Invariant foliations for stochastic partial differential equations with dynamic boundary conditions
aa r X i v : . [ m a t h . D S ] N ov Invariant foliations for stochastic partialdifferential equations with dynamic boundaryconditions
Zhongkai Guo
1. School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhan 430074, China [email protected]
March 25, 2019
Abstract
Invariant foliations are complicated random sets useful for describing andunderstanding the qualitative behaviors of nonlinear dynamical systems. Wewill consider invariant foliations for stochastic partial differential equationwith dynamical boundary condition.
Key Words:
Stochastic partial differential equation; invariant folia-tions; dynamic boundary; analytical approximations;
Invariant foliations are often used to study the qualitative properties of a flow orsemiflow nearby invariant sets. They are extremely useful because they can be usedto track the asymptotic behavior of solutions and to provide coordinates in whichsystems of differential equations may be decoupled and normal forms derived.Invariant foliations as well as invariant manifolds, provide geometric structuresfor understanding the qualitative behaviors of nonlinear dynamical systems. Thereare some works on invariant foliations for finite dimensional random dynamicalsystems by Arnold[1], Wanner[26]. For infinite dimensional, Lu and Schmalfuss[22]1onsidered the invariant foliation for stochastic partial differential equations, Chen,Duan and Zhang[10] concern the slow foliation for slow-fast stochastic evolutionarysystem, Sun, Kan and Duan[25] examined the approximation of invariant foliationfor stochastic dynamical systems. There are also some other papers about theexistence of invariant manifolds we recommend [12], [22], and so on.The intention of this article is to study the dynamics of the following class ofrandomly perturbed parabolic partial differential equations with dynamical bound-ary conditions. ∂u∂t − P nk,j =1 ∂ x k ( a kj ( x ) ∂ x j u ) + a ( x ) u = f ( u ) + ǫ ˙ W on D , ∂u∂t + P nk,j =1 ν k a kj ( x ) ∂ x j u + c ( x ) u = g ( u ) + ǫ ˙ W on ∂D ,u (0 , x ) = u ( x ) on D × ∂D , (1.1)where W , W are two independent Wiener process, 0 < ǫ ≪ ν = ( ν , · · · , ν n )is the outer normal to ∂D . Moreover, f and g are nonlinear terms, and a k,j , a , c are given coefficients.A simple example of system (1.1) is the following problem ∂u∂t − ∆ u = f ( u ) + ǫ ˙ W on D , ∂u∂t + ∂u∂ν = c ( x ) u + g ( u ) + ǫ ˙ W on ∂D ,u (0 , x ) = u ( x ) on D × ∂D . (1.2)Boundary conditions of this type are usually called stochastic dynamical boundarycondition, because on the boundary it involves the Itˆo differential of the unknownfunction u with respect to time. Deterministic parabolic systems with dynamicalboundary conditions arise in hydrodynamics and the heat transfer theory and werestudied by many authors, see [14] and [20].Boundary value problems with “classical” boundary conditions are frequentlyconverted into an abstract cauchy problem, see [16] or [19]. The abstract problemcan be seen as an evolution equation in suitable Banach space, driven by a givenoperator, and boundary values are necessary for defining the domain of this oper-ator. Here we concerned with nonclassical boundary conditions, similar problemsare already solved in the literature with different cases, see [3] and [8]. A differentapproach to evolution problem with dynamical boundary conditions were recentlyproposed in functional analysis, motivated by the study on matrix operator theory[17], it is useful to translate non-classical boundary value problems in an abstractsetting then the semigroup techniques are available. In this paper we also by thisapproach.Our main goal in this paper is to show the existence of invariant foliation forproblem (1.1) and study an approximation of it.This paper is organized as follows. In