On the weak stationary solutions to 2D g-Navier-Stokes equations

NGÀNH TOÁN HỌC 93Tạp chí Nghiên cứu khoa học, Trường Đại học Sao Đỏ, ISSN 1859-4190 Số 4 (67).2019 On the weak stationary solutions to 2D g-Navier-Stokes equations Nghiệm dừng yếu của hệ phương trình g-Navier-Stokes hai chiều Nguyen Viet Tuan, Tran Hoang Yen Email: nguyentuandhsd@gmail.com Sao Do University Received date: 02/10/2019 Accepted date: 19/12/2019 Published date: 31/12/2019 Abstract In this paper, we consider the g-Navier-Stokes equations in a two-dimensional bounded domain Ω. We study the existence and exponential stability of a stationary solution under some certain conditions. Keywords: g-Navier-Stokes equations; stationary solutions; exponential stability. Tóm tắt Trong bài báo này, chúng ta xét hệ phương trình g-Navier-Stokes trong miền hai chiều bị chặn Ω. Chúng ta nghiên cứu sự tồn tại và ổn định mũ của nghiệm dừng trong một số điều kiện nhất định. Từ khóa: Hệ phương trình g-Navier-Stokes; nghiệm dừng; ổn định mũ. 1. INTRODUCTION Let Ω be a bounded domain in with smooth boundary ∂Ω. We consider the following 2D g-Navier-Stokes equations Where: u = u (x,t) = u (u 1 ,u 2 ) is the unknown velocity vectơ; p = p (x,t) is the unknown pressure, v > 0 is the kinematic viscosity coefficient; u0 is the initial velocity. The 2D g-Navier-Stokes equations arise in a natural way when we study the standard 3D Navier- Stokes problem in a 3D thin domain Ω g = Ω ×(0,g) (see [9] ). As mentioned in [8, 9], good properties of the 2D g-Navier-Stokes equations can lead to an initial study of the 3D Navier-Stokes equations in the thin domain Ω g . In the last few years, the existence and long-time behavior of solutions in terms of existence of attractors for 2D g-Navier-Stokes equations have been studied extensively in both autonomous and non-autonomous cases (see e.g. [1, 2, 4, 5, 6, 7, 8, 10] and references therein). In this paper we will study the existence, uniqueness and stability of weak stationary solutions to problem (1). The existence of stationary solutions is proved by using the compactness method. When the viscosity is "larger" than the external force, we show that the stationary solution is unique and is globally exponentially stable. To do this, we assume that the function g satisfies the following assumption: ( ) ( )1, W ¥Î WG g such that ( )0 00 < £ £m g x M for all ( )1 2, ,= ÎWx x x and 1 20 1 ,l¥Ñ is the first eigenvalue of the g-Stokes operator in Ω (i.e. the operator A is defined in Section 2 below). This paper is organized as follows. In Section 2, for convenience of the reader, we recall some results on function spaces and operators related to 2D g-Navier-Stokes equations which will be used. In Section 3, we prove the existence, uniqueness and exponential stability of weak stationary solutions to 2D g-Navier-Stokes equations. 2. PRELIMINARY RESULTS Let ( ) ( )( )22 2,W = WL g L and ( ) ( )( )21 10 0,W = WH g H be endowed, respectively, with the inner products Người phản biện: 1. Assoc.Prof.Dr. Khuat Van Ninh 2. Dr. Dao Trong Quyet !2 ∂u ∂t −νΔu + (u.