1 Introuction Flui convection ue to ensity graients arises in geophysical ui ows in the atmosphere, oceans an the earth's mantle. The Rayleigh-Benar c

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1 Asymptotic ynamical ierence between the Nonlocal an Local Swift-Hohenberg Moels Guoguang Lin 1, Hongjun Gao 2, Jiniao uan 3 an Vincent J. Ervin 3 1. Grauate School, Chinese Acaemy of Engineering Physics P. O. BOX 2101, Beijing , an epartment of Mathematics Yunnan University, Kunming , China. 2. Laboratory of Computational Physics Institute of Applie Physics an Computational Mathematics Beijing, , China. 3. epartment of Mathematical Sciences Clemson University, Clemson, South Carolina 29634, USA. June 11, 1998 Abstract In this paper the ierence in the asymptotic ynamics between the nonlocal an local two-imensional Swift-Hohenberg moels is investigate. It is shown that the bouns for the imensions of the global attractors for the nonlocal an local Swift- Hohenberg moels ier by an absolute constant, which epens only on the Rayleigh number, an upper an lower bouns of the kernel of the nonlocal nonlinearity. Even when this kernel of the nonlocal operator is a constant function, the imension bouns of the global attractors still ier by an absolute constant epening on the Rayleigh number. Running Title: Nonlocal Swift-Hohenberg Moel Key Wors: asymptotic behavior, nonlocal nonlinearity, global attractor, imension estimates PACS Numbers: 02.30, 03.40, Author for corresponence: Professor Jiniao uan, epartment of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634, USA. uan@math.clemson.eu; Fax: (864)

2 1 Introuction Flui convection ue to ensity graients arises in geophysical ui ows in the atmosphere, oceans an the earth's mantle. The Rayleigh-Benar convection is a prototypical moel for ui convection, aiming at preicting spatio-temporal convection patterns. The mathematical moel for the Rayleigh-Benar convection involves nonlinear Navier-Stokes partial ierential euations couple with the temperature euation. When the Rayleigh number is near the onset of the convection, the Rayleigh-Benar convection moel may be approximately reuce to an amplitue or orer parameter euation, as erive by Swift an Hohenberg ([15]). In the current literature, most work on the Swift-Hohenberg moel eals with the following one-imensional euation for w(x; t), which is a localize, one-imensionalize version of the moel originally erive by Swift an Hohenberg ([15]), w t = w (1 xx ) 2 w w 3 : (1) The cubic term w 3 is use as an approximation of a nonlocal integral term. For the (local) one-imensional Swift-Hohenberg euation (1), there has been some recent research on propagating or steay patterns (e.g., [1], [6], [9]). Mielke an Schneier([10]) prove the existence of the global attractor in a weighte Sobolev space on the whole real line. Hsieh et al. ([7], [8]) remarke that the elemental instability mechanism is the negative iusion term w xx. Roberts ([12], [13]) recently re-examine the rationale for using the Swift-Hohenberg moel as a reliable moel of the spatial pattern evolution in specic physical systems. He argue that, although the localization approximation use in (1) makes some sense in the one-imensional case, this approximation is ecient in the two-imensional convection problem an one shoul use the nonlocal Swift-Hohenberg moel ([15], [12], [13]): u t = u (1 + ) 2 u u G( (x ) 2 + (y ) 2 )u 2 (; ; t); (2) where u = u(x; y; t) is the unknown amplitue function, measures the ierence of the Rayleigh number from its critical onset value, xx yy is the Laplace operator, an G(r) is a given raially symmetric function (r = p x 2 + y 2 ). The euation is ene for t > 0 an (x; y) 2, where is a boune planar omain with smooth The two-imensional version of the local Swift-Hohenberg euation for u(x; y; t) is u t = u (1 + ) 2 u u 3 : (3) Here u 3 is use to approximate the nonlocal term in (2). Roberts ([12], [13]) note that the range of Fourier harmonics generate by the nonlinearities is funamentally ierent in two-imensions than in one-imension. This ifference reuires a more sophisticate treatment of two-imensional convection problem, which leas to nonlocal nonlinearity in the Swift-Hohenberg moel. He also argue that 2

