PARADIGM FOR THE CREATION OF SCALES AND PHASES IN NONLINEAR EVOLUTION EQUATIONS

The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7, 8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard’s global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.


Introduction
The aim is to exhibit fundamental mechanisms explaining how the complexity of a nonlinear flow can suddenly increase. The phenomenon is illustrated by Theorem 1.5 and Figure 1. We explain how we could have transfer and creation of oscillations. In Subsection 1.1, we present a class of Hamilton-Jacobi equations and related assumptions. In Subsection 1.2, we state our main result. Then, in Subsection 1.3, we explain the link between this class of Hamilton-Jacobi equations and a special class of nonlinear differential equations. The main outcomes concerning the latter class of nonlinear differential equations are detailed in Section 2.
(1. 4) Then, we consider the family of local solutions u ≡ u ε (indexed by ε) to the Cauchy problems (1.5) The above equation involves fast variations in τ (due to the substitution of θ τ for τ /ε) and strong nonlinear effects linked to the substitution of εu for u ∈ R, ∇ x u for p ∈ R d and of u/ε for θ u ∈ T. The Hamilton-Jacobi equation (1.5) is highly oscillating, and therefore the same should apply to its solution u ε . It follows that the asymptotic description of u ε should require the use of several scales and phases: (i) A scale is a power ε with ∈ R + . The scale ε appears in the solution (resp. in the Cauchy problem) when it is needed (even in composite form) for the multiscale description of u ε (resp. of u 0 or H). For instance, the three scales ε 0 , ε 1 and ε 2 are needed when dealing with the function g ε : (τ, x) → sin cos(x + τ /ε) /ε 2 . (ii) A phase ϕ is a smooth real valued scalar function occurring in the solution (or in the Cauchy problem) when, after multiplication by some negative power of ε, it is substituted for a periodic variable (like θ τ or θ u ) in a profile. The phase ϕ ≡ ϕ ε may depend on ε and contain oscillations (like in the case of chirped pulses [12]) but it must be (locally) uniformly bounded with bounded first order derivatives. For example, the expression ε cos(x + τ /ε) acts as a phase (associated with the frequency ε −3 ) in the above function g ε . The weight ε is here important to recover a time bounded derivative. To tackle (1.5), restrictions on the leading term H 0 issued from (1.4) are needed. To this end, we first consider the following simplification of the nonlinear interaction in H 0 . Assumption 1.1 (Independence on the periodic variable θ u ). The leading term H 0 does not depend on the last periodic variable θ u . In other words (1. 6) We also impose the following positivity condition on H 0 .
Assumption 1.2 (Positivity condition). We have for all (θ τ , x, u, p) ∈ T × R d × R × R d (p · ∇ p )H 0 − H 0 (θ τ , x, u, p) > 0. (1.7) Note that Assumption 1.2 implies that H 0 ≡ 0. In addition, we implement an integrability condition which will serve a lot, for instance to prove the uniform (in ε) local existence of the solutions u ε . Assumption 1.3 (Complete integrability). For all position z = t (z x , z u , z p ) ∈ R d × R × R d , the solution t (Ξ 0x , Ξ 0p )(z; ·) to the Cauchy problem is globally defined in s, and it is periodic in s of period 2π.
For reasons that will become apparent later in Subsection 6.4, we need to impose the following smallness restriction on some second order derivatives of the scalar function H 0 .
The initial data in the right hand side of (1.5) is of amplitude 1. It is smooth with uniformly bounded derivatives. But the source term H is large, of size 1/ε due to (1. 7), and it implies very rapid oscillations (involving both τ and u). By this way, strong nonlinear processes are implemented when solving (1.5), see Remark 6.1. Such aspects are often studied in an isolated or partial manner. They appear for instance in the references [5,6,17,18,19,23].
1.2. Onset of smaller scales and new phases. The next result proves that the above nonlinear interactions can suddenly generate additional scales and phases. Denote by B(0, R] the closed ball in R d of center 0 and radius R ∈ R * + . Theorem 1.5 (WKB description of the flow). Under Assumptions 1.1, 1.2, 1.3, and 1.4, for all ε ∈]0, ε 0 ] with ε 0 small enough, there exists a local smooth solution u ≡ u ε to (1.5) on the product of a time interval and a spatial domain having the form [0, T ] × B(0, R] with 0 < T and R > 0. Moreover, on this uniform region, for all N ∈ N with N ≥ 3, the expression u ε can be described in the sup norm by the following multiscale and multiphase expansion u ε (τ, x) = 1 ε U −1 x; τ, τ ε + U 0 x; τ, τ ε (1. 10) where ψ ε is a phase in the sense of (ii), looking like ψ ε (x; τ ) := ε U −1 x; τ, τ ε + ε 2 U 0 x; τ, τ ε . (1.11) The two profiles U −1 (x, τ, θ τ ) and U 0 (x, τ, θ τ ) which appear both in (1.10) and (1.11) are smooth functions on the domain B(0, R] × [0, T ] × T. For j ≥ 1, the profiles U j (x; τ, θ τ ,θ • r ) are smooth with respect to the variables (x, τ ) ∈ B(0, R] × [0, T ] and smooth (periodic) with respect to the two last variables θ τ ∈ T andθ • r ∈ T. When solving (1.5), a number of new patterns are generated (creation of oscillations). When looking at smaller details that are at smaller amplitudes, the flow is growing in complexity. This cascade towards more and more scales and phases is put in concrete form at the level of (1.10) both quantitatively in terms of frequencies (larger derivatives) and qualitatively in terms of phases (extra directions of fast variations). Theorem 1.5 shows clearly that the description of the solution u ε with an incrementally precision (expressed in terms of powers of ε) is associated with an increasing agitation. This principle may also be illustrated through the Table 1. While it is commonly believed that nonlinear evolution equations can instantaneously develop microstructures (like in turbulent flows), concrete mechanisms for this and rigorous proofs are rarely exhibited (or in very specific contexts due to subsequent instabilities). Theorem 1.5 is a step forward in this direction. It is proved in Section 6, and appears as a difficult corollary of a WKB analysis which is developed in Sections 3, 4, and 5.
1. 3. From Hamilton-Jacobi equations to nonlinear differential equations. For smooth solutions, Hamilton-Jacobi equations can be solved by the method of characteristics. In the context of (1.5), this yields the following system of nonlinear differential equations (1. 12) where, given n ∈ N * , the dependent variables z ∈ R n and ν ∈ R depend on the time τ ∈ R. Here, ε ∈]0, ε 0 ] is a small positive parameter which is intended to go to zero, whereas z 0 ∈ R n and ν 0 ∈ R stand for initial data (which may depend smoothly on ε ∈ [0, ε 0 ]). The precise content of (1.12) is described in next Section 2.
The strategy to study (1.12) is inspired from [7,8]. But the works [7,8] are devoted to the characteristics of the Vlasov equation, and they rely crucially on the conservation of the kinetic energy (of charged particles). They cannot be applied directly to the system (1. 12) or in the presence of an electric field (see [9] to this end). With this in mind, the framework of [7,8] needs to be extended; and the tools of [7,8] must be revisited. In fact, in comparison to [7,8], we replace some invariant quantities by (more general) integrability conditions (which will operate at all levels to move forward). As a consequence, we have to adapt (in Sections 3, 4 and 5) our preceding arguments to the new difficulties thus generated.
The proof of Theorem 1.5 depends on studying the above system of nonlinear differential equations (1.12). It is derived from an existence result (Theorem 2.5), asymptotic descriptions (Theorem 2.7) and reduced models (Theorem 2.10) that apply to the above class of differential equations (1. 12). The contents of the latter theorems are identified in the next section.

