J Stat Phys https://doi.org/10.1007/s10955-018-2048-3
Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation Peng Gao1
Received: 5 October 2017 / Accepted: 17 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient. Keywords Higher order nonlinear Schrödinger equation · Averaging principle · Strong convergence · Fast–slow stochastic partial differential equation Mathematics Subject Classification 35Q53 · 60H15 · 70K65 · 70K70
1 Introduction In this paper, we consider the averaging principle for the higher order nonlinear Schrödinger (HNLS) equation with a fast oscillating random perturbation
This work is supported by NSFC Grant (11601073), NSFC Grant (11701078), China Postdoctoral Science Foundation (2017M611292) and the Fundamental Research Funds for the Central Universities (2412017QD002).
B 1
Peng Gao
[email protected] School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, People’s Republic of China
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⎧ ⎨ idu + (iγ u x x x + iσ u x x + μ|u|2 u)dt = f u(t), v εt dt u(0, t) = u(1, t) = u x (1, t) = 0 ⎩ u(x, 0) = u 0 (x)
in I × (0, +∞), in (0, +∞), in I,
where v(t) is governed by one dimensional stochastic reaction–diffusion equation ⎧ ⎨ dv − vx x dt = g(u, v)dt + d B in I × (0, +∞) v(0, t) = v(1, t) = 0 in (0, +∞) ⎩ v(x, 0) = v0 (x) in I. Thus, we will be concerned with the averaging principle for multiscale stochastic HNLS equation with slow and fast time-scales ⎧ idu ε = (−iγ u εx x x − iσ u εx x − μ|u ε |2 u ε + f (u ε , v ε ))dt in I × (0, T ), ⎪ ⎪ ⎪ 1 ε 1 ε ε ε ⎪ in I × (0, T ), ⎨ dv = ε (vx x + g(u , v ))dt + √ε d B ε ε ε (1.1) in (0, T ) u (0, t) = u (1, t) = u x (1, t) = 0 ⎪ ⎪ ε (0, t) = v ε (1, t) = 0 ⎪ v in (0, T ) ⎪ ⎩ ε u (x, 0) = u 0 (x), v ε (x, 0) = v0 (x) in I. In system (1.1), T > 0, I = (0, 1), γ , σ, μ are real parameters, u ε is a complex function and v ε is a real function. ε is a small and positive parameter describing the ratio of time scale between the slow component u ε and fast component v ε . f and g are external forces depending on u ε and v ε . {B(t)}t≥0 is a L 2 (I )−valued Q−Wiener process on complete filtered probability space (, F , {Ft }t≥0 , P) satisfying TrQ < ∞. The original HNLS equation was firstly derived [1] as follows, iu t + iγ u x x x + σ u x x + μ|u|2 u + iν(|u|2 u)x + iu(u 2 )x = 0.
(1.2)
This equation describes the propagation of a signal in a optic fiber [2]. γ , σ, μ, ν and mean the real parameters related to group velocity dispersion, third-order dispersion, self-phase modulation, self-steepening and delayed nonlinear response effect arising from stimulated Raman scattering, respectively. (1.2) also arose in the optical communications (see [3–5]). It can be applied to the long distance communications and ultrafast signalrouting systems. The HNLS models can describe the propagation of ultrashort light pulses which are shorter than ∼ 10−13 s. The HNLS equation has been studied from various kinds of aspects: well-posedness [6,7], unique continuation property [8,9], controllability [10], etc. In this paper, we will investigate the HNLS equation from the point of view of dynamical systems. Almost all physical systems have a certain hierarchy in which not all components evolve at the same rate, i.e., some of components vary very rapidly, while others change very slowly, see [11]. Averaging principle provides an effective tool to analyze the multiple time-scales dynamical systems. Besides, the method of averaging principle enormously reduces the computational load although computer technology is highly efficient nowadays. The theory of averaging principle has a long and rich history, which has been applied in many fields, such as, celestial mechanics, wireless communication, signal processing, oscillation theory and radiophysics. The theory of averaging principle for deterministic systems has been extensively studied for both ordinary differential equations and partial differential equations. Then, in order to consider more realistic models, it is sensible to consider some kind of stochastic perturbation represented by a noise term or a random term. The averaging principle in the stochastic ordinary differential equations was first considered by Khasminskii [12] which proved that
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an averaging principle holds in weak sense, and has been an active research field on which there is a great deal of literature. Taking into account the generalized and refined results, it is worthy quoting the paper • Convergence in probability Veretennikov [13,14], Freidlin and Wentzell [15,16]; • Mean-square type convergence Golec and Ladde [17], Givon and co-workers [18]; • Strong convergence Givon [19] and Golec [20]. In the past four decades, many problems in the natural sciences give rise to singularly perturbed systems of stochastic partial differential equations. Singularly perturbed systems have been the focus of extensive research within the framework of averaging methods. The separation of scales is then taken to advantage to derive a reduced equation, which approximates the slow components. Multiscale stochastic partial differential equations arise as models for various complex systems, such model arises from describing multiscale phenomena in, for example, nonlinear oscillations, material sciences, automatic control, fluids dynamics, chemical kinetics and in other areas leading to mathematical description involving “slow” and “fast” phase variables. We refer that, in recent years, there are many interesting results for stochastic system in infinite dimensional space: • Heat equation Cerrai and Freidlin [21], Cerrai [22,23], Bréhier [24], Wang and Roberts [25], Fu and co-workers [26–28], Xu and co-workers [29–31], Bao and co-workers [32]; • Wave equation Fu and co-workers [27,33], Pei and co-workers [34]; • Klein–Gordon equation Gao [35]; • Ginzburg–Landau equation Gao and Li [36]; • Burgers system Dong and co-workers [37]; • FitzHugh–Nagumo system Fu and co-workers [38], Xu and co-workers [29]. However, as far as we know there is no result on the averaging principle for the stochastic HNLS equation, thus, a natural question is as follows: Can we establish the averaging principle for the stochastic HNLS equation with a fast oscillation ? To be more precisely, can the slow component u ε be approximated by the solution of a reduced HNLS equation? This mathematical question arises naturally which is important from the point of view of dynamical systems from both physical and mathematical standpoints. In this paper, the main object is to establish an effective approximation for slow process u ε with respect to the limit ε → 0. The main difficulty in establishing the averaging principle for the stochastic HNLS equation is that the semigroup of higher order Schrödinger operator doesn’t have the srong smooth effect as the heat semigroup. In order to overcome this difficulty, we establish a regularity property for the semigroup of higher order Schrödinger operator in Proposition 3.1. Based on Proposition 3.1, we can prove the well-posedness of (1.1) and Hölder continuity of time variable for u ε . The rest of this paper is organised as follows: Sect. 2 contains some notations, hypotheses and useful inequalities. In Sect. 3, we prove some properties for the semigroups corresponding to (1.1). Section 4 is devoted to the well-posedness of (1.1). In Sect. 5, we introduce an auxiliary process (uˆ ε , vˆ ε ) and give the convergence of (uˆ ε , vˆ ε ) to (u ε , v ε ). In Sect. 6, we consider the averaged process u˜ and give the convergence of uˆ ε to u. ˜ The main result is given and proved in Sect. 7. Finally, we put some proofs in the Appendix at the end of this paper.
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2 Preliminaries First, we introduce the following notations and mathematical settings: • Throughout the paper, the letter C denotes positive constants whose value may change in different occasions. If it is essential, the dependence of a constant C on some parameters, say “· , will be written by C(·). • We denote by L 2 (I ) the space of all Lebesgue square integrable functions on I . The inner product on L 2 (I ) is (u, v) = Re u vd ¯ x, I
for any u, v ∈ L 2 (I ) is
L 2 (I ), where v¯
denote the complex conjugate of function v. The norm on 1
u = (u, u) 2 , for any u ∈ L 2 (I ). H s (I ) and H0s (I ) (s ≥ 0) are the classical Sobolev spaces of functions on I . The definitions of H s (I ) and H0s (I ) can be found in [39], the norm on H s (I ) and H0s (I ) is · H s . • Let Y be a Banach space, for 1 ≤ p ≤ +∞, let L p (0, T ; Y ) be the Banach space of all measurable functions u : (0, T ) → Y such that t → u(t) Y is in L p (0, T ). Let C([0, T ]; Y ) be the Banach space of all Y −valued strongly continuous functions defined on [0, T ]. We denote by L 2F (; C([0, T ]; Y )) the Banach space consisting of all Y −valued {Ft }t≥0 −adapted continuous processes X (·) such that 2 E( X (·) C([0,T ];Y ) ) < ∞.