3 nonlocal operators naturally appear in systematic erivation of simplie moels for pattern evolution, an nonlocal operators also permit symmetries which are consisitent with physical consierations. In this paper, we iscuss the ierence between nonlocal an local two-imensional Swift-Hohenberg moels (2), (3), from a viewpoint of asymptotic ynamics. We show that the bouns for the imensions of the global attractors for the nonlocal an local Swift- Hohenberg moels ier by an absolute constant, which epens only on the the Rayleigh number, an upper an lower bouns of the kernel of the nonlocal nonlinearity. Even when this kernel is a constant function, the imension bouns of the global attractors still ier by a constant epening on the Rayleigh number. In x2 an x3, we will consier the nonlocal an local Swift-Hohenberg moels, respectively. Finally in x4, we summarize the results. 2 Nonlocal Swift-Hohenberg Moel In this section, we iscuss the global attractor an its imension estimate for the nonlocal Swift-Hohenberg moel (2). In the following we use the abbreviations L 2 = L 2 (), L 1 = L 1 (), H k = H k () an H k 0 = H k 0 () (k is a non-negative integer) for the stanar Sobolev spaces. Let (; ), k k k k 2 enote the stanar inner prouct an norm in L 2, respectively. The norm for H k 0 is k k H k. ue to the Poincare ineuality, k k uk is an 0 euivalent norm in H k 0. We rewrite the two-imensional nonlocal Swift-Hohenberg euation (2) as u t + u + 2u + 2 u + u G( (x ) 2 + (y ) 2 )u 2 (; ; t) = 0; (4) where = 1. This euation is supplemente with the initial conition an the bounary conitions u(x; y; 0) = u 0 (x; y); (5) = 0; (6) where n enotes the unit outwar normal vector of the In this paper, we assume the following conitions for every t 0 an (x; y) 2, 0 < b G( x 2 + y 2 ) a; an G; rg; G 2 L 1 (); (7) where a; b > 0 are some positive constants an r = (@ x y ) is the graient operator. enote K 1 = krgk 1 an K 2 = kgk 1. To stuy the global attractor, we nee to erive some a priori estimates about solutions. 3

4 Lemma 1 Suppose u is a solution of (4)-(6). Then u is uniformly (in time) boune, an the following estimates hol for t > 0 ku(x; y; t)k 2 ku 0 (x; y)k 2 exp( 2t) + b ; (8) an thus where R = b. r lim sup t!+1 ku(x; y; t)k b R; (9) Proof. Taking the inner prouct of (4) with u, we have Note that +(u 2 ; G( = Then from (10) we get 1 2 t kuk2 + kuk 2 + 2(u; u) + kuk 2 (x ) 2 + (y ) 2 )u 2 (; )) = 0: (10) 2j(u; u)j 2kukkuk kuk 2 + kuk 2 ; (u 2 ; u 2 ( b G( (x ) 2 + (y ) 2 )u 2 (; )) G( (x ) 2 + (y ) 2 )u 2 (; ))xy u 2 (x; y)xy u 2 (; ) = bkuk 4 : t kuk2 + 2( 1)kuk 2 + 2bkuk 4 0: (11) It is easy to see that if 1, i.e., 0, then all solutions approach zero in L 2. We will not consier this simple ynamical case. In the rest of this paper we assume that > 0, i.e., < 1. Thus we have, for any constant > 0, or t kuk2 + 2kuk 2 + 2( 1 )kuk 2 + 2bkuk 4 0; (12) t kuk2 + 2kuk 2 ( 1 ) + [ p + p 2bkuk 2 ] 2 2b 4 ( 1 )2 : (13) 2b