A class of nonlinear differential equations
In this section, we present the underlying framework. In Subsection 2.1, we state the assumptions and the main results. In Subsection 2.2, we make some general comments about the content of our theorems. In Subsection 2.3, we outline the plan, we highlight some innovative ideas, and we come back to the possible applications.
As indicated, the functions A and V may imply z 0 ∈ R n . By this way, they can take into account the influence of the component z 0 in the initial condition, as it may be required in the applications. They may imply the unknown z. But they do not involve ν (the scalar variable ν appears in the right hand side only after substitution of θ r by ν/ε). Most importantly, they are periodic with respect to the two last variables θ τ and θ r , of period respectively 2π and T r (z 0 ). More precisely θ τ ∈ T := R/ 2πZ , (2.1) θ r ∈ T r,z0 := R/ T r (z 0 )Z , T r ∈ C ∞ (R n ; R * + ).
(2.2) Remark 2.1. With no loss of generality, we work with (2.1). We can also start with source terms A and V that are periodic in θ τ of period T τ (z 0 ) with T τ ∈ C ∞ (R n ; R * + ). This means to deal with θ τ ∈ T τ,z0 := R/(T τ (z 0 )Z). Suppose this is the case, then we can always replace θ τ by 2πθ τ /T τ (z 0 ). This modification has the effect to substitute A and V for T τ (z 0 ) 2π This gives rise to (2.1) without affecting the general form that has been introduced at the level of (1.12). But, when the original functions A and V do not imply z 0 , this does produce a (smooth) dependence on z 0 . This is why, care is taken to incorporate z 0 when defining A and V.
The source term inside (1.12) is: • stiff since the large weight ε −1 is put in factor of the (locally bounded) functions A and V which may satisfy A 0 ≡ 0 and V 0 ≡ 0. In fact, we will assume that V 0 is positive. • strongly oscillating due to the large amplitude oscillations which are generated by A and V after substitution of θ τ and θ r for respectively τ /ε and ν/ε; • nonlinear because A and V depend on z and θ r (and therefore ν/ε) in a non-trivial way.
Let us examine more precisely what happens at the level of ν/ε. To this end, fix ε ∈ ]0, ε 0 ] small enough, and look at the scalar component ν(τ ). Since the function V 0 will be assumed to be positive (see Assumption 2.6), by the mean value theorem, we can find some c = c(τ ) ∈ ]0, τ [ such that This indicates that the description of the solutions t (z, ν)(ε, z 0 , ν 0 ; τ ) to the system (1.12) should involve (at least) three time scales: • τ for the current time variable and normal variations (in the case of electrons in tokamaks, the value τ ∼ 1 represents a few seconds); • s := ε −1 τ for the quick time variable and for quick variations (in comparison to changes in τ ). The substitution of the periodic variable θ τ by ε −1 τ furnishes oscillations at high frequencies of size ε −1 ; • t := ε −2 τ for the rapid time variable and for rapid variations (in comparison to changes in τ and s). In view of line (2.3), the substitution of the periodic variable θ r by ε −1 ν(τ ) should produce oscillations at very high frequencies of size ε −2 . The subscript r in θ r is introduced to refer to these rapid variations.
To state our results, some basic operations on periodic functions Z(θ τ , θ r ) ∈ L 1 (T× T r,z0 ), like above A(ε, z 0 ; z; ·) or V(ε, z 0 ; z; ·), must be introduced. As a preliminary point, to sort out the different oscillating features, we need to define: • The rapid mean value of Z is the periodic function Z given by (2.4) • The rapid oscillating part of Z is the periodic function Z * defined by • The double mean value of Z (in both variables θ τ and θ r ) or the quick mean value of Z is the constant • The quick oscillating part of Z is the periodic function Z given by Recall that this induces a decomposition of Z according to Our purpose in Sections 3, 4 and 5 is to find a setting under which the system (1.12) can be solved on a time interval that is uniform in ε ∈ ]0, ε 0 ]. It is also to exhibit conditions leading to a three-scale asymptotic description (when ε goes to zero) of the flow, showing the three frequencies of size 1, ε −1 and ε −2 that are associated to the underlying presence of the time variables τ , s and t.
The mean flow associated with (2.9) is the mapping Ξ mf (z 0 ; ·) : R n × R → R n obtained by solving is derived in Subsection 3.4 from some homological equation (Lemma 3.10). Observe that when V 0 does not depend on θ r , the function A mf is extracted from A(0, z 0 ; z; ·) as it is indicated in (2.4), that is (2.12) In general, the lifespan S mf (z 0 , z) is finite, and the same holds concerning S(ε, z 0 , ν 0 ). Example 2.3. For instance, just take n = 1 and A ≡ z 2 together with ∂ θr V 0 ≡ 0 so that (2.13) In this simple case, we can see that It follows that the lifespan T (ε, z 0 , ν 0 ) = εS(ε, z 0 , ν 0 ) associated with (1.12) is finite, and shrinks to 0 like εz −1 0 when ε goes to zero. Then, there is no way to guarantee that When dealing with (2.13), the discussion about what could happen during current times τ ∼ 1 is over. We would like to avoid such situations.
To obtain (2.14), supplementary conditions on A mf are clearly needed. As a prerequisite we have to impose S mf = +∞ on R n × R n . However, this condition may not be enough. And it is certainly not sufficient to separate the quick and rapid oscillations, and then to obtain a complete description of them. To this end, we need more restrictive conditions. Assumption 2.4 (Complete integrability of the mean flow). For each position (z 0 , z) ∈ R n × R n , the function Ξ mf (z 0 , z, ·) issued from (2.11) is globally defined and periodic in s of period 2π.
The above assumption implies a strong geometrical restriction concerning (2.11). Indeed, this means that the integral curves associated with (2.11) must form (by varying the initial data z ∈ R n ) a foliation of R n by circles. After a normal form procedure (see Remark 3.4), the equation onz inside (2.9) can be transformed into a perturbed version of (2.11). From this perspective, Assumption 2.4 says that the dynamical system thus obtained is nearly integrable.
By construction, the function A mf (z 0 ; z; ·) is periodic (with respect to θ τ ) with period 2π, and therefore the same holds (with respect to the quick time variable s) for the source term inside (2.11). Assumption 2.4 requires also that all solutions to (2.11) share the same period 2π. This second condition is natural (but it is far from being systematically verified). Next we have a result on the uniform lifespan with respect to ε ∈ ]0, ε 0 ] of the flow generated by (1.12).
From there, the issue about the oscillating structure of t (z, ν) during current times τ ∼ 1 becomes meaningful. Now, to obtain a precise asymptotic description of the flow t (z, ν), we need to impose supplementary restrictions on V 0 . Assumption 2.6 (Positivity condition on the component V 0 ). The function V 0 is positive and it does not depend on θ r . Recall that This simplifies the self-interactions at the level of the equation on ν. Indeed, at leading order, the source term V is not impacted by the rapid variations (in ν/ε). Next we have a WKB expansion at all orders in ε ∈ ]0, ε 0 ] of the flow induced by the system (1.12).
which are determined through a hierarchy of well-posed modulation equations (starting from j = −1 up to any integer value of j), which are such that (2.16) and which are adjusted in such a way that, in terms of the sup norm, for all N ∈ N * , we have The two expansions (2.17) and (2.18) shed light on the time oscillations but also on the spatial oscillations (encoded in the variations with respect to z 0 and ν 0 ), thus revealing collective aspects of the motion (which are important in many applications). This is achieved through different types of phases, including mainly: • The time phase τ which is associated with quick variations.
• The exact phase ν. It has ε −1 in factor at the level of the source term of (1.12). There, the scalar component ν/ε comes to replace the periodic variable θ r and therefore it indeed plays the role of a phase. • The frozen phase ν f which is defined by It is a truncated version of ν which, like ν, operates with ε −1 in factor. • The rapid phase V −1 which is associated with rapid variations (at frequencies of size ε −2 ). As a consequence of Assumption 2.6, we will find that V −1 (τ ) > 0 for all τ > 0. Thus, the presence of rapid oscillations is sure to happen.
Remark 2.8 (About supercritical features). It is worth underlying that in (2.18), the same profiles V −1 and V 0 take part in the description of amplitudes and phases. This is typical of supercritical regimes for quasilinear equations.
Remark 2.9 (About the notion of phase). Definition (ii) of a phase ϕ (in terms of its bounded aspects) is open to various interpretations. Indeed, it is based on the dependent or independent variables that are implied. For example: • The functions ϕ ≡ ν and ϕ ≡ ν f viewed as depending on the state variable ν or ν f and from the perspective of the profiles A and V inside (1.12) can be viewed as phases. • The functions ϕ ≡ εν and ϕ ≡ εν f viewed this time as depending on (ε, z 0 , ν 0 , τ ) and from the perspective of the profiles A and V inside (1. 12) or Z j and V j inside (2.17) or (2.18) play the role of phases.
On the other hand, even if ν/ε and ν f /ε come to replace the periodic variable θ r , the functions ϕ ≡ ν and ϕ ≡ ν f viewed as depending on (ε, z 0 , ν 0 , τ ) are not (strictly speaking) phases because they are obviously not uniformly bounded. It must be clear that the above names of exact phase and frozen phase are a matter of convention.
Looking at (2.17) and (2.18), at the end, we can recognize the simultaneous presence of oscillations implying the frequencies ε −1 and ε −2 with corresponding phases τ and V −1 . There is also a nonlinear imbrication of oscillations carried by the (slightly unusual) expression V 0 (·, τ, ε −1 τ )/ε. The study of multiscale oscillations has been intensively developed in the past decades in various contexts including the topic of geometric optics [15,24,26,27,28], the theory of homogenization [1,25], two-scaled Wigner measures [16,22] or microlocal Birkhoff normal forms [22]. However, the coverage of situations which can mix oscillations of the above different types is relatively new. This is technically a difficult challenge (with potential extensions in the domain of PDEs) which apparently is not directly within reach of the aforementioned methods.
Asymptotic expansions similar to (2.17) and (2.18) already appear in the articles [7,8]. There, they were motivated by questions arising in the study of magnetized plasmas. The actual approach is much broader than in [7,8]. The purpose is indeed to achieve a comprehensive analysis in more general situations than before. It is to extend the preceding tools and also to explain them more briefly and clearly. We exhibit integrability and positivity conditions (Assumptions 2.4 and 2.6) allowing to progress. These conditions encompass and extend the framework of [7,8]. They are both intrinsic, easy to test, and suitable for many applications. They are considered in Section 6 in the case of Hamilton Jacobi equations, and they are designed [9] to incorporate the influence of an external electric field (in addition to the magnetic field) on the long time dynamical behavior of charged particles. The main outcomes of our work concerning (1.12) are the following: • A model for the leading behavior of the flow provided by • A careful analysis of the underlying stability and instability properties. This aspect is more subtle and less easy to expose since it appears throughout the text. Let us just outline some difficulties. The precise knowledge of ν (or ν f ) is essential to obtain the L ∞ -precision. As a matter of fact, a perturbation of size ε 2 at the level of V −1 may have an impact of size 1 in the calculation of the O(ε j )-terms of (2.17) and (2.18). This means that a very precise access to ν is crucial to govern the stability properties of the flow. From the above perspective, our strategy is based on two main arguments: • First, we implement a blow-up procedure. As explained in a series of remarks (see 3.4, 3.14 and 5.7), this is a kind of normal form method adapted to our context. In Section 3, this already leads to the uniform local existence of solutions (z, ν).
• Then, in Section 4, we perform a three-scale WKB calculus with supercritical attributes. The idea, as is typical in geometric optics [10,13], is to replace (1.12) by profile equations. But this time, we implement the component ν of the solution previously obtained as a phase (see Remarks 4.3 and 4.4 for this unusual trick). This means in particular that ν is viewed as oscillating with respect to itself through the oscillating implicit relation (4.1). This yields a special notion of profile equations. This is like investigating the stability issue in a quotient space: we work modulo the determination of the unknown function ν.
Because of its importance from an application standpoint, we focus below on the content of (2.20). The behavior of z is mainly governed by the profile Z 0 which reveals some kind of (large amplitude oscillating) reduced dynamics during current times τ ∼ 1. The determination of Z 0 may be achieved as indicated bellow.
Theorem 2.10 (Reduced equations). The function Z 0 (·) does not depend on ν 0 . It can be expressed in terms of Ξ mf (·) according to The function Z 0 in the right hand side of (2.22) can be determined by solving where, with ∂ −1 θr as in (3.24) (by identifying θ ≡ θ r and T ≡ T r,z0 ), we have introduced and where the access to the double mean value A 1 is furnished by (2.6).
Recall that the two symbols A 1 and A 1 are different (we have A 1 ≡ A 1 ). The notations A 1 and A 1 will be used with different meanings. Given the numerous difficulties of understanding the complex interplay between the various types of oscillations, Theorem 2.10 produces a rather easy and explicit way to determine what remains in a first approximation. In fact, it gives access to effective equations which are amenable to numerical computations.

