Singular perturbed renormalization group theory and its application to highly oscillatory problems

Renormalization group method in the singular perturbation theory, originally introduced by Chen et al, has been proven to be very practicable in a large number of singular perturbed problems. In this paper, we will firstly reconsider the Renormalization group method under some general conditions to get several newly rigorous approximate results. Then we will apply the obtained results to investigate a class of second order differential equations with the highly oscillatory phenomenon of highly oscillatory properties, which occurs in many multiscale models from applied mathematics, physics and material science, etc. Our strategy, in fact, can be also used to analyze the same problem for related evolution equations with multiple scales, such as nonlinear Klein-Gordon equations in the nonrelativistic limit regime.

1. Introduction. Renormalization group (RG) method in the singular perturbation theory was originally introduced by Chen et al [4,5] in 1980s, inspired from the classical renormalization idea in quantum mechanics [10]. The main goal of this method is to compute the effective approximate solution of different kinds of singular perturbation problems in a unified manner. So far, it has been proven to be very practicable in a large number of singular perturbed problems, such as secular problem, boundary layer problem, center manifold problem etc. [5,7,8,9,12,15,16,19].
In 1999, Ziane [21] considered a system of differential equations where A is a complex diagonalizable matrix, f(x) is a polynomial nonlinear term, and ε is a small parameter. After a typically systematic renormalization procedure, he proved two approximate results under certain assumptions.
Recently, Chiba [7] considered a more general class of singular perturbation problems as the following formẋ = εg(x, t, ε), x ∈ U, where U is an open set in C n and the closureŪ is compact, g(x, t, ε) is a vector field parameterized by ε ∈ R + , he presented a higher order RG theory for (1) with a key assumption that the nonlinear terms are almost-periodic in t, and the set of corresponding Fourier exponents having no accumulation on R. Moreover, his work turns out that, in many cases, RG theory can also lead to the existence of approximate invariant manifolds, inheritance of symmetries from those for the original equation to those for the RG equation, and unify traditional singular perturbation methods, such as the averaging method, the multiple time scale method and the center manifold reduction, etc. A natural equation is, how about the renormalization group approach to the problems with more general conditions? The first part of the present paper is to deal with this topic, in detail, we will consider equation (1) with the following assumptions on g(x, t, ε): (G 1 ): g(x, t, ε) is C 2 in x ∈ U , C 1 in t and analytic in ε ∈ I 0 , with I 0 ⊂ R an open neighborhood of the origin. Furthermore, g(x, t, ε) is Lipschitz continues in x on U , i.e., there exists a constant L U > 0 such that g(x, t, ε) − g(y, t, ε) ≤ L U x − y , for all x, y ∈ U, t ∈ R, ε ∈ I 0 .
(G 2 ): g(x, t, ε) is sufficiently smooth in (x, t, ε) and quasi-periodic in t, i.e., it can be rewritten as the following form where ω ∈ R d is a constant vector, i = √ −1, and is valid for some Γ 0 , Γ 1 , σ > 0 and any l ∈ N, x ∈ U . On the other hand, we are also interested in a class of highly oscillatory second order differential equations where˙denotes the derivatives of y respective to the time t, ε is a small real parameter, A, B ∈ R d×d are nonnegative definite matrix and positive definite matrix, respectively, f : C d → C d is a vector-valued analytic function with gauge invariance, i.e., f(e is y) = e is f(y), s ∈ R.
This model comes from a numerical study made by Bao et al [1] of problem (4) with d = 1 and B = E, here E denotes the identity matrix. They are mainly interested in the numerical methods to solve a family of highly oscillation problems like the following nonlinear Klein-Gordon equation in the nonrelativistic limit regime [2] where u = u(x, t) is a real-valued function, 0 < ε 1 is scaled to be inversely proportional to the speed of light, φ and γ are given real-valued functions, f (u) is a dimensionless real-valued function independent of ε and satisfies f (0) = 0. The model (4) was proposed from (5) by finite difference or spectral discretization with a fixed mesh size (see details in [2]). And both of these two models have a same oscillatory phenomenon that propagates high oscillatory waves with wavelength at O(ε 2 ) and amplitude at O(1), which ultimately causes many difficulties in the asymptotic analysis and severe burdens in practical computation, making the numerical approximation extremely challenging and costly in the regime of 0 < ε 1. In their recent work about (4) with d = 1, B = E, Bao et al [1] developed so called multiscale time integrators method give more effective numerical results compared with some classical numerical integrators, such as the finite difference methods and exponential wave integrators used in [2]. Meanwhile, to secure the validity of their above results, one assumption was necessarily proposed, i.e., the exact solution y(t) of (4) satisfies for some constant M > 0 and 0 < T < T * with T * the maximum time, here · represents the Euclidean norm. A similar assumption was also given in their work about equation (5).
In fact, the computational difficulties referred above are caused by the existence of secular terms in the naive approximate solution, this lead us to make a complete investigation of (4) from the point of asymptotic analysis, and it would be a lot better if we can get a rigorous statement of (6). The second part of this paper is using our new results in the first part to give a complete discussion of the highly oscillatory problem (4), and the rigorous estimation of (6) will be presented.
The rest of this paper is organized as follows. In section 2, we will introduce a typically strategy of RG method, and present some necessary estimation for the general cases (G 1 ) and (G 2 ). Highly oscillatory second order singular perturbed problem (4) will be then investigated in section 3.