All the above spaces are endowed with the canonical norm. We adopt the following hypotheses throughout this work. (H1) α is a fixed constant belongs to (1, 23 ). (H2) The functions f, g satisfy the global Lipshitz condition, explicitly, there exist positive constants L f and L g such that for any u 1 , u 2 , v1 , v2 ∈ L 2 (I ), f (u 1 , v1 ) − f (u 2 , v2 ) ≤ L f ( u 1 − u 2 + v1 − v2 ), g(u 1 , v1 ) − g(u 2 , v2 ) ≤ L g ( u 1 − u 2 + v1 − v2 ). (H3) η := λ − 2L g > 0, where L g is the Lipshitz coefficient for g in (H2) and λ > 0 is the smallest constant such that the following inequality holds
where u ∈ H01 (I ) or
u x 2 ≥ λ u 2 ,
I
ud x = 0.
Finally, we give some useful inequalities which will be used in the following proof. Lemma 2.1 (Young inequality) Let a, b ∈ (0, +∞) and ε > 0, then we have − qp
ab ≤ εa p + (εp) where 1 < p < ∞,
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1 p
+
1 q
= 1.
q −1 bq ,
Averaging Principle for the Higher...
Lemma 2.2 Let y(t) be a nonnegative function, if y ≤ −ay + f, we have y(t) ≤ y(s)e−a(t−s) +
t
e−a(t−τ ) f (τ )dτ.
s
Lemma 2.3 If s ≤ − 21 and u ∈ L 1 (I ), there exists a positive constant C such taht u H s ≤ C u L 1 . Lemma 2.4 For any u ∈ H 1 (I ), we have 1
1
u L ∞ ≤ C u 2 u H2 1 . Lemma 2.5 For any u ∈ H 4 (I ) such that u(0) = u(1) = u x (0) = 0 or u(0) = u(1) = u x (1) = 0, one has u H 3 ≤ C ∂x3 u and u H 4 ≤ C ∂x3 u H 1 . Lemma 2.6 For any ε > 0, j ∈ N+ and u ∈ H j (I ), there exists a constant C(ε) > 0 such that ∂xi u ≤ ε u H j + C(ε) u 0 < i < j.
3 Properties of Semigroups For the sake of simplicity, we will choose γ = σ = μ = 1 in the rest of this paper. All results can be extended to the general cases without difficulty. Let A1 and A2 be linear operators defined by A1 = −∂x3 + i∂x2 , A2 = ∂x2 with the domains D(A1 ) = {z ∈ H 3 (I )| z(0) = z(1) = z (1) = 0}, D(A2 ) = {z ∈ H 2 (I )| z(0) = z(1) = 0}.
It is obvious that A2 is the infinitesimal generator of a C0 -semigroup {S2 (t)}t≥0 in L 2 (I ). And it is proved in [10] that A1 also generates a C0 -semigroup {S1 (t)}t≥0 in L 2 (I ).
3.1 Property of Semigroup {S2 (t)} t≥0 The property of semigroup {S2 (t)}t≥0 is stated as follows. Lemma 3.1 [40] For any s1 , s2 ∈ R with s1 ≤ s2 , there exists a constant C such that S2 (t)z H s2 ≤ C(1 + t
(s1 −s2 ) 2
) z H s1 , z ∈ H s1 (I ), t > 0,
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3.2 Property of Semigroup {S1 (t)} t≥0 The following property for semigroup {S1 (t)}t≥0 plays an important role in the proof of the main result. Proposition 3.1 For any T > 0, β ∈ [1, 2] and γ ∈ [1, 4], there exists a positive constant C(β, γ , T ) such that β
S1 (t)z H γ ≤ C(β, γ , T )t − 2 z H γ −β , z ∈ H γ −β (I ), t ∈ (0, T ]. Proof First, according to the result in [10, p. 23], we have √ S1 (·)z L 2 (0,T ;H 1 (I )) ≤ C(1 + T ) z , z ∈ L 2 (I ), where C is independent of T . Next, we shall show that S1 (·)z L 2 (0,T ;L 2 (I )) ≤ C(1 +
√
T ) z H −1 , z ∈ H −1 (I ).
In order to prove (3.3), we consider following adjoint system ⎧ in I × (0, T ), ⎨ iwt + iwx x x − wx x = h w(0, t) = w(1, t) = wx (0, t) = 0 in (0, T ), ⎩ w(x, T ) = 0, in I. • For h ∈ L 2 (0, T ; H −1 (I )), we claim that w L ∞ (0,T ;L 2 (I )) ≤ C(1 +
√
T ) h L 2 (0,T ;H −1 (I )) .
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
Indeed, multiplying the first equation in (3.4) by −w¯ and conjugating, we have ¯ −iwt w¯ − iwx x x w¯ + wx x w¯ = −h w, ¯ i w¯ t w + i w¯ x x x w + w¯ x x w = −hw. Subtracting and integrating over I , we can obtain after some integrations by parts that d ¯ − h w)d |w(x, t)|2 d x + i|wx (1, t)|2 = (hw ¯ x. −i dt I I Taking the imaginary part and integrating over (t, T ), it follows that T 2 |h||w|d xds ≤ 2 h L 2 (0,T ;H −1 (I )) w L 2 (0,T ;H 1 (I )) . w(·, t) ≤ 2 t
I
This implies w 2L ∞ (0,T ;L 2 (I )) ≤ 2 h L 2 (0,T ;H −1 (I )) w L 2 (0,T ;H 1 (I )) .
(3.6)
Using Lemma 2.1, for any ε > 0, we have w L ∞ (0,T ;L 2 (I )) ≤ ε w L 2 (0,T ;H 1 (I )) +
C h L 2 (0,T ;H −1 (I )) . ε
Then, multiplying the first equation in (3.4) by −(1 − x)w¯ and conjugating, we have − i(1 − x)wt w¯ − i(1 − x)wx x x w¯ + (1 − x)wx x w¯ = −(1 − x)h w, ¯ ¯ i(1 − x)w¯ t w + i(1 − x)w¯ x x x w + (1 − x)w¯ x x w = −(1 − x)hw.
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(3.7)
Averaging Principle for the Higher...
Subtracting and integrating over I , we can obtain after some integrations by parts that d −i (1 − x)|w(x, t)|2 d x + 3i |wx (x, t)|2 d x + (wx w¯ − w¯ x w)d x dt I I I = (1 − x)(w h¯ − wh)d ¯ x. I
Taking the imaginary part, it is easy to obtain that d − (1 − x)|w(x, t)|2 d x + 3 wx (·, t) 2 dt I ¯ x = −2Im wx wd ¯ x + 2Im (1 − x)w hd I
I
≤ 2 wx (·, t) w(·, t) + 2 w(·, t) H 1 h(·, t) H −1 . Integrating over (0, T ) and applying Lemma 2.1, for any ε1 > 0 and ε2 > 0, we have C w 2L 2 (0,T ;L 2 (I )) ε1 C + ε2 w 2L 2 (0,T ;H 1 (I )) + h 2L 2 (0,T ;H −1 (I )) . ε2
3 w 2L 2 (0,T ;H 1 (I )) ≤ ε1 w 2L 2 (0,T ;H 1 (I )) +
Choosing ε1 = ε2 = 1, this gives w L 2 (0,T ;H 1 (I )) ≤ C w L 2 (0,T ;L 2 (I )) + C h L 2 (0,T ;H −1 (I )) √ ≤ C T w L ∞ (0,T ;L 2 (I )) + C h L 2 (0,T ;H −1 (I )) . Taking (3.7) into consideration, it follows that w L 2 (0,T ;H 1 (I )) Choosing ε =
√ C T h L 2 (0,T ;H −1 (I )) +C h L 2 (0,T ;H −1 (I )) . ≤ C T ε w L 2 (0,T ;H 1 (I )) + ε
1√ 2C T
√
, it is not difficult to see that w L 2 (0,T ;H 1 (I )) ≤ C(1 + T ) h L 2 (0,T ;H −1 (I )) .
Combining this inequality and (3.6), we have w 2L ∞ (0,T ;L 2 (I )) ≤ C(1 + T ) h 2L 2 (0,T ;H −1 (I )) , namely, w L ∞ (0,T ;L 2 (I )) ≤ C(1 +
√
• For h ∈ L 2 (0, T ; H02 (I )), we claim that w L ∞ (0,T ;H 3 (I )) ≤ C(1 +
T ) h L 2 (0,T ;H −1 (I )) .
√
T ) h L 2 (0,T ;H 2 (I )) .
(3.8)
Indeed, applying the operator P = to (3.4), we have ⎧ in I × (0, T ), ⎨ i(Pw)t + i(Pw)x x x − (Pw)x x = Ph (Pw)(0, t) = (Pw)(1, t) = (Pw)x (0, t) = 0 in (0, T ), ⎩ (Pw)(x, T ) = 0, in I. i∂x3
− ∂x2
Proceeding as in the case h ∈ L 2 (0, T ; H −1 (I )), we can deduce that √ Pw L ∞ (0,T ;L 2 (I )) ≤ C(1 + T ) Ph L 2 (0,T ;H −1 (I )) √ ≤ C(1 + T ) h L 2 (0,T ;H 2 (I )) .
