Regularity of Attractor for 3D Derivative Ginzburg-Landau Equation

Transcription

1 Dynamics of PDE, Vol.11, No.1, , 2014 Regularity of Attractor for 3D Derivative Ginzburg-Lanau Equation Shujuan Lü an Zhaosheng Feng Communicate by Y. Charles Li, receive November 26, Abstract. In this paper, we are concerne with a three-imensional erivative Ginzburg-Lanau equation with a perioic initial value conition. The smoothing property of the solution is establishe by a uniform priori estimates. The existence of the global attractors, A i H i p i = 2, 3,, for the semi-group {S i t} t 0 of the operators generate by the equation is prove by using the compactness principle. Finally, the regularity of the global attractors, namely, A 2 = A 3 = = A m, is prove by using the metho of semi-group ecomposition. Contents 1. Introuction Existence of High-Orer Attractors Decomposition of Semigroup Regularity of Attractor 106 References Introuction Many physical an chemical phenomena can be escribe by nonlinear equations, such as the Korteweg-e Vries equation for propagate waves on shallow water surfaces [1] an the Ginzburg-Lanau equation for a phase transition in superconuctivity [2]. In applie mathematics an theoretical physics, the escription of spatial pattern formation or chaotic ynamics in continuum systems, in particular biological systems or flui ynamical systems, has been a challenging task. The 1991 Mathematics Subject Classification. 35B41, 35Q56, 35B45. Key wors an phrases. Ginzburg-Lanau equation, global attractor, regularity, Höler inequality, priori estimates, semi-group ecomposition. This work is supporte by NSF of China No an c 2014 International Press

2 90 SHUJUAN LÜ AND ZHAOSHENG FENG mathematical theory behin these systems appears rich an interesting, an in the broa sense, is a topic which continuously attracts consierable attention from a variety of scientific fiels. Due to the complexity of the corresponing nonlinear evolution equations, simpler moel equations for which the mathematical issues can be solve with greater success, have been erive. The complex Ginzburg-Lanau equation GLE is one of these moels which takes the form 1.1 u t 1 + iν u iµ u 2σ u γu = 0. This equation escribes the evolution of the amplitue of perturbations to steaystate solutions at the onset of instability [3, 4]. In the past ecaes, this equation was wiely stuie for instability waves in hyroynamics, such as the nonlinear growth of Rayleigh-Bénar convective rolls [5], the appearance of Taylor vortices in the Couette flow between counter-rotating cyliners [6], the evelopment of Tollmien-Schlichting waves in plane Poiseuille flows [7], an the transition to turbulence in chemical reactions [8], The erivative Ginzburg-Lanau equation DGLE 1.2 u t = ρu iν u 1 + iµ u 2σ u + λ 1 u 2 u + λ 2 u u 2 arises as the envelope equation for a weakly subcritical to counter-propagating waves, an it is also important for a number of physical systems incluing the onset of oscillatory convection in binary flui mixture [9]. In the case of one or two imensions, finite imensional global attractors an regularity of solutions were explore in [10, 11]. When ν = 0, the equation 1.2 incorporates to the erivative nonlinear Schröinger equation [12]. In the past ecaes, equations 1.1 an 1.2 have been extensively stuie in the one or two spatial imension. For example, Ghiaglia an Héron [13], Doering et al [14], Promislow [15], Bu [16] stuie the finite-imensional attractor an relate ynamical properties for 1D or 2D GLE 1.1 with σ = 1 or 2. Lü [17] investigate the upper semi-continuity of approximate attractors of GLE 1.1 in one-imensional space with σ = 1. Guo et al [18, 19] an Gao [20, 21, 22] ealt with the 1D an 2D DGLE 1.2 an explore the existence of the global solution an the finite-imensional global attractor of DGLE 1.2 with perioic bounary conitions, Cauchy conitions or Dirichlet inhomogeneous bounary value conitions in the case of σ = 2. Nevertheless, relevant theoretical results for the case of three spatial imensions for the Ginzburg-Lanau equation appear scarce. The main reason lies in the fact that the Sobolev interpolation inequalities use in one- or two-imensional case become invali for the three-imensional case. Thus, it is necessary to make more subtle estimates for the nonlinear terms to overcome this ifficulty. Doering et al [23] an Okazawa et al [24] consiere the case of three-imensional space for GLE 1.1 with perioic bounary conitions an initial bounary value conitions, respectively. They establishe the existence an uniqueness of global solution uner certain parametric conitions. In [25, 26, 27], Lü et al also iscusse the perioic initial-value problem of GLE 1.1 in three-imensional space, an prove the existence an uniqueness of global solution uner a weaker restriction of parametric conitions. Furthermore, the existence of global attractor an exponential attractor with finite imensions as well as the upper semi-continuity of global attractor was also explore. Naer an Hatem [28] constructe a solution to DGLE 1.2 in N- imensional space. Karachlios an Zographoppulos [29] investigate a egenerate