5 So t kuk2 + 2kuk 2 By the usual Gronwall ineuality ([17]) we obtain ( 1 )2 : (14) 2b kuk 2 ku 0 k 2 ( 1 )2 exp( 2t) + : (15) 4b When = 1 =, we get the optimal or tight estimate kuk 2 ku 0 k 2 exp( 2t) + b : (16) This completes the proof of Lemma 1. Moreover, higher orer erivatives of u are also uniformly boune. Lemma 2 Suppose u is a solution of (4)-(6). Then kruk an kuk are uniformly (in time) boune. In orer to prove this lemma, we recall a few useful ineualities. Uniform Gronwall ineuality ([17]). Let g; h; y be three positive locally integrable functions on [t 0 ; +1) satisfying the ineualities y t gy + h; R R R t+1 t+1 t+1 with t gs a 1 ; t hs a 2 an t ys a 3 for t t 0 ; where the a i (i=1,2,3) are positive constants. Then y(t + 1) (a 2 + a 3 )exp(a 1 ); for t t 0 : Gagliaro-Nirenberg ineuality ([11]). Let w 2 L \ W m;r (), where 1 ; r 1. For j any integer j, 0 j m, m 1: provie k j wk p C 0 kwk 1 k m wk r 1 p = j n + (1 r m n ) + 1 ; an m j n is not a nonnegative integer If r m j n is a nonnegative integer, then the r ineuality (2) hols for = j. m Poincare ineuality ([2]). For w 2 H 1 0 (), 1 kwk 2 krwk 2 ; 5

6 where 1 is the rst eigenvalue of on the omain, with zero irichlet bounary conition Proof of Lemma 2. ue to the bounary conition (6) on ru an the Poincare ineuality, we get kruk R kuk2. Hence it is sucient to prove that kuk is boune. t+1 We rst show that t kuk 2 s is boune. In fact, using in (10), we get 2j(u; u)j 2kukkuk 1 2 kuk2 + 2kuk 2 ; Since we conclue t kuk2 + kuk 2 + 2( 2)kuk 2 + 2bkuk 4 0: (17) 2bkuk 4 + 2( 2)kuk 2 = bkuk 2 + 2(kuk kuk 2 ) 2 = bkuk 2 + 2(kuk ) 2 (2 4 )2 4 8 bkuk 2 (2 4 )2 ; 8 t kuk2 + kuk 2 + bkuk 2 (2 4 )2 8 = ( )2 : (18) 8 Integrating (18) with respect to t from t to t R + 1 an noting Lemma 1, we see that t+1 t kuk 2 s is boune. Now, multiplying (4) by 2 u an integrating over, it follows that Note that an + (u j = j 1 2 t kuk2 + k 2 uk u 2 uxy + kuk 2 G( (x ) 2 + (y ) 2 )u 2 (; )) 2 uxy = 0: (19) 2j u 2 uxyj 1 2 k2 uk 2 + 2kuk 2 ; (20) (u G( (x ) 2 + (y ) 2 )u 2 (; )) 2 uxyj (u) 2 ( G( (x ) 2 + (y ) 2 )u 2 (; ))xy 6

7 + +2 uu( ruu( (G( (x ) 2 + (y ) 2 )))u 2 (; ))xy (rg( (x ) 2 + (y ) 2 )))u 2 (; ))xyj (akuk krgk 1 kuk kgk 1kuk 2 )kuk kgk 1kuk 4 (a K K 2)kuk 2 kuk K 2kuk 4 ; where a; K 1 ; K 2 are various upper bouns of G ene in (7), an R is the L 2 boun of the solution u as in Lemma 1. Hence by (19) we get t kuk2 2[(a K K 2)kuk 2 + 2]kuk 2 + K 2 kuk 4 : (21) Finally, applying the uniform Gronwall ineuality (21) an noting Lemma 1, we conclue that kuk 2 is uniformly boune for all t 0: This proves Lemma 2. We now have the following global existence an uniueness result. Theorem 1 Let u 0 (x; y) 2 L 2 () an G satises (7), then the initial-bounary value problem (2); (5); (6) has a uniue global solution u 2 L 1 (0; 1; H 2 0 ()). Moreover, the corresponing solution semigroup S(t), ene by has a boune absorbing set u = S(t)u 0 ; B 0 = fu 2 H 2 0 () : (kuk 2 + kruk 2 + kuk 2 ) 1 2 ~ Rg; where ~ R is a postive constant which epening on the uniform boun of kuk; kruk; kuk. Finally, the solution semigroup S(t), when restricte on H 2 0 (), is continuous from H 2 0 () into H 2 0() for t > 0. Proof. The global existence, uniueness an absorbing property follow from stanar arguments (e.g., [17]) together with Lemmas 1, 2 above. The absorbing property also follows from these two lemmas. We now prove that S(t) is continuous in H 2 ()\H 1 0 (). Suppose that u 0 ; v 0 2 H 2 0 () with ku 0 k; kv 0 k 2R 1 ; we enote by u(t); v(t) the corresponing solutions, i.e., u(t) = S(t)u 0 ; v(t) = S(t)v 0. Let w(t) = u(t) v(t): Then w(t) satises w t + 2 w + 2w + w + w G( (x ) 2 + (y ) 2 )u 2 (; )+ v G( (x ) 2 + (y ) 2 )(u(; ) + v(; ))w(; ) = 0: (22) 7

