Averaging approximation to singularly perturbed nonlinear stochastic wave equations
aa r X i v : . [ m a t h . A P ] J u l Averaging approximation to singularlyperturbed nonlinear stochastic wave equations
Yan Lv ∗ A. J. Roberts † December 8, 2017
Abstract
An averaging method is applied to derive effective approximationto the following singularly perturbed nonlinear stochastic dampedwave equation νu tt + u t = ∆ u + f ( u ) + ν α ˙ W on an open bounded domain D ⊂ R n , 1 ≤ n ≤ ν > ν α ,0 ≤ α ≤ / ν , u t = ∆ u + f ( u ) + ν α ˙ W to an error of o (cid:0) ν α (cid:1) . Keywords stochastic nonlinear wave equations, averaging, tightness, mar-tingale.
Mathematics Subject Classifications (2000) ∗ School of Science, Nanjing University of Science & Technology, Nanjing, 210094,
China . mailto:[email protected] † School of Mathematics, University of Adelaide, South Australia,
Australia . mailto:[email protected] Introduction
Wave motion is one of the most commonly observed physical phenomena,and typically described by hyperbolic partial differential equations. Non-linear wave equations also have been studied a great deal in many modernproblems such as sonic booms, bottlenecks in traffic flows, nonlinear opticsand quantum field theory [13, 20, e.g.]. However, for many problems, such aswave propagation through the atmosphere or the ocean, the presence of tur-bulence causes random fluctuations. More realistic models must account forsuch random fluctuations. Hence we study stochastic wave equations [7, 9,e.g.].Here we study an effective approximation, in the sense of distribution, forthe following nonlinear wave stochastic partial differential equation ( spde ).The spde is a singularly perturbed problem on a bounded open domain D ⊂ R n , 1 ≤ n ≤ νu νtt + u νt = ∆ u ν + f ( u ν ) + ν α ˙ W , u ν (0) = u , u νt (0) = u , (1)with zero Dirichlet boundary on D . Here ν α with 0 < ν ≤ ≤ α ≤ / R n . Thenoise W ( t ) is an infinite dimensional Q-Wiener process which is detailed insection 2. The spde (1) also describes the motion of a small particle withmass ν and an infinite number of degrees of freedom [4, 5]. We are concernedwith the effective approximation of the solution to the spde (1) for small ν > α = 1 / spde (1) as ν → spde (1) are approximated by those of the deterministic pde u t = ∆ u + f ( u ) (2)as ν → α = 0 , which is an infinite dimensional version ofthe Smolukowski–Kramers approximation, is studied by analysing the struc-ture of solution of linear stochastic wave equations [4, 5]. For any T > u ( t ) to the spde (1) is approximated in probability by that ofthe stochastic system u t = ∆ u + f ( u ) + ˙ W as ν → C (0 , T ; L ( D )).Here we extend the approximating result to the case when 0 ≤ α ≤ / spde s [6, 19] in the following form u νt = ∆ u ν + f ( u ν , v ν ) + σ ˙ W ,v νt = 1 ν [∆ v ν + g ( u ν , v ν )] + σ √ ν ˙ W , where f and g are nonlinear terms, σ and σ are some constants, and W and W are Wiener processes. Notice that upon introducing v ν = u νt , the spde (1) is rewritten as u νt = v ν , u ν (0) = u ,v νt = 1 ν [ − v ν + ∆ u ν + f ( u ν )] + 1 ν − α ˙ W , v ν (0) = u , which are also in the form of slow-fast spde s. Then we can follow the stochas-tic averaging approach to derive an effective averaging approximation of u ν ,the solution of spde (1) as ν → ≤ α ≤ / α = 1 / α ∈ [0 , /