3 REGULARITY OF ATTRACTOR 91 case of DGLE 1.2 with the Dirichlet bounary value conition in N-imensional space N 2 an prove the existence of the global attractor in L 2. In this paper, we consier a perioic initial-value problem of a more general erivative 3D Ginzburg-Lanau equation: u t = 1 + iν u 1 + iµ u 2σ u + γu λ 1 u 2 u λ 2 u 2 ū, x, t R 3 R +, ux, 0 = u 0 x, x R 3, ux + e j, t = ux, t, j = 1, 2, 3, x, t R 3 R +, where ν, µ an γ > 0 are real constants, λ 1 an λ 2 are complex constant vectors, an the over-line enotes the complex conjugate. We assume that the parameters µ an σ satisfy the conition 2σ µ <, σ > 2. σ The existence of the unique solution, the finite-imensional global attractor an the exponential attractor was investigate uner the assumption 1.7 in [30] an the upper semi-continuity of global attractor was escribe in [31], but the regularity of global attractor was not iscusse therein. Since regularity an global attractor are two of the most important qualities for ynamical systems an any possible regularity of the attractor is extremely helpful for a better unerstaning of the long term behavior of the semigroup, in this paper our main purpose is to present some regularity results on the global attractor for the problem On the assumption that the issipative ynamical system associate with a partial ifferential equation possesses an global attractor A in the Sobolev space, say H 2, the regularity is to be unerstoo here in the sense of the theory of partial ifferential equations. That is, if the ata are sufficiently regular, then the global attractor lies in a set of more spatially regular functions, a Sobolev subspace H m, for an appropriate m. In other wors, the regularity of attractor means that even the system starts with a initial state u 0 x in a lower orer Sobolev space H 2, its long-time state may be a more spatially regular function in a higher orer Sobolev space H m. The ata mentione here inicate the ifferent functions an parameters appearing in the partial ifferential equation. Throughout this paper we shall use the following notions: Let = [0, 1] [0, 1] [0, 1]. We enote by, the usual inner prouct of L 2, by m the norm of Sobolev spaces H m, an = 0 an = L. Let L 2 p = {φ L 2 φx + e j = φx, j = 1, 2, 3} with the norm efine as that of L 2. Let H m p = {φ H 2 p φx + e j = φx, j = 1, 2, 3} with the norm efine as that of H 2. In our stuy, we nee the following technical three Lemmas in the proofs of our main results. Lemma 1.1 Sobolev Interpolation Inequality [32]. Let u L q, D m u L r an R n, where 1 r an 0 j m. Then there exists a constant c = cj, m,, p, q, r inepenent of u such that D j u L p c u a W m,r u 1 a L q,

4 92 SHUJUAN LÜ AND ZHAOSHENG FENG where 1 p = j 1 n + a r m + 1 a 1 n q, j m < a < 1. Lemma 1.2 Gronwall s Inequality [33]. Let yt, gt an ht be three nonnegative functions satisfying an t+r t Then we have gss α 1, y t gtyt + ht, t t 0 0. t+r t yt + r hss α 2, t+r α3 r + α 2 e α1, t t 0. t yss α 3, t t 0. Lemma 1.3 [34]. Let E be a Banach space. Suppose that {St} t 0 is a semi-group of continuous operators, i.e. St : E E, with St Sτ = St + τ, S0 = I, where I is the ientical operator. Suppose that the operator St satisfies the following three conitions. i The operator St is boune, i.e. for any given R > 0, if u E R, then there exists a constant CR such that Stu E CR, for t [0, +. ii There is a boune absorbing set B 0 E, i.e., for any given boune set B E, there exists a constant T = T B such that ST B B 0, for t T. iii St is a completely continuous operator for the sufficiently large t > 0. Then the semi-group {St} t 0 of operators has a compact global attractor A E. The rest of this paper is organize as follows. In Section 2, the smoothness of solutions is obtaine by a priori estimates uner a weaker restriction on the parameter σ. The existence of global attractors A i Hp i i = 3, 4, m for the semi-group of operators {S i t} t 0 generate by the system is prove. In Section 3, the solution operator S 2 t is ecompose as S 2 1 t+s2 2 t, where S 2 1 tu 0 is more regular than S 2 tu 0, an S 2 2 u 0 2 approaches zero as t tens to infinity uniformly for u 0 boune in Hp. 2 Section 4 is eicate to the regularity of global attractors. 2. Existence of High-Orer Attractors In this section, we prove the existence of the global attractor A m H m p m = 2, 3,,. Lemma 2.1 [30]. Suppose that the conition 1.7 hols an u 0 x H 2 p. Then the problem possesses a unique global solution ux, t L R + ; H 2 p L 2 [0, T ; H 3 p,