2.2.
A few comments on the results. The aim of this subsection is to help the reader understand the position, content and significance of the four preceding theorems. This is done below through a list of remarks.
Remark 2.11 (About the effective content of Z 0 ). The expression A 1 is issued from the double averaging procedure (2.6) which may go hand in hand with a number of cancellations. It follows that all the components of Z 0 are not necessarily activated when solving (2.23). In general, there remains a reduced number of unknowns. These are the so-called adiabatic (or guiding-center) invariants in the case of charged particles.
Remark 2.12 (About the determination of Z 0 ). The expression Z 0 consists of two distinct parts: Ξ mf and Z 0 . As explained before, the mean flow Ξ mf can be extracted from (2.9). In fact, this amounts to a multiplication of (1.12) by ε, and then to the extraction of a mean value involving A 0 and V 0 , as in Definition 2.2. Observe thatz (2.25) Thus, in coherence with what has been said before, the mean flow does furnish the leading behavior of z during quick times s near the current time τ = 0. But, near other current times τ ∈ R * + , the use of Z 0 (z 0 ; τ ) is needed to well describe Z 0 (·). The access to Z 0 is much more complicated than to Ξ mf . It involves the determination of A 1 which, in view of (2.24), is built with various derivatives and integrations of A 0 and V 0 , as well as terms of size ε inside A, like A 1 . Now, such information should be invisible (or vanishing) when performing (even multiscale) weak limits at the level of (1.12) or (2.9).
Remark 2.13 (About the geometrical interpretation of the content of Z 0 ). Under Assumption 2.4, the integral curves associated to (2.11) draw a family of circles Given z 0 ∈ R n , these circles form a foliation of R n parameterized by z. The role of z is twofold. First, since z ∈ C(z 0 ; z), the parameter z points to the circle C(z 0 ; z). Secondly, it specifies some origin on this circle. This allows to make sense of the numerical value θ τ which at the level of the expression Ξ mf (z 0 ; z; θ τ ) is comparable to the number of turns performed around C(z 0 ; z), departing from the position z. In this context, the geometrical interpretation of (2.22) is the following. The value of Z 0 (z 0 ; τ, θ τ ) is obtained by selecting the circle C z 0 ; Z 0 (z 0 ; τ ) and by carrying out on it a rotation which starts from the initial point Z 0 (z 0 ; τ ) and whose numerical value is θ τ . Now, in view of (2.23), the position z = Z 0 (z 0 ; τ ) is in general different from z 0 as soon as τ = 0. Thus, to obtain a representation formula of Z 0 which is in line with (2.22), it does not suffice to work with z = z 0 . This remark justifies the introduction at the level of (2.11) of the extra variable z. In practice, we have indeed to solve (2.11) with z other than z 0 . Remark 2.14 (About the content of the leading order term). At the end, the main limit behavior is built with the help of Ξ mf (z 0 ; ·; ·) through a current time variation corresponding to the passage from z = z 0 to z = Z 0 (z 0 ; τ ), combined with a quick time variation (made of rotations) when θ τ is replaced by τ /ε (with a number of turns that becomes very large when τ ∈ R * + is fixed and ε tends to zero). At the level of (2.26), the rapid oscillations which are associated with the phase ν f do not yet appear.
Remark 2.15 (About the structure of the whole asymptotic expansions). Look at the contribution which at the level of (2.20) has ε in factor. In general, we have ∂ θr Z 1 ≡ 0. This means that the rapid variations are activated with a small amplitude ε and corresponding frequency ε −2 . These rapid oscillations lead to many technical problems. Physically, when dealing with magnetized plasmas, they come from a fast gyromotion. At the end, the structure of z is made of the superposition of two regimes which are built with: • Large amplitude oscillations. We find that ∂ θτ Z 0 ≡ 0 when the mean flow is not constant. Then, there are quick variations (involving θ τ ) of amplitude 1; • Strong oscillations. We find that ∂ θr Z 1 ≡ 0 when A * 0 ≡ 0. Then, if we adopt the terminology of [11] (with ε replaced by √ ε), there are rapid variations (involving θ r ) of amplitude ε.
Remark 2.16 (About the imbrication between the averaging procedures). The extraction of (2.17) and (2.18) results from two averaging procedures. The first (in θ r ) is revealed by (2.12); the second (in θ τ ) occurs along the circles generated by Ξ mf . There is no evident order between these highly interconnected operations. In the blow-up Section 3, priority is given to θ τ . But, in the WKB Section 4, the integration is first in θ r and then in θ τ , as when passing from (2.4) to (2.6).
Remark 2.17 (About the origin of Theorem 1.5). The Hamilton-Jacobi equations will be solved by the method of characteristics. When doing this, the expansion (1.10) appears through a composition of the oscillations involved by (2.17) or (2.18), roughly speaking by replacing (z 0 , ν 0 ) inside (2.18) by (z 0 , ν 0 )(ε, x −1 ) where x −1 (Lemmas 6.9 and 6.10) is the inverse of the spatial characteristic x (Lemma 6.5). This is why it is very important to keep track of the dependence on the initial data (z 0 , ν 0 ) at the level of (2.18). This also explains how the complexity of the oscillating structures may increase. The chain rule indicates that the number of scales could become larger. In Subsections 6.3 and 6.4, we will show that this number does indeed increase.

2.3.
Plan and motivations. The plan of the paper is as follows.
• In Section 3, we introduce a lifting (or blow-up) procedure which may be regarded as an adapted kind of the (more classical) normal form procedure. The purpose is to remove from the right hand side of (1.12) as much non-significant singular terms as possible. The idea is to absorb some artificial oscillations by changing the unknowns. This is done by (the inverse of) a nonlinear oscillating transformation. As a corollary, in Subsection 3.2.3, we can already prove Theorem 2.5.
• In Section 4, we develop a three-scale WKB analysis involving the exact phase ν. We work at the level of the profile equations (4.3) which allow to obtain rid of ν. The idea is to seek approximate solutions t (Z a , V a ) of (4.3) in the form of expansions in powers of ε, like in (4.11) or (4.12). Formal computations lead to a hierarchy indexed by j ∈ {−1} ∪ N of well-posed equations which are highly interconnected and which allow to determine successively all the profiles Z j and V j that constitute Z a and V a . In particular, in Paragraph 4.3.1, we identify the leading profile Z 0 which can be described as in Theorem 2.10.
• In Section 5, we justify the interest of the preceding procedure (the formal calculus) by showing that the approximate solutions t (Z a , V a ) lead indeed to exact solutions of the system (1.12). This means passing from the Z j and V j of (4.11) and (4.12) to the Z j and V j of (2.17) and (2.18). This implies the freezing of the phase ν into ν f (through the implicit function theorem, see Subsection 5.2) as well as a coming back to the original field (Subsection 5.3). At the end, this yields the proof of Theorem 2.7.
• In Section 6, we implement our analysis to construct classical solutions for the Cauchy problem associated with a class of oscillating Hamilton-Jacobi equations.
Of course, weak solutions may exist [14]. But the vanishing viscosity method does not furnish a precise description of their oscillating structures. By contrast, the method of characteristics does apply and it makes such accurate information available. Still, to this end, we need to implement some specific nontrivial arguments. Indeed, the differential of x is apparently highly singular which indicates that supercritical phenomena are achieved. However, this can be overcome through the Hadamard's global inverse function theorem by exploiting transparency conditions (Paragraph 6.3.2) emanating from Assumption 2.4 (or 1.3). At the end, we achieve the proof of Theorem 1.5 in Subsection 6.5.
Hamilton-Jacobi equations like (1.5), implementing a small parameter ε → 0, can appear in many situations which inspire our interest in this topic, like: homogenization theory [1, 5,6,17,18,19,25] where they are applied to traffic flows, light propagation and optics [26,28], plasma physics [8,9], in the presence of rough domains [21], and so on. In these references, the above different multiscale aspects are often discussed separately and partially. We provide here an extensive overview and we investigate new facets. Most importantly, we achieve a better comprehension of the nonlinear specificities induced by the influence of the oscillating term u/ε inside H. Such aspects have already been raised (for instance in [23]) but without going as far as we do.
The present approach is also motivated by the need for an accurate long time (τ ∼ 1) description of the dynamics of charged particles in strongly magnetized plasmas. Recall that the gyrokinetic equations [4] deal with systems of the type (1.12) during quick times, for τ ∼ ε or s ∼ 1. In fact, the mean flow can be related to the guiding center motion, while Theorem 2.7 goes far beyond this. It significantly enhances the information content of standard ray tracing methods [29] by justifying asymptotic descriptions which prevail over longer times (namely during current times τ ∼ 1) and which are valid with any order of precision (expressed in powers of ε). Given the potential implications, there is a very abundant literature (both in physics and mathematics [2,4,29]) related to this subject. Usual approaches are however limited because they do not capture the imbrication of oscillations revealed by (2.17) and (2.18). The introduction of the preceding three-scale framework is necessary to progress. Historically, the structure of (1.12) is already implicit in [20], and it becomes more visible in the two following contributions [7,8] where it is studied in the purely magnetic case. The formulation and assumptions retained here are much more general, and they are designed to take into account the (potentially disruptive) influence of electric fields. But this requires a long preparatory work and a number of specific considerations. This is why this important aspect is developed in the separate contribution [9] with in perspective an analysis of the dynamical confinement properties inside fusion devices during long times.

Blow-up procedure
The main goal of this section is to obtain rid of the irrelevant oscillations which are put in factor of the large weight ε −1 in the source term of the system (1.12). As stated in Subsection 3.1, at the level of Proposition 3.2, we can exchange (1.12) with (3.3). In this procedure, the general form of the equations is not modified but (A, V) is replaced by (A, V), with A satisfying the simplified condition (3.4). When doing this, the crucial tool is a change of variables involving a map Ξ. The general structure of Ξ is specified in Subsection 3.2. The transformation of (1.12) under the blow-up procedure is detailed in Subsection 3.3, where it is explained how A and V can be deduced from A and V. By this way, it becomes possible to exhibit necessary and sufficient conditions on Ξ leading to (3.4). These conditions are the gateway to Assumption 2.4. The proof of Proposition 3.2 is achieved at the end, in Subsection 3.4. Remark 3.1. As commented in Remark 2.1, we can always start with some θ τ ∈ T τ,z0 := R/(T τ (z 0 )Z). It bears noting that, in such a case, Assumption 2.4 implies that the function Ξ mf (z 0 , z, ·) issued from (2.11) should be periodic in s of period T τ (z 0 ) (instead of period 2π).
3.1. Desingularization method. The aim of this subsection is to replace the original field z by some auxiliary field z. In practice, the new unknown z cannot be directly expressed in terms of z. Instead, it is revealed after a blow-up procedure on z. The term blow-up must be understood here in the sense of lifting. More precisely, given z(ε, z 0 , ν 0 ; ·) : R → R n and z or equivalently to the formula In this process, the function ν is viewed as an input. The transformation (3.2) is driven by ν. The correspondance through (3.2) between z and z (and conversely) makes sense only on condition that ν is identified and, for the moment, it is supposed to be the local solution of (1.12). Knowing this, the key tool is the map Ξ(·) which must be adjusted first. Then, we can pass from z to z by following the two arrows at the top of the preceding diagram, or equivalently by using (3.2). The interest of a lifting is to put (a part of) the singularities (of z) aside by raising the number of variables. Here, oscillations are put in factor inside Ξ(·), at the level of the periodic variables θ τ and θ r . Note again that the whole procedure is very sensitive to the choice of both ν and Ξ. The purpose is to adjust ν and Ξ in such a way that z solves a system of ODEs which is inherited from (1.12) but which is less complicated than (1.12). In practice, this should manifest as a simplification of the source term A.
which takes the same form (1.12), with new expressions A and V satisfying (for all N ∈ N) but which now involves a first component A that is subject to the crucial property Recall that A and V are likely distinct from A and V. These functions are defined up to ε = 0, and they are smooth near ε = 0. As well as for A and V from which they are issued, they do not involve ν, but only z. This nonlinearity will be sometimes marked by the notations A(z) and V(z), which focus on the dependence on z and simply dismiss the role of (ε, z 0 , θ τ , θ r ).
The proof of Proposition 3.2 is postponed to Subsection 3.4.
In other words, we have (2.14) with z 0 replaced by z 0 .
Proof. Taking into account (3.4), the source term ε −1 A in front of ∂ τ z at the level of (3.3) is of size A 1 + O(ε) = O(1) instead of being of size O(ε −1 ). As long as z remains in a compact set, say z ∈ B(0, r] where r is adjusted in such a way that z 0 ∈ B(0, r/2], the two expressions ε −1 A and V are bounded uniformly with respect to ε, z and τ by Moreover, they are Lipschitz continuous with respect to z and θ r . By Cauchy-Lipschitz theorem, there exists a unique local solution t (z, ν) to (3.3). Since A and V are periodic with respect to θ r , the explosion, if any, can only occur at the level of the component z.