2] canbe transformed to the case α = 1 / ν > ≤ α ≤ / spde (1) is approximatedin the sense of distribution by ¯ u ν which solves¯ u νt = ∆¯ u ν + f (¯ u ν ) + ν α ˙¯ W , ¯ u ν (0) = u , (3)where ¯ W ( t ) is a Wiener process distributes same as W ( t ). This result showsthat for any small ν > ≤ α ≤ / νu νt ( t ) is a higher orderterm than the random force term ν α W ( t ).Section 3 gives the approximation for the important case that α = 1 / ν = 0 , there isa small fluctuation which distributes same as √ νW ( t ), see (3). This gives amore effective approximation.Section 6 explores a parameter regime where a nonlinear coordinate trans-formation underlies the existence of a stochastic slow manifold for the case α = 0. The stochastic slow manifolds of both the spde (1) and the model (3)have the same evolution in the parameter regime and so provide evidence ofthe stronger result of pathwise approximation therein. Let D ⊂ R n , 1 ≤ n ≤ L ( D ) the Lebesgue space of square integrable real valued functions on D ,3hich is a Hilbert space with inner product h u, v i = Z D u ( x ) v ( x ) dx , u, v ∈ L ( D ) . Write the norm on L ( D ) by k u k = h u, u i / . Define the following abstractoperator Au = − ∆ u , u ∈ Dom( A ) = { u ∈ L ( D ) : Au ∈ L ( D ) , u | Γ = 0 } . Denote by { λ k } the eigenvalues of A with 0 < λ ≤ λ ≤ · · · ≤ λ k ≤ · · · , λ k → ∞ as k → ∞ . For any s ∈ R , introduce the space H s ( D ) = Dom( A s/ )endowed with the norm k u k s = k A s/ u k , u ∈ H s ( D ) . Consider the following singularly perturbed stochastic wave equation withcubic nonlinearity on D : νu νtt + u νt = ∆ u ν + βu ν − ( u ν ) + ν α ˙ W ( t ) , (4) u ν (0) = u , u νt (0) = u , (5) u ν | Γ = 0 , (6)with 0 < ν < ≤ α ≤ / { W ( t ) } t ∈ R is an L ( D )-valued twosided Wiener process defined on a complete probability space (Ω , F , {F t } t ≥ , P )with covariance operator Q such that Qe k = b k e k , k = 1 , , . . . , where { e k } is a complete orthonormal system in L ( D ), b k is a boundedsequence of non-negative real numbers. Then W ( t ) = ∞ X k =1 p b k e k w k ( t ) , where w k are real mutually independent Brownian motions [12]. Further, weassume B = ∞ X k =1 b k < ∞ and B = ∞ X k =1 λ k b k < ∞ . (7)Then by a standard method [8], for any ( u , u ) ∈ H s +10 ( D ) × H s ( D ), s ∈ R ,there is a unique solution u ν to (4)–(6), u ν ∈ L (Ω , C (0 , T ; H s +10 ( D ))) , (8) u νt ∈ L (Ω , C (0 , T ; H s ( D ))) . (9)In the following we write f ( u ) = βu − u and F ( u ) = R u f ( r ) dr .For our purpose we need the following lemma.4 emma 1 (Simon [17]) . Assume E , E and E be Banach spaces such that E ⋐ E , the interpolation space ( E , E ) θ, ⊂ E with θ ∈ (0 , and E ⊂ E with ⊂ and ⋐ denoting continuous and compact embedding respectively.Suppose p , p ∈ [1 , ∞ ] and T > , such that V is a bounded set in L p (0 , T ; E ) , and ∂ V := { ∂v : v ∈ V} is a bounded set in L p (0 , T ; E ) . Here ∂ denotes the distributional derivative. If − θ > /p θ with p θ = 1 − θp + θp , then V is relatively compact in C (0 , T ; E ) . In the following, for any
T > C T a generic positiveconstant which is independent of ν . α = 1 / We first consider the special case of α = 1 / spde s: du ν = v ν dt , u ν (0) = u , (10) dv ν = − ν [ v ν − ∆ u ν − f ( u ν )] dt + 1 √ ν dW ( t ) , v ν (0) = u . (11)Notice that the slow part u ν and fast part v ν are linearly coupled. Forsimplicity we consider ( u , u ) ∈ ( H ( D ) ∩ H ( D )) × H ( D ). Then (4)–(6)has a unique solution in L (Ω , C (0 , T ; ( H ( D ) ∩ H ( D )) × H ( D ))). Let ( u ν , v ν ) be a solution to (10)–(11) with ν > ν → Theorem 1.