5 REGULARITY OF ATTRACTOR 93 an for any R > 0 given, there exists t 2 = t 2 R such that u 2 E 2, t 0, u 2 M 2, t t 2, if u 0 2 R, where the constant E 2 epens on the parameters σ, ν, µ, γ, λ 1, λ 2 an R; an M 2 only epens on the parameters σ, ν, µ, γ, λ 1 an λ 2. Furthermore, the semi-group {St} t 0 of operators generate by the problem has a compact global attractor A 2 = A H 2 p, i.e. there exists a set A H 2 p such that a StA = A for all t 0; b iststb, A 0 for any boune set B H 2 p, where istx, Y = sup inf x y 2. y Y x X Next, we show the high-orer smoothness of the global solution for the problem Proposition 2.1. Uner [ the conitions of Lemma 2.1, suppose that m 2 is a positive integer, σ 1 m 2 2 1] or σ is a positive integer. Then there exists t m = t m R such that 2.1 u m M m, t t m, u 0 2 R, an 2.2 u m E m, t 0, u 0 m R, where the constant E m epens on the parameters σ, ν, µ, γ, λ 1, λ 2, m an R; an M m only epens on the parameters σ, ν, µ, γ, λ 1, λ 2 an m. Thus, the problem possesses the global smooth solution an the close ball u CR + ; Hp m C 1 R + ; Hp m 2, B m = {ϕ H m p ϕ m M m } is the boune absorbing set of the semi-group of operators {S m t} t 0. Proof. We prove that 2.1 an 2.2 by using the principle of mathematical inuction. When m = 2, 2.1 an 2.2 can be euce by Lemma 2.1 irectly. Suppose that 2.1 an 2.2 hol for m = 2, 3,, k 1, i.e. 2.3 u m M m for all t t m if u 0 2 R, m = 2, 3,, k 1; u m E m for all t 0 if u 0 m R, m = 2, 3,, k 1. By using the Sobolev interpolation inequality, there exist constants E m = E mr an M m such that 2.4 u W m 2, M m, t t m, 3 m k 1, an u W m 2, E m, t 0, 3 m k 1. This implies that 2.1 an 2.2 also hol for m = k.

6 94 SHUJUAN LÜ AND ZHAOSHENG FENG Setting l = [ ] k 1 2 an ifferentiating 1.4 for l times with respect to t, we have 2.5 u t l+1 1+iν u t l+1+iµ u 2σ u t l γu t l+λ 1 u 2 u t l+λ 2 u 2 ū t l = 0, If k = 2l + 1, by taking the real part of the L 2 - inner prouct of 2.5 with u t l, we have 1 2 t u t l 2 + u t l 2 = γ u t l 2 + Re 1 + iµ u 2σ u t l ū t lx 2.6 Re λ 1 u 2 u t l ū t lx Re λ 2 u 2 ū t l ū t lx. For t t k 1 = t 2l, σ 1 2 l 1 = 2 1 [ k 2 ] 1 or σ is a positive integer, in view of Höler s inequality, the Sobolev interpolation inequality an Young s inequality together with 2.3 an 2.4, the four terms on the right han sie of 2.6 can be estimate as 1 Re + iµ u 2σ u t l ū t lx 1 8 u t l 2 + cm k 1, M k 1, Re an Re So 2.6 can be simplifie as γ u t l 2 γ u t l u t l 1 8 u t l 2 + 2γ 2 u t l u t l 2 + cm k 1, M k 1, λ 1 u 2 u t l ū t lx 1 8 u t l 2 + cm k 1, M k 1, λ 2 u 2 ū t l ū t lx 1 8 u t l 2 + cm k 1, M k t u t l 2 + u t l 2 C 1 M k 1, M k 1, t t k 1. By taking the L 2 -inner prouct of 2.5 with u t l an using a similar way to the erivation of 2.7, we get 2.8 That is, Set t u t l 2 + u t l 2 C 2 M k 1, M k 1, t t k 1. t+1 t u t ls 2 s u t lt 2 + C 2 M k 1, M k 1 CM k 1 + C 2 M k 1, M k 1 = α 3, t t k 1. α 2 = C 1 M k 1, M k 1. Applying the Gronwall s inequality to 2.7, we have 2.9 u t lt α 3 + α 2, t t k 1,

