- Email: [email protected]
H1 -Uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains
Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627 www.elsevier.com/locate/na
H 1-Uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains夡 Caidi Zhaoa,∗ , Shengfan Zhoub , Xinze Liana a College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, PR China b Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, PR China
Received 12 December 2005; accepted 7 December 2006
Abstract This paper discusses the long time behavior of solutions for a two-dimensional (2D) nonautonomous micropolar fluid flow in 2D unbounded domains in which the Poincaré inequality holds. We use the energy method to obtain the so-called asymptotic compactness of the family of processes associated with the fluid flow and establish the existence of H 1 -uniform attractor. Then we prove that an L2 -uniform attractor belongs to the H 1 -uniform attractor, which implies the asymptotic smoothing effect for the fluid flow in the sense that the solutions become eventually more regular than the initial data. © 2007 Published by Elsevier Ltd. MSC: 35B41; 35Q35; 76D03 Keywords: Uniform attractor; Micropolar fluid flow; Unbounded domains; Asymptotic smoothing effect
1. Introduction Attractor is an important concept in the study of dynamical systems. There are some classical works concerning this subject (see e.g. [7,11,12,15,27,29]). In the book [29] Temam studied systematically the global attractor associated with many concrete autonomous equations arising in mathematical physics. Later, Chepyzhov and Vishik [7] presented a general theory to deal with nonautonomous equations. Generally speaking, if the nonautonomous terms in the addressed equations have some uniformity, the long time behavior of solutions could be considered with respect to (w.r.t.) all nonautonomous terms in the so-called symbol space. When the spatial domain is unbounded, there exists a considerable obstacle coming from the loss of compact embedding between usual Sobolev spaces. One remedy to this case is to consider weighted spaces (see [1,3,4,8,32,33]). Later on, Ball (see e.g. [5]) formulated the idea of energy equation to obtain some asymptotic compactness for the associated semigroup and thus could avoid the weighted spaces. In consequence, this idea was successfully developed by many authors to study the autonomous equations [10,14,23,25,31,34] and nonautonomous equations [13,21,24]. 夡 This work is supported by the NSF of China under grant number 10471086, and is supported by the NSF of Zhejiang Province under grant number M103043. ∗ Corresponding author. E-mail address: [email protected] (C. Zhao).
1468-1218/$ - see front matter © 2007 Published by Elsevier Ltd. doi:10.1016/j.nonrwa.2006.12.005
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
609
The micropolar fluid model is an essential generalization of the Navier–Stokes model in the sense that it takes into account the microstructure of the fluid [18]. The micropolar fluids were introduced in [9], which are non-Newtonian fluids and have many applications (see e.g., [9,17–21]). Mathematical theory of micropolar fluid flows were extensively studied. One can refer to [6,9,17–21,26] and the references therein. Especially, when the external force g(x, t) and moment g(x, ˜ t) are independent of time t and ⊂ R2 is bounded, [18] proved the existence of L2 -global attractor and its finite dimensionality for the semigroup associated with the 2D micropolar fluid flow, while [6] established that the L2 -global attractor is compact in H 2 space by employing an abstract result on the existence of global attractor from the viewpoint of measuring noncompactness [22]. In the present article, we consider the following 2D nonautonomous incompressible micropolar fluid flow ju − ( + r )u − 2r rot v + (u · ∇)u + ∇p = g(x, t), (x, t) ∈ × R, jt jv ˜ t), (x, t) ∈ × R, − v + 4r v − 2r rot u + (u · ∇)v = g(x, jt ju1 ju2 ∇ ·u= + =0 jx1 jx2
(1.1) (1.2) (1.3)
associated with the initial boundary conditions: u|t= = u , u|j = 0,
v|t= = v ,
∈ R,
v|j = 0,
where is an unbounded domain of R2 in which the following Poincaré inequality holds 2 (x) dx |∇(x)|2 dx, ∀(x) ∈ H01 (), 1
(1.4) (1.5)
(1.6)
where 1 is a positive constant depending only on . In Eqs. (1.1) and (1.2), the unknown vector-valued function u(x, t) = (u1 (x, t), u2 (x, t)) denotes the velocity, the scalar functions p and v represent pressure and microrotation, ˜ t) denote external force and moment, respectively, and respectively. Moreover, g(x, t) = (g1 (x, t), g2 (x, t)) and g(x, , r , are positive constants ( is usually called the Newtonian viscosity and r the microrotation viscosity). In addition, we have set ju1 jv jv ju2 . − , rot v = ,− rot u = jx1 jx2 jx2 jx1 We will assume that , r , and 1 satisfy the following condition: (A1 ) 1 · min{ + r , } > 21 . We would like to point out that condition (A1 ) plays quite an important role in this work. Factually, it allows us to prove the (uniform) asymptotic compactness of the process on space V (see (4.35)). One can see that condition (A1 ) essentially requires that some viscosity coefficients of the micropolar fluid flow have to be large enough if (and thus 1 ) is given. The main objective of the present article is to prove the existence of the uniform attractor for the family of processes ) (see Section corresponding to Eqs. (1.1)–(1.5) when G(x, t) = (g(x, t), g(x, ˜ t)) is translation compact in L2loc (R; H 0 2 2 for notations and definitions). We first show the existence of L -uniform attractor AH(G0 ) in H and H 1 -uniform for the family of processes corresponding to Eqs. (1.1)–(1.5). Then we show A0 attractor A1H(G0 ) in V H(G0 ) ⊆ 1 AH(G0 ) , which implies the asymptotic smoothing effect of the micropolar fluid flow in the sense that the solutions become eventually more regular than the initial data. We would like to remark that Łukaszewicz and Sadowski [21] studied the nonautonomous 2D magneto-micropolar fluid flow and proved the existence of L2 -uniform attractor for the family of processes corresponding to the fluid in 2D unbounded domains in which the Poincaré inequality (1.6) holds. The theory and technique used in [21] are skewproduct flow and asymptotic compactness via energy equation. The idea of this paper originates from [21]. The main
610
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
difference between [21] and the present paper (concerning the proof to the existence of the H 1 -uniform attractor) is that the term B(u, w) (see (4.27)) in the enstrophy equation does not disappear, while B(u, w) disappears in the energy equation due to its antisymmetric property (see (2.4)). Thus, this paper can be regarded as an extended application of the technique of asymptotic compactness in [21] via enstrophy equation. The paper is organized as follows. Section 2 is preliminaries. In Section 3, we show the existence of a bounded for the family of processes corresponding to the fluid flow, as well as the (H ×H(G0 ), H ) uniformly absorbing set in H . continuity, asymptotic compactness and uniformly asymptotic compactness for the family of processes defined on H and the (V × H(G0 ), V ) continuity In Section 4, we prove the existence of a bounded uniformly absorbing set in V . Then we introduce two lemmas and use them to establish the asymptotic for the family of processes defined on V . We state the main result and compactness and uniformly asymptotic compactness for the family of processes in V prove it in Section 5. 2. Preliminaries In this paper, we use the following notations and operators: Lp ()-the usual Lebesgue space with norm · Lp () ; especially, · L2 () = · , H m ()-the Sobolev space { ∈ L2 (), ∇ k ∈ L2 (), k m} with norm · H m () (see [2]), H01 ()= closure of { ∈ C0∞ ()} in H 1 (), V = { ∈ C0∞ () × C0∞ () : = (1 , 2 ), ∇ · = 0}, H = closure of V in L2 () × L2 () with norm · ; H = dual space of H , V = closure of V in H 1 () × H 1 () with norm · V ; V = dual space of V , = Hilbert space H × L2 () with norm · ; H = dual space of H , H 1
, V = Hilbert space V × H0 () with norm · V ; V = dual space of V (·, ·)-the inner product in H or in H , and V , ·, · -the dual pairing between V and V or between V 3
2 A ∈ L(V , V ) ∩ L(V ∩ (H ()) , H ) is a linear continuous operator defined by Aw, = ( + r )(∇u, ∇) + (∇v, ∇ ),
, ∀ = (, ) ∈ V , ∀ w = (u, v) ∈ V
∩ (H 2 ())3 is the domain of A, D(A) = V , V ) is a continuous operator defined by B ∈ L(V × V B(u, w), = ((u · ∇)w, ),
, ∀ ∈ V , ∀u ∈ V , w ∈ V
, H ) is a continuous operator defined by N ∈ L(V N (w) = (−2r rot v, −2r rot u + 4r v),
, ∀w = (u, v) ∈ V
), R = [, +∞), H(G0 )= closure of {G0 (· + h) : h ∈ R} in L2loc (R; H ) is called to be translation compact in L2 (R; H ) if H(G0 ) is compact in L2 (R; H ), G0 ∈ L2loc (R; H loc loc 2 2 )= the set of functions that are translation compact in L (R; H ), Lc (R; H loc )= the set of functions G ∈ L2 (R; H ) satisfying L2 (R; H b
G2L2 (R;H) = G2L2 = sup b
b
t∈R t
loc
t+1
G(s)2 ds < + ∞,
), defined by {S(t)}t 0 -the natural translation semigroup acting on L2loc (R; H S(h)w(t) = w(t + h),
), ∀h 0, ∀w ∈ L2loc (R; H
“−→” denotes convergence in the strong topology, “” denotes convergence in the weak topology, “→” denotes embedding between spaces, distM (X, Y )=supx∈X inf y∈Y dist M (x, y) denotes the Hausdorff semidistance from X ⊂ M to Y ⊂ M in the metric space M.
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
611
The following basic facts were proved in [17–19,21]. Lemma 2.1. (i) Aw, w is equivalent to ∇w2 . Moreover, for any w ∈ D(A), 1
∇w2 Aw, w wAw √ ∇wAw, 1
(2.1)
hereafter = min{ + r , } and 1 is the constant from (1.6). (ii) There exists a positive constant depending only on such that 1
1
1
1
| B(u, w), | u 2 ∇u 2 w 2 ∇w 2 ∇, 1 2
1 2
1 2
×V , (u, w, ) ∈ V × V
1 2
| B(u, w), A | u ∇u ∇w Aw A,
(u, w, ) ∈ V × D(A) × D(A).
(2.2) (2.3)
In addition, B(u, w), w = 0,
. ∀(u, w) ∈ V × V
(2.4)
(iii) There exists a positive constant 1 = min{, } such that . ∀w ∈ V
Aw, w + N(w), w 1 w2V ,
(2.5)
(iv) There exists a positive constant C(r ) depending on r such that − N(w), Aw 22r w2V + 21 Aw2 , N(w) C(r )wV ,
∀w ∈ D(A),
(2.6)
. ∀w ∈ V
(2.7)
Using the notations and operators introduced above, we can translate the weak version of the initial boundary value problem (1.1)–(1.5) into an abstract form in the sense of distributions (see [16,28,30]) as following: jw + Aw + B(u, w) + N (w) = G(x, t), jt w|t= = w = (u , v ),
, t > , w = (u, v) ∈ V
∈ R.
(2.8) (2.9)
Moreover, we have the following result. ) and any ∈ R. Lemma 2.2. Let G(x, t) = (g(x, t), g(x, ˜ t)) ∈ L2c (R; H , then Eqs. (2.8)–(2.9) possesses a unique solution w = (u, v) satisfying (a) If w ∈ H ) ∩ C(R , H ) ∩ L2 (R ; V ), w ∈ L∞ (R ; H loc Moreover,
jw ). ∈ L2loc (R ; V jt
1 1+ G2L2 , ∀t , w(t) w e
1 1 b t t 1 w(t)2 + 1 w(s)2V ds w 2 + G(s)2 ds, ∀t ,
1 2
2 − 1 1 (t−)
1 +
1 1
(2.10)
(2.11) (2.12)
hereafter the constant 1 comes from (2.5). , then Eqs. (2.8)–(2.9) admits a unique solution w = (u, v) satisfying (b) If w ∈ V ) ∩ C(R , V ) ∩ L2 (R ; D(A)), w ∈ L∞ (R ; V loc Furthermore, (t
− )w(t)2V Q1
t − , w , 2
t
jw ). ∈ L2loc (R ; H jt
(2.13)
G(s) ds , 2
∀t ,
where Q1 (z1 , z2 , z3 ) is a monotone continuous function of z1 = t − , z2 and z3 .