Assume B < ∞ . For any T > , there is a positive con-stant C T such that E (cid:20) max ≤ t ≤ T k u ν ( t ) k + max ≤ t ≤ T k v ν ( t ) k (cid:21) ≤ C T , (12)5 nd for any integer m > E Z T k u ν ( t ) k m dt ≤ C T . Proof.
The result on k u ν ( t ) k is found by a simple energy estimate [18]. Nowwe give the estimate on k v ν ( t ) k . By equation (11), v ν ( t ) = e − t/ν u + 1 ν Z t e − ( t − s ) /ν [∆ u ν ( s ) + f ( u ν ( s ))] ds + 1 √ ν Z t e − ( t − s ) /ν dW ( s ) . Noticing assumption (7), by the estimate on k u ν ( t ) k and maximal inequalityof stochastic convolution [12, Lemma 7.2], E (cid:20) max ≤ t ≤ T k v ν ( t ) k (cid:21) ≤ C T for some positive constant C T . The last inequality of the theorem is obtainedby the same method [18] and Poincar´e inequality. This completes the proof.Now by the above estimates and Lemma 1, we have the following theorem. Theorem 2.
For any
T > , {L ( u ν ) } <ν ≤ the distribution of u ν is tight inthe space C (0 , T ; H ( D )) . By the above tightness result, to determine the limit of u ν we can passto the limit ν → ϕ ∈ C ∞ ( D ), we considerthe limit of u ν,ϕ ( t ) = h u ν ( t ) , ϕ i in the space C (0 , T ) as ν → u ν,ϕ in C (0 , T ) Now we pass to the limit ν → { u ν,ϕ } in the space C (0 , T ) for any T > du ν,ϕ = v ν,ϕ dt , (13) dv ν,ϕ = − ν [ v ν,ϕ + h∇ u ν , ∇ ϕ i − h f ( u ν ) , ϕ i ] dt + 1 √ ν dW ϕ ( t ) , (14)with u ν,ϕ (0) = h u , ϕ i and v ν,ϕ (0) = h u , ϕ i where v ν,ϕ = h v ν , ϕ i and W ϕ ( t ) = h W ( t ) , ϕ i . In the following we also write v ν,ϕ as v ν,ϕ,u ( t ) which shows thedependence of v ν,ϕ on the slow part u ν .6econd, for any fixed u ∈ H ( D ) ∩ H ( D ) we consider the fast equation dv ν,u = − ν [ v ν,u − ∆ u − f ( u )] dt + 1 √ ν dW ( t ) . (15)Equation (15) has a unique stationary solution ¯ v ν,u . Moreover, the stationarysolution ¯ v ν,u is exponentially mixing and the distribution of ¯ v ν,u is the normaldistribution N (∆ u + f ( u ) , Q/
2) [4].Now for any u ∈ H ( D ) ∩ H ( D ) define H ν ( u, t ) = ν [ v ν,u ( t ) − v ν,u (0)] + Z t [ v ν,u ( s ) − ∆ u − f ( u )] ds . Then u ν,ϕ solves the following equation u ν,ϕ ( t ) = h u , ϕ i − Z t [ h∇ u ν ( s ) , ∇ ϕ i − h f ( u ν ( s )) , ϕ i ] ds + h H ν ( u ν ( t ) , t ) , ϕ i − ν h v ν,u ( t ) − v ν,u (0) , ϕ i . (16)Third, we study the behaviour of h H ν ( u ν ( t ) , t ) , ϕ i for small ν . Let H ν,ϕ ( u, t ) = h H ν ( u, t ) , ϕ i , then define M