7 REGULARITY OF ATTRACTOR 95 which leas to k u 2 = 2l+1 u 2 Cα 2, α 3, t t k Let M k = Cα 2, α 3 + Mk 1 2 an t k = t k So 2.1 hols for m = k. If u 0 H k p, by using an analogous way to the erivation of 2.7, 2.6 can be re-expresse as 2.10 t u t l 2 + u t l 2 + u t l 2 C 3 E k 1, E k 1, t 0. Multiplying 2.10 by e t an integrating it with respect to t yiels u t l 2 u t l0 2 + C 3 CR 2 + C 3 = E k, t 0. Let E k = E k + E2 k 1. So 2.2 hols for m = k too. When k = 2l + 2, by taking the L 2 -inner proucts of 2.5 with 2 u t l u, respectively, we have t u t l 2 + u t l 2 C 1M k 1, M k 1, t t k 1, an t u t l 2 + u t l 2 C 2M k 1, M k 1, t t k 1. Using an analogue iscussion to the erivation of 2.9, we have an u t lt α 3 + α 2, t t k 1, which leas to k u 2 = 2l+2 u 2 C α 2, α 3, t t k Let M k = Cα 2, α 3 + M k 1 2 an t k = t k 1 + 1, so 2.1 hols for m = k If u 0 H k p, it is euce that t u t l 2 + u t l 2 C 3E k 1, E k 1, t 0. Multiplying 2.11 by e t an integrating it with respect to t yiels u t l 2 u t l0 2 + C 1 CR 2 + C 3 = E k, t 0. Let E k = E k + E2 k 1, thus 2.2 hols for m = k too. Consequently, we conclue that 2.1 an 2.2 hol for any positive integer m 2. The proof of Proposition 2.1 is complete. In orer to prove the existence of the high-orer global attractor A m, we introuce another proposition as follows. [ Proposition 2.2. Suppose that the conitions of Proposition 2.1 hol with σ 1 m 1 ] 2 2 or σ is a positive integer. Then the semi-group of operations S m t t 0 : Hp m Hp m is uniformly compact for the sufficiently large t > 0.

8 96 SHUJUAN LÜ AND ZHAOSHENG FENG Proof. If m = 2l, we consier the real parts of the inner prouct of 2.5 with u t l an u t l, respectively. It follows from Proposition 2.1 that there exist constants C 1 = C 1 M m, M m an C 2 = C 2 M m, M m such that 2.12 t u t l 2 + u t l 2 C 1 M m, M m, t t m, an 2.13 t u t l 2 + u t l 2 C 2 M m, M m, t t m. Applying the Gronwall s inequality to 2.12 an using 2.13, we get which leas to u t l 2 C 3 M m, M m, t t m+1 = t m + 1, m+1 u 2 = 2l+1 u 2 CM m, M m u t l 2 M m+1, t t m+1 = t m + 1. If m = 2l + 1, using the same arguments as the above we have 2.14 t u t l 2 + u t l 2 C 1M m, M m, t t m, an 2.15 t u t l 2 + u t l 2 C 2M m, M m, t t m. Again, applying the Gronwall s inequality to 2.14 an using 2.15 gives m+1 u 2 = 2l+2 u 2 CM m, M m u t l 2 M m+1, t t m+1 = t m + 1. By virtue of the Sobolev compact imbeing theorem, we know that the semigroup of operators S m t is uniformly compact for t t m+1. So the proof of Proposition 2.2 is complete. [ On the other han, if σ 1 m ] 2 2 or σ > 2 is an integer, accoring to Propositions 2.1 an 2.2, one can see that S m t is strongly continuous. Making use of Lemma 1.3, we obtain the following theorem: Theorem 2.3. Suppose that all conitions of Proposition 2.2 hol. Then there exists a global attractor A m H m p of the semi-group {S m t} t 0 of the operators generate by the problem Decomposition of Semigroup In orer to prove the regularity of global attractor, it is necessary to ecompose S 2 t appropriately. In this section, we ecompose S 2 t as S 2 1 t+s2 2 t; where S 2 1 tu 0 is more regular than the solution S 2 tu 0, an S 2 2 tu 0 2 approaches zero as t tens to infinity uniformly for the boune u 0 in Hp. 2 For any given positive integer N, let S N = Span{e 2πik x : k N} an enote the orthogonal projection operator by P N : L 2 p S N an Q N = I P N [32]. Then there have

