On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors

Transcription

1 On the large time ehavior of solutions of fourth order paraolic equations and ε-entropy of their attractors M.A. Efendiev & L.A. Peletier Astract We study the large time ehavior of solutions of a class of fourth order paraolic equations defined on unounded domains. Specific examples of the equations we study are the Swift-Hohenerg equation and the Extended Fisher-Kolmogorov equation. We estalish the existence of a gloal attractor in a local topology. Since the dynamics is infinite dimensional, we use the Kolmogorov ε-entropy as a measure, deriving a sharp upper and lower ound. Résumé Sur le comportement en temps grand des solutions d équations paraoliques d ordre quattre, et l entropie de leurs attracteurs Nous étudions le comportement pour des grandes valuers du temps des solutions d une classe déquations paraolique d ordre quattre defini sûr des domaines non ornés. Les examples specifiques que nous considerons sont l équation de Swift-Hohenerg et une généralisation de l équation de Fisher-Kolmogorov. Nous démontrons l existence d une attracteur gloale dans une topologie locale, et otenons des limites supérieure et inférieure de l entropie de Kolmogorov. 1 Introduction In this paper we give a description of the large time ehavior of solutions of a family of well-known fourth order model equations of paraolic type of the form u t + 2 u + q u + f(u) = g in R 3, (1.1) where q R and f(u) and g = g(x) are given functions. Typical examples include the Extended Fisher-Kolmogorov equation u t + γ 2 u u + u 3 u = in R 3 (γ > ), (1.2) which arises in the study of i-stale systems [1] and in statistical mechanics, and the Swift- Hohenerg equation u t = αu ( 1 + x 2 ) 2u u 3 in R (α > ), (1.3) which is used as a model equation in many studies of pattern formation [7]. Both equations can e rought into the form of equation (1.1) y suitale transformations of t, x and u. GSF/ Technical University of Münich, Center for Mathematical Sciences, 85747/Garchung-Münich, Germany messoud.efendiyev@gsf.de Mathematical Institute, Leiden University, PO Box 9512, 23 RA Leiden & Centrum voor Wiskunde en Informatica (CWI), Amsterdam, The Netherlands peletier@math.leidenuniv.nl 1

2 For one-dimensional ounded domains the large time ehavior of solutions of equation (1.1) and the selection of patterns has een discussed in [5], in relation to the length of the domain. In this paper we characterize the large time ehavior in terms of the gloal attractor, which is the maximal compact invariant set which attracts all ounded sets in phase space. At present, the question of existence of gloal attractors and estimation of their dimension is well estalished for systems defined in ounded domains. Gloal attractors are then finite dimensional and their analysis can e reduced to the study of finite dimensional dynamical systems. For systems defined on unounded domains the dynamics is necessarily infinite dimensional [2], and classical tools like the (Hausdorff, fractal) dimension are no longer effective. Here the concept of Kolmogorov ε- entropy plays an important role. Recently this concept was applied to discuss dynamics descried y second order equations in unounded domains. In a series of papers in which the maximum principle was an important ingredient, the existence of the attractor was estalished and estimates of its ε-entropy were otained [2], [3]. However, for higher order equations such as (1.1) we encounter serious difficulties due to the lack of compactness, and the interplay etween different topologies will play a crucial role. We prove the existence of a gloal attractor of solutions of (1.1) in a phase space endowed with a local topology, and present sharp estimates for the Kolmogorov ε-entropy. For (1.1) the local topology is appropriate ecause of the plethora of ounded stationary solutions such as periodic solutions (cf. e.g. [6]), homoclinic and heteroclinic orits and chaotic orits, which were found for (1.2) and (1.3) in one spatial dimension. A strong topology would exclude such stationary solutions and lead to much simpler dynamics. 2 Preliminaries We shall assume that the nonlinearity f C 2 (R) in equation (1.1) satisfies the following structure hypotheses: { H1 : f(s) s c 1 + c 2 s 2+δ for s 1, s R, (2.1) H2 : f(s) s c 3 s for s 1. in which c 1, c 2, c 3 and δ are positive constants. We shall e using function spaces which are typically designed for unounded domains: W l,p (R 3 ) = {u D 1 (R 3 def ) : u,l,p = sup u, Bx 1 l,p < }, (2.2) x R 3 where Bx 1 def is the unit all centered at x. In particular we frequently use the space Φ = W 4,2 (R 3 ). In this paper we assume that H3 : g L 2 (R3 ). (2.3) We also introduce the weighted Soolev spaces with weight ϕ(x) such that ϕ(x) as x : L p φ (R3 ) = {u D (R 3 ) : u p def φ,p = φ(x) u(x) p dx < }, R 3 (2.4) W l,p φ (R3 ) = {u D (R 3 ) : D α u L p φ (R3 ) for all α l}. For the estimation of the lower ound on the Kolmogorov entropy of the attractor, we use the the properties of the space B θ,ξ, defined y B θ,ξ def = {u L (R 3 ) : supp(û) [ θ, θ] 3 + ξ, ξ R 3 }, (2.5) where û is the Fourier Transform of u. B(, ρ, B θ,ξ ) will denote the all of radius ρ aout the origin in the space B θ,ξ. 2