ν,ϕt = 1 √ ν H ν,ϕ ( u ν ( t ) , t ) . (17)By the definition of H ν,ϕ ( u, t ) and equation (15), M ν,ϕt is a martingale withrespect to {F t : t ≥ } , and the quadratic covariance is h M ν,ϕ i t = t h Qϕ, ϕ i .Now define R ν,ϕ ( t ) = − h v ν,u ( t ) − v ν,u (0) , ϕ i , then rewrite (16) as u ν,ϕ ( t ) = h u , ϕ i− Z t [ h∇ u ν ( s ) , ∇ ϕ i − h f ( u ν ( s )) , ϕ i ] ds + √ νM ν,ϕt + νR ν,ϕ ( t ) . (18)Invoking Theorem 1,lim ν → E (cid:20) max ≤ t ≤ T √ ν | R ν,ϕ ( t ) | (cid:21) = 0 . (19)Then define the process M ν,ϕt = 1 √ ν (cid:26) u ν,ϕ ( t ) − h u , ϕ i + Z t (cid:2) h∇ u ν ( s ) , ∇ ϕ i − h f ( u ν ( s )) , ϕ i (cid:3) ds (cid:27) . (20)7y the definition of H ν,ϕ ( u, t ) and (19) we have the tightness of M ν,ϕt inspace C (0 , T ) for any T > P be a limit point of the family of prob-ability measures {L ( M ν,ϕt ) } <ν ≤ and denote by M ϕt , a C (0 , T )-valued ran-dom variable with distribution P . Let Ψ be a continuous bounded functionon C (0 , T ). Set Ψ ν ( s ) = Ψ( u ν,ϕ ( s )), then noticing (19), E [( M ν,ϕt − M ν,ϕs )Ψ ν ( s )] = E (cid:2) √ ν ( R ν,ϕ ( t ) − R ν,ϕ ( s ))Ψ ν ( s ) (cid:3) → , ν → , which yields that the process {M ϕt } ≤ t ≤ T is a P -martingale with respect tothe Borel σ -filter of C (0 , T ).We consider the quadratic covariation of the martingale M ϕt . By thedefinition of M ν,ϕt , passing to the limit ν → M ϕt is asquare integrable martingale with the associated quadratic covariation pro-cess is h Qϕ, ϕ i t . Then by the representation theorem for martingales [10],without changing the distributions of M ν,ϕt and M ϕt , one extends the origi-nal probability space (Ω , F , P ) and chooses a new Wiener process ˆ W ϕ ( t ) suchthat M ϕt = √ Q ˆ W ϕ ( t ), which is unique in the sense of distribution.By the definition of M ν,ϕt , ˆ W ϕ can be chosen as h ˆ W , ϕ i where ˆ W is acylindrical Wiener process. Then from (20) we have in the sense of distribu-tion h u ν ( t ) , ϕ i = h u , ϕ i − Z t [ h∇ u ν ( s ) , ∇ ϕ i − h f ( u ν ( s )) , ϕ i ] ds + √ ν M ϕt + o (cid:0) √ ν (cid:1) = h u , ϕ i − Z t [ h∇ u ν ( s ) , ∇ ϕ i − h f ( u ν ( s )) , ϕ i ] ds + √ ν p Q h ˆ W , ϕ i + o (cid:0) √ ν (cid:1) for any ϕ ∈ C ∞ ( D ). Then by discarding the higher order term and thetightness of u ν , we have the following approximating equation d ¯ u ν = [∆¯ u ν + f (¯ u ν )] dt + √ ν d ¯ W Q , (21)where ¯ W Q is some an L ( D ) valued Q-Wiener process. Theorem 3.