∇)u +∇p = f in Ω×! + , ∇. gu( ) = 0 in Ω×!+ , u = 0 on ∂Ω, u(x,0) = u 0 (x) in Ω, ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ (1) NGHIÊN CỨU KHOA HỌC 94 Tạp chí Nghiên cứu khoa học,Trường Đại học Sao Đỏ, ISSN 1859-4190 Số 4 (67).2019 (5) ( ) ( ) ( )11 2 1 2 0, , , ,= = Î Wu u u v v v H g and norms ( ) ( )( )2 2, , , u .g gu u u u u= = Thanks to assumption (G) the norms . and . are equivalent to the usual ones in ( )( )210 .WH Let 𝒱𝒱 ( )( ) ( ){ } 2 0 : . 0 .¥= Î W Ñ =u C gu Denote by gH the closure of v in ( )2 ,WL g , and by gV the closure of v in ( )10 , .WH g It follows that ' ' ,Ì º Ìg g g gV H H V where the injections are dense and continuous. We will use * . for the norm in ' ,gV and ., . for duality pairing between gV and ' .gV We define the g-Stokes operator ': ®g gA V V by ( ) ( )( ), , ,= gAu v u v for all , .Î gu v V Then = - DgA P and ( ) ( )2 , ,= W ! gD A H g V where gP is the ortho-projector from ( )10 ,WH g onto .gV We also define the operator 𝐵𝐵: 𝑉𝑉! × 𝑉𝑉! → 𝑉𝑉!" by ( )( ) ( ), v , w , , w ,=B u b u v for all , , w ,Î gu v V where ( ) 2 , 1 , , w w .W = ¶ = ¶å ò ji ji j i vb u v u gdxx It is easy to check that if , , w ,Î gu v V then ( ) ( ) ( ), , w , w, , , , 0.= - =b u v b u v b u v v We also set ( ) ( ), u=B u B u for all .Î gu V We recall some known results which will be used in the paper. Lemma 2.1 ([1]) If n = 2 then Where , 1...3,=ic i are appropriate constants. Lemma 2.2 ([3]) Let ( )2 0, ; ,Î gu L T V then the function Cu defined by ( )( ), . , , , , ,æ öæ ö æ öÑ Ñ= Ñ = " Îç ÷ç ÷ ç ÷è ø è øè ø gg g g gCu t v u v b u v v Vg g belongs to ( )2 0, ;H ,gL T and hence also belongs to ( )2 '0, ;V .gL T Moreover, ( ) ( )0 . , ¥Ñ£ gCu t u tm for a.e. ( )0,T ,Ît and ( ) ( ) 0 . ,¥Ñ£ gCu t u tm for a.e. ( )0,T ,Ît 3. EXISTENE, UNIQUENESS AND EXPONENTIAL STABILITY OF WEAK STATIONARY SOLUTIONS First, we give the definition of the weak stationary solutions to 2D g -Navier-Stokes equations (1) Definition 3.1. Let 'Î gf V be given. A weak stationary solution to problem (1) is an element * Î gu V such that ( )* * * *,n n+ + =Au Cu B u u f in ' .gV The following theorem is our main result in this section. Theorem 3.1. Let f be given in ' .gV Then, (i) there exists a weak stationary solution * Î gu V to (1); (ii) furthermore, if the following condition holds 2 1 1 1 * 2 20 1 1 1 ,n l l ¥ é ùæ öÑê úç ÷- >ê úç ÷ç ÷ê úè øë û g c f m Where 1c is the constant in Lemma 2.1, then the weak stationary solution to is (1) unique and globally exponentially stable. That is, for any initial data 0 Î gu H and the any weak solution u (t) of (1.1) then there exists λ > 0 such that ( ) 2 2* *0 , 0.l-- £ - " ³tu t u u u e t Moreover, if *u satisfying { } 1 * 21 1 2 31 20 1 1 , ax , , ,4 n l l ¥ æ öÑç ÷£ - =ç ÷ç ÷è ø gu c m c c cc m 1* 20 0 1 1 20 1 1 , nl l ¥ æ öÑç ÷- £ -ç ÷ç ÷è ø gu u k m Where 0k is a positive real number, then there exists , 0a >k such that ( ) 2 2* * *0 , 0.a-- £ - " ³ >tu t u k u u e t t Proof. (i) Existence. Let { } 