section 2 , we will review some basicconcepts of dynamical boundary problems, random dynamical systems, invariant2oliations. The result on the existence of invariant foliations for stochastic differen-tial equation with dynamical boundary condition is described in section 3 . In thesection 4 , we present an asymptotic approximation for random invariant foliationsof equation (1.1). In this section we are going to collect some results about the theory of parabolicpartial differential equation with dynamical boundary condition. For more detailswe refer to Amann and Escher[2], Chueshov and Schmalfuss[7]. Let D ⊂ R n be a bounded C ∞ -smooth domain with the boundary ∂D , we use W sp ( D ) and W sp ( ∂D ) , s > H s ( D ) := W s ( D ) , H s ( ∂D ) := W s ( ∂D ) and L p ,q ( D ) := L p ( D ) × L q ( ∂D ) , L p ( D ) := L p,p ( D ) , p , q ≥ , V := { ( u, γu ) ∈ H ( D ) × H ( ∂D ) : u = γu on ∂D } , where the γ denote the trace operator. For the norm of L p,q ( D ) , we set k u k L p,q := k u k L p ( D ) + k γu k L q ( ∂D ) , u = ( u, γu ) ∈ L p,q ( D ) , and similarly for the other spaces. we set H := L ( D ) . Denote by k · k and ( · , · )the norm and the inner product in H , then we have V is densely and compactlyembedded in H .Now we consider the differential operators A ( x, ∂ ) := − n X k,j =1 ∂ x k ( a kj ( x ) ∂ x j ) + a ( x )and B ( x, ∂ ) := n X k,j =1 ν k a kj ( x ) ∂ x j + c ( x ) , where ν = ( ν , · · · , ν n ) is the outer normal to ∂D . We assume that a kj ( x ) and a ( x ) are C ∞ ( ¯ D ) functions and a ( x ) > x ∈ ¯ D . Let the matrix a kj ( x ) | dk,j =1 be symmetric and uniformly positive definite. The function c ( x ) ispositive and belongs to C ( ∂D ) .We consider the continuous symmetric positive bilinear form on the space V . a ( U, V ) = d X k,j =1 Z D a kj ( x ) ∂ x k u ( x ) ∂ x j v ( x ) dx + Z D a ( x ) u ( x ) v ( x ) dx + Z ∂D c ( s ) γu ( s ) γv ( s ) ds , U = ( u , γu ) , V = ( v , γv ) ∈ V . Following the Lax-Milgram theory thisbilinear form generates a positive self adjoint operator A in H . By the Greenformula this operator A is related to the pair ( A ( x, ∂ ) , B ( x, ∂ )) . Then there existsan orthonormal basis { E k } k ∈ N in H such that AE k = λ k E k , k = 1 , , · · · , < λ ≤ λ · · · , lim k →∞ λ k = ∞ , where E k is the eigenfunctions of A has the form E k = ( e k ; γ [ e k ]) , k = 1 , , , · · · ,where e k ∈ C ∞ ( ¯ D ) . and eigenvalues of A have a finite multiplicity.For the domain of definition of A , D ( A ) ⊂ H ( D ) × H ( ∂D ) , we conclude that − A is the generator of a C -semigroup ( S ( t )) t ∈ R + . More detailof A see [2].By the above analysis, the abstract form of the systems (1.1) is d ˆ X + A ˆ Xdt = F ( ˆ X ) dt + ǫdW , ˆ X = ˆ X (0) ∈ H , (2.1)where ˆ X = (cid:18) uγu (cid:19) . We will recall some basic concepts in random dynamical systems in this section(see [12]). Let (Ω , F , P ) be a probability space. A flow θ of mappings { θ t } t ∈ R isdefined on the sample space Ω such that θ : R × Ω → Ω , θ = id , θ t θ t = θ t + t . for t , t ∈ R . This flow is supposed to be ( B ( R ) ⊗ F , F )-measurable, where B ( R )is the σ -algebra of Borel sets on the real line R . To have this measurability, itis not allowed to replace F by its P -completion F P (see Arnold [1] p. 547). Inaddition, the measure P is assumed to be ergodic with respect to { θ t } t ∈ R . Then(Ω , F , P , R , θ ) is called a metric dynamical system.For our applications, we will consider a special but very important metric dy-namical system induced by the Brownian motion. Let W ( t ) be a two-sided Wienerprocess with trajectories in the space C ( R , H ) of real continuous functions definedon R , taking zero value at t = 0 . This set is equipped with the compact opentopology. On this set we consider the measurable flow θ = { θ t } t ∈ R , defined by θ t ω = ω ( · + t ) − ω ( t ) , ω ∈ Ω , t ∈ R .