2.
Renormalization group method. So far, mathematicians have developed many kinds of formulations to the renormalization group theory, each with a intrinsic interest, such as the classical form [5] from the idea of the renormalization in quantum mechanics, or envelope form [8] from the geometric point, etc. In this section, we will present a rigorous strategy of the renormalization group method to (1) in classical procedure in new conditions to get some effective approximate results.
The RG procedure can be naturally presented as flowing three steps.
Up to now, we have been able to get the formal expansion of the solution to (1). However, there remains another important task in this step, that is to figure out the singular terms. In order to deal with this point, in general, more information about g(x, t, ε) should be added. Here we follow a general assumption as referred in [6].
KBM condition: We say g(x, t, ε) satisfies the KBM condition, if for arbitrary t 0 ∈ R + , the limit exists uniformly for y ∈Ū , ε ∈ I 0 . Under the KBM condition, x (1) and x (2) can be further expressed as where According to (10), R i (ξ 0 ) and N i (ξ 0 , t) are well-defined. Then we can present the naive perturbed expansion as RENORMALIZATION GROUP THEORY AND HIGHLY OSCILLATORY PROBLEMS 1823 where · · · denotes the collection of terms whose order in ε are higher than 2.
Step 2. [Renormalization] Now, one can find that (11) are the so called secular terms [11] to cause the nonuniformly valid asymptotic results. How to deal with these terms? It is one of the main problems in the singular perturbation theory . The main idea of the RG method is to renormalize the naive expansion such that all the secular terms could be collected as an unified regular term. An usual way to illustrate this idea can be formulated as following.
Firstly, propose an ansatz free parameter σ ∈ R into x(ξ 0 , t, ε) to replace t i by Then, collect the initial constant vector ξ 0 with all the terms including to a single term ξ(σ, ε), such that x(ξ 0 , t, ε) can be rewritten as In details, expand ξ(σ, ε) = ξ 0 +εA 1 (σ)+ε 2 A 2 (σ)+· · · with A k (σ) to be determined, take it into (12) and equate the same order terms of ε, we can inductively get At last, by noting the independence of x(ξ 0 , t, ε) with respect to σ, we obtain especially, choose σ = t, we get the so called renormalization group (RG) equatioṅ with the initial condition ξ(0) = ξ 0 , and the corresponding approximate solution Step 3.
1. It is well-known that a main advantage of RG method is that it starts with a naive expansion and does not require any further a priori assumptions regarding the structure of the perturbation series, like an anticipation of scales involved in WKB and multiple scale analysis [11]. So far, the RG procedure has also been successfully developed to several kinds of evolution equations, such as singular Navier-Stokes equations [13,14], Primitive equations [17], Schrödinger equations [10], etc. 2. The smoothness assumption about g(x, t, ε) is necessary, and it is not difficult to see that one can get more stronger estimations if the smooth condition is better. For example, δ(ε) can be always obtained as δ(ε) = O(ε) if g(x, t, ε) is periodic in t, or almost-periodic with the set of corresponding Fourier exponents having no accumulation on R. We summarize the corresponding results as the following theorem, please see [7] for more details.
Theorem 2.2. Assume that g(x, t, ε) satisfies (G 1 ), and it is almost-periodic in t with the set of corresponding Fourier exponents having no accumulation on R. Let x(t) be a solution of (1), η(τ ) be the solution of (15) with η(0) = x(0) and the maximum existence interval (a, b). Then, for any closed interval [T 1 , there exists a constant ε 0 > 0, such that for any ε ∈ (0, ε 0 ), as long as T1 ε < t < T2 ε , where C is a positive constant independent to ε.
Proof. The proof can be easily concluded by noting the fact δ(ε) = O(ε) based on the assumption, we omit it here.
In what following, we consider the case (G 2 ) raised in a large number of applications as well as the model (4), it is more general rather than the one in Theorem 2.2. It is also well-known that, for general ω ∈ R d , small divisors may occur in corresponding integrals, such that the KBM condition may not be valid. Therefore, additional constraints should be proposed to overcome this difficulty. Here we introduce the general Diophantine condition.
3. Highly oscillatory problems. In this section, we turn back to consider the asymptotic properties of the highly oscillatory problem (4). Let us make the change of variables where denotes the derivation with respect to x, then (4) becomes the following singular perturbed problem , with O the d × d zero matrix. Without loss of generality, we assume that B is a diagonal matrix, and B = Λ 2 with Λ = diag{λ 1 , · · · , λ d }, λ j > 0, j = 1, · · · , d. Furthermore, we assume that (λ 1 , · · · , λ d ) satisfies the GBD condition. Make the change of variables where and we can naturally get the corresponding approximate solution with where V (t) is the solution of initial value problem with the maximum existence interval (a, b). Obviously, W(x) = V(µx).