(2.14)
612
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
Proof. (a) Let {n } be a sequence of bounded subdomains of and n → as n → +∞. One can use the Galerkin method to obtain the unique existence and regularity of solutions to Eqs. (2.8)–(2.9) on each n . Then one can use the approximation argument as that in [28] to obtain the solution of Eqs. (2.8)–(2.9) on . Since the method is classical and essentially the same as that in [21], we omit it and here we only prove (2.11) and (2.12). Let w(t) = (u(t), v(t)) , ∈ R. be the unique solution corresponding to initial data w = (u , v ) ∈ H Using w(t) to multiply Eq. (2.8), we can obtain 1 d 1 d w(t)2 + 1 1 w(t)2 w(t)2 + 1 w(t)2V 2 dt 2 dt 1 d w(t)2 + Aw, w + B(u, w), w + N (w), w 2 dt
1 1 1 = (G(t), w(t)) G(t)2 , w(t)2 + 2 2 1 1
(2.15)
where in the first inequality we used (1.6); in the second inequality we used (2.4)–(2.5); in the last inequality we used Schwartz inequality and Young inequality. By (2.15), we get d 1 G(t)2 . w(t)2 + 1 1 w(t)2 dt
1 1
(2.16)
Applying Gronwall inequality to (2.16) and taking some transformation in the integral, we have w(t)2
t 1 + e− 1 1 (t−s) G(s)2 ds w e
1 1 t t−1 1 e− 1 1 (t−s) G(s)2 ds + e− 1 1 (t−s) G(s)2 ds + · · · w 2 e− 1 1 (t−) +
1 1 t−1 t−2 t t−1 1 w 2 e− 1 1 (t−) + G(s)2 ds + e− 1 1 G(s)2 ds
1 1 t−1 t−2 t−2 −2 1 1 2 G(s) ds + · · · +e 2 − 1 1 (t−)
t−3
1 (1 + e− 1 1 + e−2 1 1 + · · ·)G2L2
1 1 b 1 w 2 e− 1 1 (t−) + (1 − e− 1 1 )−1 G2L2
1 1 b 1 1 1+ G2L2 . w 2 e− 1 1 (t−) +
1 1
1 1 b w 2 e− 1 1 (t−) +
Thus (2.11) holds. Now from (2.15) we can get 1 d
1 1 G(t)2 , w(t)2 + 1 w(t)2V w(t)2V + 2 dt 2 2 1 i.e., d 1 w(t)2 + 1 w(t)2V G(t)2 . dt
1 Integrating (2.17) w.r.t. time variable over [, t], we obtain (2.12).
(2.17)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
613
(b) Similar to the proof of (a), we only establish (2.14). For the sake of brevity, we set = 0 and t 0. Multiplying Eq. (2.8) by tAw, we can obtain
jw , tAw + Aw, tAw + B(u, w), tAw + N (w), tAw jt 1 d 1 (t Aw, w ) − Aw, w + tAw2 + t B(u, w), Aw + t N (w), Aw 2 dt 2 = t (G(t), Aw) 1 tAw2 + tG(t)2 . 4 =
(2.18)
From (2.3) and the facts u2 w2 , ∇u2 u2V w2V , we see that 1
1
1
3
t B(u, w), Aw tu 2 ∇u 2 ∇w 2 Aw 2 t Aw2 + 4 tu2 ∇u2 ∇w2 4 t Aw2 + 4 tw2 w4V . 4
(2.19)
Combining (2.6) and (2.18)–(2.19), we deduce that 1 d 1 (t Aw, w ) Aw, w + tG(t)2 + 4 tw2 w4V + 22r tw2V 2 dt 2 1 = Aw, w + tG(t)2 + tw2V 4 w2 w2V + 22r . 2
(2.20)
Now by Lemma 2.1(i), there exist two positive constants C1 and C2 such that C1 Aw, w w2V C2 Aw, w ,
. ∀w ∈ V
(2.21)
Eqs. (2.20) and (2.21) imply d (t Aw, w ) Aw, w + 2tG(t)2 + t Aw, w 2C2 4 w2 w2V + 4C2 2r . dt Thus we have d H (t)K(t)H (t) + I (t), dt
(2.22)
where H (t) = t Aw(t), w(t) , K(t) = 2C2 4 w(t)2 w(t)2V + 4C2 2r , I (t) = Aw(t), w(t) + 2tG(t)2 . Applying Gronwall inequality to (2.22), we obtain
t t
t t H (t) H (0) + I (s) ds exp K(s) ds = I (s) ds exp K(s) ds . 0
0
0
0
(2.23)
614
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
It then follows from (2.12) and (2.21) that t t I (s) ds = ( Aw(s), w(s) + 2sG(s)2 ) ds 0
0
t 1 G(s)2 ds w(s)2V ds + 2t C 1 0 0 t t 1 1 2 2 w(0) + G(s) ds + 2t G(s)2 ds C1 1
1 0 0 t 1 w(0)2 + (1 + t) C G(s)2 ds ,
t
(2.24)
0
where
1 1 1 = max C , ,2 . C1 1 C1 21
Similarly, by (2.12) we have t t K(s) ds = (2C2 4 w(s)2 w(s)2V + 4C2 2r ) ds 0
0
t
1 s 2 2C2 w(0) + G() d w(s)2V ds + 4C2 2r t
1 0 0 1 t w(0)2 1 t 4 2 2 2 2C2 w(0) + G(s) ds + 2 G(s) ds + 4C2 2r t
1 0
1
1 0
2 t 2 2 2 C w(0) + G(s) ds + t , 4
2
(2.25)
0
where
2 = max 2C2 max 1, C 4
1
1
,
21 41
, 4C2 2r
.
It follows from (2.21) and (2.23)–(2.25) that tw(t)2V C2 t Aw(t), w(t) = C2 H (t) 2 t t 2 2 2 2 C2 C1 w(0) + (1 + t) w(0) + G(s) ds exp C2 G(s) ds + t
. = Q1 t, w(0)2 ,
0
t
G(s)2 ds ,
0
(2.26)
0
1 [z2 + (1 + z1 )z3 ] exp{C 2 [(z2 + z3 )2 + z1 ]}. The proof of Lemma 2.2 is complete. where Q1 (z1 , z2 , z3 ) = C2 C
) In the sequel, we assume some field of the external force and moment G0 (x, t) = (g0 (x, t), g˜ 0 (x, t)) ∈ L2c (R; H and take H(G0 ) as the symbol space of Eq. (2.8). Obviously, H(G0 ) is strictly invariant under the acting of the natural translation semigroup {S(t)}t 0 , i.e., S(t)H(G0 ) = H(G0 ),
∀t 0.
(2.27)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
615
At the same time, we have G2L2 G0 2L2 < + ∞, b
b
∀G ∈ H(G0 ).
(2.28)
to H , via According to Lemma 2.2(a), we can define a family of processes {UG (t, )}t , G ∈ H(G0 ), from H UG (t, )w = w(t), where w(t) is the solution of Eqs. (2.8)–(2.9) with symbol G and initial data w ∈ H . Analogously, to V . Moreover, by Lemma 2.2(b), the family of processes {UG (t, )}t , G ∈ H(G0 ), can also be defined from V UG (, ) = I d (identity operator), UG (t, s)UG (s, ) = UG (t, ),
∀ ∈ R, ∀G ∈ H(G0 ),
∀t s , ∀G ∈ H(G0 ),
(2.29) (2.30)
and V , respectively. holds in H 3. Properties of the family of processes in H In this section, we present some properties of the family of processes corresponding to Eqs. (2.8)–(2.9). As mentioned in previous Introduction, these results can be derived in the same way as that in [21], which reduced the associated problem to the study of a semigroup acting in an extended phase space. At the same time, these properties can also be established by the argument similar to that in Section 4. Therefore, we just list them out. We first select some definitions from [7,13]. Definition 3.1. A set B0 is said to be a uniformly (w.r.t. G ∈ H(G0 )) absorbing set for the family of processes {U G (t, )}t , G ∈ H(G0 ), if for every bounded set B of H and any ∈ R, there exists t0 = t0 (B, ) such that G∈H (G0 ) UG (t, )B ⊆ B0 for all t t0 . × H(G0 ), H ) continuous if for Definition 3.2. The family of processes {UG (t, )}t , G ∈ H(G0 ), is said to be (H × H(G0 ) to H . any fixed t, ∈ R (t ), the mapping (w, G) → UG (t, )w is continuous from H if Definition 3.3. The family of processes {UG (t, )}t , G ∈ H(G0 ), is said to be asymptotically compact in H (n) ∞ (n) , whenever {w }∞ is bounded in H , {G(n) }∞ ⊂ H(G0 ) and {tn }∞ ⊂ R {UG(n) (tn , )w }n=1 is precompact in H n=1 n=1 n=1 with tn → +∞ as n → +∞. is said to be the uniformly (w.r.t. G ∈ H(G0 )) attracting set of {UG (t, )}t , G ∈ Definition 3.4. A set ⊂ H and any fixed ∈ R H(G0 ), in H if for any bounded set B of H lim
sup
t→+∞ G∈H(G ) 0
dist H(UG (t, )B, ) = 0.
is called The family of processes {UG (t, )}t , G ∈ H(G0 ), possessing a compact uniformly attracting set in H uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact in H . ). Then Lemma 3.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H
1 2 . : w B0 = w ∈ H 1+ G0 L2 = R0 b
1 1
1 1
(3.1)
is a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set for the family of processes {UG (t, )}t , G ∈ H(G0 ), . corresponding to Eqs. (2.8)–(2.9) in H ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.2. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H corresponding to Eqs. (2.8)–(2.9) is (H × H(G0 ), H ) continuous. ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.3. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . corresponding to Eqs. (2.8)–(2.9) is asymptotically compact in H
616
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.4. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . corresponding to Eqs. (2.8)–(2.9) is uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact in H 4. Properties of the family of processes in V ) corresponding to Eqs. (2.8)–(2.9) has In this section, we mainly prove that the family of processes (defined on V the analogous properties with those showed in Section 3. ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ) (defined on Lemma 4.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H , i.e., V ), corresponding to Eqs. (2.8)–(2.9) possesses a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set BV0 in V V V for any bounded set B of V and any fixed ∈ R, there exists a time t1 = t1 (B , ) such that
UG (t, )BV ⊆ BV0 ,
∀t t1 .
(4.1)
G∈H(G0 )
implies that BV is also bounded → H is continuous, the set BV being bounded in V Proof. Since the embedding V . We assert that the set in H UG ( + 1, )B0 (4.2) BV0 = G∈H(G0 ) ∈R
is a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set for {UG (t, )} t , G ∈ H(G 0 ), in V , where B0 is the uniformly absorbing set from Lemma 3.1. Indeed, from (3.1) we see that G∈H(G0 ) ∈R UG ( + 1, )B0 is also a . Now for any bounded set BV of uniformly (w.r.t. G ∈ H(G0 )) absorbing set for {UG (t, )}t , G ∈ H(G0 ), in H V . Thus, for any fixed ∈ R, , B is also bounded in H V G∈H(G0 )
UG (t, )BV ⊆
UG ( + 1, )B0 = BV0 ,
∀t t0 (BV , ).
(4.3)
G∈H(G0 ) ∈R
. Precisely, we have Moreover, we deduce from (2.14) and (3.1) that BV0 is bounded in V . w2V Q1 (1, R02 , G0 2L2 ) = R12 , b
∀w ∈ BV0 ,
(4.4)
where Q1 is the monotone continuous function from Lemma 2.2(b). The proof is complete.
Remark 4.1. From the proof of Lemma 4.1 we see that BV0 also absorbs uniformly (w.r.t. G ∈ H(G0 )) all bounded in the norm of V . sets of H ×H(G0 ), V ) continuity, asymptotic compactness and uniformly (w.r.t. G ∈ H(G0 )) asymptotic The definition of (V are similar to Definitions 3.2–3.4. compactness for {UG (t, )}t , G ∈ H(G0 ), in V ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ) (defined on Lemma 4.2. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H ), corresponding to Eqs. (2.8)–(2.9) is (V × H(G0 ), V ) continuous. V (n) (n) (n) (n) ∞ Proof. Let t and be fixed and t . Let {(w , G(n) )}∞ n=1 ={((u , v ), G )}n=1 ⊂ V ×H(G0 ) be a sequence that ∞ (n) (n) (n) converges to some (w , G)=((u , v ), G) ∈ V ×H(G0 ), {w (t)}n=1 ={(u (t), v (t))}∞ n=1 and w(t)=(u(t), v(t)) (n) be the corresponding solutions of Eqs. (2.8)–(2.9) with symbols {G(n) }∞ and G, and initial data {w }∞ n=1 n=1 and w , respectively. Set
(n) (t) = w(t) − w (n) (t) = UG (t, )w − UG(n) (t, )w(n) ,
n = 1, 2, . . . .
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
617
For each n we see that (n) (t) is a solution of the following equations: j (n) + A (n) + B(u, w) − B(u(n) , w(n) ) + N ( (n) ) = G − G(n) , jt
(4.5)
(n) (n) |t= = (n) = w − w .
(4.6)
Multiplying (4.5) with A (n) , we can obtain 1 d A (n) , (n) + A (n) 2 + B(u, w), A (n) − B(u(n) , w(n) ), A (n) + N ( (n) ), A (n) 2 dt 1 = (G − G(n) , A (n) ) A (n) 2 + G − G(n) 2 . 4
(4.7)
On the one hand, it follows from (2.6) that − N( (n) ), A (n) 22r (n) 2V + 21 A (n) 2 .
(4.8)
On the other hand, by (2.4) we have B(u, w), A (n) − B(u(n) , w(n) ), A (n) = B(u − u(n) , w), A (n) + B(u(n) , w), A (n) − B(u(n) , w(n) ), A (n) = B(u − u(n) , w), A (n) + B(u(n) , (n) ), A (n) .
(4.9)
Combining (2.3), (4.9) and the facts u − u(n) (n) ,
∇(u − u(n) ) u − u(n) V (n) V ,
we obtain | B(u, w), A (n) − B(u(n) , w(n) ), A (n) | = | B(u − u(n) , w), A (n) + B(u(n) , (n) ), A (n) | 1
1
1
1
1
1
1
3
u − u(n) 2 ∇(u − u(n) ) 2 ∇w 2 Aw 2 A (n) + u(n) 2 ∇u(n) 2 ∇ (n) 2 A (n) 2
18 A (n) 2 + 22 u − u(n) ∇(u − u(n) )∇wAw + 18 A (n) 2 + 24 u(n) 2 ∇u(n) 2 ∇ (n) 2 41 A (n) 2 + 22 (n) ∇ (n) ∇wAw + 24 (n) 2 ∇ (n) 2 ∇ (n) 2 41 A (n) 2 + 22 (n) 2V wV Aw + 24 (n) 2 (n) 4V .
(4.10)
Now taking (4.7)–(4.8) and (4.10) into account, we get 1 d 1 1 A (n) , (n) + A (n) 2 A (n) 2 + 22r (n) 2V + A (n) 2 + G − G(n) 2 2 dt 2 4 1 + A (n) 2 + 22 (n) 2V wV Aw + 24 (n) 2 (n) 4V . 4 Using (2.21), we have d A (n) , (n) 42r (n) 2V + 2G − G(n) 2 + 42 (n) 2V wV Aw + 44 (n) 2 (n) 4V dt 4 (n) 2V 2r + 2 wV Aw + 4 (n) 2 (n) 2V + 2G − G(n) 2 4C2 A (n) , (n) 2r + 2 wV Aw + 4 (n) 2 (n) 2V + 2G − G(n) 2 .
