Averaging principles for the Swift-Hohenberg equation

This work studies the effects of rapid oscillations (with respect to time) of the forcing term on the long-time behaviour of the solutions of the Swift-Hohenberg equation. More precisely, we establish three kinds of averaging principles for the Swift-Hohenberg equation, they are averaging principle in a time-periodic problem, averaging principle on a finite time interval and averaging principle on the entire axis.


1.
Introduction. In studies of pattern formation, the Swif t−Hohenberg equation plays a central role. It is a widely accepted model for the thermal convection in a thin layer of fluid heated from below, it describes the pattern formation in fluid layers confined between horizontal well-conducting boundaries. Proposed in 1977 by Swift and Hohenberg [35] in connection with Rayleigh-Bénard convection, it has since featured in a variety of problems, such as Taylor-Couette flow [19,30], and in the study of lasers [25]. We also refer to the surveys given in [6,7] and the recent review [3].
The study of stability of nonautonomous nonlinear systems is known as a hard problem. One of the techniques used to simplify the consideration is the averaging method consisting in replacing the original system by an averaged autonomous one which has similar property but its analysis is easier. The averaging of a partial differential equation is a process which consists in showing the convergence of the solution of an equation with rapidly varying coefficients towards an equation with simpler (e.g. constant) coefficients.
Starting from the fundamental work of Bogolyubov [2], the averaging theory for ordinary differential equations has been developed and generalized in a large number of works (see [5,11,29] and the references therein). Bogolyubov's main theorems have been generalized in [10] to the case of differential equations with bounded operator-valued coefficients. Some problems of averaging of differential equations with unbounded operator-valued coefficients have been considered in [10,20,28,36] in the framework of abstract parabolic equations. [8] obtains a general version of the Bogolyubov averaging lemma for a time-varying differential equation. The works [10,21,22] are devoted to generalization of method of averaging for dissipative partial differential equations. [8] uses the concept of pullback attractors of such systems to establish the existence of almost periodic (quasi-periodic, almost automorphic, recurrent, pseudo recurrent) solutions corresponding to time dependent coefficients of these types and a global averaging principle is derived. [17] establishes averaging principle for quasi-geostrophic motion. [18] derives an averaging principle for the 2D quasi-geostrophic flow.
• Does the averaging principle on a finite time interval for (1) hold ?
• Does the averaging principle on the entire axis for (1) hold ?
In this paper we will answer these questions.
1.1. Averaging principle in a time-periodic problem. We first study the behavior of the set of time-periodic solutions of the Swift-Hohenberg equation as the frequency of the oscillations of the right-hand side tends to infinity, where f ∈ C(R, H 1 0 (I)) and f (x, t + T ) = f (x, t). It is established that the set of periodic solutions tends to the solution set of the averaged stationary equation AVERAGING PRINCIPLES FOR THE SWIFT-HOHENBERG EQUATION   295 where f 0 (x) = 1 T T 0 f (x, t)dt. Denote by U ε the collection of εT −periodic solutions of (2) for a fixed ε > 0. Let U 0 be a collection of solutions of (3).
The first averaging principle for the Swift-Hohenberg equation is as follows: Theorem 1.1 (Averaging principle in a time-periodic problem). Let f ∈ C(R, H 1 0 (I)) and f (x, t + T ) = f (x, t). If we take β large enough, then the semideviation The classical Bogolyubov Lemma is established for the almost periodic solution for ordinary differential equations, such as in [4, P20].
2. The same problem has been established for Navier-Stokes equation in [23].
We prove the averaging principle on a finite time interval, the so-called first Bogolyubov theorem: Theorem 1.2 (Averaging principle on a finite time interval). Let T > 0 be arbitrary and fixed, u andū be the solutions of (6) and (7), B H 1 (I) (R 0 ) be an absorbing ball of (6) in H 1 (I) with radius R 0 . Suppose that the hypothesis (H1) holds, if Remark 2. This theorem gives comparison estimate, stability estimate and convergence result (as ε → 0) between (6) and (7), on finite but large time intervals.
• β is large enough. There exists a function u 0 ∈ H 4 (I) such that in I, The result in this section is the averaging principle on the entire real axis, the so-called second Bogolyubov theorem: Theorem 1.3 (Averaging principle on the entire axis). Suppose that the hypothesis (H2) holds, if ε is small enough, equation (6) has the following properties: 1. In a small neighbourhood of the stationary point u 0 , equation (6) has a unique solution u * (τ ), which is bounded on the entire axis and satisfies: 2. If the function f : R → H 1 (I) is an almost periodic function, then the solution u * is almost periodic with frequency basis contained in that of f. This paper is organized as follows. In Sec. 2, we present the framework and some preliminary results. In Sec. 3-5, we prove Theorem 1.1-Theorem 1.3.

