The ε-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

The ε-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors April 10, 2017 Mar´ıa ANGUIANO Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla. P. O. Box 1160, 41080-Sevilla (Spain). [email protected] Alain HARAUX (1, 2) 1. UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. 2- CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Boˆıte courrier 187, 75252 Paris Cedex 05, France. hara[email protected] Abstract We prove an estimation of the Kolmogorov ε-entropy in Hof the unitary ball in the space V, where His a Hilbert space and Vis a Sobolev-like subspace of H. Then, by means of Zelik’s result [5], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established. AMS classification numbers: 37L30, 35B41 Keywords: Fractal dimension; Attractors; Entropy. 1 1 Introduction Let Mbe a precompact set in a metric space X. We recall the definition of the fractal dimension of M(see, for instance, Temam [4]). According to Hausdorff criteria the set Mcan be covered by a finite number of ε-balls in Xfor every ε > 0. Denote by Nε(M, X) the minimal number of ε-balls in Xwhich cover M. Then the Kolmogorov ε-entropy of the set Min Xis defined to be the following number Hε(M, X)≡log2Nε(M, X), and the fractal dimension of Mcan be defined in the following way dimF(M) = dimF(M, X) = lim sup ε→0+ Hε(M, X) log21 ε . In the present paper, we shall be dealing with estimates of the fractal dimension of the invariant sets (attractors) of the semigroups generated by infinite-dimensional dynamical systems. The usual way of estimating the fractal dimension of invariant sets involving the Liapunov exponents and k-contraction maps (see, for instance, Temam [4]) requires the semigroup to be quasidifferentiable with respect to the initial data on the attractor. It is well known that the Hausdorff dimension is less than or equal to the fractal dimension. In this sense, in [2], Chepyzhov and Ilyin show that the Hausdorff and fractal dimension have the same upper bound generalizing to the infinite-dimensional case the method of Chen [1]. To avoid the differentiability hypothesis, Zelik, in [5], presents a new approach to estimate the dimension of invariant sets. The basic tool of his method is the following very general property. Theorem 1 (Zelik) Let Vand Hbe Banach spaces, Vbe compactly embedded in Hand let Kbe a compact subset of H. Assume that there exists a map L:K→Ksuch that L(K) = Kand the following ‘smoothing’ property is valid kL(k1)−L(k2)kV≤C|k1−k2|H,∀k1, k2∈K, C > 0.(1) Then, the fractal dimension of Kin His finite and can be estimated in the following way: dimF(K, H)≤ H 1 4C(BV(0,1), H), where Cis the same as in (1) and BV(0,1) means the unit ball centered at 0in the space V. In the present work, we show (see Theorem 2) an estimation of the Kolmogorov ε-entropy of BV(0,1) in Hwhere His a Hilbert space and Vis a Sobolev-like subspace of H. Then we deduce from Zelik’s result an estimate of the fractal dimension of the attractor of some nonlinear parabolic equations in terms of the physical parameters. This result is quite explicit and rather close from the estimate obtained in [2] under slightly different but quite related assumptions. 