(4.11)
618
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
Applying Gronwall inequality to (4.11), we obtain A
(n)
(t),
(n)
(t) A
(n)
(),
× exp 4C2
(n)
() + 2
t
2r
t
G(s) − G
2
(n)
2
(s) ds
4
+ wV Aw +
(n) 2
(n) 2V
ds .
(4.12)
Again using (2.21), we get from (4.12) that 1 UG (t, )w − UG(n) (t, )w(n) 2V C2 =
1 (n) (t)2V C2
A (n) (t), (n) (t) t 1 (n) 2 (n) 2 w − w V + 2 G(s) − G (s) ds C1
t × exp 4C2 2r + 2 wV Aw + 4 (n) 2 (n) 2V ds .
(4.13)
When t and are fixed, (2.11)–(2.14) imply
t exp 4C2 2r + 2 wV Aw + 4 (n) 2 (n) 2V ds < + ∞.
× H(G0 ), V )-continuity of the family of processes {UG (t, )}t , G ∈ H(G0 ), Therefore, (4.13) implies the (V corresponding to Eqs. (2.8)–(2.9). The proof of Lemma 4.2 is complete. , we need the following two In order to prove the asymptotic compactness of {UG (t, )}t , G ∈ H(G0 ), in V lemmas. (n) (n) ∞ Lemma 4.3. Let {w }∞ n=1 be a sequence in V converging strongly to some w ∈ V in the norm of H and {G }n=1 (n) be a sequence of H(G0 ) converging strongly to some G ∈ H(G0 ). Then for any fixed ∈ R, UG(n) (·, )w −→ ), ∀ T . UG (·, )w strongly in L2 (, T ; V
Proof. From (2.21) we have w2V C2 Aw, w C2 wAw,
∀ w ∈ D(A).
Thus for any t , UG(n) (t, )w(n) − UG (t, )w 2V C2 UG(n) (t, )w(n) − UG (t, )w A(UG(n) (t, )w(n) − UG (t, )w ).
(4.14)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
619
Integrating (4.14) w.r.t. t over [, T ], we have, using Young inequality, T UG(n) (s, )w(n) − UG (s, )w 2V ds
C2
T
UG(n) (s, )w(n) − UG (s, )w A(UG(n) (s, )w(n) − UG (s, )w ) ds
C2
T
×
T
UG(n) (s, )w − UG (s, )w ds 2
(n)
21
A(UG(n) (s, )w − UG (s, )w ) ds 2
(n)
21 .
(4.15)
T 1 (n) (n) Now (2.13) implies { A(UG(n) (s, )w − UG (s, )w ) ds} 2 < + ∞ and Lemma 3.2 implies UG(n) (s, )w as n → +∞. Therefore, we obtain from (4.15) that −→ UG (s, )w strongly in H T UG(n) (s, )w(n) − UG (s, )w 2V ds = 0. lim n→+∞
The proof is complete.
(n) ∞ Lemma 4.4. Let {w }∞ n=1 be a sequence in V converging weakly to some w in V and {G }n=1 be a sequence of (n) H(G0 ) converging strongly to some G ∈ H(G0 ). Then for any fixed t , t, ∈ R, UG(n) (t, )w UG (t, )w weakly in V and (n)
UG(n) (·, )w(n) UG (·, )w
weakly in L2 (, T ; D(A)), ∀ T .
(4.16)
Proof. For any fixed t , t, ∈ R, let UG(n) (t, )w(n) = w (n) (t),
n = 1, 2, . . . ,
UG (t, )w = w(t).
Analogous with Lemma 2.2(b), we find that {w(n) (t)}∞ n=1 is bounded in ) ∩ L2 (R ; D(A)) L∞ (R ; V loc jw(n)
2 and { jt }∞ n=1 is bounded in Lloc (R ; H ). The rest of the proof is essentially the same as that of Lemma 8.1 in [21] and we omit it.
Next we will devote to prove the following lemma, which plays the essential role when we prove the uniformly (w.r.t. . G ∈ H(G0 )) asymptotic compactness of {UG (t, )}t , G ∈ H(G0 ), in V ) and (A1 ) hold. Then the family of processes {UG (t, )}t , G ∈ H(G0 ) Lemma 4.5. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . (defined on V ), corresponding to Eqs. (2.8)–(2.9) is asymptotically compact in V (n) ∞ (n) ∞ Proof. Let {w }∞ n=1 be a bounded sequence in V , {G }n=1 ⊂ H(G0 ) and {tn }n=1 ⊂ R with tn → +∞ as n → +∞. For any fixed ∈ R, we see from Lemma 4.1 that there is a time T0 = T0 (R, ) (where R is a constant (n) (n) V V satisfying w V R for all n) such that for all tn T0 , {UG(n) (tn , )w }∞ n=1 ⊆ B0 , where B0 is the bounded . This ensures that {UG(n) (tn , )w(n) }∞ is weakly precompact in V uniformly (w.r.t. G ∈ H(G0 )) absorbing set in V n=1 and thus we have
UG(n) (tn , )w(n) w
as n → +∞, weakly in V
(4.17)
and some subsequence (still denote by) {UG(n) (tn , )w(n) }∞ . Similarly, for each T > 0 and tn T0 +T , for some w ∈ V n=1 (n) . wT = UG(n) (tn − T , )w(n) wT
as n → +∞, weakly in V
(4.18)
620
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
. Now from the definition of {S(t)}t 0 we can check the following translation identity holds for some wT ∈ V US(h)G (t, ) = UG (t + h, + h),
∀h 0, t , t, ∈ R, ∀G ∈ H(G0 ).
(4.19)
Applying (2.29)–(2.30) and (4.19), we get UG(n) (tn , ) = UG(n) (tn , tn − T )UG(n) (tn − T , ) = US(tn −T )G(n) (T , 0)UG(n) (tn − T , ),
tn − T .
(4.20)
(n)
Setting GT = S(tn − T )G(n) , we have by (4.18) and (4.20) that for ∀T > 0, tn − T , (n)
UG(n) (tn , )w(n) = UG(n) (T , 0)UG(n) (tn − T , )w(n) = UG(n) (T , 0)wT .
(4.21)
T
T
(n) (n) ∞ 2 Since {GT }∞ n=1 ⊂ H(G0 ) and H(G0 ) is compact in Lloc (R; H ), there exist a subsequence of {GT }n=1 (still denote (n) by {GT }∞ n=1 ) and some GT ∈ H(G0 ) such that (n)
GT −→ GT
) as n → +∞ for every T > 0. strongly in L2loc (R; H
(4.22)
Taking (4.17)–(4.18), (4.21)–(4.22) and Lemma 4.4 into account, we obtain w = UGT (T , 0)wT
for every T > 0,
(4.23)
where we also used the uniqueness of the limit. Now it follows from (4.17) and (4.21) that (n)
lim inf UG(n) (tn , )w(n) V = lim inf UG(n) (T , 0)wT V wV . n
n
(4.24)
T
Next we prove (n)
lim sup UG(n) (tn , )w(n) V = lim sup UG(n) (T , 0)wT V wV . n
(4.25)
T
n
. In fact, using Aw to multiply Eq. (2.8), we obtain To this end, we use the argument of enstrophy equation in V 1 1 d Aw, w + Aw, w = (G, Aw) − F (w), 2 dt 2
(4.26)
where F (w) = Aw2 − 21 Aw, w + B(u, w), Aw + N (w), w . Integrating (4.26) w.r.t. time variable over [, t], as following: we obtain the enstrophy equation in V t −(t−) +2 e−(t−s) (G(s), Aw(s)) ds Aw(t), w(t) = Aw(), w() e
−2
t
e−(t−s) F (w(s)) ds,
t .
(4.27)
(n)
Applying the enstrophy equation to UG(n) (T , 0)wT , we obtain T
AU G(n) (T , 0)wT , UG(n) (T , 0)wT = Aw T , wT e−T + 2 (n)
T
(n)
(n)
T
−2 0
By (4.4) and (2.21) we have 1 (n) (n) lim sup Aw T , wT e−T R12 e−T . C1 n
(n)
T
T 0
(n) (n) ds e−(T −s) GT (s), AU G(n) (s, 0)wT T
e−(T −s) F UG(n) (s, 0)wT
(n)
ds.
(4.28)
T
(4.29)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
621
, we infer from (4.16), (4.18) and (4.22) that Since A is a linear continuous mapping from D(A) to H ), weakly in L2 (, T ; H
(n)
AU G(n) (s, 0)wT AU GT (s, 0)wT T
∀T > .
(4.30)
Thus (4.22) and (4.30) imply
T
lim 2
n→+∞
0
(n) (n) ds = 2 e−(T −s) GT (s), AU G(n) (s, 0)wT T
T 0
e−(T −s) (GT (s), AU GT (s, 0)wT ) ds.
(4.31)
We next handle the third term of the right-hand side in (4.28). In fact, we have the following result. (n)
Lemma 4.6. Set UG(n) (s, 0)wT = (u(n) , v (n) ) and UGT (s, 0)wT = (u, v). Then T
1 (n) (n) (n) e−(T −s) AU G(n) (s, 0)wT 2 − AU G(n) (s, 0)wT , UG(n) (s, 0)wT ds n T T T 2 0 T 1 −(T −s) 2 AU GT (s, 0)wT − AU GT (s, 0)wT , UGT (s, 0)wT ds, e 2 0
T
lim inf
T
lim
n→+∞ 0 T
=
0
lim
n→+∞ 0 T
=
0
(n) (n) e−(T −s) B u(n) , UG(n) (s, 0)wT , AU G(n) (s, 0)wT ds T
T
e−(T −s) B(u, UGT (s, 0)wT ), AU GT (s, 0)wT ds, T
(4.32)
(4.33)
(n) (n) e−(T −s) N UG(n) (s, 0)wT , AU G(n) (s, 0)wT ds T
T
e−(T −s) N (UGT (s, 0)wT ), AU GT (s, 0)wT ds.
(4.34)
Proof. We first prove (4.32). Define '·, ·( : D(A) × D(A) −→ R as 'w1 , w2 ( = (Aw 1 , Aw2 ) − 21 Aw 1 , w2 ,
∀w1 , w2 ∈ D(A).
1 By (A1 ), we have 1 − 2 > 0. Thus by Lemma 2.1(i), we get 1
1 1 2 'w( = 'w, w( = Aw2 − Aw, w Aw2 − √ ∇wAw 2 2 1 1 1 2 2 Aw − Aw2 Aw = 1 − 2 1 2 1 and thus we have 1 2 1− Aw2 'w( Aw2 , 2 1
∀w ∈ D(A).
(4.35)
T 1 2 Since 0 < e−(T −s) 1 for any T s, (4.35) implies that ( 0 e−(T −s) ' · ( ds) 2 is a norm on L2 (0, T ; D(A)) that is T 1 equivalent to ( 0 A · 2 ds) 2 . At the same time we derive from (4.17)–(4.18), (4.21) and Lemma 4.4 that (n)
UG(n) (s, 0)wT UGT (s, 0)wT T
Thus we get (4.32) from (4.36).
weakly in L2 (0, T ; D(A)).
(4.36)
622
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
We now prove (4.33). Set (n)
(n)
(n)
L1 (s) = B(u(n) (s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT , T
T
L2 (s) = B(u(s), UGT (s, 0)wT ), AU GT (s, 0)wT . We have
T
e
−(T −s)
0
T 0
(n) (L1 (s) − L2 (s)) ds
(n) (n) e−(T −s) B(u(n) (s) − u(s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT ds T
T
T (n) (n) + e−(T −s) B(u(s), UG(n) (s, 0)wT − UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T T 0 T (n) −(T −s) e B(u(s), UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds + T
0
. (n) (n) (n) = M1 + M2 + M3 .
(4.37)
Using (2.3), Lemma 4.1 and the fact (n)
∇(u(n) (s) − u(s)) u(n) (s) − u(s)V UG(n) (s, 0)wT − UGT (s, 0)wT V , T
we get (n) M1
T (n) (n) −(T −s) (n) e B(u (s) − u(s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT ds = T T 0 T 1 1 (n) 1 (n) 3 u(n) (s) − u(s) 2 ∇(u(n) (s) − u(s)) 2 ∇UG(n) (s, 0)wT 2 AU G(n) (s, 0)wT 2 ds 0
T
1
R12
R1
T 0 T
1
1
3
T
1 2
(n)
(n)
3
UG(n) (s, 0)wT − UGT (s, 0)wT V AU G(n) (s, 0)wT 2 ds T
R1
(n)
u(n) (s) − u(s) 2 ∇(u(n) (s) − u(s)) 2 AU G(n) (s, 0)wT 2 ds
T
0
T
0
T
(n) UG(n) (s, 0)wT T
− UGT (s, 0)wT 2V ds
41
T 0
(n) AU G(n) (s, 0)wT 2 ds T
43 .
(4.38)
Now (2.13) implies
T
0
(n) AU G(n) (s, 0)wT 2 ds T
43
< + ∞.
