Heraklion, Crete, November 27, 2012 Dirk Blömker Amplitude Equations natural systems for SPDEs joint work with : supported by Martin Hairer (Warwick), Grigorios A. Pavliotis (Imperial College) Christian Nolde, Franz Wöhrl (Augsburg) Wael W. E. Mohammed (Mansoura), Konrad Klepel (Augsburg)
Key Idea: Complicated models near a change of stability system Dominant pattern evolve on a slow time-scale Stable pattern decay/disappear on a fast time-scale Dynamics described by simplified model Amplitude Equations Averaging
Deterministic Results for PDEs PDEs 1 on bounded : Invariant manifolds center manifold Solutions approximated by an ODE on the manifold PDEs on : Many rigorous results since the 90-th (e.g. Schneider, Uecker, Collet, Eckmann,... & many others) Validation of Amplitude or Modulation equations Our Problem: What is the impact of noise on the dominant pattern? No useful center manifold theory for stochastic PDEs yet 1 PDE=partial differential equations SPDE=stochastic PDE
Our Aim AIM: Validation of amplitude equations for SPDEs Understand effect of noise transported by the nonlinearity For the talk only one example: model noise constant in space or space-time white is a celebrated model in pattern formation. The first change of stability is a toy model for the convective instability in Rayleigh-Bénard convection.
Slow-Fast System I SPDE close to bifurcation naturally generate systems SDE coupled to a fast SPDE Example: t u = Lu + νɛ 2 u u 3 + ɛ 2 t W (SPDE) L non-positive operator - kernel N (dominant modes) νɛ 2 the distance from bifurcation ɛ 2 noise strength {W (t)} t 0 some Wiener process ( -dimensional)
Slow-Fast System II Split t u = Lu + νɛ 2 u u 3 + ɛ 2 t W u(t) = ɛv c (ɛ 2 t) + ɛv s (ɛ 2 t) (SPDE) with v c N and v s N (N kernel of L) T v c = νv c P c (v c + v s ) 3 + T Wc (SLOW) T v s = ɛ 2 Lv s + νv s P s (v c + v s ) 3 + T Ws where: P c projects onto N and P s = I P c W (T ) = ɛw (T ɛ 2 ) rescaled Wiener process (FAST)
Outline Consider equation only. 1. Brief review full noise bounded [DB, Maier-Paape, Schneider 01], [DB, Hairer 04,05] 2. Relation to random invariant manifolds 3. by degenerate noise [Hutt, et.al., 07,08], [DB, Mohammed 10], [Roberts 03], [DB, Hairer, Pavliotis 07] 4. Work in progress on 5. / Open Problems [DB, Klepel, Mohammed]
Equation
[DB, Maier-Paape, Schneider 01], [DB, Hairer, 04] equation t u = Lu + νɛ 2 u u 3 + ɛ 2 t W with L = (1 + x 2 ) 2 periodic boundary conditions on [0, 2π] dominant modes N = span{sin, cos} space-time white noise t W (SH)
Existence & Uniqueness Local solutions via fixed-point arguments: mild solutions variation of constants t u(t) = e tl u(0) + e (t s)l [ νɛ 2 u(s) u 3 (s) ] ds + W A (t) 0 where the stochastic convolution t W A (t) = e (t s)l dw (s) 0 and its spatial derivative x W A is in C 0 ([0, T ] [0, 2π]).
Attractivity & Approximation Theorem (Attractivity) [DB, Hairer 04] For all T > 0 : u c (T ɛ 2 ) = O(ɛ) and u s (T ɛ 2 ) = O(ɛ 2 ) Theorem (Approximation) [DB, Hairer 04] u(0) = ɛa(0) + ɛ 2 ψ(0) with a(0) and ψ(0) both O(1). Let a(t ) N solve T a = νa P c a 3 + T Wc and ψ(t) N solves Then for t [0, T 0 ɛ 2 ] t ψ = Lψ + t W s u(t) = ɛa(ɛ 2 t) + ɛ 2 ψ(t) + O(ɛ 3 ) Remark: Approximation remains true for invariant measures.
Approximative Center Manifold [DB, Hairer 05] N u( T ɛ 2 ) u(0) ɛa(0) ɛ 2 ψ(0) u(0) O(ɛ) N u(t) ɛa(ɛ 2 t) ɛ 2 ψ(t) u(t) O(ɛ 2 ) (with high probability) O(ɛ 3 )-ball
Random Dynamical Systems Random Invariant
Random Dynamical System () [L. Arnold, Crauel, Schmalfuß, Flandoli, Scheutzow, Chueshov, Duan, Caraballo, Kloeden, Robinson,... ] Solutions generate a random dynamical system ϕ(t, W )u 0 is solution of (SPDE) given t 0 time u 0 initial condition W whole path of the Wiener process t W (t), t R. Cocycle property: For all t, τ R + and paths W. ϕ(0, W ) = Id, ϕ(t + τ, W ) = ϕ(t, θ τ W )ϕ(τ, W ) Ergodic Shift: θ τ W = W (t + ) W (t)
Cocycle Property ϕ(t, θ τ W )ϕ(τ, W ) = ϕ(t + τ, W ) ϕ(t + τ, W ) H u 0 ϕ(τ, W ) ϕ(t, θ τ W ) H 0 τ t + τ t W (t)
Random Invariant Manifold () [Duan, Lu, Schmalfuß 03, 04], [Mohammed, Zhang, Zhao, 08] Definition (Random Invariant Manifold, ) A random set M(W ) is positive invariant, if ϕ(t, W )M(W ) M(θ t W ) for all t 0. If M(W ) = {u + ψ(w, u) u N } is the graph of a random Lipschitz mapping ψ(w, ) : N N, then M(W ) is called a Lipschitz.