1w ¥=j j be a basis of gV consisting of eigenfunctions of the g -Navier- Stokes operator .A For each m ≥ 1, let us denote { }1w ,..., w=m mV span and we would like to define an approximate weak stationary solutions mu of (1.1) by. 1 w ,g = =åmm mi j j u Such that ν u m ,v( )( ) g +ν Cum ,v( ) g + b um ,um ,v( ) ∀ ∈ To prove the existence of ,mu we define operators : ®m m mR V V by ( ) ( ) ( ) 1 1 1 1 2 2 2 21 1 1 1 1 1 2 2 2 2 22 1 1 1 2 2 23 w w , , , w , w , , , , w , w , w , , , w . ì " Îïïï " Îïï = Î Îíïï " Î Îï Îïïî g g g g g c u u v u v V c u u v Av u V b u v v D A H c u Au v u D A v V H (2) (3) (4) ( ) ( ) ( )( ) 2 2 1 , , , , , , . , W W = = Î W = Ñ Ñ ò å ò g i ig j u v uvgdx u v L g u v u v gdx ν ν = f ,v , ∀v ∈V g . 5 ( )( ) ( ) ( ), , v , , , , , , . n n= + + - " Î m g m R u v Au Cu v b u u v f v u v V , , v , , , , , , . n n - " Î m g m R u v Au Cu v b u u v f v u v V NGÀNH TOÁN HỌC 95Tạp chí Nghiên cứu khoa học, Trường Đại học Sao Đỏ, ISSN 1859-4190 Số 4 (67).2019 For all ,Î mu V If we set * 1 20 1 , 1 b n l ¥ = æ öÑç ÷-ç ÷ç ÷è ø f g m We obtain ( )( ), 0³mR u u for all Î mu V such that .b=u Consequently, by a corollary of the Brouwer fixed point theorem, for each m≥1 there exists Îm mu V such that R m (u m ) = 0 with .b£mu Taking = mv u in (5) we have ( )( ) ( ), , , u .n n+ =m m m m mggu u Cu u f By Lemma 2.2, we have 2 21* 20 1 ,nn l ¥Ñ£ +m m mgu f u u m if follows that * 1 20 1 1 . 1 mu f g m n l ¥ £ æ öÑç ÷-ç ÷ç ÷è ø Hence using (6) we deduce that the sequence { }mu is bounded in ,gV and therefore by the compactness of the embedding gV ,gH we can extract a subsequence { } { }' Ìm mu u that converges weakly in gV and strongly in gH to an element * .Î gu V It is now standard to take limits in (5) to obtain that *u is a weak stationary solution to problem (1). (ii) Uniqueness. Assume that * *,u v are two weak stationary solutions to problem (1). We have Denote v = u* - v*, we obtain Hence 12 2* * * * *21 2* * 1 20 1 , n l n l - ¥ - £ - Ñ + - u v c u v v g u v mand so This implies thanks to estimate (6) for v*. From (2) and (7), we get the desired result. (iii) Exponential stability. Let * Î gu V be the unique stationary solution to (1). From (2) and (6), it follows that We also use the notation 0 1 20 1 1 0.g l ¥Ñ = - >g m Let u (.) be any solution of problem (1). Denote w (t) = u (t) - u*, one has ( )( ) ( )( )( ) ( )( ) ( ) ( )( ) ( )* * w , w , Cw , , , , , 0, . 9 g gg g d t v t v t vdt b u t u t v b u u v v V n n+ + + - = " Î We first give an estimate for w . Replacing v by w (t) in (9) and noting that ( ) ( ) ( )( ) ( )( ) ( ) ( )( )* * *, , w , , w w , , w ,- =b u t u t t b u u t b t u t We get ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )* w , w w , w Cw , w w , u , w 0. n n + + + = g g g d t t t tdt t t b t t We can write this as ( ) ( ) ( ) ( )( ) ( ) ( )( ) 2 2 * w 2 w 2 Cw , w 2 w , u , w . n n= - - - g d t t t tdt b t t By Lemmas 2.1 and 2.2, we have By applying condition (8) we obtain ( ) ( )2 2*11 0 1 21 2w 2 w . l ng l æ öç ÷£ - +ç ÷ç ÷è ø cd