(4.39)
At the same time, from (4.18) and Lemma 3.3 we see that as n → +∞. strongly in H
(n)
wT −→ wT
(4.40)
Then it follows from Lemma 4.3 and (4.40) that (n)
UG(n) (s, 0)wT −→ UGT (s, 0)wT T
) as n → +∞. strongly in L2 (0, T ; V
(4.41)
Combining (4.38), (4.39) and (4.41), we have (n)
lim M1 = 0.
n→+∞
(4.42)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
623
By the argument similar to what used for deriving (4.42), we have T (n) (n) (n) −(T −s) M2 = e B(u(s), UG(n) (s, 0)wT − UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T
0
T
R1
(n)
UG(n) (s, 0)wT − UGT (s, 0)wT 2V ds
41
T
0
T 0
T
×
T
(n)
A(UG(n) (s, 0)wT − UGT (s, 0)wT )2 ds
(n)
AU G(n) (s, 0)wT 2 ds
21
T
1 4
T
0
and (n)
lim M2 = 0.
(4.43)
n→+∞
), by (4.30) we obtain Since e−(T −s) B(u(s), UGT (s, 0)wT ) ∈ L2 (0, T ; V T (n) −(T −s) e B(u(s), UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds lim n→+∞
T
0
(n) = lim M3 n→+∞
= 0.
(4.44)
Eqs. (4.37) and (4.42)–(4.44) imply that (4.33) holds. We next prove (4.34). Set (n)
(n)
(n)
N1 (s) = N (UG(n) (s, 0)wT ), AU G(n) (s, 0)wT , T
T
N2 (s) = N (UGT (s, 0)wT ), AU GT (s, 0)wT . We have
T
e
−(T −s)
0
T
e
−(T −s)
0
+
T
0
(n) (N1 (s) − N2 (s)) ds
(n) (n) N (UG(n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T T
(n) e−(T −s) N (UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds T
. (n) (n) = K1 + K2 .
(4.45)
Now by (2.7) and Young inequality, we obtain T (n) (n) −(T −s) e N(U (n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU (n) (s, 0)wT ds G G T
0
T
T
0
(n) N (UG(n) (s, 0)wT T
T
C(r ) 0
− UGT (s, 0)wT ) ds
(n) UG(n) (s, 0)wT T
2
21
− UGT (s, 0)wT 2V ds
T 0
(n) AU G(n) (s, 0)wT 2 ds T
21
T 0
21
(n) AU G(n) (s, 0)wT 2 ds T
21 .
(4.46)
Eqs. (4.39), (4.41) and (4.46) imply T (n) (n) (n) e−(T −s) N (UG(n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU G(n) (s, 0)wT ds lim K1 = lim n→+∞
n→+∞
= 0.
0
T
T
(4.47)
624
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
, we infer from (2.7) that At the same time, since UGT (s, 0)wT ∈ BV0 ⊂ V ). e−(T −s) N(UGT (s, 0)wT ) ∈ L2 (0, T ; H Thus it follows from (4.30) that T (n) (n) lim K2 = lim e−(T −s) N (UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds n→+∞ n→+∞ T 0
= 0.
(4.48)
Eqs. (4.45) and (4.47)–(4.48) imply that (4.34) holds. The proof of Lemma 4.6 is complete. As a corollary of Lemma 4.6, we have that T (n) −(T −s) lim sup − e F (UG(n) (s, 0)wT ) ds = − lim inf n
n
T
0
T
− 0
T
e
−(T −s)
0
(n) F (UG(n) (s, 0)wT ) ds T
e−(T −s) F (UGT (s, 0)wT ) ds.
(4.49)
We continue to prove Lemma 4.5. Combining (4.28)–(4.29), (4.31) and (4.49), we obtain T 1 (n) (n) lim sup AU G(n) (T , 0)wT , UG(n) (T , 0)wT R12 e−T + 2 e−(T −s) (GT (s), AU GT (s, 0)wT ) ds T T C n 1 0 T e−(T −s) F (UGT (s, 0)wT ) ds. (4.50) −2 0
At the same time, applying (4.27) to w = UGT (T , 0)wT , we obtain T AU GT (T , 0)wT , UGT (T , 0)wT = Aw T , wT e−T + 2 e−(T −s) (GT (s), AU GT (s, 0)wT ) ds 0
T
−2 0
e−(T −s) F (UGT (s, 0)wT ) ds.
(4.51)
It follows from (4.50)–(4.51) and (4.23) that (n)
(n)
lim sup AU G(n) (T , 0)wT , UG(n) (T , 0)wT T
n
T
AU GT (T , 0)wT , UGT (T , 0)wT + Aw, w +
1 2 −T R e , C1 1
1 2 R1 − Aw T , wT e−T C1
∀ T > 0.
(4.52)
Noticing that A·, · is equivalent to · 2V , we obtain by letting T tend to +∞ in (4.52) that (n)
lim sup UG(n) (T , 0)wT 2V w2V . T
n
is a Hilbert space, we deduce from (4.24)–(4.25), (4.17) and (4.21) that Hence (4.25) has been proved. Because V (n) limn→+∞ UG(n) (tn , )w − wV = 0. The proof of Lemma 4.5 is complete. T
) and condition (A1 ) hold. Then the family of processes {UG (t, )}t , Lemma 4.7. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H G ∈ H(G0 ) (defined on V ), corresponding to Eqs. (2.8)–(2.9) is uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact . in V
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
625
Proof. We only need to prove that {UG (t, )}t , G ∈ H(G0 ), possesses a compact uniformly (w.r.t. G ∈ H(G0 )) . We claim that the set attracting set in V
.
,H(G0 ) (BV0 ) =
UG (s, )BV0
for each ∈ R
(4.53)
t G∈H(G0 ) s t
, where the bar in is a compact uniformly (w.r.t. G ∈ H(G0 )) attracting set for {UG (t, )}t , G ∈ H(G0 ), in V V (4.53) denotes taking closure in V . In fact, the set ,H(G0 ) (B0 ) defined by (4.53) can be characterized, similarly to the semigroup case, as following: ⎧ V ⎨ w ∈ ,H(G0 ) (B0 ) ⎩
⇐⇒
V (n) ∞ there exist {w (n) }∞ n=1 ⊂ B0 , {G }n=1 ⊂ H(G0 ), {tn } ⊂ R with tn → +∞ as n → +∞ as n → +∞. such that UG(n) (tn , )w(n) −→ w in V
(4.54)
Indeed, (4.54) implies that 0,H(G0 ) (BV0 ) = ,H(G0 ) (BV0 ) for each ∈ R, in other words, ,H(G0 ) (BV0 ) is independent of . The rest of the proof to this lemma is similar to the proof of Propositions 4.1, 4.2 and 4.3(iii) in [13]. Here we only sketch the main steps and omit the detailed proof. . This assertion can be established by the uniformly (w.r.t. Step 1: ,H(G0 ) (BV0 ) is a nonempty compact set in V G ∈ H(G0 )) absorbing property (Lemma 4.1), asymptotic compactness property (Lemma 4.5) of {UG (t, )}t , G ∈ and the characterization of ,H(G ) (BV ) described by (4.54). H(G0 ), in V 0 0 and any fixed ∈ R, Step 2: For any bounded set BV of V lim
sup
t→+∞ G∈H(G ) 0
dist V (UG (t, )BV , ,H(G0 ) (BV )) = 0.
(4.55)
Eq. (4.55) could be established by contradiction, with the help of Lemma 4.5 and (4.54). , Step 3: For any bounded set BV ⊂ V
,H(G0 ) (BV ) ⊆ ,H(G0 ) (BV0 ).
(4.56)
Eq. (4.56) can be verified by using (2.30), (4.54) and Lemma 4.1. Eqs. (4.55) and (4.56) imply the uniformly (w.r.t. . G ∈ H(G0 )) attracting property of ,H(G0 ) (BV0 ) in V The proof of Lemma 4.7 is complete. 5. Existence and regularity of uniform attractor In this section, we summarize the results obtained in the previous sections. Then we state our main result and prove it. ) and {UG (t, )}t , G ∈ H(G0 ), be the family of processes corDefinition 5.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H is said to be the uniform (w.r.t. G ∈ H(G0 )) attractor of responding to Eqs. (2.8)–(2.9). A closed set ⊂ H {UG (t, )}t , G ∈ H(G0 ), in H if satisfies and any fixed ∈ R (1) (Uniformly attracting property). For any bounded set B of H lim
sup
t→+∞ G∈H(G ) 0
dist H(UG (t, )B, ) = 0.
(5.1)
(2) (Minimal property). is the minimal set (for inclusion relation) among the closed sets satisfying (1). with H Similarly, we can define the uniform (w.r.t. G ∈ H(G0 )) attractor for {UG (t, )}t , G ∈ H(G0 ), in V being replaced by V in Definition 5.1.
626
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
The main result of this paper reads as follows: ) and condition (A1 ) hold. Then the family of processes {UG (t, )}t , Main Theorem. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H G ∈ H(G0 ), corresponding to Eqs. (2.8)–(2.9) possesses a uniform (w.r.t. G ∈ H(G0 )) attractor A0H(G0 ) in H 1 , respectively. Moreover and a uniform (w.r.t. G ∈ H(G0 )) attractor AH(G0 ) in V A0H(G0 ) ⊆ A1H(G0 ) .
(5.2)
Proof. According to the well-established theory (see e.g., Theorem IV3.1 in [7]), the existence of uniform (w.r.t. and A1 G ∈ H(G0 )) attractors A0H(G0 ) in H H(G0 ) in V for {UG (t, )}t , G ∈ H(G0 ), can be derived from (see below Remark 5.1) and the Lemmas 3.4 and 4.7, respectively. We next prove (5.2). Since A1H(G0 ) is bounded in V 1 1 → H is continuous, we see A embedding V H(G0 ) is bounded in H . Also we deduce from Remark 4.1 that AH(G0 ) and thus A1 attracts uniformly (w.r.t. G ∈ H(G0 )) all bounded sets of H H(G0 ) can be regarded as a bounded uniformly . By the minimality property of A0 (w.r.t. G ∈ H(G0 )) attracting set for {UG (t, )}t , G ∈ H(G0 ), in H H(G0 ) , we obtain (5.2). The proof is complete. Remark 5.1. By the well-established theory (see e.g., [7, Theorem IV5.1, p. 89]), A0H(G0 ) and A1H(G0 ) are compact and V , respectively. in H Acknowledgments We express our sincere thanks to the anonymous reviewers for their careful reading of the manuscript and giving valuable comments and suggestions. It is their contributions that greatly improves the manuscript. We also thank the editors for their kind help. The corresponding author thanks Professors Yongsheng Li and Zhengyi Lu for their constant supports and encouragements. References [1] F. Abergel, Existence and finite dimensionality of global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108. [2] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] A.V. Babin, The attractor of a Navier–Stokes system in an unbounded channel-like domain, J. Dyn. Diff. Eqs. 4 (1992) 555–584. [4] A.V. Babin, M.I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Roy. Soc. Edinburg 116A (1990) 221–243. [5] J.M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997) 475–502; J.M. Ball, J. Nonlinear Sci. 8 (1998) 233 Erratum. [6] J. Chen, Z. Chen, B. Dong, Existence of H 2 -global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl. 322 (2006) 512–522. [7] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, vol. 49, AMS, Providence, RI, 2002. [8] M.A. Efendiev, S.V. Zelik, The attractor for nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688. [9] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [10] J.M. Ghidaglia, A note on the strong convergence towards the attractors for damped forced KdV equations, J. Differential Equations 110 (1994) 356–359. [11] J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [12] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Paris, Masson, 1991. [13] Y. Hou, K. Li, The uniform attractors for the 2D nonautonomous Navier–Stokes flow in some unbounded domain, Nonlinear Anal. 58 (2004) 609–630. [14] N. Ju, The H 1 -compact global attractor for the solutions to the Navier–Stokes equations in 2D unbounded domains, Nonlinearity 13 (2000) 1227–1238. [15] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. [16] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969. [17] G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999.
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
627
[18] G. Lukaszewicz, Long time behavior of 2D micropolar fluid flows, J. Math. Comput. Model. 34 (2001) 1787–1802. [19] G. Lukaszewicz, Asymptotic behavior of micropolar fluid flows, Inter. J. Eng. Sci. 41 (2003) 259–269. [20] G. Lukaszewicz, M.A. Rojas-Medar, M.M. Santos, Stationary micropolar fluid with boundary data in L2 , J. Math. Appl. Anal. 271 (2002) 91–107. [21] G. Lukaszewicz, W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys. 55 (2004) 247–257. [22] Q. Ma, S. Wang, C. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J. 51 (2002) 1541–1559. [23] I. Moise, R. Rosa, X.M. Wang, Attractors for non-compact semigroup via energy equations, Nonlinearity 11 (1998) 1369–1393. [24] I. Moise, R. Rosa, X.M. Wang, Attractors for non-compact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst. 10 (2004) 473–496. [25] R. Rosa, The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998) 71–85. [26] W. Sadowski, Upper bound for the number of degrees of freedom for magneto-micropolar flows and turbulence, Inter. J. Eng. Sci. 41 (2003) 789–800. [27] G. Sell, Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. [28] R. Temam, Navier–Stokes Equations (Theory and Numerical Analysis), third ed., Elsevier Science Publishers, The Netherlands, 1984. [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. [30] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, second ed., SIAM, Philadelphia, 1995. [31] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D 88 (1995) 167–175. [32] S.V. Zelik, The attractors for nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst. 7 (2001) 593–641. [33] S.V. Zelik, Attractors for reaction–diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math. 56 (2003) 584–637. [34] C. Zhao, Y. Li, H 2 -compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal. 56 (2004) 1091–1103.