(3.9)
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It follows from Lemmas 2.5 and 2.6 that w L ∞ (0,T ;H 3 (I )) ≤ C ∂x3 w L ∞ (0,T ;L 2 (I )) ≤ C Pw L ∞ (0,T ;L 2 (I )) + C ∂x2 w L ∞ (0,T ;L 2 (I )) 1 ≤ C Pw L ∞ (0,T ;L 2 (I )) + w L ∞ (0,T ;H 3 (I )) + C w L ∞ (0,T ;L 2 (I )) . 2 Then combining (3.5) and (3.9), the following holds w L ∞ (0,T ;H 3 (I )) ≤ C Pw L ∞ (0,T ;L 2 (I )) + C w L ∞ (0,T ;L 2 (I )) √ ≤ C(1 + T )( h L 2 (0,T ;H 2 (I )) + h L 2 (0,T ;H −1 (I )) ) √ ≤ C(1 + T ) h L 2 (0,T ;H 2 (I )) . According to (3.5), (3.8) and standard interpolation arguments, for any h ∈ L 2 (0, T ; L 2 (I )), we have √ w L ∞ (0,T ;H 1 (I )) ≤ C(1 + T ) h L 2 (0,T ;L 2 (I )) . Following the ideas in [41], this inequality implies (3.3). Since {S1 (t)} is the C0 semigroup in L 2 (I ), it is obvious that S1 (t)z ≤ C z , z ∈ L 2 (I ), t > 0.
(3.10)
Then, applying Lemmas 2.5 and 2.6, for any z ∈ D(A1 ), we have S1 (t)z H 3 ≤ C ∂x3 S1 (t)z ≤ C A1 S1 (t)z + C ∂x2 S1 (t)z 1 ≤ C S1 (t)A1 z + S1 (t)z H 3 + C S1 (t)z . 2 Taking (3.10) into account, we can deduce that S1 (t)z H 3 ≤ C( S1 (t)A1 z + S1 (t)z ) ≤ C( A1 z + z )
(3.11)
≤ C z H 3 . Combining (3.10), (3.11) and classical interpolation arguments, for any 0 ≤ s ≤ 3 and any z ∈ H s (I ), we have S1 (t)z H s ≤ C z H s , t > 0. (3.12) Now, we shall prove (3.1). Following the method developed in [42, Proposition 2.2], we can obtain from (3.2) and (3.12) (s = 1) that 1 (3.13) S1 (t)z H 1 ≤ C(T )t − 2 z , z ∈ L 2 (I ). Similarly, we can obtain from (3.3) and (3.12) (s = 0) that 1
S1 (t)z ≤ C(T )t − 2 z H −1 , z ∈ H −1 (I ).
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(3.14)
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Combining (3.13) and (3.14), for any z ∈ H −1 (I ), we can deduce that
t t
S1 (t)z H 1 =
S1 2 S1 2 z 1
H
t − 21
≤ C(T )t S1 z
2
(3.15)
= C(T )t −1 z H −1 . According to classical interpolation arguments, it follows from (3.13) and (3.15) that for any β ∈ [1, 2], β
S1 (t)z H 1 ≤ C(β, T )t − 2 z H 1−β , z ∈ H 1−β (I ).
(3.16)
Next, we claim that for any z ∈ H 4−β (I ), β
S1 (t)z H 4 ≤ C(β, T )t − 2 z H 4−β .
(3.17)
In fact, according to (3.16) and interpolation inequality, we can deduce that S1 (t)z H 4 ≤ C ∂x3 S1 (t)z H 1 ≤ C( A1 S1 (t)z H 1 + ∂x2 S1 (t)z H 1 ) ≤ C( S1 (t)A1 z H 1 + ∂x3 S1 (t)z + ∂x2 S1 (t)z ) 1 ≤ C S1 (t)A1 z H 1 + S1 (t)z H 4 + C S1 (t)z . 2 By (3.16), we have S1 (t)z H 4 ≤ C( S1 (t)A1 z H 1 + S1 (t)z ) β
≤ C(β, T )t − 2 A1 z H 1−β + C z β
≤ C(β, T )t − 2 z H 4−β . Applying (3.16), (3.17) and interpolation arguments, we can obtain (3.1).
4 Well-Posedness Let us explain what we mean by a solution of (1.1). Definition 4.1 A pair of functions (u ε , v ε ) is called a mild solution of (1.1) on [0, T ], if for almost each ω ∈ and t ∈ [0, T ], (u ε , v ε ) satisfies the following Itô integral form t S1 (t − s)[|u ε |2 (s)u ε (s) − f (u ε (s), v ε (s))]ds, u ε (t) = S1 (t)u 0 + i 0 t t t −s t −s 1 t 1 v0 + g(u ε (s), v ε (s))ds + √ d B. S2 S2 v ε (t) = S2 ε ε 0 ε ε ε 0 Theorem 4.1 Let T > 0. For any ε ∈ (0, 1), if (u 0 , v0 ) ∈ H α (I ) × L 2 (I ), (1.1) admits a unique mild solution (u ε , v ε ) ∈ L 2 (; C([0, T ]; H α (I )) × L 2 (; C([0, T ]; L 2 (I )) and E sup u ε (t) 2H α + E sup v ε (t) 2 ≤ C(1 + u 0 2H α + v0 2 ), 0≤t≤T
0≤t≤T
where C is a positive constant depending on ε, T and Q.
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Remark 4.1 The main idea of Proof of Theorem 4.1 comes from [43, Theorem 4.1].
4.1 Local Existence In this part, we will take ε=1 (u ε , v ε )
for the sake of simplicity, and is denoted by (u, v). All the results can be extended without difficulty to the general case. We set X τ = L 2 (; C([0, τ ]; H α (I )) × L 2 (; C([0, τ ]; L 2 (I )). Let ρ ∈ C0∞ (R) be a cut-off function such that ρ(r ) = 1 for r ∈ [0, 1] and ρ(r ) = 0 for r ≥ 2. For any R > 0, y ∈ X t and t ∈ [0, T ], we set y C([0,t];H α (I )) ρ R (y)(t) = ρ . R The truncated equation corresponding to (1.1) is the following stochastic partial differential equations: ⎧ idu + (iu x x x + u x x + ρ R (u)|u|2 u)dt = f (u, v)dt in I × (0, T ) ⎪ ⎪ ⎪ ⎪ in I × (0, T ) ⎨ dv − vx x dt = g(u, v)dt + d B (4.1) u(0, t) = u(1, t) = u x (1, t) = 0, in (0, T ) ⎪ ⎪ in (0, T ) ⎪ v(0, t) = v(1, t) = 0, ⎪ ⎩ u(x, 0) = u 0 (x), v(x, 0) = v0 (x) in I Proposition 4.1 For any (u 0 , v0 ) ∈ H α (I ) × L 2 (I ), (4.1) admits a unique mild solution (u, v) ∈ X τ∞ , where τ∞ is stopping time. Moreover, if τ∞ < +∞, then P−a.s. lim (u, v) X t = +∞.
t→τ∞
Proof We define
1R (u(t), v(t)) R (u(t), v(t)) = 2R (u(t), v(t)) t S1 (t)u 0 + 0 S 1 (t − s)[iρ R (u)(s)|u(s)|2 u(s) − i f (u(s), v(s))]ds = . t t S2 (t)v0 + 0 S2 (t − s)g(u(s), v(s))ds + 0 S2 (t − s)d B
It is easy to see that for any T0 > 0, operator R (u, v) maps X T0 into itself. First, we estimate E sup 1R (u 1 , v1 )(t) − 1R (u 2 , v2 )(t) 2H α . 0≤t≤T0
Set F(u) =
|u|2 u.
It is not difficult to obtain that
F(u 1 ) − F(u 2 ) = |u 1 |2 u 1 − |u 2 |2 u 2 = |u 1 |2 (u 1 − u 2 ) + u 1 u 2 (u¯ 1 − u¯ 2 ) + |u 2 |2 (u 1 − u 2 ) ≤ ( u 1 2L ∞ + u 2 2L ∞ ) u 1 − u 2 ≤ ( u 1 2H α + u 2 2H α ) u 1 − u 2 H α . Set FR (u) = F(u)ρ R (u).
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Averaging Principle for the Higher...