t u tdt This implies that ( )( ) ( ) 2 2 1* 20 1 2 1 * 20 1 , u , u , u , 1 . n n nn l n l ¥ ¥ = + - ѳ - - æ öÑç ÷³ - -ç ÷ç ÷è ø m gR u Au Cu f v gu f u u m g u f u m ( ) ( ) ( ) * * * * * * * * , , , , v , , 0, . n n - + - + - = " Î gg Au Av v b u u v b v v Cu Cv v v V ( ) ( ) * * * * * * * * * , , , , . n n - - = - - - g Au Av u v b v v v Cu Cv u v 2* * 1 20 1 1 2* *21 * 1 20 1 1 1 . , 7 1 g u v m c f u v g m n l l n l ¥ - ¥ æ öÑç ÷- -ç ÷ç ÷è ø £ -æ öÑç ÷-ç ÷ç ÷è ø (7) 1* 21 1 1 20 1 1 .n l l ¥ æ öÑç ÷£ -ç ÷ç ÷è ø gu c m (8) (9) (6) ↪ 12 2* * * * *211 20 1 1 .n l l -¥ æ öÑç ÷- - £ -ç ÷ç ÷è ø g u v c u v v m ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 20 1 2* 1 2*1 0 1 21 2w 2 w w 2 w w 2 2 w . 10 gd t t tdt m c u t t c u t nn l ng l ¥Ñ= - + + æ öç ÷£ - +ç ÷ç ÷è ø (10) NGHIÊN CỨU KHOA HỌC 96 Tạp chí Nghiên cứu khoa học,Trường Đại học Sao Đỏ, ISSN 1859-4190 Số 4 (67).2019 ( ) 4*41 0 1 03 3 1 0 1 1w .2 4nl g nl gln g- ³ c t (13) ( ) ( )2 2w w 0 ,l-£ tt e Where *11 0 1 21 2= 2 0. l l ng l æ öç ÷- >ç ÷ç ÷è ø c u Now, we have to give a suitable estimate on w . From (10) we have ( ) ( )2 2*10 1 21 2w 2 w 0.ng l æ öç ÷ + - £ç ÷ç ÷è ø cd t u tdt Integrating both sides from 0 to 0 0,>T we obtain ( ) ( ) ( )02 2 2*10 0 1 021 2w 2 w w 0 .ng l æ öç ÷ + - £ç ÷ç ÷è ø ò TcT u t dt This yields ( ) ( )0 2 2*10 1 021 22 w w 0 ,ng l æ öç ÷- £ç ÷ç ÷è ø ò Tc u t dt Or, equivalently, ( ) ( )0 2 2 0 *1 0 1 21 1w w 0 . 22ng l £ æ öç ÷-ç ÷ç ÷è ø ò T t dt c u Hence, by the Mean Value Theorem for Integrals, there exists ( )* 00,Ît T such that ( ) ( ) ( ) 02 2* 0 0 2 *1 0 01 21 1w w 1 w 0 . 2 ng l = £ æ öç ÷-ç ÷ç ÷è ø ò T t t dtT c u T We can take v = Aw in (10) and obtain ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) * * * * w , w w , w Cw , w , , w , , w w , w , w w , , w , w , w . n n+ + = - = - - - g gg d t A t t A t t A tdt b u u A t b u t u t A t b t t A t b t u A t b u t A tThus, we have ( ) ( ) ( ) ( ) ( ) 2 2 * * w 2 w 2 Cw, w 2 w, w, w 2 w, , w 2 , w, w . n n+ = - - - - g d t A A b Adt b u A b u A Using Lemmas 2.1 and 2.2 again, we see that ( ) 2 2 0 1 1 2 23 1 1 *2 23 1 1 1 1* *2 2 2 22 w 2 w 2 w w 2 w w w w 2 w w w 2 w w w n n ¥Ñ+ £ + + + gd t A Adt m c A A c A u A c u u A A (11) Thanks to estimate 1 21 ww l £ and 1 21 ww . l £ A Hence From which, by using hypothesis (3) we obtain ( ) 1 32 2 2 21 20 1 w 1 w 2 w w wn l ¥ æ öÑç ÷ + - £ç ÷ç ÷è ø gd t A c Adt m As ( ) 1 32 2 2 20w w 2 w w w .ng+ £d t A c Adt Then by using the Young inequality, we get ( ) 42 2 2 2 40 0 3 3 0 1 27w w w w w ,2 2ng ng n g+ £ + d ct A Adt From which we deduce ( ) 42 2 60 3 3 1 0 1 27w w w ,2 2ng ln g+ £ d ct Adt So that ( ) 2 4 241 0 3 3 1 0 1w w w 0,2nl g ln g æ ö + - £ç ÷è ø cd tdt Where 4 4 27 . 2 = cc By using (11) and assumption (4) with *0 0 01 1 4 24 1 1 ng l æ öç ÷ = -ç ÷ç ÷è ø ck u T c For some T 0 > 0, we can deduce We denote by ( )* * ,Î +¥T t the maximal existence time for the strong solution W. We define now ( ) 4* * 4 1 0 3 3 1 0 * 1 0 1sup , : w2 1 , , . 