Preliminaries.
2.1. Mathematical setting. We introduce the following mathematical setting: We denote by L 2 (I) the space of all Lebesgue square integrable real-valued functions on I. The inner product on L 2 (I) is (u, v) = I uvdx, for any u, v ∈ L 2 (I). The norm on L 2 (I) is u = (u, u) 1 2 , for any u ∈ L 2 (I). H s (I)(s ≥ 0) are the classical Sobolev spaces of real-valued functions on I. The definition of H s (I) can be found in [27], the norm on H s (I) is · H s (I) .
The letter C with or without subscripts denotes positive constants whose value may change in different occasions. We will write the dependence of constant on parameters explicitly if it is essential.
λ * > 0 is the smallest constant such that the following inequality holds where u ∈ H 1 0 (I). 2.2. Some auxiliary results. In this section, we give some auxiliary results, which will be used in the proof of main results.
We frequently use the Agmon inequality in one dimension: for any u ∈ H 1 (I), where C is a positive constant depending only on I.
We also use the following property of H 1 (I): for any u 1 , u 2 ∈ H 1 (I), where C is a positive constant depending only on I. By using the classical energy method, we can obtain the following results.
then u satisfies the following identical equation Let S(t) = e At be the semigroup corresponding to the equation Then, we have Proposition 2. If we take β large enough, there exists γ > 0 such that for any k ∈ N.
Proof. The case k = 0 can be obtained from Proposition 1. The case k ≥ 1 can be obtained by the same method as in [12,Proposition 2].
Then, we have the following result: has a unique solution y ∈ C(R; H 1 0 (I)) bounded on the entire axis: Moreover, there exists a positive constant K(γ) such that Proof. It follows from Proposition 2 that .

Proposition 4. [23, P761 Lemma 2]
Denote by χ ε (t) a periodic function with period εT defined by the formula 1 2 − t εT for 0 ≤ t ≤ εT and extended as a εTperiodic function to the whole real axis. For any absolutely continuous (on [0, εT ]) and εT -periodic numerical function η(t) the following equality holds: Here η 0 is the mean value of the function η(t) defined by the equality andη denotes the derivative with respect to s.

3.
Proof of Theorem 1.1. The proof of Theorem 1.1 is divided into several steps.
Step 1. Existence of periodic solutions {u ε } ε>0 . By the same method as in [15,14], we can obtain the following proposition: According to Proposition 5, the following result holds.
Step 2. A priori estimates of periodic solutions {u ε } ε>0 . In all subsequent arguments, the following proposition plays a crucial role. Proposition 6. If we take β large enough and u ε is the εT -periodic solution to (2), then u ε satisfies where C is a constant independent of ε. There exists ε 0 such that when 0 < ε << ε 0 , it holds that where C is a constant independent of ε.
Proof. • Multiplying (2) by u ε and integrating it over I, we get it is easy to see that 1 2 According to [15,Lemma 2.5], we obtain u ε (t) ≤ M β−2 . In view of the εT -periodicity of u ε , we can obtain • Multiplying (2) by −u εxx and integrating it over I, we get According to the interpolation inequality, it holds that According to [15,Lemma 2.5], we obtain u εx (t) ≤ M β−C .

PENG GAO
In view of the εT -periodicity of u ε , we can obtain (2) by u εt and integrating it over I, we get In view of the εT -periodicity of u ε , we can obtain (2) by u εxxxx and integrating it over I and noting that namely, it holds that Thus, we have According to Gagliardo-Nirenberg inequality, we have u x L 4 (I) ≤ C u xx Then, we have By taking β >> 1 and using the Young inequality, we have We set y = u εxx (t) 2 , it holds that d dt y ≤ Cy It follows from the comparison principle of ODE that If we take 0 < ε << 1 such that we obtain an estimate of y(t) which is uniform in [0, εT ], namely, By the εT −periodicity of the function y(t), we have an estimate uniform on the whole axis R, In view of (8) and the εT -periodicity of u ε , we can obtain Step 3. The convergence of u ε − (u ε ) 0 , where (u ε ) 0 = 1 εT εT 0 u ε (t)dt. Proposition 7. There exists ε 0 such that if 0 < ε << ε 0 , it holds that where C is a constant independent of ε.
Proof. Noting the facts that and according to the interpolation inequality, we can obtain (9).