3.2.
General structure of the lifting. The map Ξ is built as a small perturbation (of size ε) of some map Ξ 0 . More precisely Observe that the form of Ξ inside (3.1) and (3.6) is the same as the one of A and V.
For illustration purposes and to assist the reader in the understanding of the text, we will explain through a series of remarks what happens in the case of a standard normal form procedure. We start below by recalling what is meant by this.
Remark 3.4 (Normal form procedure: definition). This is when Ξ 0 (z 0 ; z; θ τ ) ≡ Ξ 0nf (z 0 ; z; θ τ ) := z and when moreover Ξ 1 does not depend on θ τ , that is when In fact, the change (3.9) is just a small perturbation of the identity map. The formula (3.9) may suffice during quick times but certainly not (always) during longer times, see Remark 3.15.
When ε = 0 inside (3.6), we recover Ξ 0 (z 0 ; z; θ τ ). In the general case, as will be seen, we need some freedom on Ξ 0 in order to absorb singular terms. Thus, we do not take Ξ 0 ≡ Ξ 0nf . Extra admissible functions Ξ 0 are presented in Paragraph 3.2.1, while basic properties of Ξ are detailed in Paragraph 3.2.2. At the end, in Paragraph 3.2.3, we explain how to pass from the initial data z 0 to z 0 . We also describe how to go from the field z to its corresponding lifting z, that is how to obtain the inverse function of (3.2).
3.2.1. Admissible functions Ξ 0 . From now on, we suppose that Ξ 0 (z 0 ; .; θ τ ) : R n → R n generates a one-to-one correspondence. Under Assumption 2.4, as a consequence of Lemma 3.10, this condition will be verified for the choice Ξ 0 = Ξ mf that we have in mind.
is a diffeomorphism from R n onto R n . The corresponding inverse is denoted by Ξ −1 0 (z 0 ; ·; θ τ ). It is a smooth function of (z 0 , z, θ τ ) on R n × R n × T.

3.2.2.
Properties of Ξ. We look here more closely at the characteristics of the map Ξ, viewed as a perturbation of Ξ 0 . Next we have a family of diffeomorphisms indexed by (ε, Lemma 3.6. Select two functions Ξ 0 and Ξ 1 satisfying respectively (3.7) and (3.8). Assume Condition 3.5 and define Ξ as it is indicated in (3.6). Fix a compact set K 0 ⊂ R n and a positive real number R ∈ R * + . Then, we can find is a diffeomorphism from B(0, R] onto its image The corresponding inverse is denoted by Moreover, given K 0 , by adjusting R large enough and ε 0 small enough, we can always ensure that Proof. Consider the auxiliary map Applying (3.6) and (3.7), the mean value theorem (in several variables) guarantees that More precisely, this is a modification of the identity map Id B(0,R] which is of size O(ε) in terms of the C 1 -norm on B(0, R]. Moreover, by compactness, this holds true uniformly with respect to z 0 ∈ K 0 , θ τ ∈ T and θ r ∈ T r,z0 . Since the set of C 1 -diffeomorphisms on B(0, R] is open, by restricting ε 0 if necessary, the map inside (3.15) is sure to be a diffeomorphism for all values of (ε, Composing (3.15) by Ξ 0 on the left, we recover as expected that Ξ(ε, z 0 ; .; θ τ , θ r ) is a diffeomorphism from B(0, R] onto its image. Now, the set is compact as the image of K 0 × K 0 by the continuous function Ξ −1 0 (·; 0). Thus, for R sufficiently large, it can be included in a ball of radius R. And thereby, we have This inclusion is none other than (3.14) when ε = 0. The general case follows by compactness and perturbative arguments, by restricting ε 0 ∈ ]0, ε 0 ] again if necessary. Moreover, by the definitions of Ξ and then Ξ −1 , we must have We can seek Ξ −1 in the form of the asymptotic expansion (3.12). Then, we can exploit the formal expansion of (3.16) in powers of ε to successively determine the Ξ −1 j with j ≥ 0. By this way, we can extract Ξ −1 0 (term with ε 0 in factor) and Ξ −1 1 (term with ε in factor) as indicated. The inverse function theorem allows to justify this calculus.

3.2.3.
Passage from the original field z to the lifting z. In practice, we fix the compact K 0 , and we consider a collection of initial conditions z 0 ∈ K 0 . Then, we adjust R and ε 0 to obtain (3.14). The property (3.14) is essential to guarantee that all positions in K 0 has a unique preimage inside B(0, R]. More precisely, for all or equivalently Note that z 0 does depend on ε (even if z 0 does not) and it does oscillate in ε due to the presence of ν 0 /ε. The change from z 0 to z 0 through Ξ −1 introduces high frequencies since in general ∂ θr Ξ −1 1 ≡ 0. However, the positions z 0 remain uniformly in ε in a compact neighborhood of . By the way of (3.18), all the initial data z 0 contained in K 0 can be converted into corresponding initial data z 0 ∈ B(0, R] for the forthcoming system of ODEs on z. Now, let K be a compact set containing K 0 in its interior (K 0 ⊂K ⊂ K R n ). By applying Lemma 3.6 with K 0 replaced by K, we can guarantee that for someε 0 ≤ ε 0 and R ≤R. By continuity, a solution z(ε, z 0 , ν 0 ; ·) of (1.12) issued from z 0 ∈ K 0 will remain in K for sufficiently small values of τ . Thus, at least locally in time, on some open time interval which may be not uniform with respect to ε ∈ ]0, ε 0 ], we can define At this stage, we have collected enough information to show Theorem 2.5.
Proof of Theorem 2.5. We can pass from the local in time solution z of (1.12) to some associated field z through (  Let t (z, ν)(ε, z 0 , ν 0 ; ·) be a local solution of (1.12). Assume Condition 3.5. Define z 0 through (3.17) and z(ε, z 0 , ν 0 ; ·) locally in time through (3.20). Then, the field t (z, ν) is the unique (local) solution of (3.3), with A and V determined as indicated below: where the point · must be replaced by (ε, z 0 ; z; θ τ , θ r ).
Proof. Recall that the component ν remains unchanged under the blow-up procedure. It is the solution to (1.12). In view of (3.2), this forces us to define V as in (3.22). Incidentally, this means that the amplitude is maintained when transferring from V to V. There remains to prove that z is a solution to the first line of (3.3) with A adjusted as in (3.21). To this end, combine (3.2) with the first equation of (1. 12) to see that we have to guarantee that In other words, exploiting again (1.12), we must impose By inverting the matrix D z Ξ, we recover exactly (3.21).

Proof of Proposition 3.2.
In what follows, we need to invert the derivative ∂ θi for i ∈ {τ, r}. To this end, define T T := R/(T Z) and introduce the sets L 1 * (T T ) made of periodic functions with zero mean, namely We can define the operators ∂ −1 Recall that Then, we apply the above arguments to define the inverse of the operators ∂ θi for i ∈ {τ, r}. We seek conditions on A and V allowing to obtain rid of the problematic term of size ε −1 which may appear when looking at the first line of (3.3). This requires to separate inside (3.21) the leading term from the terms with ε in factor. We work at the level of (3.23). Assuming as expected that A 0 ≡ 0, with Ξ as in (3.6) so that ∂ θr Ξ = ε∂ θr Ξ 1 , with A as decomposed in the beginning, we can expand (3.23) according to On the one hand, from (3.22), we have On the other hand, we have Thus, we have to deal with the condition We bring together the terms with the same power of ε in factor to obtain The first line must be zero, which is the same as . First, take the mean value with respect to θ r in order to obtain rid of Ξ 1 and to identify ∂ θτ Ξ 0 . Then, subtract the result thus obtained to deduce ∂ θr Ξ 1 . Following these lines, we can exhibit two separate conditions, namely Definition 3.8 (Homological equation). The nonlinear ordinary differential equation (3.28) in the variable θ τ is called the homological equation (or sometimes the first modulation equation).
We can complete (3.28) with some initial data z ∈ R n whose introduction has been motivated by Remark 2.13. Hence, the expression Ξ 0 becomes a function of z 0 , z and θ τ , together with As a consequence, under Assumption 2.4, the solution to the Cauchy problem (3.28)-(3.30) is global and it is periodic with respect to θ τ of period 2π.
Proof. In view of Definition 2.2, the vector fields Ξ mf and Ξ 0 are solutions to the same system of ODEs. By Cauchy-Lipschitz theorem, local solutions do exist and (by uniqueness) they must coincide. The content of Assumption 2.4 allows the conclusion.
Remark 3.11 (Impact of Assumption 2.4). Lemma 3.10 makes the connection between the notion of mean flow (Definition 2.2 appearing in the introduction after heuristic considerations) and the map Ξ 0 (derived from formal computations). The role of Assumption 2.4 is clearly to furnish global solutions to (3.28)-(3.30). It is also essential to stay in the periodic framework.
Remark 3.12 (About the verification of Condition 3.5). Since the map Ξ 0 (z 0 ; z; ·) can be viewed as a flow, Condition 3.5 is automatically verified with Ξ −1 The right-hand side of (2.11) is periodic in s of period 2π. Thus, it can be decomposed like in (2.7) into its mean value and its quick oscillating part. It follows that the solution to (2.11) is the sum of a linear function plus a periodic function. The resulting expression may indeed be periodic only if which may appear as some a posteriori condition which must be satisfied by Ξ mf .
By construction, the right-hand side of (3.29) is periodic with respect to the variable θ r , and it is of mean zero. It can be integrated as indicated in (3.24). As a consequence, the part Ξ * 1 is completely determined from (3.29). We fix Ξ 1 ≡ 0, so that At this stage, we have exhibited necessary conditions on A 0 , Ξ 0 , and Ξ 1 ≡ Ξ * 1 to obtain A 0 ≡ 0. We have now to show that these conditions are sufficient. By Lemma 3.6, the matrix D z Ξ is invertible for the data under consideration. Coming back to (3.23), we can therefore deduce the value of A, with A = εA 1 + O(ε 2 ). Looking at (3.27), we find that (3.34) Remark 3.14 (Limited framework of the normal form procedure). The restriction on Ξ which is imposed at the level of (3.9) strongly reduces the class of systems (1.12) which can be managed by the blow up procedure. Indeed, it generates compatibility conditions on A: • In view of (3.28), the selection of Ξ 0 (z 0 ; z; θ τ ) = z is coherent with the homological equation if and only if A mf ≡ 0. This requires that V −1 0 A 0 has a zero mean (in θ r ), which is very restrictive. In particular, when A 0 do not depend on θ r , this simply means that we start already with A 0 ≡ 0.
• In view of (3.29), the function Ξ 1 does in general depend on θ τ when V 0 and A 0 do depend on θ τ . Now, this would not be compatible with (3.9) which implies that ∂ θτ Ξ 1 = 0. In the normal form procedure, since D z Ξ 0 = Id and ∂ θτ Ξ 1 = 0, the expression leading to A 1 must be related to A 1 through where Ξ 1 does not depend on θ τ and is as in (3.33).
In this section, we have seen that the study of the system (1.12) is under Assumption 2.4 completely equivalent to the analysis of (3.3). The challenge now is to exploit (3.4) in order to derive a description of t (z, ν) in terms of asymptotic oscillating series in powers of ε ∈]0, ε 0 ]. In fact, this requires in addition to Assumption 2.4 imposing Assumption 2.6.