The distribution of this process generates a measure on B ( C ( R , H )) which is calledthe Wiener measure. Note that this measure is ergodic with respect to the above4ow, (see the Appendix in Arnold [1]). Later on we will consider, instead of thewhole C ( R , H ) , a { θ t } t ∈ R -invariant subset Ω ⊂ C ( R , H )) of P -measure one andthe trace σ -algebra F of B ( C ( R , H )) with respect to Ω . A set Ω is called { θ t } t ∈ R invariant if θ t Ω = Ω for t ∈ R . On F , we consider the restriction of the Wienermeasure also denoted by P .The dynamics of the system on the state space H over the flow θ is described bya cocycle. For our applications it is sufficient to assume that ( H, d ) is a completemetric space. A cocycle φ is a mapping: φ : R + × Ω × H → H, which is ( B ( R ) ⊗ F ⊗ B ( H ) , F )-measurable such that φ (0 , ω , x ) = x ∈ H ,φ ( t + t , ω , x ) = φ ( t , θ t ω , φ ( t , ω , x )) , for t , t ∈ R + , ω ∈ Ω and x ∈ H . Then φ , together with the metric dynamicalsystem θ , forms a random dynamical system. Invariant foliation is about quantifying certain sets called fibers or leaves instate space. A fiber consists of all points starting from which the dynamical orbitshave similar asymptotic behavior. These fibers are thus building blocks for under-standing dynamics. The definitions of stable fiber and unstable fiber are as follows(see[10]).Let H be a state space, and ψ ( · , · , · ) : R × Ω × H → H be a random dynamicalsystem.1. We say that W βs ( x, ω ) is a β − stablef iber passing through x ∈ H with β ∈ R − , if k ψ ( t, ω, x ) − ψ ( t, ω, e x ) k = O( e βt ) , ∀ ω ∈ Ω , as t → + ∞ for any x, e x ∈ W βs .2. We say that W βu ( x, ω ) is a β − unstablef iber passing through x ∈ H with β ∈ R + , if k ψ ( t, ω, x ) − ψ ( t, ω, e x ) k = O( e βt ) , ∀ ω ∈ Ω , as t → −∞ for any x, e x ∈ W βu .We say a foliation is invariant if it satisfy φ ( t, ω, W β ( x, ω )) ⊂ W β ( φ ( t, ω, x ) , θ t ω ) . Now we consider the existence of invariant foliation for the following equations ∂u∂t − P nk,j =1 ∂ x k ( a kj ( x ) ∂ x j u ) + a ( x ) u = f ( u ) + ǫ ˙ W on D , ∂u∂t + P nk,j =1 ν k a kj ( x ) ∂ x j u + c ( x ) u = g ( u ) + ǫ ˙ W on ∂D,u (0 , x ) = u ( x ) on D × ∂D, f, g are Lipschitz continuous function.Take the abstract form of above systems, we obtain d ˆ X + A ˆ Xdt = F ( ˆ X ) dt + ǫdW, ˆ X = ˆ X (0) ∈ H . Where ˆ X ( t ) = ( u ( t ) , γu ( t )) T , F ( ˆ X ) = ( f ( u ) , g ( γu )) T , W = ( W , W ) T ,ˆ X (0) = ( u (0) , γu (0)) T . By the Lipschitz continuous of f, g we can easily obtain F is also Lipschitz continuous. Denote the Lipschitz constant by L F .Consider the linear stochastic evolution equation dZ ǫ ( t ) + AZ ǫ ( t ) = ǫdW ( t ) . Let X ( t ) = ˆ X ( t ) − Z ǫ ( t ) then we have dX + AXdt = F ( X + Z ǫ ) dt . (3.1)In the following, we assume that Hypothesis H1
Projection operators of the linear (unbounded) operator A sat-isfies Lemma3.1. Hypothesis H2
There is constant 0 ≤ α < F : D ( A α ) → H . In addition, F satisfies the following condition k F ( u ) − F ( v ) k H ≤ L F k u − v k D ( A α ) . Take H be the space spanned by the eigenfunctions related to the first N eigenvalues of the positive symmetric operator A , we denote π as the orthogonalprojection relate to H . Similarly, we can describe the infinite dimensional space H given by the span of the eigenfunction of λ N +1 , · · · , the related projection isdenoted by π . Then we have H = H ⊕ H . For simple, we define follow notation. π A = A , π A = A ; π F = F , π F = F ; π X = X , π X = X ;Then equation (3.1) can be rewritten as (cid:26) dX = − A X + F ( X , X , θ ǫt ω ) in H ,dX = − A X + F ( X , X , θ ǫt ω ) in H . (3.2)Define e − At X = ∞ X i =1 e − λ i t ( X, E i ) E i , E i = ( e i ; γ [ e i ])and k e − At X k L ( H ,D ( A α )) = (cid:18) ∞ X i =1 e − λ i t ( X, E i ) λ αi (cid:19) , then we have the following estimate for the semigroup S ( t ) .6 emma 3.1. ([11]) Let α ∈ [0 , be a constant. Then1. for t > k e − A t k L ( H ,D ( A α )) ≤ (cid:18) αt α + λ αN +1 (cid:19) e − λ N +1 t .