3 3 Existence and uniqueness The existence of a semigroup for the Cauchy Prolem for (1.1) follows from the a-priori estimate: Theorem 3.1 Let u Φ and let (f, g) in equation(1.1) satisfy H1-3. Then equation (1.1) possesses a unique gloal solution u(t) for t <, which satisfies the following estimate: u(t) Φ Q( u() Φ )e αt + Q( g L 2 ) := Q(u, g) (3.1) in which α is a positive constant and Q a generic function of its argument(s). The proof of Theorem 3.1 is ased on successively multiplying equation (1.1) y u(t)φ ε,x (x), xi (φ ε,x xi u) and φ ε,x t u, integration y parts, and using assumptions H1-3. This leads to an estimate for the L -norm of the nonlinearity. After that, the estimate (3.1) can e otained exactly as for the linear case. Here φ ε,x (x) is a smooth weight function such that φ ε,x (x) e ε x x as x where x R 3 and ε >. We note that for every two solutions u 1 (t) and u 2 (t) of (1.1) we have u 1 (t) u 2 (t) W 1,2 Ce Kt u 1 () u 2 () φ L 2 φ, (3.2) where the constants C and K are independent of the choice of u 1 and u 2. Oviously, this estimate implies uniqueness of the solution. The existence of a gloal (in time) solution of (1.1) is a standard consequence of (3.1). Therefore, equation (1.1) generates a semigroup S t : Φ Φ, S t u = u(t), and moreover S t : L 2 (R3 ) W 4,2 (R 3 ), t >. 4 Existence of a gloal attractor Definition 4.1 A set A Φ is called a (locally compact) attractor of the semigroup S t if (1) A is ounded in Φ and compact in Φ loc := W 4,2 loc (R3 ); (2) S t A = A for all t > ; (3) A is an attracting set of a ounded set in the local topology of Φ loc. In the sequel we call an attractor as defined aove a (Φ, Φ loc )-attractor for the semigroup S t : Φ Φ. Theorem 4.1 Let the nonlinearity f, and g in equation (1.1) satisfy the hypotheses H1-3. Then the semigroup S t defined aove possesses a (Φ, Φ loc )-attractor. The proof of Theorem 4.1 is ased on the estimate (3.1), standard regularity theory and compactness of the emedding W 4+δ,2 (R 3 ) W 4,2 loc (R3 ) for any δ >. Remark We have proved that equation (1.1) has a (Φ, Φ loc )-gloal attractor A. constant it is easy to see that in general A cannot e compact in Φ. When g is Indeed, if A is compact in Φ, and u A, one can show that translation invariance implies that u must e almost periodic. Since equiliria elong to A, it follows that all equiliria must e almost periodic. It is well known however [6], that equation (1.1) for q < and f(u) = u 3 u, as well as equation (1.2), have kinks and pulse type stationary solutions. Plainly, they are not almost periodic. 5 Upper ound of the Kolmogorov entropy of the attractor We prove the following upper ound for the Kolmogorov ε-entropy of A: 3

4 Theorem 5.1 Let the nonlinearity f, and g in equation (1.1) satisfy the hypotheses H1-3. Then H ε (A ( B R, W 4,2 (B R x x )) C R + ln R ) 3 ln R ε ε, (5.1) where the constants C and R do not depend on ε, R and x. Proof Let φ = φ R,x (x) e the weight function introduced in Section 2, and let φ R,x (x) 1 in B R x. Then oviously H ε (A B R x, W 4,2 (B R x )) H ε (A, W 4,2 φ (R3 )), (5.2) so that it suffices to prove (5.1) for H ε (A, W 4,2 φ (R3 )). Let S def = S 1. Then it follows from (3.2) that for u 1, u 2 A, we have and hence that Su 1 Su 2 W 4,2 φ (R3 ) L u 1 u 2 L 2 φ (R 3 ), (5.3) H ε (A, W 4,2 φ (R3 )) H ε/l (A, L 2 φ (R3 )). (5.4) Thus, it will e enough to estimate the right hand side of (5.4), i.e., to prove that ( H ε/l (A, L 2 φ(r 3 )) C R + ln R ) 3 ln R ε ε, (5.5) where C and R do not depend on ε, R and x. In order to prove (5.5) we use the following recurrence formula: ( H ε/2 (A, L 2 φ(r 3 )) C R + ln R ) 3 + H ε (A, L 2 ε φ(r 3 )). (5.6) This relation can e estalished following [3] and using (3.2). Thus, assume that (5.6) holds. Then, plainly, H R (A, L 2 φ (R3 )) = for any R > and x (recall that A L 2 φ (R 3 ) R ). Now we iterate (5.6) k times to otain, since here the logarithm has ase 2, k 1 H k(a, R/2 L2 φ (R3 )) C (R + j) 3 C(R + k) 3 k, (5.7) j= where C does not depend on k. When we now choose k such that ε R /2 k, we otain from (5.7) that ( H ε (A, L 2 φ(r 3 )) C R + ln R ) 3 ln R ε ε, (5.8) as required. 6 Lower ound of the Kolmogorov entropy of the attractor We proceed in a more or less standard manner i.e., we aim at constructing the unstale manifold M + (z ) at some equilirium point z. Because M + (z ) A, we know that H ε (A B R x, W 4,2 (B R x )) H ε (M + (z ), W 4,2 (B R x )), (6.1) so that it suffices to compute the right hand side in (6.1). 4