Three-scale WKB calculus
We work here under Assumptions 2.4 and 2.6. The purpose is to construct approximate solutions through formal computations. The preceding work of preparation (in Section 3) allows to formulate the problem in terms of t (z, ν). Thus, we can consider (3.3) and we can benefit from (3.4). We can also exploit the content of Remark 3.15. We proceed in several stages. In Subsection 4.1, we replace t (z, ν) by some corresponding profile t (Z, V); we define a notion of profile equations with associated approximate solutions t (Z a , V a ); we also state the main result (Proposi- where ν(τ ) stands for the exact solution to (3.3). This multi-scale approach allows to separate the rapid variations (which have not yet been identified due to the presence of ν) from the slower (current and quick) variations (which must be determined first and foremost). This is like knowing z and ν modulo the action of a one-parameter group of rotations (associated with θ r ∈ R) which are aimed to be ultimately specified through the the replacement of θ r by ν/ε. At the level of (4.1), the profile t (Z, V)(τ, θ τ , θ r ) may depend on the parameters ε, z 0 and ν 0 (which will not be always indicated). Assuming (4.1), observe that where we have introduced the partial differential operator which involves the parameters (ε, z 0 ) ∈ [0, ε 0 ] × R n , is non-linear with respect to Z, and implies the coefficient V which undergoes variations in (θ τ , θ r ) ∈ T × T r,z0 . Consider the profile equations which are associated to (3.3). These are the relaxed version of (3.3) made of the transport equations together with the initial data (at time τ = 0) To recover the initial data of (3.3) with z and ν as in (4.1), in view of (3.18), we have to impose To this end, it suffices to work with the (relaxed) condition The component Z 0 may be expanded in powers of ε. The same applies to Ξ −1 .
Assuming that z 0 does not depend on ε, this yields (4.7) In particular, taking into account (3.12) and (3.30), we find that . When z 0 depends (smoothly) on ε, we have z 0 = z 00 + εz 01 + . . . , and (4.7) can be further expanded in powers of ε to obtain By the way, we can observe that the oscillations of z 0 are easily absorbed at time τ = 0 by the profile formulation, just because ν(0) = ν 0 and because Z 0 may depend on θ r . Note also that the condition (4.4) has the effect of introducing at the level of V (and therefore Z) a dependence on ν 0 . Neither Z 0 nor V 0 depend on θ τ . But the variable θ τ appears at the level of Z and V through the variations with respect to θ τ of the coefficient V inside (4.2). By looking at the Cauchy problem (4.3)-(4.4), we can see that: • The presence of ν has completely disappeared; • The equation on Z is now decoupled from the one on V.  More precisely, Definition 4.1 means that t (Z a , V a )(·, z 0 , ν 0 ; ·) is a smooth (C ∞ ) function of (ε, τ, θ τ , θ r ) ∈]0, which is such that In what follows, we seek t (Z a , V a ) through a finite series like (4.13) Moreover, in the same vein as (2.16), we impose (4.14)  4), which takes the form of (4.11)-(4.12) together with (4.13)-(4.14). The parts Z j and V j are uniquely determined by a sequence of well-posed evolution equations associated with initial data issued from (4.4), while the expressions Z j , Z * j , V j and V * j are derived from elliptic equations. We also find that Z 0 ≡ Z 0 and Z 1 ≡ Z 1 . Remark 4.3 (Meaning of the WKB hierarchy on the profiles). Readers may wonder why the formal calculus is not performed directly on the system (3.3), with an expansion of ν in powers of ε. This is because a small error on the determination of ν, even of size ε, can completely shuffle (through the substitution of θ r for ν/ε) sequences which are expressed in powers of ε. In other words, a small change of ν at the level of (4.1) can strongly modify the asymptotic representations (4.11) and (4.12) of the profiles Z and V. It can mix the terms Z j and V j and then cause intractable closure problems. This is why it is so important to work with the exact phase ν. We return to this point in Subsection 5.1.
Next we have a halfway to formal solutions of the redressed system of Proposition 3.2.
does not provide, strictly speaking, with an approximate solution to (3.3). Indeed We will see later that ∂ θr Z a = O(ε 2 ). To estimate the right hand side, we need to control the difference between the exact solution z and its model z a . However, there is no assurance for the moment that z − z a is small, of size ε N (or less). This issue is considered in Subsection 5.1.

4.2.
Three-scale analysis. The expression t (Z a , V a ) obtained through (4.11) and (4.12) is plugged into ( To this end, we perform a formal analysis at the level of (4.9) which can be expanded according to (4.16) We can resume (4.11), (4.12) and (4.14) in the form where the operations ·, · and · * are furnished by (2.4)-(2.5)-(2.6). Taking into account (4.17) when dealing with the left part of (4.16), we find that By definition, the expressions L j and M j are independent of ε ∈]0, ε 0 ]. They are obtained by collecting the terms which, for j < N , appear in factor of ε j . From this perspective, the right hand side of (4.18) is still not in convenient form. The difficulties come from the two last lines in (4.18) which involve in factor of ε k expressions that still depend on ε. We seek the profile t (Z a , V a ) to be the approximate solution to the profile equation (4.3) in the sense of preceding Definition 4.1. To this end, L j and M j should be zero. By means of (3.4) and (3.35), it is easy to see that, for j = −1, we find Integrate this with respect to θ r to deduce that ∂ θτ Z 0 = 0, and therefore (since V 0 is positive) that ∂ θr Z * 1 = 0. This implies that Z * 1 = 0. In fact, we have (4.33). These relations are used below in order to exhibit the expressions of L j for j ≥ 0.
The expression L j can also be put in the form where the expression of G j can be deduced from (4.22) as follows with S j = S j \ {(i, m, k) = (j + 2, 0, 0) and (i, m, k, l 1 ) = (2, 0, 1, j)}, In a similar fashion, we can define the expressions M j as indicated below.

The above expression can be put in the form
where again the expression K j may be deduced from (4.26). Looking at (4.16), to obtain (4.9), we have to solve for j ∈ {−1, . . . , N − 1} the following cascade of equations