2 Main results Let Hbe a separable Hilbert space with scalar product (·,·)Hand norm |·|H. Let Vbe a dense subspace of H, endowed with a Hilbert structure such that the inclusion map of Vinto His compact. Then His included in V′ with continuous imbedding. By k·kVand (·,·)Vwe denote the norm and the scalar product in V, respectively. We will denote by h·,·i the duality product between V′and V. Let A∈ L(V, V ′) be the duality map: V→V′. It is a self-adjoint monotone operator such that A−1∈ L(V′, V )⊂ L(H, H) is a compact, positive, self-adjoint operator from Hto itself. As a consequence of the Hilbert-Schmidt Theorem there exists a nondecreasing sequence of positive real numbers, 0< λ1≤λ2≤... ≤λj≤......, 2 with limj→∞ λj= +∞and there exists an orthonormal basis {wj:j≥1}of Hwith A wj=λjwjfor all j≥1. The sequence (λj) is the sequence of eigenvalues repeated according to their multiplicity. We now assume that (λj) satisfies the following growth assumption: (H1) There exist positive constants cand αsuch that λj≥cjα. Under the last assumption, the first goal in this section is to prove an estimate of the Kolmogorov ε-entropy of BV(0,1) := {u∈V, kukV≤1}. In order to do that, we shall identify Hwith l2through the identification u−→ (uj)j∈N∗ where u=Pjujwj. Theorem 2 Assume the assumption (H1). Then, the Kolmogorov ε-entropy of BV(0,1) in Hsatisfies ∀ε > 0,Hε(BV(0,1), H)<log 3 + α log 2 2 cε21/α .(2) Proof. Let u∈BV(0,1). We observe that u∈BV(0,1) ⇐⇒ ∞ X j=1 λju2 j≤1. Let W⊂Hbe the Hilbert space of vectors ufor which Pjcjαu2 j<∞with the norm kukW=cPjjαu2 j1/2. Then u∈BW(0,1) ⇐⇒ ∞ X j=1 cjαu2 j≤1. Using (H1), we have that BV(0,1) ⊂BW(0,1) and therefore Hε(BV(0,1), H)≤ Hε(BW(0,1), H).(3) If we denote µj=c−1j−α, we can write BW(0,1) as an ellipsoid given by E={(uj)∞ j=1 |∞ X j=1 u2 j µj≤1}. For a given ε > 0, let us give first an upper bound for N√2ε(E, H). Let dbe the smallest integer such that µd+1 ≤ε2. We consider the truncated ellipsoid e E={u∈ E |uj= 0 for all j≥d+ 1}. Given any ε-cover {u1, ..., uN}of e E, i.e. for each u∈e E, there exists some i∈ {1, ..., N}such that d X j=1 (uj−ui j)2≤ε2. For any u∈ E, we have ∞ X j=d+1 u2 j≤µd+1 ∞ X j=d+1 u2 j µj≤ε2, 3 and hence for some i∈ {1, ..., N}, ∞ X j=1 (uj−ui j)2= d X j=1 (uj−ui j)2+∞ X j=d+1 u2 j≤2ε2. Therefore, {u1, ..., uN}forms a √2ε-cover of the full ellipsoid E. We now view e Eas a subset of Rd, i.e. e E={u∈Rd| d X j=1 u2 j µj≤1}, and we prove the inequality Nε(e E,Rd)≤vol(e E+ε 2Bd(1)) vol(ε 2Bd(1)) ,(4) where Bd(1) = {u∈Rd|Pd j=1 |uj|2≤1}. The proof of (4) is actually simple: first of all let us consider any finite family of points A={(ai)i∈J} ⊂ e Efor which all balls B(ai,ε 2) are pairwise disjoint. Then we have [ i B(ai,ε 2)⊂e E+ε 2Bd(1), hence vol(ε 2Bd(1))card(A) = X i∈J vol(B(ai,ε 2)) ≤vol(e E+ε 2Bd(1)). To conclude, it is sufficient to remark that since the cardinality of such finite sets is bounded, we can consider such a set Awith maximal cardinality. Then for any a6∈ Ain e E, the ball B(a, ε 2) intersects at least one of the balls B(ak,ε 2),implying ||a−ak|| ≤ ε. It follows that the balls B(ai, ε) with ai∈Agive an ε-covering of e E. The result follows immediately. Since ε2< µjfor all j∈ {1, ..., d}, we can see that e Econtains the ball εBd(1) = {u∈Rd|Pd j=1 |uj|2≤ε2}, hence vol(e E+ε 2Bd(1)) ≤vol(3 2e E).(5) From (4) and (5), we deduce Nε(e E, H)≤3d εd vol(e E) vol(Bd(1)). Since the ellipsoid e Eis the image of the the unit ball by the linear transform (x1, ..., xd)−→ (√µ1x1, ..., √µdxd) it follows classically that vol(e E) vol(Bd(1)) = d Y j=1 √µj and we can deduce that Nε(e E, H)≤3d εd d Y j=1 (cjα)−1/2=3d εd1 √cd (d!)−α/2≤3d εd1 √cdd e−α 2d , 4 using the fact that µj=c−1j−αand the elementary inequality d!≥d ed . Since µd+1 ≤ε2, we deduce that Nε(e E, H)≤3dd+ 1 dα 2d eα 2d≤3deαd, and therefore Hε(e E, H)≤dlog 3 + α log 2 . Since ε2< µd=c−1d−α, we have d < c−1/αε−2/α, and we obtain Hε(e E, H)<log 3 + α log 2 1 cε21/α . Then we deduce Hε√2(E, H)<log 3 + α log 2 1 cε21/α . So that by an obvious change of notation Hε(E, H)<log 3 + α log 2 2 cε21/α . and (3) completes the proof. Remark 3 This upper bound is rather sharp: for a lower bound of the entropy, we observe that the ellipsoid E contains the truncated ellipsoid e E, which contains the ball εBd(1). Then, we have Nε 2(E, H)≥Nε 2(εBd(1), H)≥2d,(6) as a consequence of the obvious inequality card(A)vol(ε 2Bd(1)) ≥vol(εBd(1)) = 2dvol(ε 2Bd(1)) valid for any ε 2- covering of εBd(1) in Hwith centers forming the set A. Indeed the orthogonal projections in H of the covering balls on the d-dimensional space are covering balls of the projection (equal to εBd(1)) with centers in the d-dimensional space and the matter is reduced to ddimensions. When λj≤Cjα,we can deduce Hε(BV(0,1), H)≥ Hε(E, H).