2. for t ≤ k e − A t k L ( H ,D ( A α )) ≤ λ αN e − λ N t . Define a Banach space for a fixed β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) , C i, + β,α = { X : [0 , ∞ ) → D ( A αi ) , k X k C i, + β = sup t ≥ e βt k X k D ( A αi ) } < ∞ , with the norm k X k C i, + β,α = sup t ≥ e βt k X k D ( A αi ) , for i = 1 , C + β,α = C , + β,α × C , + β,α ,with norm k ( X , X ) k C + β,α = k X k C , + β,α + k X k C , + β,α , ( X , X ) ∈ C + β,α . Denote Φ( t, ω, ( X , , X , )) = (cid:0) X ( t, ω, ( X , , X , )) , X ( t, ω, ( X , , X , )) (cid:1) , the so-lution of the random system (3.2) with the initial condition Φ(0 , ω, ( X , , X , )) =( X , , X , ) . Define the difference of two dynamical orbits asΨ( t ) = Φ( t, ω, ( e X , , e X , )) − Φ( t, ω, ( X , , X , ))= ( X ( t, ω, ( e X , , e X , )) − X ( t, ω, ( X , , X , )) ,X ( t, ω, ( e X , , e X , )) − X ( t, ω, ( X , , X , ))):= ( U ( t ) , U ( t )) , with initial conditionΨ(0) = ( U (0) , U (0)) = ( e X , − X , , e X , − X , ) . Then we have X ( t, ω, ( e X , , e X , )) = U ( t ) + X ( t, ω, ( X , , X , )) ,X ( t, ω, ( e X , , e X , )) = U ( t ) + X ( t, ω, ( X , , X , )) . Thus ( U , U ) satisfies the follow system of equations (cid:26) dU dt = − A U + ∆ F ( U , U , θ t ω ) , dU dt = − A U + ∆ F ( U , U , θ t ω ) , where the nonlinearities are∆ F ( U , U , θ t ω ) = F ( U ( t ) + X ( t, ω, ( X , , X , )) , U + X ( t, ω, ( X , , X , )) , θ t ω ) − F ( X ( t, ω, ( X , , X , )) , X ( t, ω, ( X , , X , )) , θ t ω )7nd∆ F ( U , U , θ t ω ) = F ( U ( t ) + X ( t, ω, ( X , , X , )) , U + X ( t, ω, ( X , , X , )) , θ t ω ) − F ( X ( t, ω, ( X , , X , )) , X ( t, ω, ( X , , X , )) , θ t ω ) . Define W βs (( X , , X , , ω )) = { ( e X , , e X , ) ∈ D ( A α ) × D ( A α ) | Φ( t, ω, ( e X , , e X , )) − Φ( t, ω, ( X , , X , )) ∈ C + β,α } . Then we will prove W βs (( X , , X , ) , ω ) is a fiber of the foliation for the system(3.2) . Lemma 3.2.
Take β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) as a positive real number. Then ( e X , , e X , ) ∈ W β (( X , , X , , ω )) if and only if there exists a function Ψ( t ) = ( U ( t ) , U ( t )) = ( U ( t, ω, ( X , , X , ); U (0)) , U ( t, ω, ( X , , X , ); U (0))) ∈ C + β,α , such that Ψ( t ) = (cid:18) U ( t ) U ( t ) (cid:19) = (cid:18) R t ∞ e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) dse − A t U (0) + R t e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds (cid:19) , (3.3) where ∆ F and ∆ F are defined as above.Proof. Let ( e X , , e X , ) ∈ W βs (( X , , X , ) , ω ) . Using the variation of constantsformula, we have U ( t ) = e − A ( t − τ ) U ( τ ) + R tτ e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds ,U ( t ) = e − A t U (0) + R t e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds . Note that Φ( · ) ∈ C + β,α . For τ > τ > t k e − A ( t − τ ) U ( τ ) k C , + β,α ≤ sup t ≥ e βt k e − A ( t − τ ) U ( τ ) k D ( A α ) = sup t ≥ e βt k A α e − A ( t − τ ) U ( τ ) k≤ sup t ≥ e βt λ αN e − λ N ( t − τ ) k A α U ( τ ) k≤ λ αN e ( β − λ N ) t e ( λ N − β ) τ k U ( τ ) k C , + β,α → as τ → + ∞ . Then we conclude that U ( t ) = R t ∞ e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds ,U ( t ) = e − A t U (0) + R t e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds , t ) = ( U ( t ) , U ( t )) = ( U ( t, ω, ( X , , X , ); U (0)) , U ( t, ω, ( X , , X , ); U (0))) ∈ C + β,α , where Ψ( t ) = Φ( t, ω, ( e X , , e X , )) − Φ( t, ω, ( X , , X , )). Then by the definition of W βs (( X , , X , ) , ω ) we have ( e X , , e X , ) ∈ W β (( X , , X , ) , ω ),which complete our proof. Lemma 3.3.
Take β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) as a positive real number,and let U (0) = e X , − X , ∈ D ( A α ) . Then the system (3.3) has a unique solution Ψ( · ) = Ψ( · , ω, ( X , , X , ) ; U (0)) ∈ C + β,α . Proof.