4.2.2.
Problem-solving strategy. The purpose here is to explain how we can solve equations L j ≡ 0 and M j ≡ 0 to determine the profiles Z j and V j . This is a survey of the method that will be used (in next Subsection 4.3) to find approximate solutions to (4.3). In fact, this means manipulating the equations and expressions according to certain rules that we would like to emphasize and make explicit now. The well-posedness of the hierarchy of equations L j ≡ 0 and M j ≡ 0 is the gateway to the existence of the profiles t (Z j , V j ). Because of the decoupling property which is highlighted in the previous paragraph, we can primarily determine Z a . Then we can explain how to recover the remaining component V a .
Let us begin with the base case (initialization case) concerning L. This means to examine the following system (S 0 ), (4.28) We will see that solving this ssytem furnishes Z 0 ≡ Z 0 , Z 1 and Z * 2 . Now, we address the strategy of solving every single equation L j = 0 for j ≥ 1 through the following three points: (1) First we take the average of the equation L j = 0 in both variables θ τ ∈ T and θ r ∈ T r,z0 to obtain a well-posed (linear for j ≥ 1 ) Cauchy differential equation on Z j , where the profiles Z 0 up to Z j−1 , Z j , Z * j and Z * j+1 are viewed as input.
(2) Second we substitute the expression of ∂ τ Z j obtained from 1 and take the mean value of the equation L j = 0 with respect to θ r ∈ T r,z0 to obtain the expression of Z j+1 . (3) Third we subtract the expressions of ∂ τ Z j and ∂ θτ Z j+1 in L j = 0 to obtain Z * j+2 . Using such argument for j ≥ 1 gives rise to Z j by combining the three equations (S j ), L j−2 (Z 0 , . . . , Z j ) = 0 through the following steps Step 1: Apply items (1)-(3) to the equation L j−2 = 0; this leads to Z * j .
Step 3: Apply item (1) to the equation L j = 0; this allows to identify Z j .
Steps (1)-(3) allow us to determine Z * j , Z j , and Z j , and therefore Z j . This way, it becomes possible to access all the profiles by solving successively the systems (S j ) for j ∈ {0, . . . , N − 1}. Looking at (S j ) gives also access to Z j+1 (through L j ), Z * j+1 (through L j−1 ) as well as Z * j+2 (through again L j ). The arising claim (for j ≥ 2) is thus as follows: (H j ),  This means that we argue by induction. We have first to find the constraints required to validate the starting point of the induction, which is (H 2 ). Then, we proceed successively. We prove that (H j+1 ) holds true given that (H k ) is verified for k ∈ {2, . . . , j}. By this way, we can recover Z j−1 , Z j , Z * j and Z * j+1 , and so on. This program is achieved in the next section.
A similar strategy is repeated concerning the profiles V j . We start by analyzing the basic case. Then, we propagate to the higher one on the basis of the following hypothesis (H j ), (4.31) From the profiles Z j and V j thus obtained, we can build the approximate solution t (Z a , V a ) as indicated in (4.11)  To this end, we can limit ourselves to looking at the equations L −1 ≡ 0, L 0 ≡ 0 and M −1 ≡ 0. As already seen, exploiting Assumption 2.6, the restriction L −1 ≡ 0 reduces to (4.32) This amounts to Z 0 (τ, θ τ ) = Z 0 (τ ), Z 1 (τ, θ τ , θ r ) = Z 1 (τ, θ τ ). (4.33) The expression L 0 from (4.20) becomes (in view of Assumption 2.6) At this stage, we are able to complete the proof of Theorem 2.10.
Similarly, we can determine basic parts of the profile V a through the following lemma. Moreover, we have V * 1 ≡ 0. This means that V 1 ≡ V 1 . Proof. With the aid of expression (4.25), the restriction M −1 ≡ 0 reduces to Take the average in θ r to obtain rid of the last term, and then in θ τ to suppress the penultimate term. Complete with the initial data coming from (4.4). We find that which implies (4.41). Now take the mean value with respect to θ r to obtain rid of the term V 0 ∂ θr V * 1 in (4.43). Then exploit (4.44) and subtract the result thus obtained to deduce (4.45) Since V 0 (z 0 ; Z 0 (τ ); ·) ∈ L 1 * (T), we can invert the operator ∂ θτ in (4.45) to obtain (4.42). Finally, replace in (4.43) the expressions ∂ τ V −1 and ∂ θτ V 0 as indicated in (4.44) and (4.45), this gives V 0 (z 0 ; Z 0 (τ ); θ τ )∂ θr V * 1 (τ, θ τ , θ r ) = 0. Since V 0 is positive function, we must have V * 1 ≡ 0.
Recall that V * 0 ≡ 0. Thus, from Lemma 4.6, we have (H 1 ). Let us clarify here what happens at the initial time τ = 0. From (4.7), we have to impose . This means the cauchy problem (4.53) is accompanied with the following initial data Similarly for V, we have holds. From the validity of (H j ), the profiles Z k for 0 ≤ k ≤ j − 2, Z j−1 , Z * j−1 and Z * j have been identified. We still need to determine Z j−1 , Z j and Z * j+1 . To this end, consider the expression L j−1 as defined in (4.23), that is (4.49) Substituting Z j−1 = Z j−1 + Z j−1 at the level of (4.49), we obtain (4.50) We obtain a linearized version of (4.34), which is (4.52) It is obvious that the expression G j−1 ≡ G j−1 (Z 0 , Z 1 , . . . , Z j−2 , Z j−1 , Z * j−1 , Z * j ) is known function, since by induction, the profiles Z * j−1 , Z * j , Z j−1 and Z k where 0 ≤ k ≤ j − 2 are known functions on the domain [0, T ] × T × T r,z0 . With the aid of Assumption 2.6, average (4.51) in both variables θ τ and θ r to exhibit This may be completed by the initial data, namely Z 0(j−1) issued from (4.46) where, because of (H j ), the expressions Z j−1 (0, 0) and Z * j−1 (0, 0, θ r ) are known. The equation (4.53) is linear. It has therefore a solution Z j−1 on the whole interval [0, T ]. By validity of (H j ), Z j−1 and Z * j−1 are known. And thereby, in view of (2.8), this implies that the whole of Z j−1 has been identified. Substitute ∂ τ Z j−1 as indicated in (4.53) inside (4.51). Then average with respect to θ r to obtain Again substitute in (4.51) the expressions ∂ τ Z j−1 and ∂ θτ Z j as indicated in (4.53) and (4.54). This gives (4.56) This discussion determines (H j+1 ). The same strategy applies concerning the construction of the profiles V j , for j ∈ {−1, . . . , N + 1} under Assumption 2.6. Lemma 4.6 gives rise to (H 1 ). Assume now that the hypotheses (H k ) are valid for all k ∈ {1, .., j − 1}. We have to prove that (H j ) holds true. To this end, pick the expression M j−2 from (4.26) and solve the equation M j−2 ≡ 0. We find where K j−2 is known function by induction. Average (4.57) with respect to θ τ and θ r , we obtain ∂ τ V j−2 = K j−2 .
(4.58) The equation (4.58) together with the initial data coming from the second equation of (4.4) allows to determine V j−2 by time integration on the interval [0, T ]. Substitute (4.58) in the equation (4.57). Then, average with respect to θ r to obtain Again, use (4.58) and (4.59) in (4.57), we obtain Proof of Proposition 4.2. We select some N ∈ N * , and we build the profiles t (Z a , V a ) as it is indicated in (4.11) and (4.12), with the aid of the profiles t (Z j , V j ) which have been determined in the previous Paragraphs 4. 3.1 and 4.3.2. It is easy then to see that t (Z a , V a ) is an approximate solution to the profile equation (4.3). Indeed, by construction, we have reset to zero all the terms composing the sum in (4.16).

Stability estimates.
Our purpose here is to show that the formal solutions to (4.3)-(4.4) can be used to approximate through (4.15) the exact solutions of the redressed system (3.3). Note that we do not compare t (Z a , V a ) and the solution t (Z, V) to the profile equation (4.3), which would be relatively easy. Instead, we want to associate the expression t (z a , ν a ) of (4.15) and the solution t (z, ν) of (3.3). To this end, we can always consider the expression t (r z , r ν ) defined by the weighted difference or equivalently by the relation In Subsection 5.1, we prove that t (ε N r z , ε N r ν ) can indeed be viewed as a remainder, as suggested by (5.2). In Subsection 5.2, we highlight the role of the frozen phase ν f which is given by The frozen phase is a known quantity because it is built from V −1 (τ ) and V 0 (τ, θ τ ) which have been already determined. It is constructed by collecting the two first terms of the expansion (5.2) of ν. Note that the definition (5.3) coincides with (2.19) since, at the end, we will find that V −1 ≡ V −1 and V 0 ≡ V 0 . By construction, we have For j ≥ 2, in general, we have V * j ≡ 0 while θ r must be replaced by ν/ε. This means that the access to ν is necessary to construct the right hand side of (5.4), or that the knowledge of ν is required to obtain a precision of size ε or more. Now, we would like to derive a self-contained representation of z and ν (which does not call for ν). In Subsection 5.2, we explain how ν f /ε can become a substitute for ν/ε in the right part of (5.2). By this way, we end up with a justified WKB expansion of the redressed field t (z, ν) whose all components can be determined by formal computations. Finally, there remains to interpret this result in terms of the original field z. This is done in Subsection 5.3 where the proof of Theorem 2.7 is complete.
5.1. Justification of the formal computations. The aim here is to compare t (z, ν) and t (z a , ν a ). This can be done by estimating the size of the weighted difference t (r z , r ν ). To this end, the strategy is to first exhibit a non-linear differential equation satisfied by t (r z , r ν ).
Lemma 5.1. For all N ∈ N, the expression t (r z , r ν ) issued from (5.1) is subject to Proof. Taking into account (5.2), equation (3.3) can be reformulated as |εr ν (τ )| ≤C(e CT + T ). (5.9) We start with information which is helpful to estimate the size of (5.6). We have: • ∂ θr Z a = O(ε 2 ) since Z * 0 = 0 and Z * 1 = 0 according respectively to (4.14) and (4.33); • The function V(·) is locally lipschitz with respect to z. Thus, by the meanvalue theorem and because Z a , θ τ and θ r stay in compact sets, we can find some L ∈ R * + such that • For the same reasons and due to (3.4), we can find some L ∈ R * + such that (5.10) • Recall that from (4.10) we have When computing the component R z , observe in the second line of (5.6) the compensation between the loss ε −2 and the gain ε 2 brought by ∂ θr Z a . The same applies in the third line of (5.6) between the loss ε −N −1 and the gain ε N +1 given by (5.10). It follows that On the other hand, from (5.5), we know (say with the same constant C) that Then, by Gronwall's lemma, we can recover (5.8). Now, the situation is quite different concerning the other component R ν . This is due to the contribution of which can actually be of large size ε −1 . Taking this into account, we can only assert that The right-hand side does not depend on ν. It can be bounded as indicated in (5.8).
Then, after integration in time, this yields (5.9).
The drawback with (5.2) is the presence in the right hand side of the unknown function ν(τ ). This is remedied in the next subsection where ν is replaced by ν f .

5.2.
Description of the redressed field in terms of the frozen phase. The second line of (5.2) can be interpreted as an implicit relation on ν. This is not very informative because the explicit oscillating content of ν(·) remains to be clarified. Now, this may be achieved by cutting ν into the (well determined) frozen phase ν f introduced at the level of (5.3). Next we have a description of the exact phase ν through a WKB expansion involving only the frozen phase ν f .
which can be computed from the V k with k ≤ j with in particular 13) and which are adjusted in such a way that, in terms of the sup norm, we have Note a loss of precision from ε N to ε N −1 when passing from the description (5.2) to (5.14). This is coherent with the loss recorded in (5.9). By convention, we set V −1 := V −1 and V 0 := V 0 . By this way, the formulation (5.14) becomes compatible with (2.18). Moreover, from (2.15) and (4.44), we can infer that the value of V −1 (τ ) is positive as soon as τ > 0. Thus, frequencies of size ε −2 are created at time τ = 0, and then they persist.
Remark 5.4 (Exact phase vs. frozen phase). It is important to point out that neither ν nor ν f are phases in the usual sense of the term, since they both still depend on ε. The difference is that ν is (a component of) the unknown solution whereas ν f can be derived explicitly from the WKB calculus. At the end, there remains The sole use of the time phase τ and of the rapid phase V −1 (τ ) would not be consistent with an expansion of z and ν in terms of profiles (not depending on ε) due to the presence of the extra (large) shift V 0 (τ, τ ε )/ε. We had a choice of whether to make some phases or some profiles depend on ε. We have selected the first option.
Proof of Lemma 5.3. The idea is to seek an expression V e (ε, τ, θ τ , θ r , r) that is adjusted in such a way that In view of (5.2), this amounts to finding V e so that To this end, it suffices to achieve the relaxed condition H ε, τ, θ τ , θ r , r; V e (ε, τ, θ τ , θ r , r) = 0 (5.18) where we have introduced The expression H(x; y) may be viewed as a nonlinear functional in y ∈ R depending on the multidimensional parameter x = (ε, τ, θ τ , θ r , r). From this perspective, the implicit relation (5.18) may define V e as a function of x by applying the implicit function theorem. Here, it is possible (and more efficient to obtain global results) to work directly. Compute By construction, we can assert that Note that the compactness of [0, T ], T and T r,z0 , as well as the periodic behavior of V j with respect to θ r , are crucial to obtain (5.20) notwithstanding the lack of compactness concerning y ∈ R. From the bound (5.20), we can deduce that for all (ε, τ, θ τ , θ r , r, y) with ε 0 ∈ R * + small enough, we have ∂ y H(ε, τ, θ τ , θ r , r; y) > 0.
This means that we can find a unique position V e (ε, τ, θ τ , θ r , r) ∈ R leading to (5.18), with V e depending smoothly on its arguments. In particular, expanding V e near r = 0 and using (5.9), we obtain from (5.16) that Since V e (ε, τ, θ τ , θ r , 0) is also smooth in ε near ε = 0, by expanding V e in powers of ε, we can recover (5.14) in the form After comparison with (5.14), this furnishes The term with ε 0 in factor yields V e (0, τ, θ τ , θ r , 0) = V 1 (τ, θ τ ). Applying (5.22) with j = 1, we find (5.12). The next derivatives of (5.23) with respect to ε, taken at ε = 0, allow to deduce successively how the profiles V j can be expressed in terms of the V k with k ≤ j, just by applying (5.22). For instance, we find (5.13) and so on.
The preceding description (5.2) of z is not fully satisfactory. Indeed, it still involves the unknown ν. However, using Lemma 5.3, this difficulty can now easily be overcome.
for j ∈ {0, . . . , N − 2}, which can be computed from the Z k and V k (or V k ) with k ≤ j with in particular Z 1 (τ, θ τ , θ r ) = Z 1 (τ, θ τ ), (5.26) and which are adjusted in such a way that, in terms of the supremum norm, we have Proof. We substitute the phase ν as described by (5.14) in the first component of (5.2). Since Z 0 ≡ Z 0 and ∂ θr Z 1 = 0, we find that Apply Taylor expansion to the function Z j with respect to the last variable to obtain Then we plug (5.29) into (5.28), and collect the terms with the same power of ε in factor. Then, compare the result with the expansion (5.27). This allows us to determine inductively the profiles Z j from the Z k and V k with k ≤ j. In particular, the terms which have ε 0 and ε 1 in factor furnish respectively (5.25) and (5.26).