Assume that u 1 H α ≥ u 2 H α , we have without any loss of generality, FR (u 1 ) − FR (u 2 ) = (F(u 1 ) − F(u 2 ))ρ R (u 1 ) + (ρ R (u 1 ) − ρ R (u 2 ))F(u 2 ) ≤ (F(u 1 ) − F(u 2 ))ρ R (u 1 ) + (ρ R (u 1 ) − ρ R (u 2 ))F(u 2 ) = (F(u 1 ) − F(u 2 ))ρ R (u 1 ) + (ρ R (u 1 ) − ρ R (u 2 ))F(u 2 ) χ{ u 2 H α (I ) ≤2R} 1 ≤ (F(u 1 ) − F(u 2 ))ρ R (u 1 ) + ρ R L ∞ (u 1 − u 2 )F(u 2 ) χ{ u 2 H α ≤2R} R ≤ C u 1 − u 2 H α ( u 1 2H α + u 2 2H α )χ{ u 1 H α ≤2R} 1 + ρ R L ∞ u 1 − u 2 L ∞ F(u 2 ) χ{ u 2 H α ≤2R} R 1 ≤ C R 2 u 1 − u 2 H α + ρ R L ∞ u 1 − u 2 H α |u 2 |2 u 2 χ{ u 2 H α ≤2R} R 1 ≤ C R 2 u 1 − u 2 H α + ρ R L ∞ u 1 − u 2 H α u 2 3H α χ{ u 2 H α ≤2R} R ≤ C R 2 u 1 − u 2 H α . By (3.1) (β = γ = α), we have t S1 (t − s)(FR (u 1 )(s) − FR (u 2 )(s))ds 2H α E sup i 0
0≤t≤T0
≤ CE sup 0≤t≤T0
≤ CE sup ≤ CR
α
(t − s)− 2 FR (u 1 )(s) − FR (u 2 )(s) ds
0
t
α
(t − s)− 2 R 2 u 1 (s) − u 2 (s) H α ds
0
0≤t≤T0 4
t
sup
t
α
(t − s)− 2 ds
0
0≤t≤T0
2
2
2
(4.2)
E sup u 1 (t) − u 2 (t) 2H α 0≤t≤T0
≤ C R 4 T02−α E sup u 1 (t) − u 2 (t) 2H α 0≤t≤T0
and
E sup 0≤t≤T0
t
0
≤ E sup
S1 (t − s)( f (u 1 (s), v1 (s)) − f (u 2 (s), v2 (s)))ds 2H α (I )
t
0
0≤t≤T0
≤ CE sup 0≤t≤T0
≤ CE sup
0≤t≤T0
≤ C sup 0≤t≤T0
≤
C T02−α
S1 (t − s)( f (u 1 (s), v1 (s)) − f (u 2 (s), v2 (s))) H α ds t
α
(t − s)− 2 ( f (u 1 (s), v1 (s)) − f (u 2 (s), v2 (s))) ds
0 t
α
(t − s)− 2 ( u 1 (s) − u 2 (s) + v1 (s) − v2 (s) )ds
0 t
α
(t − s)− 2 ds
0
2
2
2
2 E sup u 1 (t) − u 2 (t) 2 + E sup v1 (t) − v2 (t) 2 0≤t≤T0
0≤t≤T0
E sup u 1 (t) − u 2 (t) 2 + E sup v1 (t) − v2 (t) 2 . 0≤t≤T0
0≤t≤T0
(4.3)
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P. Gao
Collecting the above estimates (4.2)–(4.3), we get E sup 1R (u 1 , v1 )(t) − 1R (u 2 , v2 )(t) 2H α 0≤t≤T0
≤ C(T02−α + R 4 T02−α ) E sup u 1 (t) − u 2 (t) 2H α + E sup v1 (t) − v2 (t) 2 . 0≤t≤T0
0≤t≤T0
(4.4) Similarly, we can deduce that E sup 2R (u 1 , v1 )(t) − 2R (u 2 , v2 )(t) 2 0≤t≤T0
≤ C T0 E sup u 1 (t) − u 2 (t) 2H α + E sup v1 (t) − v2 (t) 2 . 0≤t≤T0
(4.5)
0≤t≤T0
The estimate of R (u 1 , v1 ) − R (u 2 , v2 ) X T0 . Indeed, it follows from (4.4) and (4.5) that E sup 1R (u 1 , v1 )(t) − 1R (u 2 , v2 )(t) 2H α 0≤t≤T0
+ E sup 2R (u 1 , v1 )(t) − 2R (u 2 , v2 )(t) 2 0≤t≤T0
≤ C(T02−α + R 4 T02−α + T0 ) E sup u 1 (t)−u 2 (t) 2H α + E sup v1 (t) − v2 (t) 2 , 0≤t≤T0
0≤t≤T0
namely, we have 1
R (u 1 , v1 ) − R (u 2 , v2 ) X T0 ≤ C(T02−α + R 4 T02−α + T0 ) 2 (u 1 , v1 ) − (u 2 , v2 ) X T0 . (4.6) For a sufficiently small T0 , R (u, v) is a contraction mapping on X T0 . Hence, by applying the Banach contraction principle, R (u, v) has a unique fixed point in X T0 , which is the unique local solution to (4.1) on the interval [0, T0 ]. Since T0 does not depend on the initial value (u 0 , v0 ), this solution may be extended to the whole interval [0, T ]. We denote by (u R , v R ) this unique mild solution and let τ R = inf{t ≥ 0 : (u R , v R ) X t ≥ R}, with the usual convention that inf ∅ = ∞. Since R1 ≤ R2 , τ R1 ≤ τ R2 , we can put τ∞ = lim τ R . Set τ = τ R1 ∧ τ R2 . We define a R→+∞
local solution to (3.2) as follows u(t) = u R (t), ∀ t ∈ [0, τ R ], v(t) = v R (t), ∀ t ∈ [0, τ R ].
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Averaging Principle for the Higher...
Indeed, for any t ∈ [0, τ ] u R1 (t) − u R2 (t) =
t
0
S1 (t − s)[iρ R1 (u R1 )(s)|u R1 (s)|2 u R1 (s)
− iρ R2 (u R2 )(s)|u R2 (s)|2 u R2 (s) − i f (u R1 (s), v R1 (s)) + i f (u R2 (s), v R2 (s))]ds, t v R1 (t) − v R2 (t) = S2 (t − s)[g(u R1 (s), v R1 (s)) − g(u R2 (s), v R2 (s))]ds. 0
Proceeding as in the proof of (4.6), we can obtain that (u R1 , v R1 ) − (u R2 , v R2 ) X t ≤ C(t) (u R1 , v R1 ) − (u R2 , v R2 ) X t , where C(t) is a monotonically increasing function and C(0) = 0. If we take t sufficiently small, we can obtain u R1 (t) = u R2 (t), v R1 (t) = v R2 (t). Repeating the same argument for the interval [t, 2t] and so on yields u R1 (t) = u R2 (t), v R1 (t) = v R2 (t). for the whole interval [0, τ ]. According to this, we can know the above definition of local solution to (3.2) is well defined. If τ∞ < +∞, the definition of (u, v) yields P−a.s. lim (u, v) X t = +∞,
t→τ∞
which shows that (u, v) is the unique local solution to (3.2) on the interval [0, τ∞ ). This completes the proof of Proposition 4.1.
4.2 Some a Priori Estimates of (uε , v ε ) First, we shall prove uniform bounds with respect to ε ∈ (0, 1) for p−moment of the solution for (1.1). Proposition 4.2 For any u 0 , v0 ∈ L 2 (I ), p > 0 and T > 0, there exists a constant C( p, T ) such that sup
sup E u ε (t) 2 p ≤ C( p, T ),
sup
sup E v ε (t) 2 p ≤ C( p, T ).
ε∈(0,1) t∈[0,T ] ε∈(0,1) t∈[0,T ]
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P. Gao
Proof It is sufficient to prove this proposition when p is large enough. For p ≥ 2, applying Itô formula with v ε (t) 2 p , we have t 2p E E v ε (t) 2 p = v0 2 p + v ε (s) 2 p−2 (vxε x (s), v ε (s))ds ε 0 t 2p ε v (s) 2 p−2 (g(u ε (s), v ε (s)), v ε (s))ds + E ε 0 t p v ε (s) 2 p−2 TrQds + E ε 0 t 1 2 p( p − 1) + E v ε (s) 2 p−4 Q 2 v ε (s) 2 ds. ε 0 It is easy to obtain that (vxε x (s), v ε (s)) ≤ −λ v ε (s) 2 , (g(u ε (s), v ε (s)), v ε (s)) ≤ C v ε (s) + L g ( u ε (s) v ε (s) + v ε (s) 2 ), 1
Q 2 v ε (s) ≤ C v ε (s) . By Lemma 2.1 and recall that λ − 2L g = η, we have 2 pλ d E v ε (t) 2 p ≤ − E v ε (t) 2 p dt ε 2p + E v ε (t) 2 p−2 (C v ε (t) + L g ( u ε (t) v ε (t) + v ε (t) 2 )) ε Cp C p( p − 1) + E v ε (t) 2 p−2 + E v ε (t) 2 p−2 ε ε ηp C( p) C( p) E u ε (t) 2 p + . ≤ − E v ε (t) 2 p + ε ε ε This implies ε
E v (t)
2p
2 p − ηp ε t
≤ v0 e
C( p) + ε
t
e−
ηp ε (t−s)
(1 + E u ε (s) 2 p )ds.