8 Applying the Gagliaro-Nirenberg ineuality an the Poincare ineuality kuk 1 C 0 kuk; kwk 1 1 kwk; we obtain (similar to the proof of Lemma 2), t kwk2 C 1 kwk 2 ; which implies that kw(t)k 2 kw 0 k 2 exp(c 1 t) for some positive constant C 1. This shows that S(t) is continuous. This theorem implies that (4)-(6) enes an innite imensional nonlocal ynamical system. In the rest of this section, we consier the global attractor for the nonlocal ynamical system (4)-(6). We will establish the following result about the global attractor. Theorem 2 There exists a global attractor A for the nonlocal ynamical system (2); (5); (6). The global attractor is the! limit set of the absorbing set B 0 (as in Theorem 1), an it has the following properties: (i) A is compact an S(t)A = A; for t > 0; (ii) for every boune set B H0(), 2 lim (S(t)B; A) = 0; t!1 (iii)a is connecte in H0(); 2 where (X; Y ) = sup inf kx yk H x2x y2y 0 2 () is the Hausor istance. Moreover, the global attractor A has nite Hausor imension H (A) m, where r m C(1 + + (2a b) b ); where C > 0 is a constant epening only on the omain, an a > 0; b > 0 are the upper, lower bouns of the kernel G, respectively. Proof. The existence an properties of A are uite stanar now (see [17] an references therein). We omit this part, an only estimate the imensions below. As in [17], we may use the so-calle Constantin-Foias-Temam trace formula (which works for the semiow S(t) here) to estimate the sum of the global Lyapunov exponents of A. The sum of these Lyapunov exponents can then be use to estimate the upper bouns of A's Hausor imension, H (A). To this en, we linearize euation (4) about a solution u(t) in the global attractor to obtain an euation for v(t) an then use the trace formula to estimate the sum of the global Lyapunov exponents. oing so, we obtain v t + L(u(t))v = 0; (23) 8

9 where L(u(t))v = 2 v + 2v + v + v G( (x ) 2 + (y ) 2 )u 2 (; ) +2u G( (x ) 2 + (y ) 2 )u(; )v(; ): This euation is supplemente with v(x; y; 0) = (x; y) 2 H 2 0 (). enote by 1(x; y); : : : ; m (x; y), m linearly inepenent functions in H 2 0(), an v i (x; y; t) the solution of (23) satisfying v i (x; y; 0) = i (x; y), i = 1; : : : ; m. Let Q m (t) represent the orthogonal projection of H 2 0 () onto the subspace spanne by fv 1 (x; y; t); : : : ; v m (x; y; t)g. We nee to estimate the lower boun of T r(l(u(t)q m (t))), which gives bouns on the sum of global Lyapunov exponents. Note that in [17], the linearize euation like (23) is written as v t = L(u(t))v an in that case one nees to estimate the upper boun of T r(l(u(t)q m (t))). Suppose that 1 (t); :::; m (t) is an orthonormal basis (k j k = 1) of the subspace Q m (t)h 2 0 () for any t > 0. Now we estimate the lower boun of T r(l(u(t)q m (t))). It is easy to see that T r(l(u(t)q m )) = ( 2 j + 2 j + j + j + (2u G( (x ) 2 + (y ) 2 )u 2 (; ); j ) G( (x ) 2 + (y ) 2 )u(; ) j ; j ): Since (2 j ; j ) ( 1 k jk 2 + k j k 2 ) for any constant > 1, we get T r(l(u(t)q m )) + = 2 u j xy( [(1 1 )k jk 2 + bk j k 2 kuk 2 + ( )k j k 2 ] (1 1 )k jk 2 + G( (x ) 2 + (y ) 2 )u(; ) j ) (bkuk 2 + 2akuk 2 ) (1 1 )k jk 2 + [1 + (b 2a)kuk 2 ]m: (24) 9