stable for Swift N M(W ) for ν < 0 Z(W ) N Z(W ) random fixed point t Z(θ t W ) stable stationary solution M(W ) random invariant manifold convergence in pull-back sense (initial time to )
are moving in time! N M(θ t W ) M(W ) Z(θ t W ) Z(W ) N ϕ(t, θ t W ) ϕ(t, W ) M(θ t W ) Z(θ t W )
: vs Amplitude Eq. Random invariant manifolds: behaviour for all paths reduced SDE on a moving space Amplitude Equations: only the most likely behaviour reduced SDE on a fixed vector space Useful for the derivation of qualitative properties of & invariant measures.
due to Noise Interesting observation: Additive(!) noise may lead to stabilization (or shift of bifurcation) of dominant modes (pattern disappears)
due to Noise Well known phenomenon due to multiplicative Noise. For example: By Itô noise, due to Itô-Stratonovic correction, or Stratonovic noise due to averaging over stable and unstable directions For SDE: [Arnold, Crauel, Wihstutz 83], [Pardoux, Wihstutz 88 92]... For SPDE: [Kwiecinska 99], [Caraballo, Mao et.al. 01], [Cerrai 05], [Caraballo, Kloeden, Schmallfuß 06]... By Rotation: [Baxendale et.al. 93], [Crauel et.al. 07]... Only very few examples due to additive noise: Blow up through a small tube: e.g. [Scheutzow et.al. 93]...
Numerical Examples -model
[Hutt, Schimansky-Geier et.al. 07, 08] Space-independent (gobal) noise destroys modulated pattern in 1D--Eq.
Equation Example: Equation t u = (1 + 2 x ) 2 u + 1 20 u u3 + σ 10 tβ(t) (SH) u(t, x) R, t > 0, x [0, 2π] periodic boundary conditions Observation: 0 is is stabilized by large noise Noise only in time, constant in space
(Nolde, Wöhrl) [DB, Nolde, Mohammed, Wöhrl, 12] σ = 0.5 Semi-implicit spectral Galerkin-method using fft in Matlab
(Nolde, Wöhrl) [DB, Nolde, Mohammed, Wöhrl, 12] σ = 5 Semi-implicit spectral Galerkin-method using fft in Matlab
Results for Equation
The SPDE t u = Lu + νɛ 2 u u 3 + σɛ t β L = (1 + x 2 ) 2 periodic boundary conditions on [0, 2π] span{sin, cos} dominant pattern β is a real-valued Brownian motion σɛ 1 noise strength ν ɛ 2 1 distance from bifurcation (SH)
The Ansatz Ansatz: t u = Lu + νɛ 2 u u 3 + ɛσ t β u(t, x) = ɛa(ɛ 2 t)e ix + c.c. + ɛσ Z(t) + O(ɛ 2 ) (SH) fast process Z(t) = t 0 e (t τ) dβ(τ) complex-valued amplitude A Result: Amplitude Equation [DB, Mohammed, 10] T A = (ν 3 2 σ2 )A 3A A 2 (A)
Interesting Facts T A = (ν 3 2 σ2 )A 3A A 2 Amplitude equation is deterministic Noise leads to a stabilizing deterministic correction Formal calculation with SPDEs yields: New contribution to the amplitude equation is 3A σ 2 ɛ T β 2, (A) where β(t ) = ɛβ(t ɛ 2 ) is a Brownian motion on the slow time-scale T = ɛ 2 t
Noise 2? What is noise 2? Instead of ɛ T β consider a fast Ornstein-Uhlenbeck-process T Z (T ɛ 2 ) = Z ɛ (T ) = ɛ 1 e (T s)ɛ 2 d β(s) ɛ T β(t ), where β(t ) = ɛβ(ɛ 2 T ) is a rescaled Brownian motion 0 [DB, Hairer, Pavliotis, 07] [DB, Mohammed 08] Averaging with error bounds dx = O(ɛ r )dt + O(ɛ r )d β and X (0) = O(ɛ r ), r > 0, then T 0 X (s)z ɛ (s) 2 ds = 1 2 T 0 X (s)ds + O(ɛ 1 2r )
Averaging other averaging results For odd powers: and so on... T 0 T 0 X (s)z ɛ (s)ds = O(ɛ 1 r ) X (s)z ɛ (s) 3 ds = O(ɛ 1 3r ) Note: X = O(f ɛ ) if p > 1, T > 0 there is a C > 0 such that E sup X p Cfɛ p [0,T ]
