SOME PROPERTIES OF SOLU ONS TO 2D G NA ER STOKES EQUATIONS. We prove some important properties of solutions to the problem including the backward uniqueness property, the squeezing property. Keywords: g NavierStokes; ... In this paper, we present numerical solutions of the 2D NavierStokes equations using the fourthorder generalized harmonic polynomial cell ...
SOME PROPERT ES OF SOLUT ONS TO 2D G NA ER STOKES EQUAT ONS Cao Th Thu Trang Khoa oán Khoa h c t nh n Ema l trangctt dhhp edu Tr n Quang Th nh h c S ph m K thu t am nh Ema l tqth nh spktnd moet edu Ngày nh n bài: 18/3/2022 Ngày PB ánh giá: 17/4/2022 Ngày t ng: 27/4/2022v ABSTRACT: We consider the initial boundary value problem for 2D g-Navier-Stokes equations in bounded domains with homogeneous Dirichlet boundary conditions We prove some important properties of solutions to the problem including the backward uniqueness property, the squeezing property Keywords: g -Navier-Stokes; strong solutions; backward uniqueness property and squeezing property MỘT SỐ TÍNH CHẤT CỦA NGHIỆM ĐỐI VỚI PHƯƠNG TRÌNH g-NAVIER-STOKES HAI CHIỀU TĨM TẮT: Chúng ta xét tốn giá trị biên ban đầu cho phương trình g-Navier-Stokes chiều miền giới hạn với điều kiện biên Dirichlet Chúng tơi chứng minh số tính chất quan trọng nghiệm bao gồm tính chất lùi, tính chất ép Từ khóa: g -Navier-Stokes; nghiệm mạnh; tính chất lùi; tính chất ép INTRODUCTION Let W be a bounded domain in with smooth boundary G We consider the following two-dimensional (2D) non-autonomous g -Navier-Stokes equations: u - Du + (u t ( gu ) u u ( x,0) )u + p = f ( x, t ) in (0, T ) W, = in (0, T ) W, = on (0, T ) G, = u0 ( x), x W, 1.1 where u = u( x, t ) = (u1, u2 ) is the unknown velocity vector, p = p ( x, t ) is the unknown pressure, TR NG > is the kinematic viscosity coefficient, u0 is the initial velocity I H C H I PH NG The -Navier-Stokes equations is a variation of the standard Navier-Stokes equations More precisely, when g const we get the usual Navier-Stokes equations The 2D -Navier-Stokes equations arise in a natural way when we study the standard 3D problem in thin domains We refer the reader to for a derivation of the 2D -Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship between them As mentioned in , good properties of the 2D -Navier-Stokes equations can lead to an initiate of the study of the Navier-Stokes equations on the thin three dimensional domain W g = W (0, g ) In the last few years, the existence of both weak and strong solutions to 2D Navier-Stokes equations has been studied in 2,3 The existence of time-periodic solutions to -Navier-Stokes equations was studied recently in Moreover, the long- time behavior of solutions in terms of existence of global/uniform/pullback attractors has been studied extensively in both autonomous and non-autonomous cases, see e.g 1,5,6,7,8 and references therein However, to the best of our knowledge, little seems to be known about other properties of solutions to the 2D g -Navier-Stokes equations This is a motivation of the present paper The aim of this paper is to study some important properties of solutions to g -NavierStokes equations such as the backward uniqueness property, the squeezing property To this, we assume that the function g satisfies the following hypothesis: (G ) g W 1, (W) such that < m0 where g ( x) M for all x = ( x1 , x2 ) W, and | g | < m0 1/ , > is the first eigenvalue of the g -Stokes operator in W (i.e., the operator A defined in Section 2) The paper is organized as follows In Section 2, for convenience of the reader, we recall some auxiliary results on function spaces and inequalities for the nonlinear terms related to the g -Navier-Stokes equations Section proves a backward unique-ness result In Section 4, we prove the squeezing property for the solutions on the global attractor PRELIMINARIES Let L2 (W, g ) = ( L2 (W, g ))2 respectively, with the inner products u, v u, v g = W j =1 H 01 (W, g ) = ( H 01 (W, g )) and g be endowed, = u.vgdx, u, v L2 W, g , and W u j v j gdx, u = u1 , u2 , v = v1 , v2 H 01 W, g , T P CH KHOA H C, S 52, tháng n m 2022 and norms Thanks to assumption (G ) , the norms , 2 |.