, t ∈ R, is a one-dimensional Wiener pro ess. Theorem 2.3. If r −3/4 σ 2 α2 σ 2 α2 ν> c0 |f |λ1 + − , (2.8) 4 4 where c0 is the best onstant in Lemma 1.1, then the solution u∗ of problem (2.7) is globally exponentially stable. More pre isely, there existsN ⊂ Ω with P(N ) = 0, su h that for ω ∈ / N there is T (ω) su h that for any solution u(t) of problem (2.7), the following estimate holds for some ℓ > 0 : ku(t) − u∗ k2α ≤ ku(0) − u∗ k2α e−ℓt , ∀t ≥ T (ω). Remark 2.2. Thus, the multipli ative Ito noise stabilizes the ∗ strong stationary solution u for ν in the interval r σ 2 α2 σ 2 α2 q i −3/4 −3/4 c0 |f |λ1 + − , c0 |f |λ1 . 4 4 The larger the parameter σ , the longer the stability for the solution u∗ . Moreover, for any given ν > 0, we an always hoose a value of σ su h that (2.8) holds. 13
Chapter 3 STABILIZATION OF 2D g-NAVIER-STOKES EQUATIONS We onsider the g-Navier-Stokes equations in a two-dimensional smooth bounded domain O . First, we study the existen e and exponential stability of a strong stationary solution under some ertain onditions. Se ond, we prove that any unstable strong sta- tionary solution an be stabilized by proportional ontroller with support in an open subset ω ⊂ O su h that O\ω is suffi iently small" or by using finite-dimensional feedba k ontrols. Finally, we stabilize the long-time behavior of solutions to 2D g-Navier- Stokes equations under a tion of fast os illating-in-time external for es by showing that in this ase there exists a unique time- periodi solution and every solution tends to this periodi solution as time goes to infinity. This hapter is written based on the papers 1 and 4. 3.1. SETTING OF THE PROBLEM Let O be a bounded domain in R2 with smooth boundary ∂O. We onsider the following 2D g-Navier-Stokes equations: ∂u    − ν∆u + (u · ∇)u + ∇p = f in O × R+ ,    ∂t ∇ · (gu) = 0 O × R+ ,  in (3.1) +    u(x, t) = 0 on ∂O × R ,  u(x, 0) = u0 (x), in O.  where u = u(x, t) = (u1 , u2 ) is the unknown velo ity ve tor, p= p(x, t) is the unknown pressure, ν >0 is the kinemati vis osity oeffi ient, u0 is the initial velo ity. We assume that the fun tion g satisfies the following assump- tion: 14
(G1) g ∈ W 1,∞ (O) su h that 1/2 0 < m0 ≤g(x)≤M0 ∀x = (x1 , x2 ) ∈ O, v |∇g|∞ < m0 η1 , where η1 > 0 is the first eigenvalue of the g-Stokes operator in O (i.e. the operator Ag is defined in Chapter 1). 3.2. EXISTENCE, UNIQUENESS AND EXPONENTIAL STA- BILITY OF STATIONARY SOLUTIONS Definition 3.1. Let f ∈ L2 (Ω, g) be given. A strong stationary ∗ solution to problem (3.1) is an element u ∈ D(Ag ) su h that νAg u∗ + νCg u∗ + Bg (u∗ , u∗ ) = f in L2 (O, g). Theorem 3.1. f ∈ L2 (O, g), then problem (3.1) admits at least If one strong stationary solution u∗ satisfying 1 ku∗ kg ≤ |f |g . (3.2) 1/2 |∇g|∞ η1 ν 1 − 1/2 m0 η1 Moreover, if the following ondition holds !2 |∇g|∞ c1 |f |g ν2 1 − 1/2 > , (3.3) m0 η1 η1 where c1 is the onstant in Lemma 1.2, then the strong stationary solution to (3.1) is unique and globally exponentially stable. 3.3. STABILIZATION OF STATIONARY SOLUTIONS BY US- ING AN INTERNAL FEEDBACK CONTROL We onsider the following ontrolled 2D g-Navier-Stokes equa- tions: ∂u    − ν∆u + (u · ∇)u + ∇p ∂t    O × R+ ,     = 1ω hg + f in ∇ · (gu) = 0 in O × R+ , (3.4)   u(x, t) = 0 ∂O × R+ ,  on      u(x, 0) = u 0 in O, 15