9 REGULARITY OF ATTRACTOR 97 Lemma 3.1 [35]. If v H m p, then there exists a constant c inepenent of v an N such that P N v m cn m j P N v j, 0 < j m, an Q N v j cn j m Q N v m, j = 0, 1,, m, j Q N v cn j m m Q N v, j = 0, 1,, m. We also ecompose the solution ux, t. Let u 0 H 2 p an u = Stu 0 be the solution of the problem , then it has u = P N u + Q N u = pt + qt, where pt = P N u represents the low-frequency part of u an qt = Q N u represents the high-frequency part of u. We split the high-frequency part qt as qt = y + z, where y, z Q N L 2 p are the solutions of the following equations for t t 2 : y t 1 + iν y γy = 1 + iµq N u 2σ p + y Q N λ1 u 2 p + y + λ 2 u 2 p + ȳ, yx, t = yx + e j, t, j = 1, 2, 3; yx, t 2 = 0, z t 1 + iν z = γz 1 + iµq N u 2σ z Q N λ1 u 2 z + λ 2 u 2 z, zx, t = zx + e j, t, j = 1, 2, 3; zx, t 2 = Q N ut 2, respectively. For t t 2, yt = 0 an zt = Q N ut, where t 2 is given by Proposition 2.1. We first prove that y is smooth for t t m an z converges towar zero in H 2 p when t goes to infinity. Theorem 3.1. Uner the conition 1.7, if u 0 H 2 p satisfies u 0 2 R, then there exists a unique solution y of an a unique solution z of satisfying y, z C 1 [0, ; L 2 p C [0, ; Hp 2. Moreover, If σ is a positive integer or σ 2[ 1 m ] 2, then there exists N3 large enough, an constants K m = K m N an δ = δn > 0 such that for any given N N 3, the following estimates hol: 3.5 yt m K m, t t m, m = 2, 3,, 3.6 zt 2 ut 2 2 e δt t2 M 2 e δt t2, t t 2, where M 2, t m an R are given by Lemma 2.1 an Proposition 2.1, respectively. Proof. The existence an uniqueness can be prove irectly by using the Galerkin metho [30]. We separate our proof for estimates 3.5 an 3.6 into three steps. Step 1. Consier the estimates for y in H 2 p.

10 98 SHUJUAN LÜ AND ZHAOSHENG FENG By taking the real part of the inner prouct of 3.1 with y, we get 1 2 t y 2 + y 2 γ y 2 + Re 1 + iµ Q N u 2σ p + y ȳx = Re λ 1 Q N u 2 p + y ȳx Re λ 2 Q N u 2 p + ȳ 3.7 ȳx. For the last term in the left-han sie of the equation 3.7, using the efinition of Q N gives Re 1 + iµ [ Q N u 2σ p + y, y ] = u 2σ y 2 x+re 1 + iµ u 2σ pȳx. Furthermore, accoring to Proposition 2.1 an efinitions of P N an pt, an by using Höler s inequality an the Sobolev interpolation inequality, we euce that Re iµ u 2σ p ȳx 1 + iµ u 2σ y 2 x u 2σ y 2 x iµ 2 1 u 2σ p 2 2 x u 2σ p 2 x u 2σ y 2 x iµ 2 u 2σ u 2. Separate the first term in the right-han sie of 3.7 as Re λ 1 Q N u 2 p + y ȳx 3.9 = Re λ 1 u 2 p ȳx + Re λ 1 u 2 y ȳx. an Notice that Re λ 1 u 2 y ȳx Re λ u 2σ y 2 x 1 σ y σ 2 σ y u 2σ y 2 x y σ 2 λ1 2σ σ 2 y 2, λ 1 u 2 p ȳx λ 1 u 2 p y γ2 y 2 + λ 1 2 2γ u 4 u 2. So 3.9 can be rewritten as Re λ 1 Q N u 2 p + y ȳx 1 u 2σ y 2 x y σ 2 λ1 2σ σ 2 + γ y 2 + λ γ u 4 u 2.

11 REGULARITY OF ATTRACTOR 99 Similarly, from the secon term in the right-han sie of the equation 3.7 we have Re λ 2 Q N u 2 p + ȳ yx 1 u 2σ y 2 x y σ 2 λ2 2σ σ γ 2 y 2 + λ 1 2 2γ u 4 u 2. Substituting into 3.7, we get t y 2 + y 2 4γ + 2 σ+2 σ 2 λ1 2σ σ σ σ 2 λ2 2σ σ 2 y 2 CM 2. From Lemma 3.1, we know that So 3.12 can be simplifie as y 2 c 0 N 2 y 2. t y 2 + c 0 N 2 4γ + 2 σ+2 σ 2 λ1 2σ σ σ σ 2 λ2 2σ σ 2 y 2 CM 2. Let N 0 be large enough such that c 0 N 2 0 4γ 2 σ+2 σ 2 λ1 2σ σ 2 2 σ+2 σ 2 λ2 2σ σ 2 > 0. Then for N N 0, multiplying 3.13 by e δ0t an integrating it from t 2 to t with yt 2 = 0, we have 3.14 where 3.15 y 2 CM 2 δ 0 = K 2 0, t t 2, δ 0 = δ 0 N = cn 2 4γ 2 σ+2 σ 2 λ1 2σ σ 2 2 σ+2 σ 2 λ2 2σ σ 2. By taking the real part of the inner prouct of 3.1 with y, we have 1 2 t y 2 + y 2 = γ y 2 + Re 1 + iµ Q N u 2σ p + y ȳx +Re λ 1 Q N u 2 p + y ȳx + Re λ 2 Q N u 2 p + ȳ ȳx. Using the Sobolev interpolation inequality, the four terms in the right han sie of 3.15 can be estimate as 1 + iµ Q N u 2σ p + y ȳx 1 8 y 2 + CK 0, M 2, λ 1 γ y y 2 + 2γ 2 K0, 2 Q N u 2 p + y ȳx 1 8 y 2 + CK 0, M 2,