0
For u ε (t) 2 p , by Itô formula again, the following holds, t E u ε (t) 2 p = u 0 2 p + 2 pE u ε (s) 2 p−2 (−u εx x x (s) + iu εx x (s), u ε (s))ds 0 t u ε (s) 2 p−2 (i|u ε (s)|2 u x (s), u ε (s))ds + 2 pE 0 t u ε (s) 2 p−2 (−i f (u ε (s), v ε (s)), u ε (s))ds. + 2 pE 0
123
(4.7)
Averaging Principle for the Higher...
Direct calculation yields (−u εx x x (s) + iu εx x (s), u ε (s)) = Re [−u εx x x (x, s) + iu εx x (x, s)]u¯ ε (x, s)d x I 1 [−u εx x x (x, s) + iu εx x (x, s)]u¯ ε (x, s)d x = 2 I 1 [−u¯ εx x x (x, s) − i u¯ εx x (x, s)]u ε (x, s)d x + 2 I 1 = − |u εx (0, s)|2 , 2
(i|u ε (s)|2 u x (s), u ε (s)) = Re
i|u ε (x, s)|4 d x = 0,
I
(−i f (u ε (s), v ε (s)), u ε (s)) ≤ C u ε (s) + L f ( u ε (s) v ε (s) + u ε (s) 2 ). Thus, we have E u ε (t) 2 p ≤ u 0 2 p +2 pE u ε (s) 2 p−2 (C u ε (s) +L f ( u ε (s) v ε (s) + u ε (s) 2 )) ds. This gives
d E u ε (t) 2 p ≤ C( p) E u ε (t) 2 p + E v ε (t) 2 p + 1 . dt
Then, we have ε
E u (t)
2p
≤ u 0 e
2 p C( p)t
t
+ C( p)
eC( p)(t−s) (1 + E v ε (s) 2 p )ds.
0
Plugging this inequality into (4.7), we have E v ε (t) 2 p ≤ C( p, T )(1 + v0 2 p ) s C( p, T ) t − ηp (t−s) C( p)s e e ε u 0 2 p + C( p) eC( p)(s−τ ) (1 + E v ε (τ ) 2 p )dτ ds + ε 0 0 t s ηp C( p, T ) ≤ C( p, T )(1 + u 0 2 p + v0 2 p ) + e− ε (t−s) E v ε (τ ) 2 p dτ ds ε 0 0 t 2p 2p E v ε (τ ) 2 p dτ. ≤ C( p, T )(1 + u 0 + v0 ) + C( p, T ) 0
The Gronwall inequality implies the first inequality in Proposition 4.2, and then the second one.
Next, we will present the estimate on the slow motion u ε as a process valued in H α (I ). Proposition 4.3 Let τ = τ∞ ∧ T. For any u 0 ∈ H α (I ), v0 ∈ L 2 (I ), T > 0 and p > 0, we have sup E sup u ε (t) H α ≤ C( p, T )
(4.8)
E sup v ε (t) 2 p ≤ C(ε, p, T ).
(4.9)
2p
ε∈(0,1)
t∈[0,τ ]
t∈[0,τ ]
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P. Gao
Proof It is sufficient to prove (4.8) and (4.9) when p is large enough. We can write the first equation in system (1.1) in its integral form t t u ε (t) = S1 (t)u 0 + i S1 (t − s)(|u ε |2 u ε )(s)ds − i S1 (t − s) f (u ε (s), v ε (s))ds 0
0
= I1 + I2 + I3 . Using (3.12), we can show that 2p
2p
I1 H α ≤ C( p) u 0 H α . It follows from (3.1)(β = α2 + 45 ) that t I2 H α ≤ S1 (t − s)(|u ε |2 u ε )(s) H α ds 0 t 2α+5 ≤ C(T ) (t − s)− 8 (|u ε |2 u ε )(s) 2α−5 ds H 4 0 t 2α+5 ≤ C(T ) (t − s)− 8 (|u ε |2 u ε )(s) L 1 ds 0 t 2α+5 ≤ C(T ) (t − s)− 8 u ε (s) L ∞ u ε (s) 2 ds 0 t 1 2α+5 5 ≤ C(T ) (t − s)− 8 u ε (s) H2 1 u ε (s) 2 ds 0 t 1 2α+5 5 ≤ C(T ) (t − s)− 8 u ε (s) H2 α u ε (s) 2 ds 0
where we have used Lemma 2.3 (s =
2α−5 4 )
and Lemma 2.4. Then, we can deduce that
2p
E sup I2 H α t∈[0,τ ]
t
2 p 1 2α+5 5 ≤ C( p, T )E sup (t − s)− 8 u ε (s) H2 α u ε (s) 2 ds t∈[0,τ ] 0
t t 2 p−1 p − (2α+5) p sup ≤ C( p, T )E sup (t − s) 4(2 p−1) ds u ε (s) H α u ε (s) 5 p ds t∈[0,τ ] 0
≤ C( p, T )
T
s
p − (2α+5) 4(2 p−1)
ds
2 p−1
0
0
t∈[0,τ ] 0
T
E u ε (s) H α ds + 2p
T
E u ε (s) 10 p ds .
0
Taking p>
4 3 − 2α
and using Proposition 4.2, we have
2p E sup I2 H α ≤ C( p, T ) 1 + t∈[0,τ ]
According to (3.1) for β = γ = α, if p > 1/(2 − α),
123
T 0
E u ε (s) H α ds 2p
Averaging Principle for the Higher...
we can obtain that 2p
E sup I3 H α t∈[0,τ ]
t
2 p α ≤ C( p, T )E sup (t − s)− 2 f (u ε (s), v ε (s)) ds t∈[0,τ ] 0
t
2 p α ≤ C( p, T )E sup (t − s)− 2 (1 + u ε (s) + v ε (s) )ds
≤ C( p, T )
t∈[0,τ ] 0 T
s
αp − 2 p−1
ds
2 p−1
0
T
(1 + E u ε (s) 2 p + E v ε (s) 2 p )ds
0
≤ C( p, T ). Consequently, we have E sup u
ε
t∈[0,τ ]
2p (t) H α
≤ C( p, T ) 1 + E
T 0
2p u ε (s) H α ds .
Applying Gronwall’s inequality, we complete the proof of (4.8). To prove (4.9), noting that t t t −s t −s 1 t 1 v ε (t) = S2 S2 S2 v0 + g(u ε (s), v ε (s))ds + √ dB ε ε 0 ε ε ε 0 = I1 + I2 + I3 . It is clear that
E sup I1 2 p ≤ C( p, T ) v0 2 p ≤ C( p, T ), 0≤t≤τ
E sup I3 2 p ≤ C(ε, p, T ). 0≤t≤τ
Now, it is sufficient to estimate I2 . Applying Proposition 4.2, we can deduce that E sup 0≤t≤τ
I2 2 p
≤ C(ε, p, T )E sup
0≤t≤T T
≤ C(ε, p, T )
t
ε
ε
2 p
(1 + u (s) + v (s) )ds
0
(1 + E u ε (s) 2 p + E v ε (s) 2 p )ds
0
≤ C(ε, p, T ). With the help of the above estimates, we arrive at E sup v ε (t) 2 p ≤ C(ε, p, T ). 0≤t≤τ
We complete the proof of (4.9).
4.3 Proof of Theorem 4.1 Now, we prove Theorem 4.1. Proof of Theorem 4.1 By Chebyshev inequality, Proposition 4.3 and the definition of (u ε , v ε ), we have
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P. Gao
P({ω ∈ | τ∞ (ω) < +∞}) = lim P({ω ∈ | τ∞ (ω) ≤ T }) T →+∞
= lim P({ω ∈ | τ (ω) = τ∞ (ω)}) T →+∞
= lim
lim P({ω ∈ | τ R (ω) ≤ τ (ω)})
= lim
lim P({ω ∈ | (u ε , v ε ) X τ ≥ (u ε , v ε ) X τ R })
= lim
lim P({ω ∈ | (u ε , v ε ) X τ ≥ R})
≤ lim
lim
T →+∞ R→+∞ T →+∞ R→+∞ T →+∞ R→+∞
E (u ε , v ε ) 2X τ R2
T →+∞ R→+∞
= 0,
this shows that P({ω ∈ | τ∞ (ω) = +∞}) = 1, namely, τ∞ = +∞ P-a.s.
