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Averaging of highly-oscillatory transport equations

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature and re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.


Introduction
In a large variety of situations, one is confronted to the resolution of a family of transport equations of the form ∂ t f (t, y) + F ε (y) · ∇ y f (t, y) = 0, f (0, y) = f 0 (y) ∈ R, t ∈ R, y ∈ R n , (1.1) indexed by a small positive parameter ε, whose occurrence in real-life models often lies at the core of numerous theoretical and numerical difficulties encountered in obtaining a(-n) (approximate-) solution. The nature of the difficulties triggered by the presence of ε may vary according to the form of the vector field y → F ε (y) ∈ R n . In this article, we shall address the specific situation where it can be split into two parts F ε (y) = 1 ε ω(y) G(y) + K(y) (1.2) where the flow (t, y 0 ) → Φ t (y 0 ) associated with the differential equatioṅ y(t) = G(y(t)), y(0) = y 0 , is assumed to be periodic, regardless of the specific trajectory (i.e. independently of the initial condition y 0 at time t = 0) and where y → ω(y) is a scalar function bounded from below by a positive constant. Owing to the 1/ε-term in front of the vector field G, the solution of the transport equation evolves in a highly-oscillatory regime as soon as ε becomes small, which is specifically the regime under investigation here. Given their relevance in this context, we shall pay special attention to geometric aspects, such as volume-or energy-preservation, and for this reason, we will also discuss into more details the case where both G and K are divergence-free and/or Hamiltonian. Examples of highly-oscillatory equations of the form (1.1) are numerous [2,3,4,5,18,19,20,21]. It is obviously out of the scope of this introductory paper to treat all of them: we will rather concentrate on the following model that will constitute hereafter our target application, namely the Vlasov equation with strong magnetic field: where x ∈ R 3 and v ∈ R 3 denote respectively the position and the velocity of a particle, where f : R × R 3 × R 3 → R is the distribution function, i.e. the density of particles at time t, position x and velocity v, and where E : R 3 → R 3 and B : R 3 → R 3 are respectively the electric and magnetic fields, assumed to be external at this stage (i.e. not coupled with f through Maxwell equations for instance). Our first objective is this paper is to derive asymptotic models for equation (1.1) with F ε satisfying (1.2) and ω ≡ 1. Rather then merely obtain the limit equation where ε tends to zero, we strive after higher-order terms in powers of ε. The methodology we propose relies on the theory of averaging for highly-oscillatory ordinary differential equations [27,28], and more precisely on normal forms obtained as ε-expansions. Such series have been derived with B-series in [7,8,10] or somehow more simply in [24,25,26] with word-series. The underlying results we shall lean onto will be presented in Section 3. Prior to that, we shall show in Section 2 how the splitting of the vector field F ε into two commuting vector fields naturally leads to two independent transport equations. The first result of this paper (for constant ω) will be stated in Section 4. In Section 5, we will address the much more involved situation of a varying frequency (ω non-constant in (1.2)), which requires to work in an augmented space. In particular, the main result of this paper will be stated there. It allows to rewrite the original transport equation (1.1) as a set of four non-stiff equations for a phase-function (S) and a profile-function (h). The two equations for the profile function are the counterpart of the averaged equation obtained elsewhere in the literature. However, solving the equation for the phase-function S allows to recover exactly the complete solution of (1.1). This part is up to our knowledge completely new. Since we use series-expansions, it is possible to write down explicitly and in a systematic way the terms appearing in the four equations for S and h. In Section 6, we shall eventually envisage our target application (1.4) and show how to obtain the terms of these developments. Firstly, in Section 6.1, we will consider the case of a constant magnetic field B(x) ≡ B in two dimensions (1.4), as it appears to be a simple application of the results of Section 4. Secondly, in Section 6.2, we will address the more complicated situation of a varying magnetic field (B non-constant in (1.4)), which requires a preliminary treatment of the transport equation, as exposed in Section 5. At last, we shall treat equation (1.4) in full generality, i.e. in three dimensions and with a non-constant magnetic field, and compare the equations we obtain with our methodology to results previously published in the literature.

