Fine scales of decay rates of operator semigroups

Transcription

1 Operator Semigroups meet everything else Herrnhut, 5 June 2013

2 A damped wave equation 2 u u + 2a(x) u t2 t = 0 (t > 0, x Ω) u(x, t) = 0 (t > 0, x Ω) u(, 0) = u 0 H0 1 (Ω), u t (, 0) = u 1 L 2 (Ω). Here, Ω is a (smooth) bounded domain in R n, and a : Ω [0, ) (continuous).

3 A damped wave equation 2 u u + 2a(x) u t2 t = 0 (t > 0, x Ω) u(x, t) = 0 (t > 0, x Ω) u(, 0) = u 0 H0 1 (Ω), u t (, 0) = u 1 L 2 (Ω). Here, Ω is a (smooth) bounded domain in R n, and a : Ω [0, ) (continuous). Energy decreasing in t. E(u, t) = 1 2 Ω ( u 2 + u 2) dx, t

4 Energy decay Except in degenerate cases, 1 the energy E(u, t) 0 as t ; 2 if the domain of damping {x : a(x) > 0} satisfies the geometric optics condition then the decay occurs at an exponential rate (Bardos-Lebeau-Rauch, 1992); 3 in other cases, the decay occurs at a polynomial rate or a logarithmic rate, uniformly for smooth initial data.

5 Operator formulation Reformulate the damped wave equation: X = H0 1 (Ω) L 2 (Ω), ( ) 0 1 A =, 2a(x) D(A) = (H 2 H 1 0 ) H 1 0. ( ) u(t) U(t) = u X, E(u, t) = 1 2 U(t) 2 H 1 L, 2 t ( ) U u0 (t) = AU(t), U(0) =. u 1

6 PDE methods establish that σ(a) {λ C : a Re λ < 0}, and (is A) 1 grows (at most) polynomially as s. Moreover, A generates a C 0 -semigroup of contractions {T (t) : t 0} on X. Decay of E(u, t) for initial data (u 0, u 1 ) D(A) corresponds to decay of T (t)(λ A) 1 for any λ ρ(a). One establishes such decay by some form of complex inversion of Laplace transforms.

7 Exponential decay Assume that A generates a C 0 -semigroup {T (t) : t 0} on a Banach space. If (λ A) 1 is bounded for {Re λ 0}, then it is also bounded for {Re λ δ } for some δ > 0. If T (t) Ce δt for some δ > 0, then (λ A) 1 C whenever Re λ 0. If T (t)(µ A) 1 Ce δt for some δ > 0 and any µ ρ(a), then (λ A) 1 C (1 + λ ) whenever Re λ 0.

8 Theorem (Weis-Wrobel 1996; Gearhart, Prüss, ) 1 Assume that (λ A) 1 C (Re λ 0). (1) Then T (t)a 1 C e δt (t > 0). (2) 2 Assume that X is a Hilbert space. If (1) holds then T (t) C e δt (t > 0). If (λ A) 1 C(1 + λ ) whenever Re λ δ, then (2) holds whenever δ < δ. Here it is not necessary to assume that T is bounded. The Gearhart-Prüss Theorem does not extend to L p -spaces (p < ).

9 Necessary condition From now on, assume that T is bounded, so σ(a) {Re λ 0}. Theorem Let m : (0, ) (0, ) be decreasing with lim t m(t) = 0. Assume that T (t)(1 A) 1 m(t) (t > 0). Then σ(a) ir is empty, and, for each c (0, 1), (is A) 1 = O ( m 1 (c/ s ) ) ( s ).

10 Converse question Question: For which (if any) increasing functions M is it true that (is A) 1 M( s ) = T (t)a 1 C/M 1 (ct)?

11 Converse question Question: For which (if any) increasing functions M is it true that (is A) 1 M( s ) = T (t)a 1 C/M 1 (ct)? Consider X = C, A = 1, T (t) = e t. For a positive answer, M 1 cannot increase faster than exponentially, so M increases at least logarithmically.