8 cT t t T s s t t nl g ln g nl g ìï é ù= Î -í ë ûïî üé ù³ " Î ýë ûþ By continuity and (13), we see that T > t*. We show that T = +∞, which implies in particular that T *= +∞. Assume to the contrary that T = +∞. This implies that *<T T because the solution remains bounded 1 32 2 21 20 1 1 3 1 3* *2 2 2 21 1 4 41 1 1 32 2*2 21 1 2 20 1 1 2 w 2 w w w 2 w w 2 w w 2 w 2 w w w 4 w , n l l l n l l ¥ ¥ Ñ£ + + Ñ£ + + g A c A m c cu A u A g cA c A u A m ( ) 2 2* 1 1 2 20 1 1 1 3 2 2 2w 2 1 w 2 w w w . n l l ¥ é ùæ öÑê úç ÷ + - -ê úç ÷ç ÷ê úè øë û £ gd ct u Adt m c A (12) 1 32 2 21 20 1 1 3 1 3* *2 2 2 21 1 4 41 1 1 32 2*2 21 1 2 20 1 1 2 w 2 w w w 2 w w 2 w w 2 w 2 w w w 4 w , n l l l n l l ¥ ¥ Ñ£ + Ñ£ g A c A m c cu A u A g cA c A u A m NGÀNH TOÁN HỌC 97Tạp chí Nghiên cứu khoa học, Trường Đại học Sao Đỏ, ISSN 1859-4190 Số 4 (67).2019 When t → T. By definition of T, and by (12) we see that ( ) 2 2 *1 01w w 0, , . 8nl g é ù+ £ " Î ë û d t A t t Tdt This deduces that ( ) ( ) 22 * *w w , , . é ù£ " Î ë ût t t t T In particular we have ( ) 441 0 1 03 3 1 0 1 1w ,2 8nl g nl gln g- ³ c T and then, by continuity, there is e> 0, such that *e+ <T T and ( ) 4 *41 0 1 03 3 1 0 1 1w , , .2 8nl g nl g eln g é ù- ³ " Î +ë û c t t t T This is a contradiction with the maximality property of T. Finally, we have shown that * ,= = +¥T T so that the strong solution under consideration is global. Moreover, since T = +∞ we deduce from (14) that ( ) ( ) 22 * *w w , ,a-£ " ³tt t e t t Where 1 01 .8a nl g= Now combining (11) and (15), we arrive at ( ) ( )2 2 *w w 0 , ,a-£ " ³tt k e t t Where 1 2 2 2 02 4 0 1 1 . 3 2 2 = =k c k c k The proof is complete. (14) REFERENCES [1] C.T. Anh and D.T. Quyet (2012), Long-time behavior for 2D non-autonomous g-Navier-Stokes equations, Ann. Pol. Math., 103, 277-302. [2] C.T. Anh, D.T. Quyet and D.T. 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Roh (2006), Derivation of the g-Navier- Stokes equations, J. Chungcheon Math. Soc., 19, 213-218. [10] D. Wu and J. Tao (2012), The exponential attractors for the g-Navier-Stokes equations, J. Funct. Spaces Appl., Art. ID 503454, 12 pp. (15) Nguyen Viet Tuan - Education: + 2006: Bachelor of Mathematics and Informatics, Hanoi Pedagogican University 2 + 2011: Master of Mathematical, Hanoi Pedagogican University 2 + 2019: PhD of Mathematics, Hanoi Pedagogican University 2 - Working Experience: + 2007 to date: Lecturer at Faculty of Basic Sciences, Sao Do University - Research interests: + Stability and stabilization for some evolution equations in fluid mechanics - Email: nguyentuandhsd@gmail.com - Phone: 0978 235 234 AUTHORS BIOGRAPHY Tran Hoang Yen - Training, reseaching process: + In 2004: Graduated BA from English Faculty of Ha Noi University + In 2011: Graduated MA from University of Languages and International Studies, Ha Noi National University majored in Foreign Language Teaching Method - Current job: Lecturer of Tourism and Foreign Language Faculty, Sao Do University - Interests: Language, Foreign Language Teaching Methods - Email: yendhsd@gmail.com - Tel: 0986596586