12 100 SHUJUAN LÜ AND ZHAOSHENG FENG an λ 2 Q N u 2 p + ȳ ȳx 1 8 y 2 + CK 0, M 2, respectively. Thus, 3.15 can be simplifie as t y 2 + y 2 CK 0, M 2. Since y 2 cn 2 y 2, we further have t y 2 + cn 2 y 2 CK 0, M 2. Multiplying the above inequality by e cn 2t an integrating it from t to t 2 with yt 2 = 0 yiels y 2 CM 0, K 0 = K cn 2 1, t t 2. Using the same proceure, by consiering the real part of the inner prouct of 3.1 with 2 y, we have t y 2 + y 2 = Re 1 + iµ Q N u 2σ p + y 2 ȳx Re λ 1 Q N u 2 p + y 2 ȳx +γ y 2 Re λ 2 Q N u 2 p + ȳ 2 ȳx. We estimate each term of the right sie in 3.17 for σ 1 2. By using Höler s inequality, the Sobolev interpolation inequality, as well as , we have 1 + iµ Q N u 2σ p + y 2 ȳx 1 8 y 2 + CK 0, K 1, M 2, λ 1 an λ 2 So, 3.17 can be simplifie as γ y y 2 + 2γ 2 K1, 2 Q N u 2 p + y 2 ȳx 1 8 y 2 + CK 0, K 1, M 2, Q N u 2 p + ȳ 2 ȳx 1 8 y 2 + CK 0, K 1, M 2. t y 2 + y 2 CK 0, K 1, M 2. Since y 2 cn 2 y 2, we further have 3.18 t y 2 + cn 2 y 2 CK 0, K 1, M 2. Multiplying 3.18 by e cn 2t an integrating it from t from t 2 with yt 2 = 0 yiels y 2 CK 0, K 1, M 2 cn 2 = K2, t t 2.

13 REGULARITY OF ATTRACTOR 101 Set K 2 = K0 2 + K2 1 + K 2 2. So it is easy to see that 3.5 hols for m = 2. Step 2. Estimates for y in Hp m m 3. We prove that 3.5 hols for any m 3 by using the mathematical inuction. For m = 3, ifferentiating 3.1 with respect to t an using the real part of the L 2 -inner prouct with y t, we have t y t 2 + y t 2 = Re λ 1 Q N u 2 p + y ȳ t tx +Re λ 2 Q N u 2 p + ȳ ȳ t tx +γ y t 2 + Re 1 + iµ Q N u 2σ p + y, ȳ t tx. For t t 2, we estimate the four terms in the right han sie of By using Höler s inequality, the Sobolev interpolation inequality an Proposition 2.1, we euce that 1 + iµ Q N u 2σ p + y, ȳ t tx 1 8 y t 2 + CK 2, M 2, λ 1 γ y t y t 2 + CK 2, M 2, Q N u 2 p + y ȳ t tx 1 8 y t 2 + CK 2, M 2, an λ 2 Q N u 2 p + ȳ ȳ t tx 1 8 y t 2 + CK 2, M 2. Thus, 3.19 can be simplifie as 3.20 t y t 2 + y t 2 CK 2, M 2. Since y t 2 cn 2 y t 2, we further have t y t 2 + cn 2 y t 2 CK 2, M 2. Multiplying 3.20 by e cn 2t, integrating it from t from t 2 an using the inequality yt 2 c ut 2 2σ ut CM 2, we have y t 2 e cn 2 t t 2 y t t CK 2, M 2 cn 2 e cn 2 t t 2 CM 2 + CK 2, M 2 cn 2 = ρ 2 3, t t 2.

14 102 SHUJUAN LÜ AND ZHAOSHENG FENG That is, y y t + C 0 M 2, K 2 ρ 3 + CM 2, K 2 = K 3, t t 3 = t Set K 3 = K2 2 + K 3 2. So 3.5 hols for m = 3. Suppose that 3.5 hols for any 3 < m k 1 k is a positive integer, namely, there exists a constant K m such that 3.21 y m K m, t t m, m k 1. It follows from the Sobolev interpolation inequality that 3.22 y W m 2, K m, t t m, m k 1, where the constant K m epens on K m. From Proposition 2.1 an the efinition of P N we have 3.23 p m M m, p W m 2, CK m, t t m, m k 1, where t m in is the same as the one given in Proposition 2.1. Now, we consier the case of m = k. Let l = [ ] k 1 2. Differentiating 3.1 for l times with respect to t gives 3.24 y t l iν y t l iµq N u 2σ p + y t l γy t l = Q N λ1 u 2 p + y + λ 2 u 2 p + ȳ t l. If k = 2l + 1, by consiering the real part of the inner prouct of 3.24 with y t l, we have 1 2 t y t l 2 + y t l 2 = γ y t l 2 + Re 1 + iµ Q N u 2σ p + y t l ȳ t lx +Re Q N λ1 u 2 p + y ȳ t l t lx +Re Q N λ2 u 2 p + ȳ 3.25 ȳ t l t lx. We estimate each term in the right han sie of 3.25 by applying Höler s inequality, the Sobolev interpolation inequality, Young s inequality, as well as When t t k 1 = t 2l an σ is a positive integer or σ l 2, we have 1 + iµ u 2σ p + y t l ȳ t lx 1 8 y t l 2 + CK k 1, K k 1, M k 1, M k 1, an γ y t l y t l 2 + CK k 1, K k 1, M k 1, M k 1, Q N λ1 u 2 p + y ȳ t l t lx 1 8 y t l 2 + CK k 1, K k 1, M k 1, M k 1, Q N λ2 u 2 p + ȳ ȳ t l t lx 1 8 y t l 2 + CK k 1, K k 1, M k 1, M k 1.