10 We introuce notation f(x; y) = m P R j j j 2. Note that m = f(x; y)xy. generalize Sobolev-Lieb-Thirring ineuality ([17], page 462), X m f 3 (x; y)xy K 0 k j k 2 ; By the where K 0 > 0 epening only on the omain. Moreover, ue to the fact that L 3 (),! L 1 (), m 3 = ( f(x; y)xy) 3 C 2 f 3 (x; y)xy K 0 C 2 m X = C k j k 2 k j k 2 for some constants C 2 > 0; C > 0 epening only on the omain. Thus (1 1 ) m X k j k 2 (1 1 ) 1 C m3 : (25) Therefore, by (24)-(25) we have T r(l(u(t)q m )) 1 1 C m3 ( (2a b)kuk 2 )m 1 1 C m3 ( (2a b) b )m > 0 (26) whenever m > s [ (2a b) b ] C : (27) 1 1 The right han sie of (27) has the minimal value of r m C(1 + + (2a b) b ) (28) when = (2a b). b As in [17], we conclue that the Hausor imension of A is estimate as in (28). This proves Theorem 2. 10

11 3 Local Swift-Hohenberg Moel Similarly, for the two-imensional local Swift-Hohenberg euation (3), we can obtain the existence of the global attractor ~ A. We omit this part an will only estimate the imension of ~ A. Theorem 3 There exists the global attractor ~ A for the local ynamical system (3), (5), (6). The Hausor imension of ~ A is nite, an H ( ~ A) m1 C(1 + p ), where C is a constant epening only on the omain. Proof. As in the proof of Ttheorem 2, we consier the linearize euation of (3), ene by v t + L 1 (u(t))v = 0; where Then we estimate = = L 1 (u(t))v = 2 v + 2v + v + 3u 2 v: T r(l 1 (u(t)q m )) ( 2 j + 2 j + j + 3u 2 j ; j ) [k j k 2 + 2( j ; j ) + k j k 2 + 3(u 2 j ; j )] (1 1 )k jk 2 + ( ); P where we have use the fact that 3(u 2 j ; j ) 0. Noting again that m 3 C m k j k 2 an = 1, we have whenever T r(l 1 (u(t)q m )) 1 1 C m3 ( 1 + )m > 0 (29) m > s C ( 1 + ) : (30) 1 1 The right han sie of (30) has the minimal value of m C(1 + p ) (31) when = 1 + p. This completes the proof. 11

12 4 iscussions In this paper, we have iscusse the Hausor imension estimates for the global attractors of the two-imensional nonlocal an local Swift-Hohenberg moel for Rayleigh-Benar convection. The Hausor imension for the global attractor of the nonlocal moel is estimate as r m C(1 + + (2a b) b ); while for the local moel this estimate is m C(1 + p ); where C > 0 is an absolute constant epening only on the ui convection omain, an > 0 measures the ierence of the Rayleigh number from its critical convection onset value. Note that a; b > 0 are the upper an lower bouns, respectively, of the kernel G of the nonlocal nonlinearity in (2). The two imension estimates above ier by an absolute constant (2a b) b, which epens only on the the Rayleigh number through, an upper an lower bouns of the kernel G of the nonlocal nonlinearity. Moreover, if the kernel G is a constant function (thus, a = b = G), then the imension estimate for the nonlocal moel becomes m C(1 + p 2); which still iers from the imension estimate for the local moel by a constant epening on the Rayleigh number through. Acknowlegement. Part of this work was one while Jiniao uan was visiting the Institute for Mathematics an its Applications (IMA), Minnesota, an the Center for Nonlinear Stuies, Los Alamos National Laboratory. This work was supporte by the Nonlinear Science Program of China, the National Natural Science Founation of China Grant , the Science Founation of Chinese Acaemy of Engineering Physics Grant , an the USA National Science Founation Grant MS References [1] J. P. Eckmann an C. E. Wayne, Propagating fronts an the center manifol theorem, Comm. Math. Phys. 136 (1991), [2] A. Frieman, Partial ierential Euations, Holt, Reinhart an Winston, [3] J. M. Ghialia, M. Marion an R. Temam, Generalization of the Sobolev-Lieb- Thirring ineualities an Applications to the imension of Attractors, i. an Int. Es., 1 (1988),

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