H 1-Uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains夡 Caidi Zhaoa,∗ , Shengfan Zhoub , Xinze Liana a College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, PR China b Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, PR China
Received 12 December 2005; accepted 7 December 2006
Abstract This paper discusses the long time behavior of solutions for a two-dimensional (2D) nonautonomous micropolar fluid flow in 2D unbounded domains in which the Poincaré inequality holds. We use the energy method to obtain the so-called asymptotic compactness of the family of processes associated with the fluid flow and establish the existence of H 1 -uniform attractor. Then we prove that an L2 -uniform attractor belongs to the H 1 -uniform attractor, which implies the asymptotic smoothing effect for the fluid flow in the sense that the solutions become eventually more regular than the initial data. © 2007 Published by Elsevier Ltd. MSC: 35B41; 35Q35; 76D03 Keywords: Uniform attractor; Micropolar fluid flow; Unbounded domains; Asymptotic smoothing effect
1. Introduction Attractor is an important concept in the study of dynamical systems. There are some classical works concerning this subject (see e.g. [7,11,12,15,27,29]). In the book [29] Temam studied systematically the global attractor associated with many concrete autonomous equations arising in mathematical physics. Later, Chepyzhov and Vishik [7] presented a general theory to deal with nonautonomous equations. Generally speaking, if the nonautonomous terms in the addressed equations have some uniformity, the long time behavior of solutions could be considered with respect to (w.r.t.) all nonautonomous terms in the so-called symbol space. When the spatial domain is unbounded, there exists a considerable obstacle coming from the loss of compact embedding between usual Sobolev spaces. One remedy to this case is to consider weighted spaces (see [1,3,4,8,32,33]). Later on, Ball (see e.g. [5]) formulated the idea of energy equation to obtain some asymptotic compactness for the associated semigroup and thus could avoid the weighted spaces. In consequence, this idea was successfully developed by many authors to study the autonomous equations [10,14,23,25,31,34] and nonautonomous equations [13,21,24]. 夡 This work is supported by the NSF of China under grant number 10471086, and is supported by the NSF of Zhejiang Province under grant number M103043. ∗ Corresponding author. E-mail address: [email protected] (C. Zhao).
1468-1218/$ - see front matter © 2007 Published by Elsevier Ltd. doi:10.1016/j.nonrwa.2006.12.005
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
609
The micropolar fluid model is an essential generalization of the Navier–Stokes model in the sense that it takes into account the microstructure of the fluid [18]. The micropolar fluids were introduced in [9], which are non-Newtonian fluids and have many applications (see e.g., [9,17–21]). Mathematical theory of micropolar fluid flows were extensively studied. One can refer to [6,9,17–21,26] and the references therein. Especially, when the external force g(x, t) and moment g(x, ˜ t) are independent of time t and ⊂ R2 is bounded, [18] proved the existence of L2 -global attractor and its finite dimensionality for the semigroup associated with the 2D micropolar fluid flow, while [6] established that the L2 -global attractor is compact in H 2 space by employing an abstract result on the existence of global attractor from the viewpoint of measuring noncompactness [22]. In the present article, we consider the following 2D nonautonomous incompressible micropolar fluid flow ju − ( + r )u − 2r rot v + (u · ∇)u + ∇p = g(x, t), (x, t) ∈ × R, jt jv ˜ t), (x, t) ∈ × R, − v + 4r v − 2r rot u + (u · ∇)v = g(x, jt ju1 ju2 ∇ ·u= + =0 jx1 jx2
(1.1) (1.2) (1.3)
associated with the initial boundary conditions: u|t= = u , u|j = 0,
v|t= = v ,
∈ R,
v|j = 0,
where is an unbounded domain of R2 in which the following Poincaré inequality holds 2 (x) dx |∇(x)|2 dx, ∀(x) ∈ H01 (), 1
(1.4) (1.5)
(1.6)
where 1 is a positive constant depending only on . In Eqs. (1.1) and (1.2), the unknown vector-valued function u(x, t) = (u1 (x, t), u2 (x, t)) denotes the velocity, the scalar functions p and v represent pressure and microrotation, ˜ t) denote external force and moment, respectively, and respectively. Moreover, g(x, t) = (g1 (x, t), g2 (x, t)) and g(x, , r , are positive constants ( is usually called the Newtonian viscosity and r the microrotation viscosity). In addition, we have set ju1 jv jv ju2 . − , rot v = ,− rot u = jx1 jx2 jx2 jx1 We will assume that , r , and 1 satisfy the following condition: (A1 ) 1 · min{ + r , } > 21 . We would like to point out that condition (A1 ) plays quite an important role in this work. Factually, it allows us to prove the (uniform) asymptotic compactness of the process on space V (see (4.35)). One can see that condition (A1 ) essentially requires that some viscosity coefficients of the micropolar fluid flow have to be large enough if (and thus 1 ) is given. The main objective of the present article is to prove the existence of the uniform attractor for the family of processes ) (see Section corresponding to Eqs. (1.1)–(1.5) when G(x, t) = (g(x, t), g(x, ˜ t)) is translation compact in L2loc (R; H 0 2 2 for notations and definitions). We first show the existence of L -uniform attractor AH(G0 ) in H and H 1 -uniform for the family of processes corresponding to Eqs. (1.1)–(1.5). Then we show A0 attractor A1H(G0 ) in V H(G0 ) ⊆ 1 AH(G0 ) , which implies the asymptotic smoothing effect of the micropolar fluid flow in the sense that the solutions become eventually more regular than the initial data. We would like to remark that Łukaszewicz and Sadowski [21] studied the nonautonomous 2D magneto-micropolar fluid flow and proved the existence of L2 -uniform attractor for the family of processes corresponding to the fluid in 2D unbounded domains in which the Poincaré inequality (1.6) holds. The theory and technique used in [21] are skewproduct flow and asymptotic compactness via energy equation. The idea of this paper originates from [21]. The main
610
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
difference between [21] and the present paper (concerning the proof to the existence of the H 1 -uniform attractor) is that the term B(u, w) (see (4.27)) in the enstrophy equation does not disappear, while B(u, w) disappears in the energy equation due to its antisymmetric property (see (2.4)). Thus, this paper can be regarded as an extended application of the technique of asymptotic compactness in [21] via enstrophy equation. The paper is organized as follows. Section 2 is preliminaries. In Section 3, we show the existence of a bounded for the family of processes corresponding to the fluid flow, as well as the (H ×H(G0 ), H ) uniformly absorbing set in H . continuity, asymptotic compactness and uniformly asymptotic compactness for the family of processes defined on H and the (V × H(G0 ), V ) continuity In Section 4, we prove the existence of a bounded uniformly absorbing set in V . Then we introduce two lemmas and use them to establish the asymptotic for the family of processes defined on V . We state the main result and compactness and uniformly asymptotic compactness for the family of processes in V prove it in Section 5. 2. Preliminaries In this paper, we use the following notations and operators: Lp ()-the usual Lebesgue space with norm · Lp () ; especially, · L2 () = · , H m ()-the Sobolev space { ∈ L2 (), ∇ k ∈ L2 (), k m} with norm · H m () (see [2]), H01 ()= closure of { ∈ C0∞ ()} in H 1 (), V = { ∈ C0∞ () × C0∞ () : = (1 , 2 ), ∇ · = 0}, H = closure of V in L2 () × L2 () with norm · ; H = dual space of H , V = closure of V in H 1 () × H 1 () with norm · V ; V = dual space of V , = Hilbert space H × L2 () with norm · ; H = dual space of H , H 1
, V = Hilbert space V × H0 () with norm · V ; V = dual space of V (·, ·)-the inner product in H or in H , and V , ·, · -the dual pairing between V and V or between V 3
2 A ∈ L(V , V ) ∩ L(V ∩ (H ()) , H ) is a linear continuous operator defined by Aw, = ( + r )(∇u, ∇) + (∇v, ∇ ),
, ∀ = (, ) ∈ V , ∀ w = (u, v) ∈ V
∩ (H 2 ())3 is the domain of A, D(A) = V , V ) is a continuous operator defined by B ∈ L(V × V B(u, w), = ((u · ∇)w, ),
, ∀ ∈ V , ∀u ∈ V , w ∈ V
, H ) is a continuous operator defined by N ∈ L(V N (w) = (−2r rot v, −2r rot u + 4r v),
, ∀w = (u, v) ∈ V
), R = [, +∞), H(G0 )= closure of {G0 (· + h) : h ∈ R} in L2loc (R; H ) is called to be translation compact in L2 (R; H ) if H(G0 ) is compact in L2 (R; H ), G0 ∈ L2loc (R; H loc loc 2 2 )= the set of functions that are translation compact in L (R; H ), Lc (R; H loc )= the set of functions G ∈ L2 (R; H ) satisfying L2 (R; H b
G2L2 (R;H) = G2L2 = sup b
b
t∈R t
loc
t+1
G(s)2 ds < + ∞,
), defined by {S(t)}t 0 -the natural translation semigroup acting on L2loc (R; H S(h)w(t) = w(t + h),
), ∀h 0, ∀w ∈ L2loc (R; H
“−→” denotes convergence in the strong topology, “” denotes convergence in the weak topology, “→” denotes embedding between spaces, distM (X, Y )=supx∈X inf y∈Y dist M (x, y) denotes the Hausdorff semidistance from X ⊂ M to Y ⊂ M in the metric space M.
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
611
The following basic facts were proved in [17–19,21]. Lemma 2.1. (i) Aw, w is equivalent to ∇w2 . Moreover, for any w ∈ D(A), 1
∇w2 Aw, w wAw √ ∇wAw, 1
(2.1)
hereafter = min{ + r , } and 1 is the constant from (1.6). (ii) There exists a positive constant depending only on such that 1
1
1
1
| B(u, w), | u 2 ∇u 2 w 2 ∇w 2 ∇, 1 2
1 2
1 2
×V , (u, w, ) ∈ V × V
1 2
| B(u, w), A | u ∇u ∇w Aw A,
(u, w, ) ∈ V × D(A) × D(A).
(2.2) (2.3)
In addition, B(u, w), w = 0,
. ∀(u, w) ∈ V × V
(2.4)
(iii) There exists a positive constant 1 = min{, } such that . ∀w ∈ V
Aw, w + N(w), w 1 w2V ,
(2.5)
(iv) There exists a positive constant C(r ) depending on r such that − N(w), Aw 22r w2V + 21 Aw2 , N(w) C(r )wV ,
∀w ∈ D(A),
(2.6)
. ∀w ∈ V
(2.7)
Using the notations and operators introduced above, we can translate the weak version of the initial boundary value problem (1.1)–(1.5) into an abstract form in the sense of distributions (see [16,28,30]) as following: jw + Aw + B(u, w) + N (w) = G(x, t), jt w|t= = w = (u , v ),
, t > , w = (u, v) ∈ V
∈ R.
(2.8) (2.9)
Moreover, we have the following result. ) and any ∈ R. Lemma 2.2. Let G(x, t) = (g(x, t), g(x, ˜ t)) ∈ L2c (R; H , then Eqs. (2.8)–(2.9) possesses a unique solution w = (u, v) satisfying (a) If w ∈ H ) ∩ C(R , H ) ∩ L2 (R ; V ), w ∈ L∞ (R ; H loc Moreover,
jw ). ∈ L2loc (R ; V jt
1 1+ G2L2 , ∀t , w(t) w e
1 1 b t t 1 w(t)2 + 1 w(s)2V ds w 2 + G(s)2 ds, ∀t ,
1 2
2 − 1 1 (t−)
1 +
1 1
(2.10)
(2.11) (2.12)
hereafter the constant 1 comes from (2.5). , then Eqs. (2.8)–(2.9) admits a unique solution w = (u, v) satisfying (b) If w ∈ V ) ∩ C(R , V ) ∩ L2 (R ; D(A)), w ∈ L∞ (R ; V loc Furthermore, (t
− )w(t)2V Q1
t − , w , 2
t
jw ). ∈ L2loc (R ; H jt
(2.13)
G(s) ds , 2
∀t ,
where Q1 (z1 , z2 , z3 ) is a monotone continuous function of z1 = t − , z2 and z3 .
(2.14)
612
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
Proof. (a) Let {n } be a sequence of bounded subdomains of and n → as n → +∞. One can use the Galerkin method to obtain the unique existence and regularity of solutions to Eqs. (2.8)–(2.9) on each n . Then one can use the approximation argument as that in [28] to obtain the solution of Eqs. (2.8)–(2.9) on . Since the method is classical and essentially the same as that in [21], we omit it and here we only prove (2.11) and (2.12). Let w(t) = (u(t), v(t)) , ∈ R. be the unique solution corresponding to initial data w = (u , v ) ∈ H Using w(t) to multiply Eq. (2.8), we can obtain 1 d 1 d w(t)2 + 1 1 w(t)2 w(t)2 + 1 w(t)2V 2 dt 2 dt 1 d w(t)2 + Aw, w + B(u, w), w + N (w), w 2 dt
1 1 1 = (G(t), w(t)) G(t)2 , w(t)2 + 2 2 1 1
(2.15)
where in the first inequality we used (1.6); in the second inequality we used (2.4)–(2.5); in the last inequality we used Schwartz inequality and Young inequality. By (2.15), we get d 1 G(t)2 . w(t)2 + 1 1 w(t)2 dt
1 1
(2.16)
Applying Gronwall inequality to (2.16) and taking some transformation in the integral, we have w(t)2
t 1 + e− 1 1 (t−s) G(s)2 ds w e
1 1 t t−1 1 e− 1 1 (t−s) G(s)2 ds + e− 1 1 (t−s) G(s)2 ds + · · · w 2 e− 1 1 (t−) +
1 1 t−1 t−2 t t−1 1 w 2 e− 1 1 (t−) + G(s)2 ds + e− 1 1 G(s)2 ds
1 1 t−1 t−2 t−2 −2 1 1 2 G(s) ds + · · · +e 2 − 1 1 (t−)
t−3
1 (1 + e− 1 1 + e−2 1 1 + · · ·)G2L2
1 1 b 1 w 2 e− 1 1 (t−) + (1 − e− 1 1 )−1 G2L2
1 1 b 1 1 1+ G2L2 . w 2 e− 1 1 (t−) +
1 1
1 1 b w 2 e− 1 1 (t−) +
Thus (2.11) holds. Now from (2.15) we can get 1 d
1 1 G(t)2 , w(t)2 + 1 w(t)2V w(t)2V + 2 dt 2 2 1 i.e., d 1 w(t)2 + 1 w(t)2V G(t)2 . dt