Introduce two operators F and F satisfying F (Ψ)[ t ] = R t ∞ e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds , F (Ψ)[ t ] = e − A t U (0) + R t e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds . It is easily to verify that F i maps C i, + β, α into itself respectively, i = 1 , F : C + β,α → C + β,α , F (Ψ) := ( F (Ψ) , F (Ψ)) . Then F is well-defined in C + β,α .For every Ψ = ( U , U ) ∈ C + β, α and e Ψ = ( e U , e U ) ∈ C + β,α , we have k F (Ψ) − F ( e Ψ) k C , + β,α = k R t ∞ e − A ( t − s ) [∆ F ( U ( s ) , U ( s ) , θ s ω ) ds − ∆ F ( e U ( s ) , e U ( s ) , θ s ω ) ds ] k C , + β,α = k R t ∞ e − A ( t − s ) [ F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s ) + X ( s, ω, ( X , , X , ))) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , )))] ds k C , + β,α ≤ sup t ≥ e βt k R t ∞ e − A ( t − s ) [ F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s ) + X ( s, ω, ( X , , X , ))) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , )))] ds k D ( A α ) ≤ sup t ≥ e βt R ∞ t k A α e − A ( t − s ) [ F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s ) + X ( s, ω, ( X , , X , )) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , ))] k ds ≤ sup t ≥ L F λ αN R ∞ t e ( − λ N + β )( t − s ) ( k U ( s ) − e U ( s ) k C , + β,α + k U ( s ) − e U ( s ) k C , + β,α ) ds ≤ sup t ≥ L F λ αN R ∞ t e ( − λ N + β )( t − s ) ds k Ψ − e Ψ k C + β,α ≤ L F λ αN β − λ N k Ψ − e Ψ k C + β,α , k F (Ψ) − F ( e Ψ) k C , + β,α = k R t e − A ( t − s ) [∆ F ( U ( s ) , U ( s ) , θ s ω ) ds − ∆ F ( e U ( s ) , e U ( s ) , θ s ω ) ds ] k C , + β,α = k R t e − A ( t − s ) [ F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s ) + X ( s, ω, ( X , , X , ))) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , )))] ds k C , + β,α ≤ sup t ≥ e βt k R t e − A ( t − s ) [ F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s ) + X ( s, ω, ( X , , X , ))) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , )))] ds k D ( A α ) ≤ sup t ≥ e βt R t ( α α ( t − s ) α + λ αN +1 ) e − λ N +1 ( t − s ) k F ( U ( s ) + X ( s, ω, ( X , , X , )) , U ( s )+ X ( s, ω, ( X , , X , ))) − F ( e U ( s ) + X ( s, ω, ( X , , X , )) , e U ( s ) + X ( s, ω, ( X , , X , )))] k ds ≤ sup t ≥ R t ( α α ( t − s ) α + λ αN +1 ) e ( − λ N +1 + β )( t − s ) ( k U ( s ) − e U ( s ) k C , − β,α + k U ( s ) − e U ( s ) k C , + β,α ) ds ≤ sup t ≥ R t ( α α ( t − s ) α + λ αN +1 ) e ( − λ N +1 + β )( t − s ) ds k Ψ − e Ψ k C + β,α ≤ L F ( λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) k Ψ − e Ψ k C + β,α , where C α = α α · Γ(1 − α ) , with Γ the Gamma function.By the definition of F (Ψ) we have k F (Ψ) − F ( e Ψ) k C + β,α = k F (Ψ) − F ( e Ψ) k C , + β,α + k F (Ψ) − F ( e Ψ) k C , + β,α ≤ L F ( λ αN β − λ N + λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) k Ψ − e Ψ k C + β,α . Denote k = L F ( λ αN β − λ N + λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) < . (3.4)Then the maping F (Ψ) is contractive in C + β,α uniformly. From the uniform contrac-tion mapping principle, for each U (0) ∈ D ( A α ), the mapping F (Ψ) has a uniquefixed point, we still denoted it by Ψ( · ) = Ψ( · , ω, ( X , , X , ) , U (0)) ∈ C + β,α . SoΨ( · , ω, ( X , , X , ) , U (0)) ∈ C + β,α is a unique solution of the system (3.2). Lemma 3.4.