5.3.
Back to the original field. The final stage is to provide a WKB expansion concerning the original field z.
Lemma 5.6 (Asymptotic description of the original field). Fix N ≥ 2. Then there exist profiles with j ∈ {0, . . . , N − 2} which can be computed from the Z k and V k (or Z k and V k ) with k ≤ j with in particular and which are adjusted in such a way that, in terms of the sup norm, we have Remark 5.7 (Normal form procedure: implication). When Ξ is as in (3.9), we have ∂ θτ Ξ 0 ≡ 0, and therefore the dependence of Z 0 on θ τ is not activated. Then, there remains Z 0 ≡ Z 0 .

(5.34)
Recall that Ξ 0 is the mean flow (Lemma 3.10) and that Ξ 1 can be deduced from (3.37). Taylor's Theorem in both variables z and θ r can be applied to develop the right hand side of (5.34) in powers of ε. Then, by identifying the terms with the same power of ε in factor, we can obtain explicit formulas yielding the Z j in terms of the Z k and V k . For ε 0 , we obviously obtain (5.31). The expression having ε in factor at the level of (5.34) is composed of two contributions. The first coming from Ξ 0 yields the first line of (5.32); the second issued from Ξ 1 leads to the second line of (5.32).
Proof of Theorem 2.7. To conclude, it suffices to compile what has been done before. The expansion (2.17) is the same as (5.33) in Lemma 5.6. On the other hand, the description of the exact phase ν is achieved in (5.14), at the le vel of Lemma 5.3. The O(ε ∞ ) in (2.17) and (2.18) can be obtained by just varying the choice of N , with an arbitrary remainder of size ε N −1 which is controlled at the level of Proposition 5.2.

Application to Hamilton-Jacobi equations
Let d ∈ N * . We work with the scalar function H : H(τ, x, u, p) .
Given some initial data u 0 : R d → R, the Hamilton-Jacobi equation built with H and u 0 takes the following form The study of evolution equations like (6.1) is fundamental in classical mechanics. It is a long-standing concern [3] which has motivated many contributions. The discussion depends heavily on the functional framework retained on H and u. Roughly speaking: (GW) Global weak solutions u can be constructed by compactness arguments (CA), see [14] and subsequent works. But uniqueness and stability require additional constraints on both H (typically convexity conditions with respect to p) and u (entropic conditions); (LS) Local smooth solutions u are available by the method of characteristics (MC). But this approach can work only under minimal smoothness conditions (say H and u 0 in C 2 ) and as long as the spatial projections of the (phase space) characteristics do not cross.
In Subsection 6.1, we introduce a small parameter ε ∈]0, ε 0 ] (going to zero) at the level of the system (6.1) in order to obtain a family of Cauchy problems of the type (1.5). In addition, we comment the effects of introducing such parameter in this system. In Subsection 6.2, we connect the resolution of the PDE (1.5) to the one of the ODE (1.12). Then, we show Theorem 1.5 in three steps: in Subsection 6.3, we detail the behavior of the spatial characteristic x(ε, ·); in Subsection 6.4, we construct its inverse map x −1 (ε, ·); and in the last Subsection 6.5, we complete the proof.
6.1. Oscillating framework. To go beyond the standard results, a natural strategy is to implement a parameter (here ε ∈]0, ε 0 ]) whose effect is to break (asymptotically when ε goes to zero) the usual assumptions. By this way, it can be possible to better target some underlying difficulties, and then to solve them. This is done in Paragraph 6.1.1 by introducing oscillations. In Paragraph 6.1.2, we describe the strategy which allows to achieve the proof of Theorem 1.5.
6.1.1. Data adjustment. Recall that we have introduced the variables u and p in the introduction which serve to replace respectively the terms εu and ∇ x u. In addition, for ∈ {τ, u}, we have denoted by θ the periodic variable which is aimed to be replaced with the quotient /ε. We also had that the expression H introduced at the level of (1.2) is indeed a function of the variables (ε, θ τ , x, u, p, θ u ), which is assumed to be smooth on its domain of definition. As indicated in (1.4), the function H(ε, ·) can be expanded in powers of ε near ε = 0. Now, replace H and u 0 inside (6.1) by H ε and u 0ε as indicated below: • The hamiltonian H ε (·) ≡ H(ε, ·) may depend on ε ∈ ]0, ε 0 ] according to When H 0 ≡ 0, the source term H ε is of large amplitude ε −1 , and it does imply oscillations (in both τ and u) at frequencies of size ε −1 . The role of ε is precisely to impact the C 1 -estimates (on H) which become non uniform in ε ∈]0, ε 0 ]; • The initial data u 0 inside (6.1) may depend on ε. More precisely, with U 0 smooth and as in (1.1)-(1.3), we impose u 0ε (x) = U 0 (ε, x).
And thus, the initial value problems (6.1) with H ≡ H ε as in (6.2) and u ε (0, ·) = u 0ε as above is exactly the Hamilton-Jacobi equation (1.5). From now on, we consider the smooth solutions u ε ≡ u ε (τ, x) ≡ u(ε, τ, x), sometimes simply denoted by u, to the the initial value problems (1.5).
Remark 6.1 (Return to a more standard regime). Applying the method of characteristics, a smooth solution u ε does exist (locally in space) on a maximal time interval [0, T ε [, with T ε ∈ R * + that may shrink to zero when ε goes to 0. Assuming that U 00 ≡ 0, changing τ into ε 2τ and u into εũ, we obtain It is clear that the lifespan associated with (6.3) is uniform in ε ∈]0, ε 0 ], and therefore that ε 2 T ε . When U 00 ≡ 0, such a lower bound is no more evident. This means that, in comparison with the first well understood situation (6.3), the study of (1.5) for τ ∼ 1 corresponds to a very long time investigationτ ∼ ε −2 for large initial data of size ε −1 . The latter difficulty is of course partly offset by the (nonlinear) periodic behavior of H with respect to θ u .

Strategy, intermediate results and notations.
To construct solutions u ε (uniformly in ε) and to justify asymptotic results (when ε → 0), the above-mentioned approaches (GW) and (LS) face two significant barriers: (i) Compactness arguments (CA) are not accompanied by a (strong form of) stability allowing to compare exact and approximate solutions; (ii) The method of characteristics (MC) may be subjected (after spatial projection) to crossing problems at times T ε vanishing when ε goes to zero.
The first challenge (i) seems difficult to overcome. Indeed, the potential lack of control makes it impossible to compare the weak and approximate solutions with each other. In particular, in the continuation of the indent (i), a method relying on the absorption of small error terms cannot be implemented (with quantitative estimates). On the other hand, for reasons that have been already discussed in Remark 2.12, other more direct tools like homogenization [25] or multiscale young measures [1] are not amenable to capture the leading behavior of u ε . And they cannot be implemented to compare the exact solution to the approximate one with a high degree of precision. To obtain accurate descriptions of the solution u ε , we must stick to the approach (ii).
The link between (1.5) and (1.12) is achieved through Lemma 6.2 together with the representation formula (6.8). Then, we have to detail inside (6.8) the content of u(ε, ·) and x −1 (ε, ·). To this end, our strategy is to: • Exhibit (Lemma 6.5) the asymptotic expansions of the spatial projections x(ε, τ, x) of the (phase space) characteristics. This requires to check the validity of Assumptions 2.4 and 2.6 (of Subsection 2.1) in the contexts inherited from (1.12); τ at the position x(ε, τ, x). When the map x →x = x(ε, τ, x) is a local diffeomorphism, the inverse x −1 (ε, τ,x) does exist and it furnishes a unique feedback allowing to solve (1.5). Then, we can simply recover u(ε, ·) through u(ε, τ,x) = u ε, τ, x −1 (ε, τ,x) . (6.8) The inversion formula (6.8) makes the transition from the Lagrangian point of view in (6.6), where u and x are functions of (τ, x), to the Eulerian perspective where u is a function measuring a quantity at the locationx through which the motion flows as time passes. In view of (6.8), to determine u(ε, ·) at the position (τ,x), we need to compose the (oscillating) quantity u(ε, τ, x) with the (oscillating) position x ≡ x −1 (ε, τ,x). In doing so, we must face a composition of oscillations, where it is crucial to know precisely how the functions u(·) and x −1 (·) depend respectively on (ε, τ, x) and (ε, τ,x).
Proof of Lemma 6.2. In view of (6.5), the first equation of (6.6) is just a reformulation of (6.4). Now, the definition (6.5) leads tȯ Compute the spatial gradeant of (1.5) to obtain Taking into account (6.4), the first two terms of (6.10) coincide withṗ. By this way, we can recognize the second equation inside (6.6). Finally, combining the second part of (6.9) with (1.5), (6.4) and (6.5), we find the third equation of (6.6).