1 Integrating (2.17) w.r.t. time variable over [, t], we obtain (2.12).
(2.17)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
613
(b) Similar to the proof of (a), we only establish (2.14). For the sake of brevity, we set = 0 and t 0. Multiplying Eq. (2.8) by tAw, we can obtain
jw , tAw + Aw, tAw + B(u, w), tAw + N (w), tAw jt 1 d 1 (t Aw, w ) − Aw, w + tAw2 + t B(u, w), Aw + t N (w), Aw 2 dt 2 = t (G(t), Aw) 1 tAw2 + tG(t)2 . 4 =
(2.18)
From (2.3) and the facts u2 w2 , ∇u2 u2V w2V , we see that 1
1
1
3
t B(u, w), Aw tu 2 ∇u 2 ∇w 2 Aw 2 t Aw2 + 4 tu2 ∇u2 ∇w2 4 t Aw2 + 4 tw2 w4V . 4
(2.19)
Combining (2.6) and (2.18)–(2.19), we deduce that 1 d 1 (t Aw, w ) Aw, w + tG(t)2 + 4 tw2 w4V + 22r tw2V 2 dt 2 1 = Aw, w + tG(t)2 + tw2V 4 w2 w2V + 22r . 2
(2.20)
Now by Lemma 2.1(i), there exist two positive constants C1 and C2 such that C1 Aw, w w2V C2 Aw, w ,
. ∀w ∈ V
(2.21)
Eqs. (2.20) and (2.21) imply d (t Aw, w ) Aw, w + 2tG(t)2 + t Aw, w 2C2 4 w2 w2V + 4C2 2r . dt Thus we have d H (t)K(t)H (t) + I (t), dt
(2.22)
where H (t) = t Aw(t), w(t) , K(t) = 2C2 4 w(t)2 w(t)2V + 4C2 2r , I (t) = Aw(t), w(t) + 2tG(t)2 . Applying Gronwall inequality to (2.22), we obtain
t t
t t H (t) H (0) + I (s) ds exp K(s) ds = I (s) ds exp K(s) ds . 0
0
0
0
(2.23)
614
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
It then follows from (2.12) and (2.21) that t t I (s) ds = ( Aw(s), w(s) + 2sG(s)2 ) ds 0
0
t 1 G(s)2 ds w(s)2V ds + 2t C 1 0 0 t t 1 1 2 2 w(0) + G(s) ds + 2t G(s)2 ds C1 1
1 0 0 t 1 w(0)2 + (1 + t) C G(s)2 ds ,
t
(2.24)
0
where
1 1 1 = max C , ,2 . C1 1 C1 21
Similarly, by (2.12) we have t t K(s) ds = (2C2 4 w(s)2 w(s)2V + 4C2 2r ) ds 0
0
t
1 s 2 2C2 w(0) + G() d w(s)2V ds + 4C2 2r t
1 0 0 1 t w(0)2 1 t 4 2 2 2 2C2 w(0) + G(s) ds + 2 G(s) ds + 4C2 2r t
1 0
1
1 0
2 t 2 2 2 C w(0) + G(s) ds + t , 4
2
(2.25)
0
where
2 = max 2C2 max 1, C 4
1
1
,
21 41
, 4C2 2r
.
It follows from (2.21) and (2.23)–(2.25) that tw(t)2V C2 t Aw(t), w(t) = C2 H (t) 2 t t 2 2 2 2 C2 C1 w(0) + (1 + t) w(0) + G(s) ds exp C2 G(s) ds + t
. = Q1 t, w(0)2 ,
0
t
G(s)2 ds ,
0
(2.26)
0
1 [z2 + (1 + z1 )z3 ] exp{C 2 [(z2 + z3 )2 + z1 ]}. The proof of Lemma 2.2 is complete. where Q1 (z1 , z2 , z3 ) = C2 C
) In the sequel, we assume some field of the external force and moment G0 (x, t) = (g0 (x, t), g˜ 0 (x, t)) ∈ L2c (R; H and take H(G0 ) as the symbol space of Eq. (2.8). Obviously, H(G0 ) is strictly invariant under the acting of the natural translation semigroup {S(t)}t 0 , i.e., S(t)H(G0 ) = H(G0 ),
∀t 0.
(2.27)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
615
At the same time, we have G2L2 G0 2L2 < + ∞, b
b
∀G ∈ H(G0 ).
(2.28)
to H , via According to Lemma 2.2(a), we can define a family of processes {UG (t, )}t , G ∈ H(G0 ), from H UG (t, )w = w(t), where w(t) is the solution of Eqs. (2.8)–(2.9) with symbol G and initial data w ∈ H . Analogously, to V . Moreover, by Lemma 2.2(b), the family of processes {UG (t, )}t , G ∈ H(G0 ), can also be defined from V UG (, ) = I d (identity operator), UG (t, s)UG (s, ) = UG (t, ),
∀ ∈ R, ∀G ∈ H(G0 ),
∀t s , ∀G ∈ H(G0 ),
(2.29) (2.30)
and V , respectively. holds in H 3. Properties of the family of processes in H In this section, we present some properties of the family of processes corresponding to Eqs. (2.8)–(2.9). As mentioned in previous Introduction, these results can be derived in the same way as that in [21], which reduced the associated problem to the study of a semigroup acting in an extended phase space. At the same time, these properties can also be established by the argument similar to that in Section 4. Therefore, we just list them out. We first select some definitions from [7,13]. Definition 3.1. A set B0 is said to be a uniformly (w.r.t. G ∈ H(G0 )) absorbing set for the family of processes {U G (t, )}t , G ∈ H(G0 ), if for every bounded set B of H and any ∈ R, there exists t0 = t0 (B, ) such that G∈H (G0 ) UG (t, )B ⊆ B0 for all t t0 . × H(G0 ), H ) continuous if for Definition 3.2. The family of processes {UG (t, )}t , G ∈ H(G0 ), is said to be (H × H(G0 ) to H . any fixed t, ∈ R (t ), the mapping (w, G) → UG (t, )w is continuous from H if Definition 3.3. The family of processes {UG (t, )}t , G ∈ H(G0 ), is said to be asymptotically compact in H (n) ∞ (n) , whenever {w }∞ is bounded in H , {G(n) }∞ ⊂ H(G0 ) and {tn }∞ ⊂ R {UG(n) (tn , )w }n=1 is precompact in H n=1 n=1 n=1 with tn → +∞ as n → +∞. is said to be the uniformly (w.r.t. G ∈ H(G0 )) attracting set of {UG (t, )}t , G ∈ Definition 3.4. A set ⊂ H and any fixed ∈ R H(G0 ), in H if for any bounded set B of H lim
sup
t→+∞ G∈H(G ) 0
dist H(UG (t, )B, ) = 0.
is called The family of processes {UG (t, )}t , G ∈ H(G0 ), possessing a compact uniformly attracting set in H uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact in H . ). Then Lemma 3.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H
1 2 . : w B0 = w ∈ H 1+ G0 L2 = R0 b
1 1
1 1
(3.1)
is a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set for the family of processes {UG (t, )}t , G ∈ H(G0 ), . corresponding to Eqs. (2.8)–(2.9) in H ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.2. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H corresponding to Eqs. (2.8)–(2.9) is (H × H(G0 ), H ) continuous. ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.3. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . corresponding to Eqs. (2.8)–(2.9) is asymptotically compact in H
616
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
). Then the family of processes {UG (t, )}t , G ∈ H(G0 ), Lemma 3.4. Assume that G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . corresponding to Eqs. (2.8)–(2.9) is uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact in H 4. Properties of the family of processes in V ) corresponding to Eqs. (2.8)–(2.9) has In this section, we mainly prove that the family of processes (defined on V the analogous properties with those showed in Section 3. ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ) (defined on Lemma 4.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H , i.e., V ), corresponding to Eqs. (2.8)–(2.9) possesses a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set BV0 in V V V for any bounded set B of V and any fixed ∈ R, there exists a time t1 = t1 (B , ) such that
UG (t, )BV ⊆ BV0 ,
∀t t1 .
(4.1)
G∈H(G0 )
implies that BV is also bounded → H is continuous, the set BV being bounded in V Proof. Since the embedding V . We assert that the set in H UG ( + 1, )B0 (4.2) BV0 = G∈H(G0 ) ∈R
is a bounded uniformly (w.r.t. G ∈ H(G0 )) absorbing set for {UG (t, )} t , G ∈ H(G 0 ), in V , where B0 is the uniformly absorbing set from Lemma 3.1. Indeed, from (3.1) we see that G∈H(G0 ) ∈R UG ( + 1, )B0 is also a . Now for any bounded set BV of uniformly (w.r.t. G ∈ H(G0 )) absorbing set for {UG (t, )}t , G ∈ H(G0 ), in H V . Thus, for any fixed ∈ R, , B is also bounded in H V G∈H(G0 )
UG (t, )BV ⊆
UG ( + 1, )B0 = BV0 ,
∀t t0 (BV , ).
(4.3)
G∈H(G0 ) ∈R
. Precisely, we have Moreover, we deduce from (2.14) and (3.1) that BV0 is bounded in V . w2V Q1 (1, R02 , G0 2L2 ) = R12 , b
∀w ∈ BV0 ,
(4.4)
where Q1 is the monotone continuous function from Lemma 2.2(b). The proof is complete.
Remark 4.1. From the proof of Lemma 4.1 we see that BV0 also absorbs uniformly (w.r.t. G ∈ H(G0 )) all bounded in the norm of V . sets of H ×H(G0 ), V ) continuity, asymptotic compactness and uniformly (w.r.t. G ∈ H(G0 )) asymptotic The definition of (V are similar to Definitions 3.2–3.4. compactness for {UG (t, )}t , G ∈ H(G0 ), in V ). Then the family of processes {UG (t, )}t , G ∈ H(G0 ) (defined on Lemma 4.2. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H ), corresponding to Eqs. (2.8)–(2.9) is (V × H(G0 ), V ) continuous. V (n) (n) (n) (n) ∞ Proof. Let t and be fixed and t . Let {(w , G(n) )}∞ n=1 ={((u , v ), G )}n=1 ⊂ V ×H(G0 ) be a sequence that ∞ (n) (n) (n) converges to some (w , G)=((u , v ), G) ∈ V ×H(G0 ), {w (t)}n=1 ={(u (t), v (t))}∞ n=1 and w(t)=(u(t), v(t)) (n) be the corresponding solutions of Eqs. (2.8)–(2.9) with symbols {G(n) }∞ and G, and initial data {w }∞ n=1 n=1 and w , respectively. Set
(n) (t) = w(t) − w (n) (t) = UG (t, )w − UG(n) (t, )w(n) ,
n = 1, 2, . . . .
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
617
For each n we see that (n) (t) is a solution of the following equations: j (n) + A (n) + B(u, w) − B(u(n) , w(n) ) + N ( (n) ) = G − G(n) , jt
(4.5)
(n) (n) |t= = (n) = w − w .
(4.6)
Multiplying (4.5) with A (n) , we can obtain 1 d A (n) , (n) + A (n) 2 + B(u, w), A (n) − B(u(n) , w(n) ), A (n) + N ( (n) ), A (n) 2 dt 1 = (G − G(n) , A (n) ) A (n) 2 + G − G(n) 2 . 4
(4.7)
On the one hand, it follows from (2.6) that − N( (n) ), A (n) 22r (n) 2V + 21 A (n) 2 .
(4.8)
On the other hand, by (2.4) we have B(u, w), A (n) − B(u(n) , w(n) ), A (n) = B(u − u(n) , w), A (n) + B(u(n) , w), A (n) − B(u(n) , w(n) ), A (n) = B(u − u(n) , w), A (n) + B(u(n) , (n) ), A (n) .
(4.9)
Combining (2.3), (4.9) and the facts u − u(n) (n) ,
∇(u − u(n) ) u − u(n) V (n) V ,
we obtain | B(u, w), A (n) − B(u(n) , w(n) ), A (n) | = | B(u − u(n) , w), A (n) + B(u(n) , (n) ), A (n) | 1
1
1
1
1
1
1
3
u − u(n) 2 ∇(u − u(n) ) 2 ∇w 2 Aw 2 A (n) + u(n) 2 ∇u(n) 2 ∇ (n) 2 A (n) 2
18 A (n) 2 + 22 u − u(n) ∇(u − u(n) )∇wAw + 18 A (n) 2 + 24 u(n) 2 ∇u(n) 2 ∇ (n) 2 41 A (n) 2 + 22 (n) ∇ (n) ∇wAw + 24 (n) 2 ∇ (n) 2 ∇ (n) 2 41 A (n) 2 + 22 (n) 2V wV Aw + 24 (n) 2 (n) 4V .
(4.10)
Now taking (4.7)–(4.8) and (4.10) into account, we get 1 d 1 1 A (n) , (n) + A (n) 2 A (n) 2 + 22r (n) 2V + A (n) 2 + G − G(n) 2 2 dt 2 4 1 + A (n) 2 + 22 (n) 2V wV Aw + 24 (n) 2 (n) 4V . 4 Using (2.21), we have d A (n) , (n) 42r (n) 2V + 2G − G(n) 2 + 42 (n) 2V wV Aw + 44 (n) 2 (n) 4V dt 4 (n) 2V 2r + 2 wV Aw + 4 (n) 2 (n) 2V + 2G − G(n) 2 4C2 A (n) , (n) 2r + 2 wV Aw + 4 (n) 2 (n) 2V + 2G − G(n) 2 .