Decomposition of a transport equation
Let us consider the Liouville equation associated to a split vector field of the form and let us make the fundamental assumption that the Lie bracket of F 1 and F 2 vanishes, that is to say that

A few elementary relations from differential geometry
This commutation of vector fields further manifests itself as the commutation of the two flows 1 associated with F 1 and F 2 . More precisely, denoting respectively by ϕ t , Φ t , and Ψ t , the t-flows corresponding to respectively F , F 1 and F 2 , as defined by the differential equations then the following relations hold true In other words and equivalently, the pullback of F 1 through Φ t and Ψ t is F 1 itself, i.e.
∀t ∈ R, ∀y ∈ R n , ∂ y Φ t (y) F 1 (y) = F 1 • Φ t (y) and ∂ y Ψ t (y) F 1 (y) = F 1 • Ψ t (y), (2.3) and the pullback of These relations can be found in most textbooks on dynamical systems or easily re-derived. 1 These flows are assumed to be defined for all t ∈ R and all y ∈ R n without further notice.

Characteristics and separation of variables
The method of characteristics immediately gives for any smooth solution of (1.1) and given the initial condition at time t = 0, the solution of (1.1) may be written as Now, owing to relations (2.2), a somehow natural step forward consists in separating the two times in Φ and Ψ and defining the new functionf with additional variable τ We are now in position to state a few elementary results: The solution (t, y) → f (t, y) of equation (1.1) may be obtained as the value on the diagonal τ = t of any solutionf (it is not unique) of the equation Proof. Letf be any solution of (2.7). It is straightforward to see that, for any given fixed value y ∈ R n , one has (∂ tf )(t, t, y) + (∂ τf )(t, t, y) = ∂ t (f (t, t, y)) so thatf (t, t, y) satisfies equation (1.1) with the same initial condition as f (t, y). The result follows by uniqueness of the solution of (1.1).
This result is of little interest per se. More interesting is the following lemma, which shows that, as far as the specific solutionf of (2.7) is concerned, it satisfies two independent equations.
Proof. We have on the one hand where we have used the second relation of (2.1) with τ replaced by −τ and y by Ψ −t (y), and on the other hand where we have used the second relation of (2.3). Adding up the two terms, we get equation (2.8). Now, repeating similar arguments, using the third relation of (2.1) with t replaced by −t and the first relation of (2.4), we get and on the other hand, using the second relation of (2.4) and again, upon adding up these two terms, we obtain equation (2.9).
It is worth emphasizing that, as stated in previous lemma, the initial conditioñ is enough to determine a unique solution: in order to getf at (t, τ ), the two equations (2.8) and (2.9) have to be solved successively. Nevertheless, the order in which they are solved does not play any role.

Averaging of ordinary differential equations in a nutshell
Since our approach for averaging the transport equation (1.1) consists in averaging first the characteristics and then rewrite the corresponding Liouville equations, we hereafter recall the main results upon which we shall lean. In this expository paper, we content ourselves with formal expansions, thus neglecting at this stage the occurence of error terms. This is justified by the fact that these errors actually become exponentially small in terms of ε (they are bounded by Ce −C/ε for some positive constant C). A completely rigorous treatment of this part can be found for in [8] and the consequences in our situation will be analysed in a forthcoming paper [11], together with the numerical counterpart of the technique presented here.

A normal form theorem
Consider the highly-oscillatory differential equatioṅ i.e. equation (1.2) with ω ≡ 1, where both vector fields G and K are assumed to be smooth 2 . As already alluded to in the Introduction section, the fundamental assumption (H) required to go any further is that: (H) G generates a periodic flow, regardless of the specific trajectory (i.e. with a period which remains independent of the initial value). By convention, we will suppose here that this period is 2π.
Now, in accordance with previous notations, let us write ϕ ε t the t-flow of the differential equation and Φ τ and Ψ t the τ -and t-flows of the two differential equations Since the Lie-bracket of G and K has here no reason to vanish, the two flows Φ τ and Ψ t do not commute and we can not reproduce right away the analysis conducted in previous section. It is precisely the aim of averaging to rewrite F ε as the sum of two commuting fields 3 . As already emphasized, this is in general possible only up to small error terms, so that the theorem stated below is to be understood in a formal sense.
Theorem 3.1 Suppose that the vector fields F ε can be split according to equation (3.1) and that G satisfies assumption (H). Then there exist two vector fields G ε and K ε such that (ii) the vector field G ε generates a flow Φ ε t which is 2π-periodic, regardless of the specific trajectory, i.e.
Remark 3.2 As a consequence of the first point of Theorem 3.1, we have where Ψ ε t denotes the t-flow associated with vector field K ε .
This result brings us back to Section 2 and indeed allows to split equation (1.1) into two equations of the form (2.8, 2.9); details will be given in Section 4. We conclude this subsection with a few additional statements related to the conservation of geometric properties by stroboscopic averaging.
Theorem 3.3 Suppose that the vector fields F ε can be split according to equation (3.1) and that G satisfies assumption (H). Then the two vector fields G ε and K ε of Theorem 3.1 have the following properties: (i) if both G and K are divergence-free vector fields, then so are G ε and K ε ; (ii) if both G and K are Hamiltonian vector fields, then so are G ε and K ε ; Remark 3.4 The properties of Theorem 3.3 are intimately linked to the choice of stroboscopic averaging (see [10,14]), which is the only averaging procedure preserving geometric properties of the initial vector field F ε .