5 Auxiliary Process (uˆ ε , vˆ ε ) First, we introduce an auxiliary process (uˆ ε (t), vˆ ε (t)). Fix a positive number δ and do a partition of time interval [0, T ] of size δ. We construct a process vˆ ε (t) by means of the equation t 1 t ε 1 t 1 ε ε ε ε vˆ (t) = v (kδ) + vˆ (s)ds + g(u (kδ), vˆ (s))ds + √ dB ε kδ x x ε kδ ε kδ for t ∈ [kδ, min((k + 1)δ, T )). Also, define the process uˆ ε (t) by a linear equation t t t ε ε ε ε 2 ε |uˆ (sδ )| uˆ (sδ )ds−i f (u ε (sδ ), vˆ ε (s))ds uˆ (t) = u 0 + (−uˆ x x x (s)+i uˆ x x (s))ds+i 0
0
0
[ δs ]δ
is the nearest breakpoint proceeding s. for t ∈ [0, T ], where sδ = Proceeding as in Proposition 4.2, we have Proposition 5.1 For any u 0 ∈ H α (I ), v0 ∈ L 2 (I ), p > 0 and T > 0, there exists a constant C( p, T ) such that 2p sup E sup uˆ ε (t) H α ≤ C( p, T ), ε∈(0,1)
sup
t∈[0,T ]
sup E vˆ ε (t) 2 p ≤ C( p, T ).
ε∈(0,1) t∈[0,T ]
To show the convergence of (u ε , v ε ) to (uˆ ε , vˆ ε ), we need to provide a Hölder continuity of time variable for u ε . Proposition 5.2 For any u 0 ∈ H α (I ), v0 ∈ L 2 (I ), T > 0, 0 < t ≤ t + h ≤ T and p > 0, sup E u ε (t + h) − u ε (t) 2 p ≤ C( p, T )h (α−1) p ,
ε∈(0,1)
where C( p, T ) is a positive constant independent of ε.
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Averaging Principle for the Higher...
Proof It is not difficult to show that
u ε (t + h) − u ε (t) = (S1 (h) − I )u ε (t) + i −i
t+h
S1 (t + h − s)(|u ε |2 u ε )(s)ds
t t+h
S1 (t + h − s) f (u ε (s), v ε (s))ds.
t
It follows from (3.1)(β = 3 − α, γ = 3) that for any z ∈ H α (I ),
h
d S1 (s)z
(S1 (h) − I )z =
ds
ds 0
h
=
A S (s)zds 1 1
0 h ≤C S1 (s)z H 3 ds 0
≤C(T )
h
s−
3−α 2
z H α ds
0
≤C(T )h
α−1 2
z H α .
Thus, we can obtain from Propositions 4.2 and 4.3 that E u ε (t + h) − u ε (t) 2 p
2 p
≤ C( p)E (S1 (h) − I )u (t) S1 (t + h − s)(|u | u )(s)ds
t
t+h
2 p
+ C( p)E
S1 (t + h − s) f (u ε (s), v ε (s))ds
t
t+h 2 p 2p ≤ C( p, T )h (α−1) p E u ε (t) H α + C( p)E S1 (t + h − s)(|u ε |2 u ε )(s) ds ε
+ C( p)E
+ C( p)E
2p
t+h
ε 2 ε
t
t+h
S1 (t + h − s) f (u ε (s), v ε (s)) ds
2 p
t
≤ C( p, T )h (α−1) p + C( p)E
+ C( p)E
t+h
(|u ε |2 u ε )(s) ds
t t+h
f (u ε (s), v ε (s)) ds
2 p
t
≤ C( p, T )h (α−1) p + C( p)E
t+h
t
t+h
+ C( p)h 2 p−1 E
2 p
u ε (s) 3H 1 ds
2 p
f (u ε (s), v ε (s)) 2 p ds
t
≤ C( p, T )h (α−1) p + C( p)h 2 p E sup u ε (s) H α 6p
s∈[0,T ]
+ C( p)h
2 p−1
t+h
E
(1 + u ε (s) 2 p + v ε (s) 2 p )ds
t
≤ C( p, T )h (α−1) p . This ends the proof of Proposition 5.2.
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P. Gao
Now we can establish convergence of uˆ ε to u ε and vˆ ε to v ε , respectively. Proposition 5.3 For any u 0 ∈ H α (I ), v0 ∈ L 2 (I ), p ≥ 2, ε ∈ (0, 1) and T > 0, there exists a constant C( p, T ) such that ε
2p
≤ C( p, T ) δ
ε
ε
2p
δ (α−1) p+1 ≤ C( p, T ) . ε
E sup u (t) − uˆ (t) t∈[0,T ]
sup E v (t) − vˆ (t)
t∈[0,T ]
(α−1) p
δ (α−1) p+1 + ε
ε
, (5.1)
Proof Since the proof is similar to that of [37, Lemma 4.7], we just sketch it. It follows from Itô formula and Proposition 5.2 that d E v ε (t) − vˆ ε (t) 2 p dt p C( p) E u ε (t) − u ε (kδ) 2 p ≤ − (λ − L g )E v ε (t) − vˆ ε (t) 2 p + ε ε p C( p, T ) ≤ − (λ − L g )E v ε (t) − vˆ ε (t) 2 p + (t − kδ)(α−1) p . ε ε This implies E v ε (t) − vˆ ε (t) 2 p ≤
C( p, T ) ε
t
p
e− ε (λ−L g )(t−s) (s − kδ)(α−1) p ds
kδ
δ (α−1) p+1 ≤ C( p, T ) . ε This proves the first inequality in (5.1). To obtain the second inequality in (5.1), we recall that t u ε (t) − uˆ ε (t) = i S1 (t − s)[|u ε (s)|2 u ε (s) − |u ε (sδ )|2 u ε (sδ )]ds 0 t S1 (t − s)[ f (u ε (s), v ε (s)) − f (u ε (sδ ), vˆ ε (s)]ds. −i 0
Similar as in [37], we can deduce that E sup u ε (t) − uˆ ε (t) 2 p t∈[0,T ]
≤ C( p, T )
T
0
E u ε (s) − u ε (sδ ) 4 p ds
1 2
0
T
E u ε (s) H 1 + E u ε (sδ ) H 1 ds 8p
8p
1 2
E u ε (s) − u ε (sδ ) 2 p + E v ε (s) − vˆ ε (s) 2 p ds 0 δ (α−1) p+1 (α−1) p + . ≤ C( p, T ) δ ε + C( p, T )
T
The proof is complete.
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Averaging Principle for the Higher...
6 Averaged Process u˜ We first consider the frozen equation associated to fast motion for fixed slow component ⎧ ⎨dv = (vx x + g(u, v))dt + d B in I × (0, +∞) v(0, t) = v(1, t) = 0 in (0, +∞) (6.1) ⎩ v(x, 0) = v0 (x) in I. We denote v u,v0 the solution to (6.1), we now discuss the asymptotic behavior of the fast equation (6.1). By the same method as in [36, Lemma 3.1] and [44], we can obtain the existence of the invariant measure for (6.1), namely, we have Proposition 6.1 For u, X, Y ∈ L 2 (I ), (i) There exists a positive constant C such that v u,X and v u,Y satisfy: E v u,X (t) 2 ≤ e−2ηt X 2 + C( u 2 + 1), E v u,X (t) − v u,Y (t) 2 ≤ X − Y 2 e−2ηt ,
(6.2)
for t ≥ 0. (ii) There is unique invariant measure μu for the Markov semigroup Ptu associated with system (6.1) in L 2 (I ). Moreover, we have z 2 μu (dz) ≤ C(1 + u 2 ). L 2 (I )
(iii) There exists a positive constant C such that v u,X satifies: E f (u, v u,X ) − f˜(u) 2 ≤ C(1 + X 2 + u 2 )e−2ηt for t ≥ 0. The proof of Proposition 6.1 can be found in Appendix. Now we consider the averaged equation: ⎧ ˜ in I × (0, T ) ⎨u˜ t + u˜ x + u˜ x x x + u˜ u˜ x = f˜(u) u(0, ˜ t) = u(1, ˜ t) = u˜ x (1, t) = 0 in (0, T ) ⎩ u(x, ˜ 0) = u 0 (x) in I, where f˜(u) =
(6.3)
L 2 (I )
f (u, v)μu (dv)
and μu is an invariant measure for (6.1). Proceeding as in Proposition 4.2, we have Proposition 6.2 For any u 0 ∈ H α (I ), p > 0 and T > 0, there exists a constant C( p, T ) such that 2p E sup u(t) ˜ H α ≤ C( p, T ), t∈[0,T ]
where u˜ is the solution to (6.3). Next, In order to prove that the slow component process u ε converges strongly to the averaged process u, ˜ we have to give the error between uˆ ε and u. ˜
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P. Gao
Proposition 6.3 For any u 0 ∈ H α (I ), v0 ∈ L 2 (I ), p ≥ 2, ε ∈ (0, 1) and T > 0, there exists a constant C( p, T ) such that
2p E sup uˆ ε (t) − u(t) ˜ t∈[0,T ]
1 δ (α−1) p+1 δ (α−1) p+ 2 ε ≤ C( p, T ) √ + C( p, T ) δ (α−1) p + + + δ 2 p−1 + + √ ε δ ε n 1
ε C( p,T )n 4 p . e δ
Proof We define the stoping time for any n ≥ 1 and ε > 0: ˜ τnε = inf{t > 0 : uˆ ε (t) H α + u(t) H α > n}. First, we intend to show that
2p E sup uˆ ε (t) − u(t) ˜ · χ{T ≤τnε } t∈[0,T ]
1
δ (α−1) p+1 δ (α−1) p+ 2 ε C( p,T )n 4 p + + δ 2 p−1 + e . ≤ C( p, T ) δ (α−1) p + √ ε δ ε
(6.4)
In fact, it is not difficult to see that t ˜ =i S1 (t − s)[|u ε (sδ )|2 u ε (sδ ) − |u ε (s)|2 u ε (s)]ds uˆ ε (t) − u(t) 0 t +i S1 (t − s)[|u ε (s)|2 u ε (s) − |uˆ ε (s)|2 uˆ ε (s)]ds 0 t 2 S1 (t − s)[|uˆ ε (s)|2 uˆ ε (s) − |u(s)| ˜ u(s)]ds ˜ +i 0 t S1 (t − s)[ f (u ε (sδ ), vˆ ε (s)) − f˜(u ε (s))]ds −i 0 t S1 (t − s)[ f˜(u ε (s)) − f˜(uˆ ε (s))]ds −i 0 t −i S1 (t − s)[ f˜(uˆ ε (s)) − f˜(u(s))]ds ˜ 0
:=
6
Jk (t).
