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Universit¨ at Augsburg Amplitude Equation for stoch. SH Equation Konrad Klepel Amplitude Equation for the generalized Swift Introduction Hohenberg Equation with Noise Setting Bounded domains Main theorem Konrad Klepel Differences to the deterministic case Short Overview of the proof Unbounded domains (work in progress) 04. Oktober 2012

Introduction The generalized Swift Hohenberg equation Amplitude Equation for ∂ t u = ru − (1 + ∂ x 2 ) 2 u + α u 2 − u 3 stoch. SH (SH) Equation Konrad Klepel is a qualitative model for Rayleigh Benard convection. Introduction Setting Bounded domains Main theorem Differences to the deterministic case Short Overview of the proof Unbounded domains (work in progress) � User:WikiRigaou / Wikimedia Commons / CC-BY-SA-3.0 c

Introduction Amplitude Equation for stoch. SH Equation Konrad Klepel It is well known (Cross/Hohenberg 93, Hilali 95, Burke/Knobloch 06) that Introduction � � Setting | r | · A ( | r | t ) · e ix + | r | · A ( | r | t ) · e − ix . u ( t , x ) ≈ Bounded domains Main theorem where the complex amplitude A ( T ) of the dominant mode e ix Differences to the deterministic case is the solution of Short Overview of the proof 27 α 2 − 1) | A | 2 A , Unbounded ∂ T A = sign( r ) A + 3( 38 domains (work in progress)

Setting Amplitude Equation for We consider the following stochastic version of (SH) stoch. SH Equation ∂ t u = νε 2 u − (1 + ∂ x 2 ) 2 u + α u 2 − u 3 + εσ∂ t β, (SSH) Konrad Klepel Introduction where Setting ◮ β ( t ) is a real valued standard Brownian motion, Bounded domains ◮ α , σ and ν are real-valued constants, Main theorem Differences to the deterministic ◮ the small parameter ε > 0 relates the distance from case Short Overview bifurcation to the noise strength. of the proof Unbounded We estimate (SSH) by a similar amplitude equation as in the domains (work in progress) deterministic case: 27 α 2 − 1) A | A | 2 + 3( α 2 − 1 2 ) σ 2 A ) dT + 2 ασ Ad ˜ dA = ( ν A + 3( 38 β . (AE)

Result on bounded domains Amplitude Theorem Equation for stoch. SH Equation Let T 0 > 0 , α ∈ ❘ with α 2 < 27 38 and 0 < κ . Let u be a mild Konrad Klepel solution of (SSH) with � u (0) � ∞ = O ( ε 1 − κ ) . Let A ( T ) ∈ C , T ∈ [0 , T 0 ] solve (AE), then ∀ p ∈ ◆ : ∃ C p such that Introduction Setting � x ) 2 u s (0) � ∞ > ε 2 − 19 κ � � u ( t ) − u A ( t ) − e − t (1+ ∂ 2 ≤ C p ε p Bounded sup P domains t ∈ [0 , T 0 ] Main theorem Differences to the deterministic case with the approximation Short Overview of the proof u A ( t , x ) = ε A ( ε 2 t ) e ix + ε ¯ A ( ε 2 t ) e − ix + ε Z ε ( ε 2 t ) Unbounded domains (work in progress) where Z ε is the Ornstein-Uhlenbeck process defined by � T e − ε − 2 ( T − s ) d ˜ Z ε ( T ) := ε − 1 σ β ( s ) . 0

Comparing the two Amplitude equations Amplitude Equation for stoch. SH Deterministic: Equation Konrad Klepel 27 α 2 − 1) | A | 2 A ∂ T A = rA + 3( 38 Introduction Setting With added noise: Bounded domains 27 α 2 − 1) A | A | 2 ) dT Main theorem dA = ( ν A + 3( 38 Differences to the deterministic case + 3( α 2 − 1 2 ) σ 2 A ) dT + 2 ασ Ad ˜ β . Short Overview of the proof Unbounded domains (work Where does the difference come from? in progress) ◮ Noise - Nonlinearity interaction, � � aZ 2 ◮ Averaging ( ε dt ≈ a σ dt ).