Take β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) as a positive real number. Let Ψ( · ) = Ψ( · , ω, ( X , , X , ); U (0)) ∈ C + β,α , be the unique solution of the system (3.3). Then for every U (0) , ˜ U (0) ∈ D ( A α ) ,we have the following estimate k Ψ( · , ω, ( X , , X , ) , U (0)) − Ψ( · , ω, ( X , , X , ) , ˜ U (0)) k C + β,α ≤ C − k k U (0) − ˜ U (0) k D ( A α ) . roof. By the same arguments as in Lemma (3.3), we have k F (Ψ) k C , + β,α ≤ L F λ αN β − λ N k Ψ k C + β,α , and k F (Ψ) k C , + β,α ≤ L F ( λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) k Ψ k C + β,α + C k U (0) k D ( A α ) . By the definition of F (Ψ) we have k F (Ψ) k C + β,α ≤ C − k k U (0) k D ( A α ) , here k is defined as (3.4).Using the same argument as above we can easily deduce our result as k Ψ( · , ω, ( X , , X , ) , U (0)) − Ψ( · , ω, ( X , , X , ) , ˜ U (0)) k C + β,α ≤ C − k k U (0) − ˜ U (0) k D ( A α ) . This complete our proofFor every ζ ∈ D ( A α ) , we define f ( ζ , ( X , , X , ) , ω ) := X , + R ∞ e − A s ∆ F ( U ( s, ω, ( X , , X , ); ( ζ − X , )) ,U ( s, ω, ( X , , X , ); ( ζ − X , )) , θ s ω ) ds . (3.5) Theorem 3.1. (Invariant foliation) Take β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) as apositive real number. Then the foliation of the system (3.2) exists. Moreover,1. A fiber is a graph of a Lipschitz function, that is W βs (( X , , X , ) , ω ) = { ( ζ , f ( ζ , ( X , , X , ) , ω )) | ζ ∈ D ( A α ) } , (3.6) where ( X , , X , ) ∈ D ( A α ) × D ( A α ) , the function f ( ζ , ( X , , X , ) , ω ) definedas (3.5) is Lipschitz continuous respect to ζ and the Lipschitz constant
Lip f satisfies Lip f ≤ L F λ αN β − λ N C − k , where k is defined as (3.4). . Exponentially approaching for positive time. That is , for any two points ( X , , X , ) and ( X , , X , ) in the same fiber W βs (( X , , X , ) , ω ) . Then wehave following estimate. k Ψ( t, ω, ( X , , X , )) − Ψ( t, ω, ( X , , X , )) k D ( A α ) × D ( A α ) ≤ Ce − βt − k k X , − X , k D ( A α ) = O ( e − βt ) , ∀ t → + ∞ . Here C is a positive constant.3. The foliation is invariant, that is Ψ( t, ω, W βs (( X , , X , ) , ω )) ⊂ W βs (Ψ( t, ω, ( X , , X , )) , θ t ω ) . Proof.
1. Note that e X , − X , e X , − X , = R ∞ e − A s ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds e X , − X , . This implies that e X , = X , + R ∞ e − A s ∆ F ( U ( s, ω, ( X , , X , ); U (0)) ,U ( s, ω, ( X , , X , ); U (0) , θ ǫt ω ) ds = X , + R ∞ e − A s ∆ F ( U ( s, ω, ( X , , X , ); e X , − X , ) ,U ( s, ω, ( X , , X , ); e X , − X , ) , θ ǫt ω ) ds, which is just f ( ζ , ( X , , X , ) , ω ) if we take e X , as ζ ∈ D ( A α ) . Then fromabove discussion we have W βs (( X , , X , ) , ω ) = { ( ζ , f ( ζ , ( X , , X , ) , ω )) | ζ ∈ D ( A α ) } . ζ , e ζ in D ( A α ) , we have k f ( ζ , ( X , , X , ) , ω ) − f ( e ζ, ( X , , X , ) , ω ) k C , + β,α = k U ( t, ω, ( X , , X , ); ζ − X , ) − U ( t, ω, ( X , , X , ); e ζ − X , ) k C , + β,α | t =0 ≤ k U ( t, ω, ( X , , X , ); ζ − X , ) − U ( t, ω, ( X , , X , ); e ζ − X , ) k C , + β,α ≤ L F λ αN β − λ N k Ψ( · , ω, ( X , , X , ); ζ − X , ) − Ψ( · , ω, ( X , , X , ); e ζ − X , ) k C + β,α ≤ L F λ αN β − λ N C − k k ζ − e ζ k D ( A α ) . This completes the proof of this part.2. In this part we show that any two orbits in a given fiber are exponentiallyapproaching each other in positive time. k Ψ( · ) k C + β,α = k U ( · ) k C , + β,α + k U ( · ) k C , + β,α ≤ k e − A t U (0) k C , + β,α + k R t e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds k C , + β,α + k R t ∞ e − A ( t − s ) ∆ F ( U ( s ) , U ( s ) , θ s ω ) ds k C , + β ≤ C k U (0) k D ( A α ) + L F λ αN β − λ N k Ψ( · ) k C , + β,α + L F ( λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) k Ψ( · ) k C , + β,α ≤ C k U (0) k D ( A α ) + k k Ψ( · ) k C + β,α , where k is defined as (3.4) and Ψ( · ) as (3.3) . Then we have k Ψ( · ) k C + β,α ≤ C − k k U (0) k D ( A α ) . By the definition of Ψ( · ) and U (0) , we get k Φ( t , ω , ( e X , , e X , )) − Φ( t , ω , ( X , , X , )) k D ( A α ) × D ( A α ) ≤ Ce − βt − k k U (0) k D ( A α ) , ∀ t ≥ . Take two point ( X , , X , ) and ( X , , X , ) in the same fiber W βs (( X , , X , ) , ω ) ,alongwith the Triangle Inequality we have k Φ( t , ω , ( X , , X , )) − Φ( t , ω , ( X , , X , )) k D ( A α ) × D ( A α ) ≤ Ce − βt − k k U (0) k D ( A α ) , ∀ t ≥ . W βs (( X , , X , ) , ω ) ,we will show that Φ( τ , ω , · ) maps it into the fiber W βs (Φ( τ , ω , ( X , , X , ) , θ τ ω )) .Let ( e X , , e X , ) ∈ W βs (( X , , X , ) , ω ) . ThenΦ( · , ω , ( e X , , e X , )) − Φ( · , ω , ( X , , X , )) ∈ C + β ,α , which implies thatΦ( · + τ , ω , ( e X , , e X , )) − Φ( · + τ , ω , ( X , , X , )) ∈ C + β ,α . By the cocycle propertyΦ( · + τ , ω , ( e X , , e X , )) = Φ( · , θ τ ω , Φ( τ, ω , ( e X , , e X , ))) , Φ( · + τ , ω , ( X , , X , )) = Φ( · , θ τ ω , Φ( τ , ω , ( X , , X , ))) , and the definition of foliation, we haveΦ( τ , ω , ( X , , X , )) ∈ W βs (Φ( τ , ω , ( X , , X , )) , θ τ ω ) . This completes the proof of this part.