Assume B < ∞ and α = 1 / . For small ν > , thereis a new probability space ( ¯Ω , ¯ F , ¯ P ) , an extension of the original probabil-ity space (Ω , F , P ) , such that for any T > , the solution u ν to (10)–(11) is approximated by ¯ u ν which solves (21), to an error of o (cid:0) √ ν (cid:1) , in thespace C (0 , T ; H ( D )) for almost all ω ∈ ¯Ω . The above spde (21) is more effective than the limit pde (2) [11] as itincorporates fluctuations for small ν > νu νt ( t ) is a higher order term than √ νW ( t ) for small ν > νu νt ( t )is always a higher order term than ν α W ( t ) for any 0 ≤ α ≤ / The case of α = 0 Next we consider the case of α = 0 ; that is, consider the following spde νu νtt + u νt = ∆ u ν + βu ν − ( u ν ) + ˙ W ( t ) , (22) u ν (0) = u , u νt (0) = u , (23) u ν | Γ = 0 . (24)First we have the following a priori estimates on u ν in the space C (0 , T ; H ( D )). Theorem 4 (Cerrai & Freidlin [5]) . Assume B < ∞ . For any T > , thereis a positive constant C T such that E (cid:20) max ≤ t ≤ T k u ν ( t ) k (cid:21) ≤ C T . We follow the approach for the case of α = 1 / u ν = √ νu ν and ˜ v ν = √ νu νt . Then d ˜ u ν = ˜ v ν dt , ˜ u ν (0) = √ νu ,d ˜ v ν = − ν (cid:20) ˜ v ν − ∆˜ u ν − √ νf (cid:18) ˜ u ν √ ν (cid:19)(cid:21) dt + 1 √ ν dW ( t ) , ˜ v ν (0) = √ νu . By standard energy estimates [18], by a similar discussion to that in Section 3,and by Theorem 4, we have the following theorem.
Theorem 5.
Assume B < ∞ . For any T > , there is a positive con-stant C T such that E (cid:20) max ≤ t ≤ T k ˜ u ν ( t ) k + max ≤ t ≤ T k ˜ v ν ( t ) k (cid:21) ≤ C T , and for any integer m > E Z T k ˜ u ν ( t ) k m dt ≤ C T . Moreover, the distribution of ˜ u ν is tight in space C (0 , T ; H ( D )) . We consider the asymptotic approximation of ˜ u ν for small ν > ϕ ∈ C ∞ ( D ), let ˜ u ν,ϕ = h ˜ u ν , ϕ i , ˜ v ν,ϕ = h ˜ v ν , ϕ i and W ϕ ( t ) = h W ( t ) , ϕ i .Then d ˜ u ν,ϕ = ˜ v ν,ϕ dt ,d ˜ v ν,ϕ = − ν (cid:2) ˜ v ν,ϕ + h∇ ˜ u ν , ∇ ϕ i − √ ν h f (˜ u ν / √ ν ) , ϕ i (cid:3) dt + 1 √ ν dW ϕ ( t ) , u ν,ϕ (0) = h ˜ u ν (0) , ϕ i and ˜ v ν,ϕ (0) = h ˜ v ν (0) , ϕ i .We also consider the following fast spde for fixed ν and ˜ u ∈ H ( D ) ∩ H ( D ): d ˜ v ν, ˜ u = − ν (cid:2) ˜ v ν, ˜ u − ∆˜ u − √ νf (cid:0) ˜ u/ √ ν (cid:1)(cid:3) dt + 1 √ ν dW ( t ) . (25)For fixed ν ∈ (0 ,
1] and ˜ u ∈ H ( D ) ∩ H ( D ), spde (25) has a unique station-ary solution with the normal distribution N (∆˜ u + √ νf (˜ u/ √ ν ) , Q/
2) [4].Now for any ˜ u ∈ H ( D ) ∩ H ( D ) define e H ν (˜ u, t ) = ν (cid:2) ˜ v ν, ˜ u ( t ) − ˜ v ν, ˜ u (0) (cid:3) + Z t (cid:2) ˜ v ν, ˜ u ( s ) − ∆˜ u − √ νf (˜ u/ √ ν ) (cid:3) ds . Thus we can follow the same discussion in last section for the case of α = 1 / u ν,ϕ ( t ) = √ ν h u , ϕ i − Z t h∇ ˜ u ν ( s ) , ∇ ϕ i ds + √ ν Z t h f (cid:0) ˜ u ν ( s ) / √ ν (cid:1) , ϕ i ds + √ ν ˜ M ν,ϕt , (26)where √ ν ˜ M ν,ϕt is the remainder term. By a similar discussion to that of thelast section, ˜ M ν,ϕt is tight in space C (0 , T ) for any T > P be a limitpoint of the family of probability measures L{ ˜ M ν,ϕt } <ν ≤ in space C (0 , T ).Let ˜ M ϕt be a C (0 , T )-valued random variable with distribution ˜ P . Then wehave the following lemma. Lemma 2.