15 REGULARITY OF ATTRACTOR 103 Substituting the above four inequalities into 3.25 yiels 3.26 t y t l 2 + y t l 2 C 1 K k 1, K k 1, M k 1, M k 1, t t k 1. Consiering the inner prouct of 3.24 with y t l an using a similar argument as the erivation of 3.26, we euce 3.27 t y t l 2 + y t l 2 C 2 K k 1, K k 1, M k 1, M k 1, t t k 1. When t t k 1, it follows from 3.27 that t+1 t y t ls 2 s y t lt 2 + C 2 K k 1, K k 1, M k 1, M k 1 CK k 1 + C 2 K k 1, K k 1, M k 1, M k 1 = α 3. On the other han, setting α 2 = C 1 K k 1, K k 1, M k 1, M k 1 an applying the Gronwall s inequality to 3.26, we have 3.28 y t lt α 3 + α 2, t t k 1, which leas to k y 2 = 2l+1 y 2 Cα 2, α 3, t t k Set K k = Cα 2, α 3 + Kk 1 2 an t k = t k So 3.5 hols for m = k. Similarly, if k = 2l+2, consiering the real parts of the inner proucts of 3.24 with 2 y t l an y t l, respectively, an using an analogous way to the erivation of 3.26 an 3.27, we erive that t y t l 2 + y t l 2 C 1K k 1, K k 1, M k 1, M k 1, t t k 1, t y t l 2 + y t l 2 C 2K k 1, K k 1, M k 1, M k 1, t t k 1. Following the erivation of 3.28, we get which implies y t lt α 3 + α 2, t t k 1, k y 2 = 2l+2 y 2 C α 2, α 3, t t k By setting K k = C α 2, α 3 + K2 k 1 an t k = t k 1 + 1, we see that 3.5 hols for m = k. Consequently, the proof of 3.5 is complete. Step 3. Estimates for z in Hp. 2 By consiering the real part of the inner prouct of the equation 3.3 with z, we have t z 2 + z 2 + u 2σ z 2 x γ z 2 = Re λ1 u 2 z + λ 2 u 2 z zx.

16 104 SHUJUAN LÜ AND ZHAOSHENG FENG We estimate the right han sie of 3.29 by applying Höler s inequality an Young s inequality: 1 σ 2 Re λ 1 u 2 z zx λ 1 z u 2σ z 2 σ x z 2 2σ x 1 4 z u 2σ z 2 x + σ 2 λ 1 2σ 4 σ σ 2 2 σ 2 z 2, 2 an Re λ 2 u 2 z zx 1 4 z By Lemma 3.1, we know that Thus, 3.29 can be rewritten as z 2 c 0 N 2 z 2. u 2σ z 2 x + σ 2 λ 2 2σ σ σ σ 2 z 2. t z 2 + c 0 N γ 2 4 σ σ 2 σ 2 λ1 2σ σ 2 + λ2 2σ σ 2 z 2 0. Choose N 1 N 0 sufficient large such that δ 1 N 1 = c 0 N1 2 2γ 2 4 σ σ 2 σ 2 λ1 2σ σ 2 + λ2 2σ σ 2 > 0. For N N 1, multiplying 3.30 with e δ1t an integrating it for from t 2 to t yiels where zt 2 zt 2 2 e δ1t t2, t t 2, δ 1 = δ 1 N = c 0 N 2 2γ 2 4 σ σ 2 σ 2 λ 1 2σ σ 2 + λ2 2σ σ 2. Then, we consier the real part of the inner prouct of 3.3 with z an obtain 1 2 t z 2 + z 2 γ z 2 = Re 1 + iµ u 2σ 3.31 z zx + Re λ1 u 2 z + λ 2 u 2 z zx. For the right han sie of 3.31, using Höler s inequality an Young s inequality again yiels 3.32 Re 1 + iµ Q N u 2σ z zx 1 4 z 2 + u 4σ z 2, an λ1 u 2 z + λ 2 u 2 z zx z λ λ 2 4 u 8 z 2. Substituting 3.32 an 3.33 into 3.31, we have 3.34 t z 2 + z 2 2γ z 2 CM 2 z 2 0. Thanks to Lemma 3.1, we know that z 2 c 1 N 2 z 2, z 2 c 2 N 2 z 2.