The Theorem t u = Lu + νɛ 2 u u 3 + σɛ t β T A = (ν 3 2 σ2 )A 3A A 2 Theorem Approximation [DB, Mohammed 10] u is solution of (SH) in H 1 A is solution of (A) u(0) = ɛ A(0)e ix + c.c. + ɛψ 0 with ψ 0 e ±ix. Then for κ, T 0, p > 0 there is C > 0 such that ( P sup u(t) ɛv(ɛ 2 t) H 1 > ɛ 2 κ) t [0,T 0 /ɛ 2 ] < Cɛ p + P( u(0) H 1 > ɛ 1 κ/2 ) with v(t ) = A(T )e ix + c.c. + σz ɛ (T ) + e T ɛ 2L ψ 0. Remark: Theorem holds in a much more general setting. (SH) (A)
Numerical justification (Nolde / Wöhrl) [DB, Nolde, Mohammed, Wöhrl, 12] Numerical approximation of R sup (T ) = sup s [0,T ] R(s) and R(T ) = u(t ɛ 2 ) ɛ [ Z ɛ (T ) + A(T )e ix + c.c. ] 0.025 0.02 0.015 0.01 ν = 1, σ = 1 15, ɛ = 1 10 R sup R 0.005 0 10 20 30 40 50 T mean over a few hundred realizations
Open Problem If we consider higher order corrections [DB, Mohammed, 10] or Burgers equation & degenerate noise [DB, Hairer, Pavliotis, 07] we need to treat terms like T X (s)z ɛ (s) 2 d β(s) 0 Problem: In order to derive error estimates we rely on Martingal representation and Levy characterization. Thus dim(n ) = 1 is necessary For dim(n ) > 1 weak convergence results available = Averaging (well known)
Martingale Approximation Result in 1D For T 0 f (a, Z ɛ)d β with Z ɛ (t) = 1 t ɛ 0 e (t s)λɛ 2 d β(s). Lemma [DB, Hairer, Pavliotis, 07] M(t) continuous martingale with quadratic variation f g arbitrary adapted increasing process with g(0) = 0 Then (with respect to an enlarged filtration) there exists a continuous martingale M(t) with quadratic variation g such that γ < 1/2 C > 0 with E sup M M p C E sup f g p/2 [0,T ] [0,T ] +C ( Eg(T ) 2p) 1/4( E sup f g p) γ. [0,T ]
Unbounded or Large Domains
Modulated Pattern Problem: A full band of eigenvalues changes stability Modulation of dominant pattern Periodic Pattern u(x) ɛae ix + c.c. where A C Modulated Pattern (many modes near ±1 contribute) u(x) ɛa(ɛx)e ix + c.c. where A : R C
Results on Large Domains t u = Lu + ɛ 2 νu u 3 + ɛ 3 2 ξ (SH) Results for deterministic PDE (on R): [Kirrmann, Mielke, Schneider, 92] and many more... Result for SPDE (on [ L/ɛ, L/ɛ]): Only on large [DB, Hairer, Pavliotis 05] u(t, x) = ɛa(ɛ 2 t, ɛx) e ix + c.c. + O ( ɛ 2) A(T, X ) C solves stochastic Ginzburg-Landau equation: T A = 4 X 2 A + νa 3 A 2 A + ση (GL) space-time white noise η(t, X ) C, even if ξ colored in space!
Degenerate Noise on R Ansatz: t u = Lu + νɛ 2 u u 3 + σɛ t β on R (SH) u(t, x) = ɛa(ɛ 2 t, ɛx)e ix + c.c. + ɛσz(t) + O(ɛ 2 ) fast OU-process Z(t) = t 0 e (t τ) dβ(τ) Result: Amplitude Equation [DB, Mohammed, 11] T A = 4 2 X A + (ν 3 2 σ2 )A 3A A 2 (GL)
Open Problems Regularity of A Kirrmann-Mielke-Schneider needed A C 1,4 b ([0, T ] R) DB-Hairer-Pavliotis needed A = B + Z with B C 0 ([0, T ], H 1 [ L, L]) and Z C 0 ([0, T ] [ L, L]) Gaussian DB-Mohammed needed A C 0 ([0, T ], H α (R)), α > 1/2 But: For space-time white noise is the Amplitude A only Hölder continuous (Exponent < 1 2 ) And: Bounds for terms like require some regularity of A. and A(T, ) = for all T > 0. e tl [A(ɛx)e ix ] [e 4T 2 X A](ɛx) e ix
Further results / Work in Progress Higher order corrections Martingale terms [Roberts, Wei Wang, 09], [DB, Mohammed 11] Large Domains Modulated Pattern [DB, Klepel, Mohammed] Levy noise [DB, Hausenblas] Local shape of random invariant manifolds [DB, Wei Wang, 09], [Duan, et.al.] Approximation of invariant measures [DB, Hairer, 04]
SPDEs near a change of stability Transient dynamics via amplitude equations Stabilisation due to additive noise Effect of noise on dominant modes Noise transported by nonlinearity between Fourier-modes