Expansions in powers of ε of the vector fields G ε and K ε
Since we wish in particular to identify the asymptotic behaviour of (1.1) in the limit when ε tends to zero as well as higher-order terms in ε, it is essential to consider ε-expansions of the various functions appearing in Theorem 3.1. Since this was precisely the point of view adopted in [10,14], where B-series were used to this aim, we shall again quote the following result 4 : Theorem 3.5 Consider the Fourier series of The averaged vector field K ε admits the following formal ε-expansion where the coefficientsβ are universal (problem-independent). Similarly, the vector field G ε admits the following formal ε-expansion Remark 3.6 The fact that geometric properties of G ε and K ε are inherited from F ε may also be seen as a direct consequence of the form of previous expansions, which are linear combinations of embedded Lie-brackets of theK k 's. As a matter of fact, the process of changing the variables in K τ and forming Lie-brackets of theK k 's commute. For instance, if both G and K are Hamiltonian, then K τ is of the form and all Fourier coefficientsK k (y) = J −1 ∇ yĤk (y) are also Hamiltonian. Since where {·, ·} denotes the Poisson bracket operation, it is then immediate to see that both G ε and K ε are Hamiltonian with Hamiltonians given by formulas (3.3) and (3.4) where Lie brackets are replaced by Poisson brackets and theK k 's by theĤ k 's. Similarly, if div (G) = div (K) = 0, then div (K k ) = 0 for all k ∈ Z and a standard computation shows that so that again both G ε and K ε are divergence-free.
In order to be able to derive the expansions of G ε and K ε , it still remains to give the value of the coefficientsβ appearing in formula (3.3). This is the purpose of next proposition.
Proposition 3.7 The coefficientsβ can be computed recursively from the following formulas, which hold for all values of j ∈ Z * , r, s ∈ N * and (l 1 , . . . , l s ) ∈ Z s : For the sake of illustration and later use, we now give the first terms of , as stated in [7]: The following expressions of the first three terms of the averaged equation have also been derived in various places and do not use Fourier coefficients: Further terms can be formally obtained by using a non-linear Magnus expansion [1]. Each of these is a linear combination of iterated integrals of iterated brackets of K τ .

Averaging for highly-oscillatory kinetic equations
Compiling the arguments of the two previous sections, it is now straightforward to obtain the following corollary, which establishes in particular the formal existence of an averaged transport equation for problems of the form (1.1, 1.2).
Equation (ii) is the so-called averaged kinetic equation. Moreover, the ε-expansions of G ε and K ε are given by formulas (3.3, 3.4) of Theorem 3.5. If in addition G and K are both divergence-free, then so are G ε and K ε , and similarly, if G and K are both Hamiltonian, then so are G ε and K ε , with Hamiltonians that can be obtained again from formulas (3.3, 3.4) by replacing Lie brackets by Poisson brackets.
Proof. The statements follow immediately by taking F 1 = 1 ε G ε and F 2 = K ε in Lemma 2.2.
As a first illustration of this corollary, we consider the very simplified case of a set of particles evolving in a constant electric field (independent of time and phase-space variables) and submitted to a constant magnetic field. The corresponding equation Due to the extreme simplicity of the vector field F ε , the solution f (t, x, v) can be written explicitly asf (t, τ, x, v) for τ = t/ε and In order to apply the result of this section, we write The corresponding flow Φ τ (associated with G) simply reads where we have used the relation e θJ = (cos θ)I + (sin θ)J and have written cos θ = 1 2 (e iθ + e −iθ ) and sin θ = 1 2i (e iθ − e −iθ ). Formula (3.3) then gives By direct differentiation of formula (4.2) w.r.t. t and then τ , it can be seen thatf satisfies both equations (i) and (ii), as expected.