5 To this end, we assume that g = and f() =, so that u = is an equilirium solution. Linearizing equation (1.1) aout u =, we otain the equation w t + 2 w + q w + f ()w = in R 3. (6.2) We now assume that u = is exponentially unstale, i.e., H4 : σ( 2 q f (), L 2 (R 3 )) {Reλ > }, (6.3) where σ(l, W ) denotes the spectrum of the linear operator L in the space W. The characteristic equation corresponding to equation (6.2) has een studied in [5] (for the Swift-Hohenerg equation). From this analysis conditions on q can readily e identified for which H4 is satisfied. We use the following lemma from [3]: Lemma 6.1 Assume that g = and f() =, and that H1-H4 are satisfied. Then there exist constants θ >, ξ R 3 and ρ >, and a C 1 -map V : B(, ρ, B θ,ξ ) M + A such that for u 1, u 2 B(, ρ, B θ,ξ ) we have V(u 1 ) V(u 2 ) W 4,2 (B R ) u 1 u 2 L (B R ) Cρ 2, (6.4) where C is a constant which is independent of R. Let ε > e small enough, and ρ def = (ε/2c) 1/2 ρ. We choose u 1, u 2 B(, ρ, B θ,ξ ) such that u 1 u 2 L (B R ) > ε Then we conclude from (6.4) that Comining (6.4) and (6.5) we otain V(u 1 ) V(u 2 ) W 4,2 (B R ) ε/2. (6.5) H ε/4 (A B R x, W 4,2 (B R x )) H ε (B(, (ε/2c) 1/2, B θ,ξ ), L (R 3 )). (6.6) By a scaling argument we then have H ε (B(, (ε/2c) 1/2, B θ,ξ ), L (R 3 )) = H (2Cε) 1/2(B(, 1, B θ,ξ ), L (R 3 )) C R 3 ln R ε, (6.7) where the last estimate is due to Kolmogorov [4]. independent of R and ε. Thus we have proved: Here the positive constants C and R are Theorem 6.1 Assume that g = and f() =, and that H1-H4 are satisfied. Then, the entropy of the restrictions A B R satisfies the following lower ound: H ε (A B R, L (B R )) C R 3 ln R ε, (6.8) where the positive constants C and R are independent of R and ε. Remark The estimate (6.8) shows that the restrictions A B R have usually infinite fractal dimension. Moreover, estimates (5.1) and (6.8) confirm the fact that the upper ounds (5.1) are in a x sense sharp for every fixed values of R, ε and x. We emphasize that in the case g = const, we have an additional structure on A, namely we have an action of a (1 + 3) - parametric semigroup {S (t,h), t, h R 3 } on A, that is S (t,h) A = A, S (t,h) := S t T h, S (t,h) : Φ Φ with T h S t = S t T h. (6.9) Here {T h, h R 3 } is a group of translation, (T h u)(x) := u(x + h). In a forthcoming paper, we will study dynamical properties of S (t,h) in order to descrie spatio-temporal complexity (for some values of q in (1.1)) of the attractor. 5

6 References [1] Dee, G.T. & W. van Saarloos, Bistale systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 6 (1988) [2] Efendiev, M.A. & S. Zelik,The attractor for a nonlinear reaction-diffusion system in an unounded domain, Comm. Pure Appl. Math. LIV (21) [3] Efendiev, M.A. & S. Zelik, Upper and lower ounds for the Kolmogorov entropy of the attractor for an RDE in an unounded domain, J. Dyn. Diff. Equ. 14 (22) [4] Kolmogorov, A.N. & V.M. Tikhomirov, E-entropy and ε-capacity of sets in functional spaces In: Selected works of A.N. Kolmogorov, Vol. III, Kluwer, Dordrecht, [5] Peletier, L.A. & V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenerg equation, Physica D 194 (24) [6] Peletier, L.A. & W.C. Troy, Spatial Patterns: Higher order models in Physics and Mechanics, Birkhäuser, Boston, 21. [7] Swift, J.B. & P.C. Hohenerg, Hydrodynamic fluctuations at the convective instaility, Phys. Rev. A 15 (1977)