(4.11)
618
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
Applying Gronwall inequality to (4.11), we obtain A
(n)
(t),
(n)
(t) A
(n)
(),
× exp 4C2
(n)
() + 2
t
2r
t
G(s) − G
2
(n)
2
(s) ds
4
+ wV Aw +
(n) 2
(n) 2V
ds .
(4.12)
Again using (2.21), we get from (4.12) that 1 UG (t, )w − UG(n) (t, )w(n) 2V C2 =
1 (n) (t)2V C2
A (n) (t), (n) (t) t 1 (n) 2 (n) 2 w − w V + 2 G(s) − G (s) ds C1
t × exp 4C2 2r + 2 wV Aw + 4 (n) 2 (n) 2V ds .
(4.13)
When t and are fixed, (2.11)–(2.14) imply
t exp 4C2 2r + 2 wV Aw + 4 (n) 2 (n) 2V ds < + ∞.
× H(G0 ), V )-continuity of the family of processes {UG (t, )}t , G ∈ H(G0 ), Therefore, (4.13) implies the (V corresponding to Eqs. (2.8)–(2.9). The proof of Lemma 4.2 is complete. , we need the following two In order to prove the asymptotic compactness of {UG (t, )}t , G ∈ H(G0 ), in V lemmas. (n) (n) ∞ Lemma 4.3. Let {w }∞ n=1 be a sequence in V converging strongly to some w ∈ V in the norm of H and {G }n=1 (n) be a sequence of H(G0 ) converging strongly to some G ∈ H(G0 ). Then for any fixed ∈ R, UG(n) (·, )w −→ ), ∀ T . UG (·, )w strongly in L2 (, T ; V
Proof. From (2.21) we have w2V C2 Aw, w C2 wAw,
∀ w ∈ D(A).
Thus for any t , UG(n) (t, )w(n) − UG (t, )w 2V C2 UG(n) (t, )w(n) − UG (t, )w A(UG(n) (t, )w(n) − UG (t, )w ).
(4.14)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
619
Integrating (4.14) w.r.t. t over [, T ], we have, using Young inequality, T UG(n) (s, )w(n) − UG (s, )w 2V ds
C2
T
UG(n) (s, )w(n) − UG (s, )w A(UG(n) (s, )w(n) − UG (s, )w ) ds
C2
T
×
T
UG(n) (s, )w − UG (s, )w ds 2
(n)
21
A(UG(n) (s, )w − UG (s, )w ) ds 2
(n)
21 .
(4.15)
T 1 (n) (n) Now (2.13) implies { A(UG(n) (s, )w − UG (s, )w ) ds} 2 < + ∞ and Lemma 3.2 implies UG(n) (s, )w as n → +∞. Therefore, we obtain from (4.15) that −→ UG (s, )w strongly in H T UG(n) (s, )w(n) − UG (s, )w 2V ds = 0. lim n→+∞
The proof is complete.
(n) ∞ Lemma 4.4. Let {w }∞ n=1 be a sequence in V converging weakly to some w in V and {G }n=1 be a sequence of (n) H(G0 ) converging strongly to some G ∈ H(G0 ). Then for any fixed t , t, ∈ R, UG(n) (t, )w UG (t, )w weakly in V and (n)
UG(n) (·, )w(n) UG (·, )w
weakly in L2 (, T ; D(A)), ∀ T .
(4.16)
Proof. For any fixed t , t, ∈ R, let UG(n) (t, )w(n) = w (n) (t),
n = 1, 2, . . . ,
UG (t, )w = w(t).
Analogous with Lemma 2.2(b), we find that {w(n) (t)}∞ n=1 is bounded in ) ∩ L2 (R ; D(A)) L∞ (R ; V loc jw(n)
2 and { jt }∞ n=1 is bounded in Lloc (R ; H ). The rest of the proof is essentially the same as that of Lemma 8.1 in [21] and we omit it.
Next we will devote to prove the following lemma, which plays the essential role when we prove the uniformly (w.r.t. . G ∈ H(G0 )) asymptotic compactness of {UG (t, )}t , G ∈ H(G0 ), in V ) and (A1 ) hold. Then the family of processes {UG (t, )}t , G ∈ H(G0 ) Lemma 4.5. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H . (defined on V ), corresponding to Eqs. (2.8)–(2.9) is asymptotically compact in V (n) ∞ (n) ∞ Proof. Let {w }∞ n=1 be a bounded sequence in V , {G }n=1 ⊂ H(G0 ) and {tn }n=1 ⊂ R with tn → +∞ as n → +∞. For any fixed ∈ R, we see from Lemma 4.1 that there is a time T0 = T0 (R, ) (where R is a constant (n) (n) V V satisfying w V R for all n) such that for all tn T0 , {UG(n) (tn , )w }∞ n=1 ⊆ B0 , where B0 is the bounded . This ensures that {UG(n) (tn , )w(n) }∞ is weakly precompact in V uniformly (w.r.t. G ∈ H(G0 )) absorbing set in V n=1 and thus we have
UG(n) (tn , )w(n) w
as n → +∞, weakly in V
(4.17)
and some subsequence (still denote by) {UG(n) (tn , )w(n) }∞ . Similarly, for each T > 0 and tn T0 +T , for some w ∈ V n=1 (n) . wT = UG(n) (tn − T , )w(n) wT
as n → +∞, weakly in V
(4.18)
620
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
. Now from the definition of {S(t)}t 0 we can check the following translation identity holds for some wT ∈ V US(h)G (t, ) = UG (t + h, + h),
∀h 0, t , t, ∈ R, ∀G ∈ H(G0 ).
(4.19)
Applying (2.29)–(2.30) and (4.19), we get UG(n) (tn , ) = UG(n) (tn , tn − T )UG(n) (tn − T , ) = US(tn −T )G(n) (T , 0)UG(n) (tn − T , ),
tn − T .
(4.20)
(n)
Setting GT = S(tn − T )G(n) , we have by (4.18) and (4.20) that for ∀T > 0, tn − T , (n)
UG(n) (tn , )w(n) = UG(n) (T , 0)UG(n) (tn − T , )w(n) = UG(n) (T , 0)wT .
(4.21)
T
T
(n) (n) ∞ 2 Since {GT }∞ n=1 ⊂ H(G0 ) and H(G0 ) is compact in Lloc (R; H ), there exist a subsequence of {GT }n=1 (still denote (n) by {GT }∞ n=1 ) and some GT ∈ H(G0 ) such that (n)
GT −→ GT
) as n → +∞ for every T > 0. strongly in L2loc (R; H
(4.22)
Taking (4.17)–(4.18), (4.21)–(4.22) and Lemma 4.4 into account, we obtain w = UGT (T , 0)wT
for every T > 0,
(4.23)
where we also used the uniqueness of the limit. Now it follows from (4.17) and (4.21) that (n)
lim inf UG(n) (tn , )w(n) V = lim inf UG(n) (T , 0)wT V wV . n
n
(4.24)
T
Next we prove (n)
lim sup UG(n) (tn , )w(n) V = lim sup UG(n) (T , 0)wT V wV . n
(4.25)
T
n
. In fact, using Aw to multiply Eq. (2.8), we obtain To this end, we use the argument of enstrophy equation in V 1 1 d Aw, w + Aw, w = (G, Aw) − F (w), 2 dt 2
(4.26)
where F (w) = Aw2 − 21 Aw, w + B(u, w), Aw + N (w), w . Integrating (4.26) w.r.t. time variable over [, t], as following: we obtain the enstrophy equation in V t −(t−) +2 e−(t−s) (G(s), Aw(s)) ds Aw(t), w(t) = Aw(), w() e
−2
t
e−(t−s) F (w(s)) ds,
t .
(4.27)
(n)
Applying the enstrophy equation to UG(n) (T , 0)wT , we obtain T
AU G(n) (T , 0)wT , UG(n) (T , 0)wT = Aw T , wT e−T + 2 (n)
T
(n)
(n)
T
−2 0
By (4.4) and (2.21) we have 1 (n) (n) lim sup Aw T , wT e−T R12 e−T . C1 n
(n)
T
T 0
(n) (n) ds e−(T −s) GT (s), AU G(n) (s, 0)wT T
e−(T −s) F UG(n) (s, 0)wT
(n)
ds.
(4.28)
T
(4.29)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
621
, we infer from (4.16), (4.18) and (4.22) that Since A is a linear continuous mapping from D(A) to H ), weakly in L2 (, T ; H
(n)
AU G(n) (s, 0)wT AU GT (s, 0)wT T
∀T > .
(4.30)
Thus (4.22) and (4.30) imply
T
lim 2
n→+∞
0
(n) (n) ds = 2 e−(T −s) GT (s), AU G(n) (s, 0)wT T
T 0
e−(T −s) (GT (s), AU GT (s, 0)wT ) ds.
(4.31)
We next handle the third term of the right-hand side in (4.28). In fact, we have the following result. (n)
Lemma 4.6. Set UG(n) (s, 0)wT = (u(n) , v (n) ) and UGT (s, 0)wT = (u, v). Then T
1 (n) (n) (n) e−(T −s) AU G(n) (s, 0)wT 2 − AU G(n) (s, 0)wT , UG(n) (s, 0)wT ds n T T T 2 0 T 1 −(T −s) 2 AU GT (s, 0)wT − AU GT (s, 0)wT , UGT (s, 0)wT ds, e 2 0
T
lim inf
T
lim
n→+∞ 0 T
=
0
lim
n→+∞ 0 T
=
0
(n) (n) e−(T −s) B u(n) , UG(n) (s, 0)wT , AU G(n) (s, 0)wT ds T
T
e−(T −s) B(u, UGT (s, 0)wT ), AU GT (s, 0)wT ds, T
(4.32)
(4.33)
(n) (n) e−(T −s) N UG(n) (s, 0)wT , AU G(n) (s, 0)wT ds T
T
e−(T −s) N (UGT (s, 0)wT ), AU GT (s, 0)wT ds.
(4.34)
Proof. We first prove (4.32). Define '·, ·( : D(A) × D(A) −→ R as 'w1 , w2 ( = (Aw 1 , Aw2 ) − 21 Aw 1 , w2 ,
∀w1 , w2 ∈ D(A).
1 By (A1 ), we have 1 − 2 > 0. Thus by Lemma 2.1(i), we get 1
1 1 2 'w( = 'w, w( = Aw2 − Aw, w Aw2 − √ ∇wAw 2 2 1 1 1 2 2 Aw − Aw2 Aw = 1 − 2 1 2 1 and thus we have 1 2 1− Aw2 'w( Aw2 , 2 1
∀w ∈ D(A).
(4.35)
T 1 2 Since 0 < e−(T −s) 1 for any T s, (4.35) implies that ( 0 e−(T −s) ' · ( ds) 2 is a norm on L2 (0, T ; D(A)) that is T 1 equivalent to ( 0 A · 2 ds) 2 . At the same time we derive from (4.17)–(4.18), (4.21) and Lemma 4.4 that (n)
UG(n) (s, 0)wT UGT (s, 0)wT T
Thus we get (4.32) from (4.36).
weakly in L2 (0, T ; D(A)).
(4.36)
622
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
We now prove (4.33). Set (n)
(n)
(n)
L1 (s) = B(u(n) (s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT , T
T
L2 (s) = B(u(s), UGT (s, 0)wT ), AU GT (s, 0)wT . We have
T
e
−(T −s)
0
T 0
(n) (L1 (s) − L2 (s)) ds
(n) (n) e−(T −s) B(u(n) (s) − u(s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT ds T
T
T (n) (n) + e−(T −s) B(u(s), UG(n) (s, 0)wT − UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T T 0 T (n) −(T −s) e B(u(s), UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds + T
0
. (n) (n) (n) = M1 + M2 + M3 .
(4.37)
Using (2.3), Lemma 4.1 and the fact (n)
∇(u(n) (s) − u(s)) u(n) (s) − u(s)V UG(n) (s, 0)wT − UGT (s, 0)wT V , T
we get (n) M1
T (n) (n) −(T −s) (n) e B(u (s) − u(s), UG(n) (s, 0)wT ), AU G(n) (s, 0)wT ds = T T 0 T 1 1 (n) 1 (n) 3 u(n) (s) − u(s) 2 ∇(u(n) (s) − u(s)) 2 ∇UG(n) (s, 0)wT 2 AU G(n) (s, 0)wT 2 ds 0
T
1
R12
R1
T 0 T
1
1
3
T
1 2
(n)
(n)
3
UG(n) (s, 0)wT − UGT (s, 0)wT V AU G(n) (s, 0)wT 2 ds T
R1
(n)
u(n) (s) − u(s) 2 ∇(u(n) (s) − u(s)) 2 AU G(n) (s, 0)wT 2 ds
T
0
T
0
T
(n) UG(n) (s, 0)wT T
− UGT (s, 0)wT 2V ds
41
T 0
(n) AU G(n) (s, 0)wT 2 ds T
43 .
(4.38)
Now (2.13) implies
T
0
(n) AU G(n) (s, 0)wT 2 ds T
43
< + ∞.
(4.39)
At the same time, from (4.18) and Lemma 3.3 we see that as n → +∞. strongly in H
(n)
wT −→ wT
(4.40)
Then it follows from Lemma 4.3 and (4.40) that (n)
UG(n) (s, 0)wT −→ UGT (s, 0)wT T
) as n → +∞. strongly in L2 (0, T ; V
(4.41)
Combining (4.38), (4.39) and (4.41), we have (n)
lim M1 = 0.
n→+∞
(4.42)
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
623
By the argument similar to what used for deriving (4.42), we have T (n) (n) (n) −(T −s) M2 = e B(u(s), UG(n) (s, 0)wT − UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T
0
T
R1
(n)
UG(n) (s, 0)wT − UGT (s, 0)wT 2V ds
41
T
0
T 0
T
×
T
(n)
A(UG(n) (s, 0)wT − UGT (s, 0)wT )2 ds
(n)
AU G(n) (s, 0)wT 2 ds
21
T
1 4
T
0
and (n)
lim M2 = 0.