k=1
• For J1 (t), we can obtain from (3.14) that
2 p S1 (t − s)[|u ε (sδ )|2 u ε (sδ ) − |u ε (s)|2 u ε (s)] ds 0
t 2 p ≤ C( p, T ) |u ε (sδ )|2 u ε (sδ ) − |u ε (s)|2 u ε (s) ds 0
t 2 p ≤ C( p, T ) u ε (sδ ) − u ε (s) ( u ε (sδ ) 2H 1 + u ε (s) 2H 1 )ds
J1 (t) 2 p ≤
t
0
t 1 t 1 2 2 8p 8p ≤ C( p, T ) u ε (sδ ) − u ε (s) 4 p ds ( u ε (sδ ) H α + u ε (s) H α )ds . 0
123
0
Averaging Principle for the Higher...
It follows from Propositions 4.3 and 5.2 that E
sup
t∈[0,T ∧τnε ]
J1 (t) 2 p
≤ C( p, T )
T
0
ε
ε
E u (sδ ) − u (s) ds 4p
1
T
2
0
(E u ε (sδ ) H α + E u ε (s) H α )ds 8p
8p
1 2
≤ C( p, T )δ (α−1) p . • For J2 (t), similar as in the estimate of J1 , it follows from Proposition 5.3 that E
sup
t∈[0,T ∧τnε ]
J2 (t) 2 p
T 1 T 1 2 2 8p 8p ≤ C( p, T ) E u ε (s) − uˆ ε (s) 4 p ds (E u ε (s) H α + E uˆ ε (s) H α )ds 0 0 (α−1) p+ 21 δ ≤ C( p, T ) δ (α−1) p + . √ ε
• For J3 (t), recalling the definition of stop time τnε , we can deduce that E
sup
t∈[0,T ∧τnε ]
J3 (t) 2 p
≤ C( p, T )E
t
sup
t∈[0,T ∧τnε ] 0
≤ C( p, T )n 4 p E ≤ C( p, T )n 4 p
2 uˆ ε (s) − u(s) ( ˜ uˆ ε (s) 2H α + u(s) ˜ H α )ds
t
sup
t∈[0,T ∧τnε ] 0
T
E
0
sup
r ∈[0,s∧τnε ]
uˆ ε (s) − u(s) ds ˜
2 p
2 p
uˆ ε (r ) − u(r ˜ ) 2 p ds.
• For J4 (t), we can prove that
E
sup
t∈[0,T ∧τnε ]
J4 (t)
2p
ε ≤ C( p, T ) + δ
ε + δ (α−1) p + δ 2 p−1 . δ
(6.5)
The proof of (6.5) can be found in Appendix. • For J5 (t), using Propositon 5.3, we have E
sup
t∈[0,T ∧τnε ]
T
J5 (t) 2 p ≤ C( p, T )E
u ε (s) − uˆ ε (s) 2 p ds
0
δ (α−1) p+1 ≤ C( p, T ) δ (α−1) p + ε
.
• For J6 (t), it is easy to show that E
sup
t∈[0,T ∧τnε ]
T
J6 (t) 2 p ≤ C( p, T ) 0
E
sup
r ∈[0,s∧τnε ]
uˆ ε (r ) − u(r ˜ ) 2 p ds.
Consequently, it is not difficult to see that (6.4) can be obtained from the above estimates.
123
P. Gao
On the other hand, according to Propositions 5.1 and 6.2, we have
2p ˜ · χ{T >τnε } E sup uˆ ε (t) − u(t) t∈[0,T ]
1 1
2 4p 2 P(T > τnε ) ˜ ≤ E sup uˆ ε (t) − u(t) t∈[0,T ]
1
4p 2 ≤ C( p) E sup uˆ ε (t) 4 p + E sup u(t) ˜ t∈[0,T ]
t∈[0,T ]
(6.6)
1 1 2 × √ E sup uˆ ε (t) H α + E sup u(t) ˜ Hα n t∈[0,T ] t∈[0,T ] 1 ≤ C( p, T ) √ . n
Combining (6.4) and (6.6), the proof is complete.
7 Main Result Now, we are in a position to present the main result in this paper. Theorem 7.1 Assume (H1)–(H3) hold, then, for any u 0 ∈ H α (I ), v0 ∈ L 2 (I ) and T > 0, (u ε , v ε ) is the solution of (1.1) and u˜ is the solution of the averaged equation (6.3), then for p > max 2,
5 , 4(α − 1)
we have
1 ˜ ) ≤ C( p, T ) E( sup u (t) − u(t) − ln ε 0≤t≤T ε
2p
1 8p
.
Proof Combining the results in Propositions 5.3 and 6.3, we have
2p E sup u ε (t) − u(t) ˜ t∈[0,T ]
2p ≤ C( p)E sup u ε (t) − uˆ ε (t) 2 p + C( p)E sup uˆ ε (t) − u(t) ˜ t∈[0,T ]
t∈[0,T ]
1 ≤ C( p, T ) √ + C( p, T ) n 1
δ (α−1) p+1 ε C( p,T )n 4 p δ (α−1) p+ 2 ε e + . × δ (α−1) p + + δ 2 p−1 + + √ ε δ δ ε Taking
1 2
δ = ε and n =
123
4p
−
1 ln ε, 8C( p, T )
Averaging Principle for the Higher...
we obtain that
2p E sup u ε (t) − u(t) ˜ t∈[0,T ]
1 81p − ln ε 1
(α−1) p (α−1) p 1 (α−1) p 1 1 1 1 + C( p, T ) ε 2 + ε 2 − 4 + ε 2 − 2 + ε p− 2 + ε 2 + ε 4 ε − 8 .
≤ C( p, T )
If p > max 2, we have
E
5 , 4(α − 1)
ε
sup u (t) − u(t) ˜
2p
≤ C( p, T )
0≤t≤T
1 − ln ε
1 8p
.
This ends the proof of Theorem 7.1.
8 Appendix 8.1 Proof of Proposition 6.1 (i) By applying the generalized Itô formula with 21 v u,X 2 , we can obtain that t t t 1 u,X 2 1 2 u,X u,X u,X v = X + (v , vx x + g(u, v ))ds + (v u,X , d B)+ TrQ 2 2 2 0 0 t t t t 1 2 u,X 2 u,X u,X vx ds + (v , g(u, v ))ds + (v u,X , d B) + TrQ. = X − 2 2 0 0 0 Taking mathematical expectation from both sides of above equation, we have t t 1 t 1 E vxu,X 2 ds + E(v u,X , g(u, v u,X ))ds + TrQ, E v u,X 2 = E X 2 − 2 2 2 0 0 namely, d E v u,X 2 = −2E vxu,X 2 + 2E(v u,X , g(u, v u,X )) + TrQ. dt It follows from Young’s inequality that d E v u,X 2 dt ≤ −2E vxu,X 2 + 2E(v u,X , g(u, v u,X )) + TrQ ≤ −2E vxu,X 2 + 2E( v u,X g(u, v u,X ) ) + C ≤ −2E vxu,X 2 + 2E[ v u,X (L g ( u + v u,X ) + C)] + C ≤ −2λE v u,X 2 + 2L g E v u,X 2 + 2L g E( v u,X u ) + CE v u,X + C ≤ −2λE v u,X 2 + 4L g E v u,X 2 + C u 2 + C = −2ηE v u,X 2 + C u 2 + C.