| and ||.|| are equivalent to the usual ones in ( L (W)) and in ( H 01 (W )) Let (C0 (W)) : = {u Denote by H g the closure of H 01 (W, g ) It follows that Vg ( gu ) = in L (W, g ) , and by Vg the closure of Hg in Hg Vg , where the injections are dense and continuous We will use || ||* for the norm in Vg , and , for duality pairing between Vg and Vg We now define the trilinear form b by b(u , v, w) = i , j =1 u W i vj w j gdx, xi whenever the integrals make sense It is easy to check that if u, v, w Vg , then Hence b(u, v, v) = 0, "u, v Vg Set A : Vg Vg by Au, v = ((u, v)) g , B : Vg Vg Vg by B (u, v), w = b(u, v, w) and put Bu = B (u , u ) Denote D( A) = {u Vg : Au H g , then D( A) = H (W, g ) Vg and Au = - Pg Du , "u D ( A), where Pg is the ortho-projector from L (W, g ) onto H g Using the H o lder inequality and the Ladyzhenskaya inequality (when n = ) | u |L c | u |1/ | u |1/2 , "u H 01 (W), and the interpolation inequalities, as in one can prove the following result Lemma 2.1 If n = , then c1 | u |1/2 || u ||1/2 || v ||| w |1/2 || w ||1/2 , "u , v, w Vg , | b(u, v, w) | c2 | u |1/2 || u ||1/2 || v ||1/ | Av |1/2 | w |, "u Vg , v c3 | u |1/2 | Au |1/2 || v ||| w |, "u D ( A), v Vg , w Hg, c4 | u ||| v ||| w |1/2 | Aw |1/2 , "u H g , v Vg , w D( A), and TR NG D ( A), w I H C H I PH NG Hg, 2.1 | B(u, v) | + | B(v, u) | c5 || u |||| v ||1- | Av | , "u Vg ; v D( A) where (0,1) ; ci , i = 1, For every u , v 2.2 ,5, are appropriate constants D ( A) , then | B (u , v ) | c6 | Au ||| v ||, 2.3 || u ||| Av | L2 (0, T ;Vg ) , then the function Cu defined by Lemma 2.2 Let u ((Cu (t ), v) g = g g u, v g , u , v , "v Vg , g =b g belongs to L2 (0, T ; H g ) , and therefore also belongs to L2 (0, T ;Vg ) Moreover, | Cu(t ) | | g| || u(t ) ||, for a.e t (0, T ), m0 and || Cu(t ) ||* | g| || u(t ) ||, for a.e t (0, T ) m0 11/ Since g - ( g )u = -Du - ( )u, g g we have (-Du, v) g = ((u, v)) g + (( g g )u, v) g = ( Au, v) g + (( )u, v) g , "u, v Vg g g We recall the definition of strong solutions to problem 1.1 Def n t on 2.1 A function u is called a strong solution to problem 1.1 on the interval (0, T ) if u C ( 0, T ;Vg ) L2 (0, T ; D( A)), du / dt L2 (0,T ; H g ), d u (t ) + Au (t ) + Cu (t ) + B(u (t ), u (t )) = f (t ) in H g , for a.e t (0, T ), dt u (0) = u0 Theorem 2.1 For any T > 0, u0 Vg , and f L2 (0, T ; H g ) given, problem 1.1 has a unique strong solution u on (0, T ) Moreover, the strong solutions depend continuously on the initial data in Vg T P CH KHOA H C, S 52, tháng n m 2022 We recall here some a priori estimates of strong solutions frequently used later BACKWARD UNIQUENESS PROPERTY Let u, v solve respectively the -Navier-Stokes equations du + Au + Cu + B(u, u ) = f , dt u = u0 3.1 dv + Av + Cv + B (v, v ) = f , dt v = v0 3.2 Two solutions u, v are called a backward uniqueness property if u(t1 ) = v(t1 ) then u (t ) = v (t ) for all time t < t1 Lema 3.1 [9 If a function w L (0, T ;V ) L2 (0, T ; D( A)) satisfy dw + Aw = h(t , w(t )), t (0, T ), dt where h is function from (0, T ) V into H such that | h(t , w(t )) | k (t ) || w(t ) ||, for a.e, t (0, T ), k L2 (0, T ), and w(T ) = then w(t ) = 0,0 < t < T Theorem 3.1 Under the assumptions of Theorem 2.1 , then the strong solutions of -Navier-Stokes have a backward uniqueness property Proof Denote , we have dw + Aw = - B (u , u ) + B (v, v ) - Cw dt = - B (u , w) - B ( w, v ) - Cw