where 1ω is the hara teristi fun tion of the subset ω ⊂ O with 2 smooth boundary ∂ω , f ∈ L (O, g) and u0 ∈ Hg are given, hg = hg (x, t) is the ontrol. Let us define Oω = O\ω, Vgω = u ∈ (C0∞ (Oω ))2 : ∇ · (gu) = 0 . Let Agω be the g -Stokes operator defined on Oω . We denote by η1∗ (ω) the first eigenvalue of the operator Agω . Consider the feedba k ontroller hg = −k(u − u∗ ), k ∈ R+ , and the orresponding losed loop system ∂u    − ν∆u + (u · ∇)u ∂t    +1ω k(u − u∗ ) + ∇p = f O × R+ ,  in    ∇ · (gu) = 0 in O × R+ , (3.5)   u(x, t) = 0 ∂O × R+ ,     on   u(x, 0) = u (x) in O. 0 We set γg∗ (u∗ ) = sup {|bg (u, u, u∗ )| : |u|g = 1} ≤ γg ku∗ kD(Ag ) . Theorem 3.2. Let u∗ ∈ D(Ag ) be any strong stationary solution to (3.4) su h that ! |∇g|∞ ν 1− 1/2 η1∗ (ω) > γg∗ (u∗ ). (3.6) m0 η1 Then for ea h u0 ∈ Hg and k ≥ k0 suffi iently large but inde- pendent of u0 , there is a weak solution u ∈ C([0, +∞); Hg ) ∩ L2loc (0, +∞; Vg ) to (3.5) su h that |u(t) − u∗ |g ≤ e−δt |u0 − u∗ |g , ∀t ≥ 0, for some δ > 0. 16
Remark 3.1. By the Poin ar² inequality, we have −2 η1∗ (ω) ≥C sup dist(x, ∂O) . x∈Oω Hen e η1∗ (ω) an be made arbitrarily large by making the annular domain Oω = O \ ω ¯ is thin enough. Therefore, it follows from ∗ Theorem 3.2, that any steady-state u is exponentially stabilizable if Oω is suffi iently thin". 3.4. STABILIZATION OF STATIONARY SOLUTIONS BY US- ING FINITE-DIMENSIONAL FEEDBACK CONTROLS We onsider the following ontrolled 2D g-Navier-Stokes equa- tions with interpolant operator Ih : ∂u    − ν∆u + (u · ∇)u + ∇p ∂t    = −µIh (u − u∗ ) + f O × R+ ,  in    ∇ · (gu) = 0 in O × R+ , (3.7)   u(x, t) = 0 ∂O × R+ ,     on   u(x, 0) = u in O, 0 where f = f (x) ∈ Hg is given. We assume that the feedba k ontroller Ih : Vg → H g is an interpolant operator that approximates the identity with error of order h, i.e., it satisfies the following estimate M0 2 2 |ϕ − Ih (ϕ)|2g ≤ c0 h kϕk2g , ∀ϕ ∈ Vg . (3.8) m0 Theorem 3.3. f ∈ Hg and let u∗ be any strong stationary Let solution to (3.1) obtained in Theorem 3.1. Suppose that Ih satisfies (3.8), µ and h are positive parameters su h that M0 2 2 2ν|∇g|2∞ 2c21 |f |2g µc h < ν and µ > + 2 . (3.9) m0 0 m20 η1 ν 3 1 − |∇g|1/2 ∞ m0 η1 17
Then for ea h u0 ∈ Hg given, there exists a unique weak solution u to system (3.7) su h that for any T > 0, du u ∈ C([0, T ]; Hg ) ∩ L2 (0, T ; Vg ), ∈ L2 (0, T ; Vg′ ), dt and |u(t) − u∗ |2g ≤ e−ηt |u0 − u∗ |2g , ∀t ≥ 0, (3.10) |∇g|2∞ c21 |f |2g where η = µ − 2ν −2 2 > 0 due to on- m20 |∇g|∞ η1 ν 3 1 − 1/2 m0 η1 dition (3.9). 3.5. STABILIZATION BY USING FAST OSCILLATING-IN- TIME EXTERNAL FORCES In this se tion, we onsider the following system  ∂u  ∂t − ν∆u + (u · ∇)u + ∇p = F (x, ω0 t)   in O × R+ , ∇ · (gu) = 0 in O × R+ , (3.11)   u(x, t) = 0 ∂O × R+ .  on We give the following assumption on the external for e: (F1) For any positive onstant ω0 > 0, we assume that the for e term F (x, ω0 t) is a time periodi fun tion with period Tper having the following stru ture: There exists a time periodi fun tion h(x, ω0 t) with period Tper su h that 1  ω ht (x, ω0 t) = −F (x, ω0 t)   in O × R+ , ∇ · (gh) = 0 in O × R+ , (3.12)   h=0 ∂O.  on We also assume that 18