Proposition 8.
There exists a constant C > 0 such that the time-periodic solutions u ε satisfy sup Proof. Now let us multiply both sides of (1) by the function χ ε (t − s) used in Proposition 4 and integrate the result over s from 0 to εT . Then, in view of the εT -periodicity of the function u ε and by Proposition 4, we have According to Proposition 6, it holds that Step 4. The convergence subsequence of {(u ε ) 0 } ε>0 .
Step 5. Proof of Theorem 1.1. Suppose that the assertion of Theorem 1.1 does not hold. Then there exist numbers δ 0 , ε n → 0, and t n ∈ R such that Therefore, for some subsequence of functions u εn k and for numbers t n k ∈ R, we have But this contradicts the fact (see Proposition 9) that the sequence u εn k , contains a subsequence converging to an element from U 0 . Thus, Theorem 1.1 is proved.

Proof of Theorem 1.2.
Proof of Theorem 1.2. It was shown in [16,24], that the nonlinear semigroup F (t) corresponding to equation (4) possesses absorbing sets in the space H 1 (I), in particular, the ball B H 1 (I) (R 0 ), where R 0 is large enough. This means that for every bounded set B in H 1 (I), there exists t(B, R 0 ) > 0 such that In addition, the semigroup is uniformly bounded in this space, that is, given any ball, in particular, the ball B H 1 (I) (R 0 ), there exists a ball B H 1 (I) (R) such that By increasing R we may assume that We suppose that the trajectory of (6) with u(0) ∈ B H 1 (I) (R 0 ) stays in the ball . This will be proved in the end of this section.
Given a point u 0 in B H 1 (I) (R 0 ), let trajectories of systems (6) and (7) start from this point. Set z = u −ū, then z satisfies By the semigroup theory, we can obtain the following equivalent integral equation It is easy to check that re −εγr min (K, σ(r))dr.
By the same method as in [21, P659], according to the above inequalities, we have where G(ε) → 0 as ε → 0. It follows from Gronwall inequality that Namely, by assuming that the trajectory u(t) with initial value u(0) ∈ B H 1 (I) (R 0 ) stays in the ball B H 1 (I) (R) on the interval [0, T ε ], we have proved the proximity of solutions of (6) and (7) in H 1 (I).
Next, we prove the trajectory u(t) with initial condition u(0) ∈ B H 1 (I) (R 0 ) stays in the ball B H 1 (I) (R) on the interval [0, T ε ]. Indeed, let ε be so small that the right-hand side of (13) is less than ρ 2 , where ρ is defined earlier in this section when we discuss absorbing sets. Suppose that the trajectory u(t) leaves the ball B(R) during the interval [0, T ε ] and let τ * be the first moment when u(τ * ) H 1 (I) = R. However, on the interval τ ∈ [0, τ * ] both trajectories of (6) and (7) stay in the ball B H 1 (I) (R) and what we have proved so far shows that the inequality u(τ ) −ū(τ ) H 1 (I) ≤ ρ 2 is valid. In particular, it is valid for τ = τ * . This together with the inequality ū(τ * ) H 1 (I) ≤ R − ρ, which holds by the hypothesis of the theorem, gives the contradiction: and we set F (h, ε, τ ) = N (u 0 + h − εv) − N (u 0 ).

Study of v.
We consider equation (14). If f is almost periodic with value in H 1 (I), then v is almost periodic in H 1 (I) with frequency basis contained in that of f .
Proof. The desired solution is given by the formula Letting δ → 0 and then ε → 0 we obtain lim ε→0 sup τ ∈R εv(τ, ε) H 1 (I) = 0. Let us prove the last statement of the proposition. It is easy to check that it is sufficient to show that every m-recurrent sequence {τ m } is also v-recurrent.
If h 1 , h 2 ∈ B H 1 (I) (ρ) and choosing ρ, ε, u 0 H 1 (I) , m C b (R,H 1 (I)) small enough, F is a contraction map taking a ball in C b (R, H 1 (I)) of radius ρ into itself. By the Banach contraction principle, F has a unique fixed point in other words, h * is a H 1 (I)-bounded solution of equation (16). Proof of Theorem 1.3. According to Proposition 10 and Proposition 11, noting the representation u = u 0 + h − εv, we can prove Theorem 1.3.