6.2.2.
Comments on assumptions and main result. Starting from (6.6), there are different ways of falling within the context of (1.12). Indeed, the connection between (1.5) and (1.12) can be achieved through (6.6) by specifying the values of z and ν in terms of x, p and u. When doing this, care must be taken to recover the special structure of (1.12). The selection of ν := u is a natural choice. Now, one is tempted to simply take z = t (x, p). But, to ensure that the profiles A and V do not depend on ν ≡ u as required in (1.12), we must incorporate εu as a component of z. For this reason, we work with ν := u, n = 2d + 1. (6.11) We want to be consistent with the notation used in Sections 3, 4, and 5. To this end, we compare the system on (z, ν) that is issued from (6.6)-(6.11) with (1.12). With A = t (A x , A p , A u ) as well as θ u ≡ θ r , these two systems can be identified on condition that A x (ε; z; θ τ , θ r ) := ∇ p H(ε, θ τ , z x , z u , z p , θ r ), V(ε; z; θ τ , θ r ) := (z p · ∇ p )H − H (ε, θ τ , z x , z u , z p , θ r ). (6.12) The function A p must be smooth near ε = 0. This could be inconsistent with the weight ε −1 remaining in the second line of (6.12). On the other hand, in coherence with Assumption 2.6, the expression V 0 must be positive. These considerations lead to Assumptions 1.1 and 1.2. Then, with the convention H −1 ≡ 0 and A j = t (A jx , A jp , A ju ), for all j ∈ N, we find that . Since V 0 is defined in terms of H 0 , in view of Assumption 1.1, the function V 0 does not depend on θ r . Thus, we can apply (2.12) to see that

This implies that the mean flow (Definition 2.2) which is denoted by
where t (Ξ 0x , Ξ 0p ) satisfies (with θ τ ≡ s as in the introduction) the Hamiltonian system (1.8). Then, Assumption 2.4 amounts to the same thing as Assumption 1.3.

Remark 6.3 (Common situations leading to Assumption 1.3).
In general, it is not easy to test the periodic condition presented in Assumption 1.3. We furnish below a list of situations where Ξ 0 (z; ·) is indeed periodic. Suppose that the function H 0 does not depend on x. Then, the mean flow is given by When ∇ p H 0 ≡ 0, it is obvious that the mean flow Ξ 0 is periodic in s of period 2π. When moreover ∇ p H 0 ≡ 0, the function Ξ 0 is non constant (the dynamic is nontrivial). For similar reasons, if we assume that H 0 does not depend on p and is such that ∇ x H 0 ≡ 0, the function Ξ 0 (z; ·) is periodic in s of period 2π. When ∂ s H 0 ≡ 0, the system (1.8) is autonomous. Then, Assumption 1.3 is satisfied on condition that the level curves of H 0 (·, z u , ·) are (locally) diffeomorphic to a family of circles (existence of Liouville torus and thereby of action-angle variables).
In line with (3.37), we find that the lifting Ξ is given by From definition (6.11) of z and ν together with (6.5) and the initial data of (6.4), at time τ = 0, we must start with ε j (z 0j , ν 0j )(x). (6.17) Observe in particular that ν 00 (x) = U 00 (x). (6.18) From now on, we select x in a ball B(0, R] for some R ∈ R * + . Knowing what A, V and Ξ 0 ≡ Ξ mf (Lemma 3.10) are, we can deduce the value of A 1 through (3.34), and then we have access to Z 0 through (4.34). Now, consider Since Ξ 0 (z; ·) is periodic, the set K (which is presented in Assumption 1.4) is compact.
The main purpose of Section 6 is to prove Theorem 1.5 which is an important consequence of Theorem 2.7. Looking at the asymptotic description (1.10) of the solution u ε , it bears noting that: • The function ψ ε is a phase in the sense (ii) of Section 1: it is smooth scalar function and its first derivatives are uniformly bounded of size at most O(1); • The rapid variableθ 0 r is activated at the level of the profiles U j as soon as j ≥ 1; • Recall that V −1 ≡ V −1 and look at (4.44). Since V 0 is a positive function, in view of (6.23), (6.45) and (6.61), we can assert that U −1 is not zero for τ > 0. Thus (time) oscillations at frequency ε −3 do occur inside (1.10); • Recall that V 0 ≡ V 0 = V 0 (τ ) + V 0 (τ, θ τ ). At time τ = 0, taking into account (4.47), this is just ν 00 = U 00 which may be chosen non-zero. The same remains true for τ ∈ R * + (small enough). In view of (6.24), (6.46) and (6.62), we find in general that U 0 ≡ 0. This means that the O(ε 2 ) terms inside (1.11) is also essential. The construction of the phase ψ ε appearing in (1.11) is explained in what follows. In the next Section 6.3, we start the proof of Theorem 1.5 by looking at the component x(ε, ·) of (6.6), which is the spatial projection of z.
6.3. Spatial component of the characteristics x. The first thing to check is the uniform local existence of x. Below, we prove that the map x(ε; ·) exists locally uniformly in ε. Lemma 6.4. Select any R ∈ R * + . Under Assumptions 1.1, 1.2 and 1.3, we can find some T ∈ R * + such that the solution (z, ν)(ε, τ, x) to (1.12) with A and V as in (6.12) and initial data z 0 and ν 0 as in (6.17) is, for all ε ∈]0, ε 0 ], defined on This means that the initial data coming from (6.17) remain uniformly in a compact set. On the other hand, we have seen that Assumptions 1.1, 1.2 and 1.3 imply Assumption 2.4 when dealing with the system on (z, ν) which is issued from (6.6)-(6.11). Thus, it suffices to apply Theorem 2.5.
From there, the aspects about the oscillating structure of x make sense. To this end, we clarify the asymptotic expansion of x in Paragraph 6.3.1. Then, in Paragraph 6.3.2, we explore the effect of a transparency condition emanating from Assumption 1.3 on the differential of x: D x x(·). Indeed, the latter condition furnishes a control on the size of the Jacobian matrix D x x(·).
6.3.1. Asymptotic expansion of x. Note that Assumptions 1.1 and 1.2 also imply Assumption 2.6. We can apply Theorems 2.7 to obtain a description of (z, ν) which is coherent with (2.17) and (2.18). We find that x ≡ x(ε, τ, x) is given by where the Z jx , V −1 and V 0 are issued from the procedure of Section 5. Before proceeding, the expansion of x(ε, τ, x) must be further simplified. Next we a result on the asymptotic oscillating description of the spatial component of the characteristics.
Lemma 6.5. Under Assumptions 1.1, 1.2, and 1.3, the map (τ, x) → x(ε, τ, x) can be expressed according to the following asymptotic expansion (which is valid for all N ∈ N with N ≥ 2) x(ε, τ, x) where, with z 00 , z 01 and ν 00 as in (6.18), we have introduced , ν 00 (x); τ, θ τ , (6.24) and where, for j ≥ 0, the profiles Z • jx (x; τ, θ τ , θ • r ) are periodic with respect to the two last variables θ τ ∈ T and θ • r ∈ T. Recall that the three couples of phases • 0 in (6.44) are (in general) distinct from one another. This is why the symbols θ r , θ • r , andθ • r are not the same. This is aimed at highlighting the difference between the various phases that are involved.
Proof of Lemma 6.5. Comparing (6.21) with (6.22), there are two improvements: (A) The first two terms of the expansion -that is the first line of (6.22) have been clarified; (B) The structure of the phase -that is what comes to replace θ • r in the second line of (6.22) has been reduced. We consider one item at the time.

(6.36)
In view of (6.35), the expression D x Z • 0x (x; τ, s) must be computed at the position s = τ /ε. This means that s must be replaced by τ /ε in (6.36). At first sight, the integral (in r) from 0 up to τ /ε should furnish a contribution of size ε −1 , which would indicate that D x x(ε, τ, x) is very large (and therefore out of control). However, as mentioned in Remark 3.13, Assumption 1.3 implies that This identity can be viewed as a transparency condition: nonlinear terms that should contribute (without Assumption 1.3) disappear in practice. As a matter of fact, denoting by s the integer part of the real number s, we can assert that ∇ p H 0 r, Ξ 0x ( Z 0 (z 00 (x); τ ); r), Z 0u (z 00 (x); τ ), Ξ 0p ( Z 0 (z 00 (x); τ ); r) dr.

(6.38)
We see here why the cancelation property (6.37) is crucial. It allows to reduce the long time integration (when s is replaced by τ /ε with τ > 0 and ε 1) in the second line of (6.36) to an integration over some interval of uniformly bounded size (in s), as indicated in (6.38). We have the following interesting expression of the differential D x x.
(6.39) Formula (6.39) with Assumption 1.4 are the gate to prove that the map x → x(ε; ·) is uniformly invertible. We will prove this fact in the next Section 6.4.
Proof. The proof is based on the expansion (6.39). We take τ ≤ T and ε ≤ ε 0 with T and ε 0 small enough. We work under Assumption 1.4 with δ small enough. In view of (6.41) and (6.39), both D x Z • 0x (x; τ, s) and D x x(ε, τ, x) are of the form Id+B with B < 1. Thus, for all τ ∈ [0, T ], the maps x → Z • 0x (x; τ, s) and x → x(ε, τ, x) are locally invertible (uniformly in s ≡ θ τ for the first map and in ε ∈]0, ε 0 ] for the second).
The next step is to find the asymptotic expansion of the inverse x −1 . We have the following asymptotic description of the inverse map of the spatial component of the characteristics. Lemma 6.10. Under Assumptions 1.1, 1.2, 1.3, and 1.4, for |τ | small enough, for all ε ∈]0, ε 0 ] and for all N ≥ 2, the inverse map x −1 (ε, τ,x) can be expanded according to
As foreseen, the rapid variableθ 0 r can be activated at the level of the profilesẐ • jx only for j ≥ 2. Now, in coherence with (6.45) and (6.46), we can assert that . We may see (6.48) as a consequence of a relaxed condition involving τ ,x, θ τ and θ • r . Then, we can work withx, θ τ andθ • r fixed in compact sets. In what follows, these variables are mentioned only when it is necessary to avoid confusion. In this perspective, the contribution which has ε 2 in factor inside (6.48) can be written with by construction where D 0 and D 1 are entirely determined (since they depend on the already known functionsẐ • 0x andẐ • 1x ). Remark that (6.49) is a nonlinear equation. This means that the actual asymptotic calculus is critical. We come back to this point in Paragraph 6.4.2.
For |η| chosen small enough, the Jacobian matrix of F η,R is bounded and the corresponding Jacobian determinant is nonzero at each point.