(7) Since ε2≥µd+1 =C−1(d+ 1)−α, we have d≥C−1/αε−2/α −1, and from (6), we obtain Hε 2(E, H)≥1 Cε21/α −1, and by an obvious change of notation Hε(E, H)≥1 4Cε21/α −1. Therefore, from (7), we obtain Hε(BV(0,1), H)≥1 4Cε21/α −1. which is not so far from (2). 5 Now, consider Au +g(u) = λu, (8) where g:V→V′is a continuous nondecreasing function and λis a positive constant. We define by Cthe set of equilibria of (8) and we consider the identity map I:C → C. The second goal in this section is to estimate the fractal dimension of C. First, we prove the following result. Proposition 4 For all u, v ∈ C, ku−vkV≤√λ|u−v|H. Proof. Let uand vbelong to Cand set u−v, where uand vare solutions to (8). Then, we obtain hAu −Av, u −vi+hg(u)−g(v), u −vi=λ|u−v|2 H. Since gis non-decreasing, the conclusion follows easily. Finally, using Theorems 1 and 2 together with Proposition 4, we deduce the following result. Theorem 5 Assume the assumption (H1). Then, any compact subset K⊂ C has a finite fractal dimension with dimFK < log 3 + α log 2 32λ c1/α . Remark 6 As we shall see in the next section, in the applications to concrete elliptic equations, the function N(λ) = min {n∈N, λn+1 ≥λ}behaves like some positive power of λfor large values of λ. The following example now shows that for general monotonic maps g, the estimate given by Theorem 5 is optimal up to a multiplicative constant in such a case, therefore essentially optimal as far as the growth as a function of λis concerned and a general monotone map gis allowed. Let us consider λ > 0,n∈Nsuch that λn< λ ≤λn+1 and set g(u) := n X j=1 (λ−λj)Pj(u), where Pjis the orthogonal projection from Hto the eigenspace of Acorresponding to the eigenvalue λj. Now the equation Au +g(u) = λu reduces to Au = n X j=1 λjPj(u) + λ(u− n X j=1 Pj(u)), so that Xn= n M j=1 Pj(H) = n M j=1 ker(A−λjI)⊂ C. Consequently, in this case Ccontains a vector space of dimension d=card {j∈N∗, λ > λj}=N(λ). In particular for the unit ball Kof this finite dimensional space, which is a compact subset of Cwe find dimFK≥N(λ). When λk≤Ckα,then N(λ)≥λ C1/α −1.This confirms the optimality of the upper estimate up to a constant for λlarge. 6 3 Application to some elliptic equations Let Ω ⊂RN,N≥1, be a bounded open domain with sufficiently smooth boundary. We denote by (·,·) the inner product in L2(Ω), and by |·|L2(Ω) the associated norm. By ||·||H1 0(Ω) we denote the norm in H1 0(Ω), which is associated to the inner product ((·,·)) := (∇·,∇·).We will denote by h·,·i the duality product between H−1(Ω) and H1 0(Ω). By k·kL∞(Ω) we denote the norm in L∞(Ω). Let A=−∆Dwith ∆Dthe Dirichlet Laplacian on Ω. We denote by λ1the first eigenvalue of the A. Let λj denote the jth eigenvalue of Ω for the Dirichlet boundary problem. We use the estimate (see Li and Yau [3] for more details) λj≥NCN N+ 2j2/N V−2/N ,(9) where Vis the volume of Ω, and CN= (2π)2B−2/N N, with BN= volume of the unit N-ball. Taking into account (9), in particular, we have (H1) with α=2 Nand c=4πN N+ 2(Γ(1 + N 2))2/N |Ω|−2/N , where |Ω|denotes the N-dimensional measure of Ω. As a consequence we find the following result. Corollary 7 Let gbe any nondecreasing continuous function of the real variable swith super-linear growth at infinity. Let Cbe the set of