Theorem 3.2.
Under the same conditions of Theorem 3.1, the foliation of thesystem (1.1) is ˆ W βs (( X , , X , ) , ω ) = W βs (( X , , X , ) , ω ) + Z ǫ ( t ) . Proof.
By the relationship between solutions of systems (1.1) and (3.2), the system(1.1) has a Lipschitz continuous invariant foliation under the same conditions ofTheorem 3.1, which is represented byˆ W βs (( X , , X , ) , ω ) = W βs (( X , , X , ) , ω ) + Z ǫ ( t ) . Now we consider an approximation of invariant foliation of the system (1.1)when ǫ is sufficiently small ∂u∂t − P nk,j =1 ∂ x k ( a kj ( x ) ∂ x j u ) + a ( x ) u = f ( u ) + ǫ ˙ W on D , ∂u∂t + P nk,j =1 ν k a kj ( x ) ∂ x j u + c ( x ) u = g ( u ) + ǫ ˙ W on ∂D,u (0 , x ) = u ( x ) on D × ∂D. d ˆ X + A ˆ Xdt = F ( ˆ X ) dt + ǫdW, ˆ X = ˆ X (0) ∈ D ( A α ) , (4.1)where ˆ X = ( u, γu ) T , F ( ˆ X ) = ( f ( u ) , g ( γu )) T , W = ( W , W ) and X (0) = ( u (0) , γu (0)) T .Consider a linear stochastic evolution equation dZ ǫ ( t ) + AZ ǫ ( t ) = ǫdW ( t ) . Then Z ǫ ( ω ) = ǫZ ( ω ) , where Z ( ω ) is the stationary solution of dZ + AZ = dW ( t ) . Set X ( t ) = ˆ X ( t ) − Z ǫ ( t ) . Then X ( t ) satisfies dX + AXdt = F ( X + Z ǫ ) dt . (4.2)We propose an approach to approximate the random invariant foliation byasymptotic analysis when ǫ sufficiently small. The stable fiber of the invariantfoliation for (4.2) passing through X = ( X , , X , ) is denoted by W βs (( X , , X , ) , ω ) = { ζ + f ǫ ( ζ , ( X , , X , ) , ω ) | ζ ∈ D ( A α ) } , where f ǫ ( ζ , ( X , , X , ) , ω )= X , + R ∞ e A s [ F ( U ( s, ω, ( X , , X , ); ( ζ − X , )) + X ( s, ω, ( X , , X , )) ,U ( s, ω, ( X , , X , ); ( ζ − X , )) + X ( s, ω, ( X , , X , )) , θ s ω ) − F ( X ( s, ω, ( X , , X , )) , X ( s, ω, ( X , , X , )) , θ s ω )] . (4.3)For the deterministic fiber ( i.e. ǫ = 0) represented as { ζ + f d ( ζ ) | ζ ∈ D ( A α ) } , where f ǫ ( · , ( X , , X , ) , ω ) : D ( A α ) → D ( A α ) and f ( · , ( X , , X , )) : D ( A α ) → D ( A α ) are Lipschitz mappings. We expand f ǫ ( ζ , ( X , , X , ) , ω ) = f d ( ζ ) + ǫ f ( ζ , ( X , , X , ) , ω ) + ǫ f ( ζ , ( X , , X , ) , ω ) + · · · + ǫ k f k ( ζ , ( X , , X , ) , ω ) + · · · . (4.4)As section 2 we take U = e X − X then for U we have follow equation dU + AU = F ( U + X + ǫZ ) − F ( X + ǫZ ) . (4.5)15rite the solution of (4.5) in the form U ( t ) = U ( d ) ( t ) + ǫU (1) ( t ) + · · · + ǫ k U ( k ) ( t ) + · · · , (4.6)with the initial condition U (0) = ζ + f ǫ ( ζ , ω ) − X = ζ − X + f ( ζ ) + ǫ f ( ζ , ω ) + · · · . We expand X ( t ) = X ( d ) ( t ) + ǫX (1) ( t ) + · · · + ǫ k X ( k ) + · · · (4.7)and F ( U ( t ) + X ( t )) = F ( U ( d ) ( t ) + X ( d ) ( t )) + ǫF ′| U ( d )( t )+ X ( d )( t ) × ( U (1) ( t ) + X (1) ( t )) + · · · . (4.8)Substituting (4.6), (4.7), (4.8), into equation (4.5), and equating the terms withthe same power of ǫ , we get ( dU ( d ) ( t ) dt = − AU ( d ) ( t ) + F ( U ( d ) ( t ) + X ( d ) ( t )) − F ( X ( d ) ( t )) ,U ( d ) (0) = ζ + f ( ζ ) − X , and dU (1) ( t ) dt = − AU (1) ( t ) + F ′| U ( d )( t )+ X ( d )( t ) × ( U (1) ( t ) + X (1) ( t ) + Z ( t )) − F ′| X ( d )( t ) × ( X (1) ( t ) + Z ( t )) ,U (1) (0) = f ( ζ , ω ) . Then dU (1) ( t ) dt = ( − A + F ′| U ( d )( t )+ X ( d )( t ) ) U (1) ( t ) + ( F ′| U ( d )( t )+ X ( d )( t ) − F ′| X ( d )( t ) )( X (1) ( t ) + Z ( t )) ,U (1) (0) = f ( ζ , ω ) , here X ( d ) ( t ) and X (1) ( t ) satisfy the following equations dX ( d ) ( t ) dt = − AX ( d ) ( t ) + F ( X ( d ) ( t )) ,X ( d ) (0) = X , and dX (1) ( t ) dt = − AX (1) ( t ) + F ′| X ( d )( t ) ( X (1) ( t ) + Z ( t )) ,X (1) (0) = 0 . F ′| U ( d )( t )+ X ( d )( t ) as the first order Fr´echet derivative of the functionof F evaluated at U ( d ) ( t ) + X ( d ) ( t ) .With (4.8), the right hand side of (4.3) can be written as f ǫ ( ζ , ω ) = I + ǫI + R , (4.9)where R represents the remainder term and the other two terms are as follows. I = π X + Z ∞ e As π ( F ( U ( d ) ( s, ζ − π X , X ) + X ( d ) ( s )) − F ( X ( d ) ( s ))) ds , (4.10) I = R ∞ e As π [ F ′| U ( d )( t )+ X ( d )( t ) × ( U ( s, ζ − π X , X ) + X ( s ) + Z ) − F ′| X ( d )( t ) × ( X (1) ( s ) + Z )] ds . (4.11)Substituting (4.4) into (4.9), and matching the powers in ǫ , we get f d ( ζ ) = I = π X + Z ∞ e As π ( F ( U ( d ) ( s, ζ − π X , X ) + X ( d ) ( s )) − F ( X ( d ) ( s ))) ds and f ( ζ , ω, X ) = I = R ∞ e As π [ F ′| U ( d )( t )+ X ( d )( t ) × ( U ( s, ζ − π X , X ) + X (1) ( s ) + Z ) − F ′| X ( d )( t ) × ( X (1) ( s ) + Z )] ds . As a summary, we obtain the following result about approximating invariant foli-ation for the random evolutionary equation with dynamic boundary condition.
Theorem 4.1. (Approximation of invariant foliation) Let W βs ( X , ω ) = { ( ζ , f ( ζ , ( X , , X , ) , ω )) | ζ ∈ D ( A α ) } represent a stable fiber, passing through a point X of the invariant foliation forthe random equation dX + AXdt = F ( X + ǫZ ) dt . Assume that1. F is twice continuously Fr´echet differentiable.2. For β = λ N + L F k λ αN ∈ ( λ N , λ N +1 ) and C α = α α · Γ(1 − α ) , the following gapcondition is satisfied L F ( λ αN β − λ N + λ αN +1 λ N +1 − β + C α ( λ N +1 − β ) − α ) < . hen for ǫ sufficiently small, a fiber of the random invariant foliation can beapproximated as W βs ( X , ω ) = { ζ + f d ( ζ ) + ǫ f ( ζ , ω, X ) + R | ζ ∈ D ( A α ) } . here k R k ≤ C ( ω ) ǫ with C ( ω ) < ∞ , a.s. , f d ( ζ ) = π X + Z ∞ e As π ( F ( U ( d ) ( s, ζ − π X , X ) + X ( d ) ( s )) − F ( X ( d ) ( s ))) ds and f ( ζ , ω, X ) = R ∞ e As π [ F ′| U ( d )( t )+ X ( d )( t ) × ( U (1) ( s, ζ − π X , X ) + X (1) ( s ) + Z ) − F ′| X ( d )( t ) × ( X (1) ( s ) + Z )] ds . Acknowledgments
The author would like to Thank G. Chen for helpful discussions and suggestions.
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