Idea of Proof / Rescaling Amplitude Equation for We rescale the the solutions of (SSH) to their natural stoch. SH Equation timescale: Konrad Klepel v ( T ) := ε − 1 u ( ε 2 T ) . Introduction Since we are on bounded domains we can write v as Setting � Bounded v k e ikx + e − T ε − 2 (1+ ∂ 2 x ) 2 v s (0) v = ae ix + ε Φ e i 2 x + c . c . + ε Ψ+ Z ε + domains Main theorem | k |≥ 3 Differences to the deterministic case Short Overview The mild solution of v k looks as follows of the proof Unbounded domains (work v k ( T ) = in progress) � T e − ε − 2 (1 − k 2 ) 2 ( T − s ) � � ν v k ( s ) + ε − 1 α ( � v 2 ) k ( s ) − ( � v 3 ) k ( s ) ds , 0

Idea of Proof / Reduction Amplitude Equation for stoch. SH With this we can show that until a stopping time τ ∗ Equation ( � v ( T ) � ∞ = O ( ε 0 − ) for T ∈ [0 , τ ∗ ]) we have Konrad Klepel � v − a − Z − e − T ε − 2 (1+ ∂ 2 x ) 2 v s (0) � ∞ = O ( ε 1 − ) Introduction Setting Bounded and by calculating ( � v 2 ) i and ( � v 3 ) i ( i ∈ 1 , 2 , 3) we get domains Main theorem Differences to a Φ + 2 α a Ψ − 3 a | a | 2 − 3 aZ 2 ε + ε − 1 2 α aZ ε + R 1 ) dT the deterministic da = ( ν a + 2 α ¯ case Short Overview d Φ = ( − 9 ε − 2 Φ + ε − 2 α a 2 + R 2 ) dT of the proof Unbounded d Ψ = ( − ε − 2 Ψ + ε − 2 α | a | 2 + ε − 2 α Z 2 domains (work ε + R 3 ) dT in progress) We exchange the a φ , a ψ and eps − 1 2 α aZ ε terms by applying Itˆ o differentiation on these terms.

The rest of the proof Amplitude Equation for stoch. SH Equation Konrad Klepel The rest of the proof consists of three parts: Introduction � � Setting aZ 2 ◮ Show that ε dt ≈ a σ dt (Averaging Lemma) Bounded domains ◮ Show that a is approximately A Main theorem Differences to ◮ Show that the stopping time τ ∗ is long enough (i.e. bigger the deterministic case Short Overview then a fixed time independent of ε ) of the proof Unbounded domains (work in progress)

The SSH equation on unbounded domains Amplitude Equation for stoch. SH Equation Konrad Klepel The solution to the amplitude equation A ( T ) has values in the Introduction Sobolev space H α , α > 1 / 2 defined by Setting H α := { u ∈ L 2 ( ❘ ; ❈ ) : F − 1 ((1 + k 2 ) α/ 2 F u ) ∈ L 2 ( ❘ ; ❈ ) } . Bounded domains Main theorem Differences to The solution to (SSH) is approximated by the deterministic case Short Overview of the proof u ( t , x ) ≈ ε A ( ε 2 t , ε x ) e ix + ε ¯ A ( ε 2 t , ε x ) e − ix + Z ε Unbounded domains (work in progress)

Problems on unbounded domains Amplitude Equation for stoch. SH Equation Konrad Klepel ◮ No Fourier series, but Fourier transform with bands of Introduction Eigenvalues. Setting ◮ SDEs for the modes feature a full linear operator instead Bounded domains of a scalar, which makes the exchanging of mixed products Main theorem Differences to ( a Φ , a Ψ , .. ) much more difficult. the deterministic case ◮ Bounds still depend a lot on A being in H 1 / 2+ which Short Overview of the proof Unbounded prohibits more general noise (which is at most H 1 / 2 − ). domains (work in progress)

Thank you. Amplitude Equation for stoch. SH Equation Konrad Klepel Introduction Setting Thank you for your attention! Bounded domains Main theorem Differences to the deterministic case Short Overview of the proof Unbounded domains (work in progress)