Using 2.2 and Lemma 2.2, we obtain | h(t , w(t )) |:=| - B(u, w) - B( w, v) - Cw | c5 || u ||1- | Au | +c5 || v ||1- | Av | + | g| m0 Applying Lemma 3.1 with k (t ) = c5 || u ||1- | Au | +c5 || v ||1- | Av | + we have the proof SQUEEZING PROPERTY TR NG I H C H I PH NG | g| , m0 || w || We write for the orthogonal projection onto the finite-dimensional subspace, and for the projection onto its orthogonal complement Then by 12 , we can define a continuous -Navier-Stokes equations and it has global attractor in Vg semigroup S (t ) of Def n t on 4.1 Write S = S (1) Then the squee ing property holds if, for each < < 1, there exists a finite rank orthogonal projection P ( ) , with orthogonal complement Q ( ) , such that for every u , v either | Q( Su - Sv) | | P( Su - Sv) | 4.1 | Su - Sv |< | u - v | 4.2 or, if not, then Theorem 4.1 If f H g then the squee ing property holds for the 2D g - Navier- Stokes equations Proof The equation for the difference w(t ) = u (t ) - v (t ) is dw + Aw + Cw + B (u , w) + B( w, v) = 0, dt 4.3 and we will write p = Pn w, q = Qn w, First, we take the inner product of 4.3 with w = p + q , using b(u , w, p ) = b(u, p + q, p ) = b(u, q, p), we have d | p |2 + || p ||2 + (Cw, p) g = -b(u, q, p) - b( w, v, p) dt Using the bounds on b in Lemma 2.1 and the existence of an absorbing set in H g ,Vg and D( A) , we can obtain d | p |2 + || p ||2 + (Cw, p) g dt -C (| u |1/ | Au |1/2 | q ||| p || - | w ||| p ||| v |1/ | Av |1/ ) C (| q | where = n 1/ | p|+| w| 1/ | p |), Using Lemma 2.2, we obtain d | p |2 + || p ||2 -C | q | dt -C 1/ 1/ | p | + 1+ | g| | w| m0 1/ | p| | p | (| q | + | p |), so that T P CH KHOA H C, S 52, tháng n m 2022 | p| d | p| dt -| p|( | p | +C 1/2 1/2 - | p|( 1/2 4.4 | p | +C | q | +C | p |) 4.3 with Now take the inner product of (| q | + | p |)) , we have d | q |2 + || q ||2 + (Cw, q) g = -b(u, p, q) - b( w, v, q ) dt C | p ||| q || +C | w ||| q || Using Lemma 2.2, we obatain d | q |2 + || q ||2 dt C | p ||| q || + C + | g| m0 | w ||| q || C || q || (| p | + | q |), so that |q| d |q| dt || q || (- 1/2 | q | +C | p | +C | q |) Provided that the expressionin the parentheses is negative, 1/2 ( - C ) | q |> C | p |, then we have |q| d |q| dt 1/ | q | (- 1/ | q | +C | p | +C | q |) 4.5 We now choose n large enough that 1/2 - C > 2C 4.6 Now, either 4.1 holds, and so there is nothing to prove, or it does not, in which case 4.7 | Qw(1) |>| Pw(1) | In this case, using 4.6 , we have ( 1/ - C ) | Qw(t ) |> 2C | Pw(t ) |, holds for t = Since w(t ) is continuous into H g , then 4.8 4.8 holds in a neighbourhood of t = We consider two possibilities: If 4.8 holds for all t then we have, by 4.6 , ( TR NG 1/2 - C ) | q | -C | p |> ( I H C H I PH NG 1/2 - C ) | q |> C | q |, ,1 , for t ,1 , and so 4.5 becomes d | q | - C | q |, dt which gives | q(1) | e - C | |q Since 4.7 holds, this implies that | w(1) | 2e - C 1 |q | 2e 2 C and using the Lipschitz property of strong solutions, | w | w(1) | L = This gives 4.2 , provided that hold on all of t n ,1 , then it holds on t ( 1/ - 12 e C |w |, 1 | L | w(0) |, we have 2 | w(0) | is chosen large enough If 4.8 does not t0 ,1 , with - C ) | Qw(t0 ) |= 2C | Pw(t0 ) | 4.9 In this case we take F (t ) = F p (t ), q(t ) = | p | + | q | exp 1/2 |q| C (| p | + | q |) From 4.4 , 4.5 holds, we have dF dt Thus F(1) 0; " t t0 ,1 F(t0 ) However, at t = we have 4.8 , so that F (1) | q(1) | e 1/2 /C , and at t = t0 the equality 4.9 hold, which gives and so 62| TRƯỜNG ĐẠI HỌC HẢI PH NG T P CH KHOA H C, S 52, tháng n m 2022 1/2 +C | q(t0 ) | e 2C F(t0 ) = 1/2 /( 1/2 +C ) It follows that | q(1) | e - 1/2 1/2 +C e | q(t0 ) |, 2C /C and using once more the Lipschitz property of strong solutions, we obtain | q(1) | e - 1/2 /C 1/ +C e L(1) | w(0) | 2C Since | p(1) |