Pullback Exponential Attractors for Evolution Processes in Banach Spaces 1

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1 Pullback Exponential Attractors for Evolution Processes in Banach Spaces 1 Stefanie Sonner BCAM NUMERIWAVES MEETING March 15, joint work with Alexandre N. Carvalho

2 Plan Exponential Attractors for Semigroups Nonautonomous Attractors Exponential Pullback Attractors Properties and Previous Work A General Existence Result Applications Chafee Infante Equation Damped Wave Equation

3 Exponential Attractors for Semigroups Autonomous initial value problems { t u = A(u) in Banach space X u t=0 = u 0 u 0 X (1) Assuming global well-posedness of (1) = generates a semigroup in X, S(t) : X X, t 0, S(t) S(s) = S(t + s) t, s 0 S(0) = Id (t, x) S(t)x continuous

4 Exponential Attractors for Semigroups Longtime Dynamics Many dissipative systems posses a global attractor A (a) A X (b) S(t)A = A, t 0, compact, invariant, (c) attracts every bounded set D X, lim dist H(S(t)D, A) = 0. t Properties: A unique, minimal closed set attracting all bounded sets, in most cases, the fractal dimension dim f (A) = k < = there exists a continuous embedding A R 2 k +1

5 Exponential Attractors for Semigroups Drawbacks of Global Attractors Example: ẋ = x(x 1) 2 x t=0 = x o, x o R = A = [0, 1], but ɛ > 0 ẋ = (x(x 1) 2 + ɛ) x t=0 = x o, x o R = A ɛ = {y ɛ }. In general, rate of convergence unknown, can be arbitrarily slow A not stable under perturbations complex structure, not computable

6 Exponential Attractors for Semigroups Exponential Attractors 1, 2 M is an exponential attractor for S(t), t 0, if (a) M X (b) dim f (M) < compact, (c) S(t)M M, t 0, finite dimensional, semi-invariant, (d) ω > 0 such that for every bounded set D X lim t eωt dist H (S(t)D, M) = 0. Recently, extension to non-autonomous problems... 1 Eden, Foias, Nicolaenko, Temam (1994). 2 Efendiev, Miranville, Zelik (2000).

7 Nonautonomous Attractors Evolution Processes Nonautonomous initial value problems { t u = A(t)u in Banach space X u t=s = u s s R, u s X (2) Assuming global well-posedness of (2) = generates an evolution process in X, U(t, s) : X X, t s, family of operators such that U(t, s) U(s, r) = U(t, r) t s r U(t, t) = Id t R (t, s, x) U(t, s)x continuous

8 Nonautonomous Attractors Nonautonomous Attractors Different concepts: uniform attractors, pullback and forwards attractors {A(t) t R} global pullback attractor for U(t, s), t s, if (a) A(t) X compact for all t R, (b) U(t, s)a(s) = A(t), t s (c) for all bounded D X and t R invariant, lim dist H(U(t, t s)d, A(t)) = 0, s (d) {A(t) t R} minimal within the family of closed subsets, that pullback attract all bounded sets.

9 Exponential Pullback Attractors Properties and Previous Work Exponential Pullback Attractors Global pullback attractors have similar drawbacks as global attractors in the autonomous case. {M(t) t R} exponential pullback attractor for U(t, s), t s, if (a) M(t) X compact for all t R, (b) sup t R dim f (M(t)) < (c) U(t, s)m(s) M(t), t s finite dimensional, semi-invariant, (d) ω > 0 such that for every bounded D X and t R lim s eωs dist H (U(t, t s)d, M(t)) = 0.

10 Exponential Pullback Attractors Properties and Previous Work Exponential Pullback Attractors 3,4,5,6 exponential rate of convergence, robust under perturbations finite dimensional, A(t) M(t) = existence and finite fractal dimension of {A(t), t R} However, not unique! Assumptions in 4,5 : process satisfies smoothing property Hölder continuous in time strongly pullback bounded dissipative = M(s) is bounded t R s t A(s) s t 3 Efendiev, Miranville, Zelik (2005). 4 Langa, Miranville, Real (2010). 5 Czaja, Efendiev (2011). 6 Efendiev, Yamamoto, Yagi (2011).

11 Exponential Pullback Attractors A General Existence Result Assumptions U(t, s), t s, evolution process in Banach space X, U = S + C (A 1 ) Y normed space, X Y compact, dense (A 2 ) {B(t) t R} X bounded, semiinvariant, pullback absorbing sets: For every bounded D X, t R exists T D,t > 0: U(r, r s)d B(r) s T D,t, r t (A 3 ) (S smoothing) There exists t > 0 and κ > 0: S(t + t, t)u S(t + t, t)v X κ u v Y u, v B(t) (A 4 ) (C contraction) There exists 0 λ < 1 2 : C(t + t, t)u C(t + t, t)v X λ u v X u, v B(t) (A 5 ) (U Lipschitz continuous) For all s t there exists L t,s > 0: U(s, t)u U(s, t)v X L t,s u v X u, v B(t)

12 Exponential Pullback Attractors A General Existence Result Existence of Pullback Exponential Attractors Theorem Let U(t, s), t s, be an evolution process in X and (A 1 )-(A 5 ) be satisfied. If the diameter of the absorbing sets {B(t), t R} grows at most sub-exponentially in the past, then, there exists for every 0 < ν < 1 2 λ a pullback exponential attractor {M ν (t) t R} and dim V f (M ν (t)) log 1 2(ν+λ) ( N Y ( ν B X κ 1 (0) )) t R, where N Z ɛ (D) is the minimal number of ɛ-balls in a metric space Z with centers in D, needed to cover D Z.

13 Exponential Pullback Attractors A General Existence Result Properties It follows the existence of the global pullback attractor and { ( dim V f (A(t)) inf log ν (0, 1 1 N Y ( ν B X 2 λ) 2(ν+λ) κ 1 (0) )) }. Remark: Hypothesis (H 4 ) can be omitted. Relation with the global pullback attractor M(t) = Λ(ˆB, t) n N E n (t), {E n (t)} <. If the pullback absorbing time T D,t is independent of t, the pullback exponential attractor M(t) is also a forwards exponential attractor.

14 Applications Chafee Infante Equation Applications Nonautonomous Chafee Infante Equation u t (x, t) = u(x, t) + λu(x, t) β(t) ( u(x, t) ) 3 u(x, t) = 0 ν x u(x, s) = u s (x) x Ω, t > s Ω, t s x Ω, s R Ω R n bounded, smooth domain, λ R, u s C 0 (Ω). β C 1 (R; R + ), 0 < β(t) β 0, lim β(t) = 0, lim t β (t) β(t) t β 1 t R, e γt = 0 γ > 0. β(t)

15 Applications Damped Wave Equation Applications Nonautonomous Damped Wave Equation u tt (x, t) + β(t)u t (x, t) = u(x, t) + f (u(x, t)) x Ω, t > s u(x, t) = 0 x Ω, t s u(x, s) = u s (x), u t (x, s) = v s (x) x Ω, s R Ω R n bounded, smooth domain, u s H0 1(Ω), v s L 2 (Ω). β C 1 (R; R + ), f C 1 (R; R), lim sup z f (z) z = 0, f (z) c(1 + z ρ ), 0 < ρ < 2 n 2, 0 < β 0 β(t) β 1, t R.