123
P. Gao
Hence, by applying Lemma 2.2 with E v u,X (t) 2 , we have E v u,X (t) 2 ≤ e−2ηt X 2 + C( u 2 + 1). It is easy to see that ⎧ ⎨ d(v u,X − v u,Y ) = [(v u,X − v u,Y )x x + g(u, v u,X ) − g(u, v u,Y )]dt (v u,X − v u,Y )(0, t) = (v u,X − v u,Y )(1, t) = 0 ⎩ u,X (v − v u,Y )(x, 0) = X (x) − Y (x)
in I × (0, T ) in (0, T ) in I,
thus, it follows from the energy method that 1 u,X v − v u,Y 2 2 t 1 = X − Y 2 + (v u,X − v u,Y , (v u,X − v u,Y )x x + g(u, v u,X ) − g(u, v u,Y ))ds 2 0 t t 1 (v u,X − v u,Y )x 2 ds + (v u,X − v u,Y , g(u, v u,X ) − g(u, v u,Y ))ds, = X − Y 2 − 2 0 0
namely, d u,X − v u,Y 2 = −2 (v u,X − v u,Y )x 2 + 2(v u,X − v u,Y , g(u, v u,X ) − g(u, v u,Y )). v dt Thus, we have d u,X − v u,Y 2 ≤ − 2 (v u,X − v u,Y )x 2 + 2L g v u,X − v u,Y 2 v dt ≤ − 2λ v u,X − v u,Y 2 + 2L g v u,X − v u,Y 2 ≤ − 2λ v u,X − v u,Y 2 + 4L g v u,X − v u,Y 2 = − 2(λ − 2L g ) v u,X − v u,Y 2 = − 2η v u,X − v u,Y 2 , this yields Thus, we have
v u,X − v u,Y 2 ≤ X − Y 2 e−2ηt . E v u,X − v u,Y 2 ≤ X − Y 2 e−2ηt .
(ii) (6.2) implies that for any u ∈ L 2 (I ) that there is unique invariant measure μu for the Markov semigroup Ptu associated with the system (8.2) in L 2 (I ) such that u u Pt ϕdμ = ϕdμu , t ≥ 0 L 2 (I )
L 2 (I )
for any ϕ ∈ Bb (L 2 (I )) the space of bounded functions on L 2 (I ). Then by repeating the standard argument as in [23, Proposition 4.2] and [21, Lemma 3.4], the invariant measure satisfies z 2 μu (dz) ≤ C(1 + u 2 ). L 2 (I )
123
Averaging Principle for the Higher...
(iii) According to the invariant property of μu , (ii) and (6.2), we have
2
u,X u
E f (u, v ) − f (u, Y )μ (dY ) E f (u, v u,X ) − f˜(u) 2 =
L 2 (I )
u,X E f (u, v =
) − E
L 2 (I )
2
f (u, v u,Y )μu (dY )
2
u,X u,Y u
=
E[ f (u, v ) − f (u, v )]μ (dY )
2 L (I ) ≤C E v u,X − v u,Y 2 μu (dY ) 2 L (I ) ≤C X − Y 2 e−2ηt μu (dY ) L 2 (I )
≤ C(1 + X 2 + u 2 )e−2ηt .
8.2 Proof of (6.5) set n t = [ δt ], then for t ∈ [n t δ, (n t + 1)δ ∧ T ], we can write J4 (t) =
n t −1 (k+1)δ
+
n t −1 (k+1)δ k=0 t
+
S1 (t − s)[ f (u ε (kδ), vˆ ε (s)) − f˜(u ε (kδ))]ds
kδ
k=0
nt δ
S1 (t − s)[ f˜(u ε (kδ)) − f˜(u ε (s))]ds
kδ
S1 (t − s)[ f (u ε (n t δ), vˆ ε (s)) − f˜(u ε (s))]ds
:= J41 (t) + J42 (t) + J43 (t). For J41 (t), by a time shift transformation, we can obtain that for any k and t ∈ [0, δ), kδ+s 1 kδ+s ε 1 kδ+s 1 vˆ ε (s + kδ) = v ε (kδ) − vˆ x x (r )dr + g(u ε (kδ), vˆ ε (r ))dr + √ dB ε kδ ε kδ ε kδ s s s 1 1 1 ˜ vˆ ε (r + kδ)dr + g(u ε (kδ), vˆ ε (r + kδ))dr + √ d B, = v ε (kδ) − ε 0 xx ε 0 ε 0 ˜ where B(t) := B(t + kδ) − B(kδ) is the shift version of B(t) and hence they have the same distribution. ¯ Let B(t) be a Q−Wiener process defined on the same stochastic basis and independent ε ε of B(t). We construct a process v u (kδ),u (kδ) by means of
s ε ε v u (kδ),v (kδ) ε s s s ε ε ε ε ε ε ε = v ε (kδ) − vxux(kδ),v (kδ) (r )dr + g(u ε (kδ), v u (kδ),v (kδ) (r ))dr + d B¯ 0 0 0
1 s ε 1 s u ε (kδ),v ε (kδ) r u ε (kδ),v ε (kδ) r dr + dr = v (kδ) − v g u (kδ), vx x ε 0 xx ε ε 0 ε s 1 ¯¯ d B, +√ ε 0 ε
123
P. Gao
¯¯ is the scaled version of B(t). ¯ where B(t) By the uniqueness of the solution, we have
s
ε ε , (u ε (kδ), vˆ ε (s + kδ)) u ε (kδ), vxux(kδ),v (kδ) ε where denotes a coincidence in distribution sense. Then, proceeding as in [26], according to Proposition 6.1 and the method in [26–31,38], we can prove that E
sup
t∈[0,T ∧τnε ]
J41 (t) 2 ≤ C(T )
where
1
k (s, τ ) = E
ε2 δ2
δ ε
max
k∈[0,[ Tδ ]−1] 0
δ ε
τ
k (s, τ )dsdτ,
S1 (δ − sε)[ f (u ε (kδ), vˆ ε (sε + kδ)) − f˜(u ε (kδ))]
0
× S1 (δ − τ ε)[ f (u ε (kδ), vˆ ε (τ ε + kδ)) − f˜(u ε (kδ))]d x ≤ C( p)e−(s−τ )ρ (ρ is a positive constant). Thus we have ε2 E sup ≤C(T ) 2 δ t∈[0,T ∧τnε ] 2 ε ε ≤ C(T ) 2 + . δ δ
J41 (t) 2
δ 1 δ 1 1 · − 2 + e−ρ ε · 2 ρ ε ρ ρ
(8.1)
Besides, applying Propositions 4.2 and 5.1, we can obtain that E
sup
t∈[0,T ∧τnε ]
J41 (t) 2 p ≤ E
s s 2 p
δ , vˆ ε (s) + f˜ u ε δ ds
f uε δ δ 0
≤ C( p, T ) 1 + sup E u ε (s) 2 p + sup E vˆ ε (s) 2 p T
s∈[0,T ]
s∈[0,T ]
≤ C( p, T ). (8.2) Combining (8.1) and (8.2), we have E
sup
t∈[0,T ∧τnε ]
J41 (t) 2 p ≤ E
sup
t∈[0,T ∧τnε ]
J41 (t) 2(2 p−1)
ε ≤C( p, T ) + δ
1
ε . δ
2
E
sup
t∈[0,T ∧τnε ]
J41 (t) 2
1 2
(8.3)
For J42 (t), by Proposition 5.2, we can show that E
sup
t∈[0,T ∧τnε ]
J42 (t) 2 p
≤ C( p, T ) 0
≤ C( p, T )δ
123
T
E u ε (sδ ) − u ε (s) 2 p ds
(α−1) p
.
(8.4)
Averaging Principle for the Higher...
For J43 (t), by Propositions 4.2 and 5.1, we can show that E
sup
t∈[0,T ∧τnε ]
J43 (t) 2 p ≤ C( p)δ 2 p−1 E
T
(1 + u ε (n t δ) 2 p + vˆ ε (s) 2 p + u ε (s) 2 p )ds
0
≤ C( p, T )δ 2 p−1 . (8.5) We conclude from (8.3)–(8.5) that (6.5) holds. Acknowledgements I sincerely thank Professor Yong Li for many useful suggestions and help.
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