For any ϕ ∈ C ∞ ( D ) , the process ˜ M ϕt defined on the probabil-ity space ( C (0 , T ) , B ( C (0 , T )) , ˜ P ) is a square integrable martingale with theassociated quadratic covariation process h Qϕ, ϕ i t . By the representation theorem for martingales [10], without changingthe distributions of ˜ M ν,ϕt and ˜ M ϕt one can extend the original probabilityspace (Ω , F , P ) and choose a new cylindrical Wiener process ˜ W ( t ) such that˜ M ϕt = √ Q h ˜ W , ϕ i , which is unique in the sense of distribution.Then in the sense of distribution by (26) we write out h ˜ u ν ( t ) , ϕ i = √ ν h u , ϕ i − Z t h∇ ˜ u ν ( s ) , ∇ ϕ i ds + √ ν Z t h f (cid:0) ˜ u ν ( s ) / √ ν (cid:1) , ϕ i ds + √ ν ˜ M ϕt + o (cid:0) √ ν (cid:1) = √ ν h u , ϕ i − Z t h∇ ˜ u ν ( s ) , ∇ ϕ i ds + √ ν Z t h f (cid:0) ˜ u ν ( s ) / √ ν (cid:1) , ϕ i ds + √ ν p Q h ˜ W , ϕ i + o (cid:0) √ ν (cid:1) (27)10or any ϕ ∈ C ∞ ( D ). Then we have, noticing that ˜ u ν = √ νu ν , the followingapproximating spde for small ν > d ¯ u ν = [∆¯ u ν + f (¯ u ν )] dt + d ¯ W Q , ¯ u ν (0) = u , (28)where ¯ W Q is some L ( D ) valued Q-Wiener process. Then we infer the fol-lowing result. Theorem 6.
Assume B < ∞ and α = 0 . Then for small ν > , thereis a new probability space ( ¯Ω , ¯ F , ¯ P ) which is an extension of the originalprobability space (Ω , F , P ) such that for any T > , the solution u ν to (22)–(24) is approximated by ¯ u ν which solves (28), to an error of o (cid:0) (cid:1) , in thespace C (0 , T ; H ( D )) for almost all ω ∈ ¯Ω . < α < / Now we consider the case of 0 < α < / spde νu νtt + u νt = ∆ u ν + βu ν − ( u ν ) + ν α ˙ W ( t ) , (29) u ν (0) = u , u νt (0) = u , (30) u ν | Γ = 0 . (31)First, by the same analysis as Theorem 4, we also have the followingresult on the a priori estimates on u ν . Theorem 7 (Cerrai & Freidlin [5]) . Assume B < ∞ . For any T > , thereis a positive constant C T such that E (cid:20) max ≤ t ≤ T k u ν ( t ) k (cid:21) ≤ C T . We also apply the method in Section 3. Make the following scaling trans-formation ˜ u ν = ν / − α u ν and ˜ v ν = ν / − α v ν . Then d ˜ u ν = ˜ v ν dt ,d ˜ v ν = − ν (cid:20) ˜ v ν − ∆˜ u ν − ν / − α f (cid:18) ˜ u ν ν / − α (cid:19)(cid:21) dt + 1 √ ν dW ( t ) , ˜ u ν (0) = ν / − α u , ˜ v ν (0) = ν / − α u . By a direct energy estimate or the scaling transformation and Theorem 7 wededuce the following theorem. 11 heorem 8.