12 Normal semigroups Theorem Let (T (t)) t 0 be a semigroup of normal operators on a Hilbert space X with generator A. Assume that σ(a) {λ C : Re λ < 0} and sup{re λ : λ σ(a)} = 0, M(s) = sup{ (ir A) 1 : r s}, s 0. Let c > 0. The following are equivalent: (i) There exists C such that T (t)a 1 C M 1 (ct), t c 1 M(0); (ii) There exists B such that M(τ) ( τ ) M(s) c log B, τ 0; s 1. s

13 Logarithmic decay Now T is any bounded semigroup and σ(a) ir is empty. Theorem (Lebeau 1996, Burq 1998) Suppose that X is a Hilbert space and that (is A) 1 C exp(c s ) (s R). Then T (t)a 1 ( ) 1 = O log t (t ).

14 Polynomial decay Theorem (Batkai et al. 2006; Liu-Rao, 2005) Assume that Then, for each ε > 0, (is A) 1 C(1 + s ) α T (t)a 1 = O ( 1 t 1 α ε If X is a Hilbert space, then ( ) T (t)a 1 (log t) 1 α +1 = O t 1 α ) (s R). (t ). (t ).

15 Qualitative result The following has been known since 1986, although it follows from a Tauberian theorem of Ingham (1935). Theorem If T is bounded and σ(a) ir is empty then lim T (t)a 1 = 0. (3) t

16 Outline proof (Newman-Korevaar) Main estimate T (t)a 1 = 1 ) (1 + z2 2πi γ R 2 (z A) 1 T (t) dz z. T (t)a 1 2K R + 1 2π γ ) (1 + z2 R 2 (z A) 1 tz dz e z Choose R and γ, and estimate the integral (crudely)..

17 Quantified version Theorem (B.-Duyckaerts, 2008) Suppose that (is A) 1 M( s ) (s R). Then, for any c (0, 1), T (t)a 1 = O ( ) 1 M 1 log (ct) (t ), where M log (s) = M(s) (log(1 + M(s)) + log(1 + s)).

18 Examples 1. M(s) = C: one obtains exponential decay, i.e. the Weis-Wrobel Theorem, but only for bounded semigroups. 2. M(s) = C exp(α s ): one obtains logarithmic decay of the same form as Lebeau and Burq, without restriction to Hilbert space. 3. M(s) = C(1 + s ) α : one obtains polynomial stability: (is A) 1 C(1 + s ) α = T (t)a 1 = O ( (log t t ) 1 ) α similar form to Liu and Rao, with a smaller exponent in the logarithmic term and without restriction to Hilbert space. Can one get rid of the logarithmic correction completely?

19 Theorem (Borichev-Tomilov 2010) Assume that (is A) 1 C(1 + s ) α 1 If X is a Hilbert space, then T (t)a 1 ( ) 1 = O t 1 α (s R). (t ). 2 For Banach spaces, it is not possible to improve the conclusion that ( (log T (t)a 1 ) 1 ) t α = O (t ). t

20 Outline of proof Hilbert space, M(s) = C(1 + s) α. Use complex analysis to show (λ A) 1 ( A) α C (Re λ 0). Use Plancherel s Theorem to show T (t)( A) α Ct 1. Use semigroup property and interpolation (moment inequality) to show T (t)( A) 1 Ct 1/α.

21 Fine scales What if M(s) s α log s or M(s) Then M 1 (t) sα log s (for example)? ( ) t 1/α or M 1 (t) (t log t) 1/α. log t

22 Fine scales What if M(s) s α log s or M(s) Then M 1 (t) sα log s (for example)? ( ) t 1/α or M 1 (t) (t log t) 1/α. log t A measurable function l : (a, ) (0, ) is slowly varying if l(γs) lim s l(s) = 1 for all γ > 0. Equivalently, ( s l(s) = c(s) exp a ε(u) u ) du where c(s) c > 0 and ε(s) 0 as s.

23 Fine scales What if M(s) s α log s or M(s) Then M 1 (t) sα log s (for example)? ( ) t 1/α or M 1 (t) (t log t) 1/α. log t A measurable function l : (a, ) (0, ) is slowly varying if l(γs) lim s l(s) = 1 for all γ > 0. Equivalently, ( s l(s) = c(s) exp a ε(u) u ) du where c(s) c > 0 and ε(s) 0 as s. Examples: (log s) β (β R); exp ( (log s) β) (0 < β < 1).