(4.43)
n→+∞
), by (4.30) we obtain Since e−(T −s) B(u(s), UGT (s, 0)wT ) ∈ L2 (0, T ; V T (n) −(T −s) e B(u(s), UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds lim n→+∞
T
0
(n) = lim M3 n→+∞
= 0.
(4.44)
Eqs. (4.37) and (4.42)–(4.44) imply that (4.33) holds. We next prove (4.34). Set (n)
(n)
(n)
N1 (s) = N (UG(n) (s, 0)wT ), AU G(n) (s, 0)wT , T
T
N2 (s) = N (UGT (s, 0)wT ), AU GT (s, 0)wT . We have
T
e
−(T −s)
0
T
e
−(T −s)
0
+
T
0
(n) (N1 (s) − N2 (s)) ds
(n) (n) N (UG(n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU G(n) (s, 0)wT ds T T
(n) e−(T −s) N (UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds T
. (n) (n) = K1 + K2 .
(4.45)
Now by (2.7) and Young inequality, we obtain T (n) (n) −(T −s) e N(U (n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU (n) (s, 0)wT ds G G T
0
T
T
0
(n) N (UG(n) (s, 0)wT T
T
C(r ) 0
− UGT (s, 0)wT ) ds
(n) UG(n) (s, 0)wT T
2
21
− UGT (s, 0)wT 2V ds
T 0
(n) AU G(n) (s, 0)wT 2 ds T
21
T 0
21
(n) AU G(n) (s, 0)wT 2 ds T
21 .
(4.46)
Eqs. (4.39), (4.41) and (4.46) imply T (n) (n) (n) e−(T −s) N (UG(n) (s, 0)wT ) − N (UGT (s, 0)wT ), AU G(n) (s, 0)wT ds lim K1 = lim n→+∞
n→+∞
= 0.
0
T
T
(4.47)
624
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
, we infer from (2.7) that At the same time, since UGT (s, 0)wT ∈ BV0 ⊂ V ). e−(T −s) N(UGT (s, 0)wT ) ∈ L2 (0, T ; H Thus it follows from (4.30) that T (n) (n) lim K2 = lim e−(T −s) N (UGT (s, 0)wT ), AU G(n) (s, 0)wT − AU GT (s, 0)wT ds n→+∞ n→+∞ T 0
= 0.
(4.48)
Eqs. (4.45) and (4.47)–(4.48) imply that (4.34) holds. The proof of Lemma 4.6 is complete. As a corollary of Lemma 4.6, we have that T (n) −(T −s) lim sup − e F (UG(n) (s, 0)wT ) ds = − lim inf n
n
T
0
T
− 0
T
e
−(T −s)
0
(n) F (UG(n) (s, 0)wT ) ds T
e−(T −s) F (UGT (s, 0)wT ) ds.
(4.49)
We continue to prove Lemma 4.5. Combining (4.28)–(4.29), (4.31) and (4.49), we obtain T 1 (n) (n) lim sup AU G(n) (T , 0)wT , UG(n) (T , 0)wT R12 e−T + 2 e−(T −s) (GT (s), AU GT (s, 0)wT ) ds T T C n 1 0 T e−(T −s) F (UGT (s, 0)wT ) ds. (4.50) −2 0
At the same time, applying (4.27) to w = UGT (T , 0)wT , we obtain T AU GT (T , 0)wT , UGT (T , 0)wT = Aw T , wT e−T + 2 e−(T −s) (GT (s), AU GT (s, 0)wT ) ds 0
T
−2 0
e−(T −s) F (UGT (s, 0)wT ) ds.
(4.51)
It follows from (4.50)–(4.51) and (4.23) that (n)
(n)
lim sup AU G(n) (T , 0)wT , UG(n) (T , 0)wT T
n
T
AU GT (T , 0)wT , UGT (T , 0)wT + Aw, w +
1 2 −T R e , C1 1
1 2 R1 − Aw T , wT e−T C1
∀ T > 0.
(4.52)
Noticing that A·, · is equivalent to · 2V , we obtain by letting T tend to +∞ in (4.52) that (n)
lim sup UG(n) (T , 0)wT 2V w2V . T
n
is a Hilbert space, we deduce from (4.24)–(4.25), (4.17) and (4.21) that Hence (4.25) has been proved. Because V (n) limn→+∞ UG(n) (tn , )w − wV = 0. The proof of Lemma 4.5 is complete. T
) and condition (A1 ) hold. Then the family of processes {UG (t, )}t , Lemma 4.7. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H G ∈ H(G0 ) (defined on V ), corresponding to Eqs. (2.8)–(2.9) is uniformly (w.r.t. G ∈ H(G0 )) asymptotically compact . in V
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
625
Proof. We only need to prove that {UG (t, )}t , G ∈ H(G0 ), possesses a compact uniformly (w.r.t. G ∈ H(G0 )) . We claim that the set attracting set in V
.
,H(G0 ) (BV0 ) =
UG (s, )BV0
for each ∈ R
(4.53)
t G∈H(G0 ) s t
, where the bar in is a compact uniformly (w.r.t. G ∈ H(G0 )) attracting set for {UG (t, )}t , G ∈ H(G0 ), in V V (4.53) denotes taking closure in V . In fact, the set ,H(G0 ) (B0 ) defined by (4.53) can be characterized, similarly to the semigroup case, as following: ⎧ V ⎨ w ∈ ,H(G0 ) (B0 ) ⎩
⇐⇒
V (n) ∞ there exist {w (n) }∞ n=1 ⊂ B0 , {G }n=1 ⊂ H(G0 ), {tn } ⊂ R with tn → +∞ as n → +∞ as n → +∞. such that UG(n) (tn , )w(n) −→ w in V
(4.54)
Indeed, (4.54) implies that 0,H(G0 ) (BV0 ) = ,H(G0 ) (BV0 ) for each ∈ R, in other words, ,H(G0 ) (BV0 ) is independent of . The rest of the proof to this lemma is similar to the proof of Propositions 4.1, 4.2 and 4.3(iii) in [13]. Here we only sketch the main steps and omit the detailed proof. . This assertion can be established by the uniformly (w.r.t. Step 1: ,H(G0 ) (BV0 ) is a nonempty compact set in V G ∈ H(G0 )) absorbing property (Lemma 4.1), asymptotic compactness property (Lemma 4.5) of {UG (t, )}t , G ∈ and the characterization of ,H(G ) (BV ) described by (4.54). H(G0 ), in V 0 0 and any fixed ∈ R, Step 2: For any bounded set BV of V lim
sup
t→+∞ G∈H(G ) 0
dist V (UG (t, )BV , ,H(G0 ) (BV )) = 0.
(4.55)
Eq. (4.55) could be established by contradiction, with the help of Lemma 4.5 and (4.54). , Step 3: For any bounded set BV ⊂ V
,H(G0 ) (BV ) ⊆ ,H(G0 ) (BV0 ).
(4.56)
Eq. (4.56) can be verified by using (2.30), (4.54) and Lemma 4.1. Eqs. (4.55) and (4.56) imply the uniformly (w.r.t. . G ∈ H(G0 )) attracting property of ,H(G0 ) (BV0 ) in V The proof of Lemma 4.7 is complete. 5. Existence and regularity of uniform attractor In this section, we summarize the results obtained in the previous sections. Then we state our main result and prove it. ) and {UG (t, )}t , G ∈ H(G0 ), be the family of processes corDefinition 5.1. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H is said to be the uniform (w.r.t. G ∈ H(G0 )) attractor of responding to Eqs. (2.8)–(2.9). A closed set ⊂ H {UG (t, )}t , G ∈ H(G0 ), in H if satisfies and any fixed ∈ R (1) (Uniformly attracting property). For any bounded set B of H lim
sup
t→+∞ G∈H(G ) 0
dist H(UG (t, )B, ) = 0.
(5.1)
(2) (Minimal property). is the minimal set (for inclusion relation) among the closed sets satisfying (1). with H Similarly, we can define the uniform (w.r.t. G ∈ H(G0 )) attractor for {UG (t, )}t , G ∈ H(G0 ), in V being replaced by V in Definition 5.1.
626
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
The main result of this paper reads as follows: ) and condition (A1 ) hold. Then the family of processes {UG (t, )}t , Main Theorem. Let G0 = (g0 , g˜ 0 ) ∈ L2c (R; H G ∈ H(G0 ), corresponding to Eqs. (2.8)–(2.9) possesses a uniform (w.r.t. G ∈ H(G0 )) attractor A0H(G0 ) in H 1 , respectively. Moreover and a uniform (w.r.t. G ∈ H(G0 )) attractor AH(G0 ) in V A0H(G0 ) ⊆ A1H(G0 ) .
(5.2)
Proof. According to the well-established theory (see e.g., Theorem IV3.1 in [7]), the existence of uniform (w.r.t. and A1 G ∈ H(G0 )) attractors A0H(G0 ) in H H(G0 ) in V for {UG (t, )}t , G ∈ H(G0 ), can be derived from (see below Remark 5.1) and the Lemmas 3.4 and 4.7, respectively. We next prove (5.2). Since A1H(G0 ) is bounded in V 1 1 → H is continuous, we see A embedding V H(G0 ) is bounded in H . Also we deduce from Remark 4.1 that AH(G0 ) and thus A1 attracts uniformly (w.r.t. G ∈ H(G0 )) all bounded sets of H H(G0 ) can be regarded as a bounded uniformly . By the minimality property of A0 (w.r.t. G ∈ H(G0 )) attracting set for {UG (t, )}t , G ∈ H(G0 ), in H H(G0 ) , we obtain (5.2). The proof is complete. Remark 5.1. By the well-established theory (see e.g., [7, Theorem IV5.1, p. 89]), A0H(G0 ) and A1H(G0 ) are compact and V , respectively. in H Acknowledgments We express our sincere thanks to the anonymous reviewers for their careful reading of the manuscript and giving valuable comments and suggestions. It is their contributions that greatly improves the manuscript. We also thank the editors for their kind help. The corresponding author thanks Professors Yongsheng Li and Zhengyi Lu for their constant supports and encouragements. References [1] F. Abergel, Existence and finite dimensionality of global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108. [2] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] A.V. Babin, The attractor of a Navier–Stokes system in an unbounded channel-like domain, J. Dyn. Diff. Eqs. 4 (1992) 555–584. [4] A.V. Babin, M.I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Roy. Soc. Edinburg 116A (1990) 221–243. [5] J.M. Ball, Continuity properties of global attractors of generalized semiflows and the Navier–Stokes equations, J. Nonlinear Sci. 7 (1997) 475–502; J.M. Ball, J. Nonlinear Sci. 8 (1998) 233 Erratum. [6] J. Chen, Z. Chen, B. Dong, Existence of H 2 -global attractors of two-dimensional micropolar fluid flows, J. Math. Anal. Appl. 322 (2006) 512–522. [7] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, vol. 49, AMS, Providence, RI, 2002. [8] M.A. Efendiev, S.V. Zelik, The attractor for nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688. [9] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [10] J.M. Ghidaglia, A note on the strong convergence towards the attractors for damped forced KdV equations, J. Differential Equations 110 (1994) 356–359. [11] J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [12] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Paris, Masson, 1991. [13] Y. Hou, K. Li, The uniform attractors for the 2D nonautonomous Navier–Stokes flow in some unbounded domain, Nonlinear Anal. 58 (2004) 609–630. [14] N. Ju, The H 1 -compact global attractor for the solutions to the Navier–Stokes equations in 2D unbounded domains, Nonlinearity 13 (2000) 1227–1238. [15] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. [16] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969. [17] G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999.
C. Zhao et al. / Nonlinear Analysis: Real World Applications 9 (2008) 608 – 627
627
[18] G. Lukaszewicz, Long time behavior of 2D micropolar fluid flows, J. Math. Comput. Model. 34 (2001) 1787–1802. [19] G. Lukaszewicz, Asymptotic behavior of micropolar fluid flows, Inter. J. Eng. Sci. 41 (2003) 259–269. [20] G. Lukaszewicz, M.A. Rojas-Medar, M.M. Santos, Stationary micropolar fluid with boundary data in L2 , J. Math. Appl. Anal. 271 (2002) 91–107. [21] G. Lukaszewicz, W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys. 55 (2004) 247–257. [22] Q. Ma, S. Wang, C. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J. 51 (2002) 1541–1559. [23] I. Moise, R. Rosa, X.M. Wang, Attractors for non-compact semigroup via energy equations, Nonlinearity 11 (1998) 1369–1393. [24] I. Moise, R. Rosa, X.M. Wang, Attractors for non-compact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst. 10 (2004) 473–496. [25] R. Rosa, The global attractor for the 2D Navier–Stokes flow on some unbounded domains, Nonlinear Anal. 32 (1998) 71–85. [26] W. Sadowski, Upper bound for the number of degrees of freedom for magneto-micropolar flows and turbulence, Inter. J. Eng. Sci. 41 (2003) 789–800. [27] G. Sell, Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. [28] R. Temam, Navier–Stokes Equations (Theory and Numerical Analysis), third ed., Elsevier Science Publishers, The Netherlands, 1984. [29] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, 1988. [30] R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, second ed., SIAM, Philadelphia, 1995. [31] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D 88 (1995) 167–175. [32] S.V. Zelik, The attractors for nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst. 7 (2001) 593–641. [33] S.V. Zelik, Attractors for reaction–diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math. 56 (2003) 584–637. [34] C. Zhao, Y. Li, H 2 -compact attractor for a non-Newtonian system in two-dimensional unbounded domains, Nonlinear Anal. 56 (2004) 1091–1103.