Assume B < ∞ . For any T > , there is a positive con-stant C T such that E (cid:20) max ≤ t ≤ T k ˜ u ν ( t ) k + max ≤ t ≤ T k ˜ v ν ( t ) k (cid:21) ≤ C T , and for any integer m > E Z T k ˜ u ν ( t ) k m dt ≤ C T . Moreover, the distribution of ˜ u ν is tight in space C (0 , T ; H ( D )) . Then we can follow the same discussion of Section 4 and have the followingresult.
Theorem 9.
Assume B < ∞ and < α < / . For small ν > , thereis a new probability space ( ¯Ω , ¯ F , ¯ P ) which is an extension of the originalprobability space (Ω , F , P ) such that for any T > , the solution u ν to (29)–(31) is approximated by ¯ u ν which solves d ¯ u ν = [∆¯ u ν + f (¯ u ν )] dt + ν α d ¯ W Q , ¯ u ν (0) = u , (32) to an error of o (cid:0) ν α (cid:1) , in the space C (0 , T ; H ( D )) for almost all ω ∈ ¯Ω . α = 0 This section shows the long time effectiveness of the averaged model by com-paring it to the original via their stochastic slow manifolds.We compare the spde (22) and its model spde (28) in a parameterregime where both have an accessible stochastic slow manifold. Considerthe spde (22) restricted to one spatial dimension as νu tt + u t = u xx + f ( u ) + σ ˙ W where f = (1 + β ′ ) u − u . (33)Consider this spde on the non-dimensional domain D = (0 , π ) with bound-ary conditions u = 0 on x = 0 , π . The parameter σ here explicitly measuresthe overall size of the Q-Wiener process W ( t ) which by (7) is finite. Thesmall parameter β ′ measures the distance from the stochastic bifurcationthat occurs near β ′ = 0 . In this domain there will be a stochastic slow man-ifold of the spde (33) that matches the slow dynamics in the approximating spde (28). This section compares the stochastic slow manifolds.12he spde (33) has a technically challenging spectrum. However, the con-struction of its stochastic slow manifold is easiest by embedding the spde (33)as the γ = 1 case of the following slow-fast system of spde s u t = u xx + u + v , (34) νv t = − v − γν ( ∂ xx + 1) u t + β ′ u − u + σ ˙ W . (35)The parameter γ controls the homotopy: from a tractable base when γ = 0as then all linear modes in the very fast v equation (35) decay at the samerate 1 /ν (and the slow u modes of sin kx have decay rates 1 − k ); to theoriginal spde (33) when γ = 1 (upon eliminating v ). A stochastic slow manifold appears
On the non-dimensional inter-val (0 , π ), with Dirichlet boundary conditions on u , the eigenmodes mustbe proportional to sin kx for integer wavenumber k . Neglecting noise tem-porarily, σ = 0 in this sentence, for all ν < ≤ γ ≤ u, v ) ∝ (sin x,
0) (local in ( u, v, σ ), but global in ν and γ ).By stochastic center manifold theory [1, 3], and supported by stochastic nor-mal form transformations [2, 15, 16], when the noise spectrum truncates andthe nonlinearity is small enough, the dynamics of the spde s (34)–(35) areessentially finite dimensional and a stochastic slow manifold exists which isexponentially quickly attractive to all nearby trajectories. Computer algebra constructs the stochastic slow manifold