24 Regularly varying case M is regularly varying if M(s) sα l(s) where l is slowly varying. Consider the case α = 1 (purely for simplicity), l increasing. Theorem (B-Chill-Tomilov, 2012+?) Assume that X is a Hilbert space, and l is slowly varying and increasing, and (is A) 1 ( ) s = O ( s ). l( s ) Then T (t)a 1 ( ) 1 = O tl(t) (t ). For many (but not all) l, this gives the optimal result T (t)a 1 = O(1/M 1 (t)).

25 Theorem (B-Chill-Tomilov, 2012+?) Assume that X is a Hilbert space, and l is slowly varying and decreasing, and (is A) 1 ( ) s = O ( s ). l( s ) Then, for every ε > 0, T (t)a 1 = O ( ) (logt) ε t l(t) (t ). Here l is a slowly varying function which is sometimes, but not always, the same as l. However the optimal result would have ε = 0.

26 Outline of proof Hilbert space, M(s) s l( s ), l increasing. Find a complete Bernstein function f l, as large as possible, such that (λ A) 1 f l ( A 1 ) C (Re λ 0). For example, f log ( A 1 ) = A 1 log(i A). Use Plancherel s Theorem to show T (t)f l ( A 1 ) Ct 1. Use an interpolation inequality for complete Bernstein functions of semigroup generators, together with an Abelian/Tauberian theorem for Stieltjes transforms (Karamata, 1930s), to remove the log term in the M log result.

27 Complete Bernstein functions f : (0, ) (0, ) is a complete Bernstein function if s f (s) = a + bs + dν(u) (s > 0) s + u (0, ) for some a, b 0 and positive measure ν on (0, ). Equivalently, f extends analytically to C \ (, 0] mapping the upper half-plane to itself, and lim s 0+ f (s) exists and is real [i.e, f is a Nevanlinna-Pick function and is positive on the positive axis.] Equivalently, f (s) = S(1/s), where S(s) = a + b s + (0, ) 1 s + u dν(u). This integral is the Stieltjes transform of the measure ν (or of its distribution function).

28 Interpolation inequality Theorem Let A be the generator of a bounded C 0 -semigroup (T (t)) t 0 on a Banach space, and assume that A is invertible. There exists a constant c > 0 such that the following holds for all complete Bernstein functions f and all t 0: T (t)f ( A 1 ) c T (2t)A 1 T (t)a 1 f ( T (t)a 1 ).

29 One singularity Katznelson-Tzafriri theorem for semigroups (Esterle-Strouse-Zouakia, Vu) gives: Let A be the generator of a bounded C 0 -semigroup (T (t)) t 0 on a Banach space. The following are equivalent: σ(a) ir {0}; T (t)a(i A) 2 0. There is a quantified version (Martinez, 2011).

30 Theorem (B-Chill-Tomilov, 2012 (sofa in Kraków) +?) Let A be the generator of a bounded C 0 -semigroup (T (t)) t 0 on a Hilbert space. The following are equivalent: σ(a) ir {0} and (is A) 1 is bounded for large s ; T (t)a(i A) 1 0.

31 Rates for one singularity A generates bounded C 0 -semigroup on Hilbert space, σ(a) ir = {0}, { (is A) 1 M( s 1 ), s 1, C s 1. We hope for ( T (t)a(i A) 1 = O 1 M 1 (ct) ). This works in general, in a limited way (the correction is bigger than logarithmic); exactly for M(s) s α ; well for M(s) sα l(s), where l is increasing; poorly for M(s) sα l(s), where l is decreasing.

32 Other extensions 1. There is a version of the M log -theorem for Laplace transforms. Due to lack of Neumann series one has to assume that ˆf extends to a a region of known shape with known estimates. It has applications to considerations of local energy of waves when Ω is unbounded, with f (t) = B 1 T (t)b 2 where B 1 and B 2 are cut-off operators. 2. Higher order versions. Estimates for T (t)a k follow trivially from the semigroup property, and one can extend M log results to Laplace transforms.