solutions of the equation u∈H1 0(Ω) ∪L∞(Ω); −∆u+g(u) = λu. Then Cis compact with a finite fractal dimension such that dimFC<32N/2log 3 + 2 N log 2 N+ 2 4πN N/21 Γ (1 + N/2) |Ω|λN/2. Proof. Compactness is an immediate consequence of super-linear growth at infinity. Then it is sufficient to apply Theorem 5 with K=C 4 Application to parabolic equations Now, we consider the following problem ut−∆u+g(u) = λu in Ω,(10) with the zero Dirichlet boundary condition, u= 0 on ∂Ω,(11) and the initial condition u(x, 0) = u0(x),for x∈Ω,(12) where λis a positive constant and g∈C1(R) is a non-decreasing function. We assume that the non-linear term gsatisfies a dissipativity assumption of the form g(s)s≥β|s|p∀s∈R,(13) and the following growth restriction of the derivative |g(s1)−g(s2)| ≤ γ Mp−2|s1−s2|for |s1|,|s2| ≤ M, (14) for some β > 0, γ > 0, M > 0 and p > 2. A typical example of a function satisfying the previous conditions is g(s) = β|s|p−2s, with p > 2.In this case we may take γ=β(p−1). 7 We define a semigroup {S(t), t ≥0}in L2(Ω) by S(t)u0=u(t; 0, u0)∀u0∈L2(Ω),∀t≥0,(15) where u(t; 0, u0) is the unique solution of (10)-(12). We denote by Athe global attractor associated with the semigroup Sdefined by (15). Our aim is to estimate the fractal dimension of A. First, we need the following results. Proposition 8 Assume (13). Then the attractor Aassociated with (10)-(12) is bounded in L2(Ω). More concretely, there exists a positive constant C(p, β, λ, Ω) such that |a|L2(Ω) ≤C, ∀a∈ A. Proof. Multiplying (10) by u, 1 2 d dt |u|2 L2(Ω)+|∇u|2 L2(Ω) = (λu −g(u), u). Using (13) and Young’s inequality applied with the conjugate exponents p 2and p p−2, we have (λu −g(u), u)≤ZΩλu2−β|u|pdx ≤2(p−2) p2 1 β|Ω|λp p−2, then, using the Poincar´e inequality, we obtain d dt |u|2 L2(Ω)+ 2λ1|u|2 L2(Ω) ≤4(p−2) p2 1 β|Ω|λp p−2. Multiplying by e2λ1tand integrating between 0 and t, we obtain |u(t)|2 L2(Ω) ≤e−2λ1t|u0|2 L2(Ω) +2(p−2) p2 1 βλ1|Ω|λp p−21−e−2λ1t. We observe that if u(t)∈ A, then there exists u0∈ A such that u(t) = S(t)u0. Then, we have |S(t)u0|2 L2(Ω) ≤e−2λ1t|u0|2 L2(Ω) +2(p−2) p2 1 βλ1|Ω|λp p−21−e−2λ1t. Fix t > 0, and consider a∈ A. Then, there exists u0∈ A such that a=S(t)u0, and we have |a|2 L2(Ω) ≤e−2λ1t|u0|2 L2(Ω) +2(p−2) p2 1 βλ1|Ω|λp p−21−e−2λ1t. If ttends to +∞, we obtain |a|2 L2(Ω) ≤2(p−2) p2 1 βλ1|Ω|λp p−2∀a∈ A. Taking into account Proposition 8, we prove the following result Proposition 9 Assume (13). Then the attractor Aassociated with (10)-(12) is uniformly bounded in L∞(Ω). More precisely, kakL∞(Ω) ≤λ β1 p−2 ,∀a∈ A.(16) 8 Proof. Using (13), we observe g(s)−λs ≥0 if s≥M:= λ β1 p−2 .(17) Let u∈L2(Ω), we define u+(x) = u(x), if u(x)>0, 0, in other case, and u−(x) = u(x), if u(x)<0, 0, in other case. Multiplying (10) by (u(x)−M)+, taking into account (17) and using the Poincar´e inequality, we have d dt (u−M)+2 L2(Ω)≤ −2λ1(u−M)+2 L2(Ω) . Multiplying by e2λ1tand integrating between 0 and t, we obtain (u(t)−M)+2 L2(Ω) ≤e−2λ1t(u0−M)+2 L2(Ω) . We observe that if u(t)∈ A, then there exists u0∈ A such that u(t) = S(t)u0. Then, we have ZΩ (S(t)u0−M)2 +dx ≤e−2λ1tZΩ (u0−M)2 +dx. As Ais bounded in L2(Ω), then we can deduce that there exists a positive constant b C(p, β, λ, Ω), which is independent of u0, such that ZΩ (u0−M)2 +dx ≤ZΩ (u0−M)2dx ≤2|u0|2 L2(Ω) +M2|Ω|≤b C, and, we have ZΩ (S(t)u0−M)2 +dx ≤b Ce−2λ1t. Fix t > 0, and consider a∈ A. Then, there exists u0∈ A such that a=S(t)u0, and we have 0≤ZΩ (a−M)2 +dx ≤b Ce−2λ1t. If ttends to +∞, we obtain ZΩ (a−M)2 +dx = 0 ∀a∈ A, then (a−M)+= 0 =⇒a≤M∀a∈ A. We use a similar reasoning for (u+M)−, and then we have kakL∞(Ω) ≤M∀a∈ A. Now, we define Cγ,β := 1 2 + γ β ,(18) and we consider the map SCγ,β λ:A → A. Taking into account Proposition 9, we prove the following result. 9