We seekthe stochastic slow manifold as a systematic perturbation of the slow sub-space u = a sin x . The intricate algebra necessary to handle the multitude ofnonlinear noise interactions is best left to a computer [14, 16, e.g.]. However,the following expressions may be checked by substituting into the govern-ing spde s (34)–(35) and confirming the order of the residuals is as small asquoted—albeit tedious, this check is considerably easier than the derivation.The evolution on the stochastic slow manifold may be written˙ a = β ′ a − a + (cid:2) − νβ ′ + νa − a (cid:3) b ˙ w + (cid:2) ( + β ′ ) a − a (cid:3) b ˙ w + a b ˙ w + o (cid:0) ν + β ′ + a , σ (cid:1) (36)The stochastic slow manifold itself involves Ornstein–Uhlenbeck processeswritten as convolutions over the past history of the noise processes: define13 − µt ⋆ ˙ w = R t −∞ exp[ − µ ( t − s )] dw s for decay rates µ k = k − k th mode. Then the stochastic slow manifold is u = a sin x + a sin 3 x − a (cid:2) b e − t ⋆w sin x + b e − t ⋆w sin 3 x (cid:3) + X k ≥ b k (cid:2) µ k ν + γν ( µ k − µ k e − µ k t ⋆ ) (cid:3) e − µ k t ⋆ ˙ w k sin kx − X k ≥ b k e − t/ν ⋆ ˙ w k sin kx + β ′ X k ≥ b k e − µ k t ⋆e − µ k t ⋆ ˙ w k sin kx + X k ≥ (cid:8) b k +2 e − µ k t ⋆e − µ k +2 t ⋆ ˙ w k +2 sin kx − b k e − µ k t ⋆e − µ k t ⋆ ˙ w k sin kx + b k e − µ k +2 t ⋆e − µ k t ⋆ ˙ w k sin[( k + 2) x ] (cid:9) + O (cid:0) ν + β ′ + a , σ (cid:1) , (37)and a correspondingly complicated expression for the field v ( x, t ). Observethat the slow sde (36) does not contain any fast time convolutions fromthe Ornstein–Uhlenbeck processes: it would be incongruous to have suchfast processes in a supposedly slow model. We keep fast time convolutionsout of the slow sde (36) by introducing carefully crafted terms in the slowmode sin x in the parametrization of the stochastic slow manifold (37): herethe amplitude of the slow mode sin x is approximately a − a b e − t ⋆w − b e − t/ν ⋆ ˙ w . Other methods which do not adjust the slow mode either averageover such adjustments and so are weak models, or invoke fast processes inthe slow model.Note that the homotopy parameter γ affects the stochastic slow manifoldshape (37), but only weakly. To this order the homotopy has no effect onthe evolution on the stochastic slow manifold (36). Compare with SPDE (28) The corresponding stochastic slow manifoldof the spde (28), in this parameter regime, is straightforward to construct,via the web server [16] for example. For stochastic slow manifold ¯ u ≈ ¯ a sin x one finds the corresponding slow sde ˙¯ a = β ′ ¯ a − ¯ a + (cid:2) − a (cid:3) ¯ b ˙¯ w + (cid:2) ( + β ′ )¯ a − a (cid:3) ¯ b ˙¯ w + a ¯ b ˙¯ w + o (cid:0) β ′ + ¯ a , σ (cid:1) . (38)This slow sde is symbolically identical with the sde (36), one just removesthe overbars. We conclude that these stochastic slow manifolds confirm themodeling of the spde (22) by its model spde (28); at least in the regime ofone space dimension with small amplitude a , bifurcation parameter β ′ , andfinite truncation to the noise. 14 cknowledgements The research was supported by the NSF of Chinagrant No. 10901083, Zijin star of Nanjing University of Science and Technol-ogy, and the ARC grant DP0988738.
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