17 REGULARITY OF ATTRACTOR 105 Substituting the above expressions into 3.34, we further have 3.35 t z 2 + c 1 N 2 2γ 2 + c 2 CM 2 N 2 z 2 0. Choose N 2 N 1 large enough such that δ 2 N 2 = c 1 N 2 2 2γ 2 + c 2 CM 2 N 2 2 > 0. For N N 2, multiplying 3.35 by e δ2t an integrating it from t 2 to t yiels z 2 zt 2 2 e δ2t t2, t t 2. where δ 2 = δ 2 N = c 1 N 2 2γ 2 + c 2 CM 2 N 2 > 0. By consiering the real part of the inner prouct of equation 3.1 with 2 z an using the same iscussion as the erivation of 3.34, we have 3.36 t z 2 + z 2 2γ z 2 CM 2 z From Lemma 3.1, we know that Thus, 3.36 can be rewritten as z 2 c 3 N 2 z 2, z 2 1 c 4 N 2 z t z 2 + c 3 N 2 2γ c 4 CM 2 N 2 z 2 0. Similarly, take N 3 N 2 large enough such that δ 3 N 3 = c 3 N 2 3 2γ + c 4 CM 2 N 2 3 > 0. For N N 3, multiplying 3.37 with e δ3t an integrating it from t 2 to t leas to where z 2 zt 2 2 e δ3t t2, t t 2, δ 3 = δ 3 N = c 3 N 2 2γ + c 4 CM 2 N 2. Let δ = 1 2 minδ 1, δ 2, δ 3. It is easy to see that 3.6 hols. Consequently, the proof of Theorem 3.1 is complete. In virtue of Theorem 3.1, the solution operator St = S 2 t : Hp 2 generate by the problem can be ecompose as where S 2 1 t an S S 2 1 tu 0 = an S 2 t = S S 2 2 tu 0 = t + S2 2 t, t 0, t are efine by { PN ut + yt = pt + yt, t 2, P N ut = pt, t t 2, { zt, t t2, Q N ut = qt, t t 2. H 2 p Here ut = S 2 u 0 for t t 2, an yt an zt are solutions of systems an , respectively. Hence, for every u H 2 p, we have 3.40 S 2 tu = S 2 1 tu + S2 2 tu.

18 106 SHUJUAN LÜ AND ZHAOSHENG FENG 4. Regularity of Attractor Theorem 4.1. Suppose that the conition 1.7 hols an σ is a positive integer or σ 1 2 [ m 2 ] for any positive integer m 2. Let A m be the global attractors of the semi-group of operators, an {S m t} t 0 generate by the problem Then we have i For any m 3, A 2 is a boune an close set in H m p. ii A 2 = A m for m 3. Proof. i Suppose that u A 2. We shall prove u H m p for any m 3. Owing to the well-known characterization of the ω-limit set [29], there exists a sequence of elements u n in B 2 an a sequence of positive real numbers t n which approaches infinity as n tens to infinity such that 4.1 S 2 t nu n u in H 2 p, as n +. From 3.40 it hols 4.2 S 2 t nu n = S 2 1 t nu n + S 2 2 t nu n, n N. Base on the efinitions of S 2 1 Theorem 3.1, if N is large enough, then we have 4.3 an 4.4 t an S2 2 t i.e an 3.39 an using S 2 1 t nu n m CN, m, n N, S 2 2 t nu n 2 ce δt n un 2, n N. From 4.3, there exists a subsequences {t n } n >0 an w H m p such that 4.5 S 2 1 t n u n w weakly in Hm p, as n, an 4.6 w m lim n inf S 2 1 t n u n m CN, m. Taking account of ϕ L 2 p, from 4.2 we get S 2 t n u n, ϕ = S 2 1 t n u n, ϕ + S 2 2 t n u n, ϕ. Letting n tens to + in the above expression an using 4.1, 4.4 an 4.5, we fin u, ϕ = w, ϕ, ϕ L 2 p. Setting ϕ = m u an using 4.6, we have m u w m CN, m, which shows u H m p. In other wors, A 2 is a boune set in H m p. ii We show that A 2 A m. Since A m attracts all boune sets in H m p an A 2 is boune in H m p, we get ist H m p S 2 ta 2, A m = ist H m p S m ta 2, A m 0, as t. In aition, A 2 is an invariant set of S 2 t, namely, S 2 ta 2 = A 2, which leas to ist H m p A 2, A m = 0.

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