High-oscillations with varying frequency
In this section, we again consider the transport equation where the vector field F ε is now of the form with G still generating a 2π-periodic flow Φ τ , independently of the initial condition. In this form, Theorem 4.1 does not directly apply, owing to the non-existence of a common frequency for all trajectories (if ω varies). In order to rewrite (5.1) in a more amenable form, we thus divide it by ω 1 ω(y) ∂ t f (t, y) + 1 ω(y) F ε (y) · ∇ y f (t, y) = 0. where DF ε [f ] is the Lie derivative of f in the direction of the augmented vector fielď whereǦ still generates a 2π-periodic flow. Denotingφ ε s the s-flow associated with the vector fieldF ε , we may write from (5.4) so that ∀s ∈ R, f •φ ε −s (t, y) = f (t, y).

Immersion as the stationary solution of an extended equation
The extended function g(s, Y ) = f (φ ε s (Y )) can be regarded as the (stationary) solution of the transport equation Proceeding as in Section 2, we then get two equations for which in principle can be solved one after another in any order. Note the usual relatioñ g(s, s/ε, Y ) = g(s, Y ) = f (Y ). However, there is here no initial condition at s = τ = 0, sinceg(0, 0, Y ) = g(0, Y ) = f (t, y), i.e. precisely the unknown of the original problem.
Remark 5.2 Several remarks are in order: • The result in previous theorem is at this stage only formal. It will be the subject of a forthcoming paper to prove error estimates for the defects in (5.10) and (5.11).
• It can be easily seen from the two second equations of (5.10) and (5.11) that both S and h are periodic in τ . This indeed follows from Theorem 3.1 and the method of characteristics.
• The two equations (5.13, 5.14) and (5.16, 5.17) can be solved in arbitrary order. As a matter of fact, recalling that the fieldsǨ ε (Y ) andǦ ε (Y ), used in the averaged models (5.7) and (5.8), have a vanishing Lie-bracket, the following relations hold true: The commutation of the flows of (5.16) and (5.17) now follows provided that which an easy consequence of relations (5.19) and of the following equation, which holds for any scalar function α, any vector field F and any vector u ∈ R n : ∂ y (α(y)F(y)) u = (∇ y α(y) · u)F(y) + α(y)(∂ y F(y)) u.

An illustrative elementary example
Our aim here is to illustrate the result of Section 5 on an elementary example for which exact solutions can be easily obtained. Consider the following transport equation (5.20) where y ∈ R 2 , and where ω(y) = 1 + |y| 2 = 1 + y 2 1 + y 2 2 and J = 0 1 −1 0 .
This equation can be solved as follows: let ϕ ε t (y) be the flow of the characteristics equatioṅ y = 1 ε ω(y)Jy + y.
By taking its inner product by y, we have immediately |ϕ ε t (y)| = exp(t)|y|, so that ϕ ε t (y) = exp(t) exp As a consequence, the explicit solution of (5.20) reads Now, we observe that the two fields ω(y)Jy and K(y) = y do not commute, and in order to transform the problem into a highly-oscillatory problem with y-independent frequency, one has to divide equation (5.20) by ω and immerse the equation on f into an augmented one for the unknown g(s, t, y) Jy + y ω(y) · ∇ y g = 0, g(0, t, y) = f (t, y). .

The Vlasov equation with a strong magnetic field
In this section, we consider the case of particles submitted to a strong magnetic field and evolving in an electric field E(x) depending on the position. We recall hereinafter the corresponding equation (1.4) which is closely related to the illustrative example of Section 4, though with the additional difficulty that E and B may vary. We further assume here that E derives from a potential U , i.e. that E(x) = −∇ x U (x).

Constant magnetic field
Over a first phase, we assume that the magnetic field is constant. This means that, up to constant rotation, we have B(x) = (0, 0, b(x)) T and b(x) ≡ b. Upon rescaling the time x, v) we may even assume that b = 1. This hypothesis allows to assert that equation n = 4 and .
We now repeat the steps followed for the example of Section 4, starting first with the flow Φ τ (associated with G) The time-dependent vector field K τ then writes

Formula (3.3) then gives
where we used computed successively At second order, equation (i) of Corollary 4.1 forf (t, τ, x, v) thus has the following form

Magnetic field with varying intensity and constant direction
Over this second phase, we address the case of a magnetic field with varying intensity b(x) and constant direction B(x) = (0, 0, b(x)) T . In order to handle this new difficulty, one has to re-parametrize the time as in Section 5. Equivalently, this amounts to dividing equation (1.4) by b(x). In order to do so, b(x) should not vanish for any x in R 2 and we will make this assumption for the remaining of this section. The "new" equation writes Denoting Y = (t, x 1 , x 2 , v 1 , v 2 ) ∈ R 5 the now extended phase-space variable, we may also write previous equation as the Lie-derivative of f in the direction of an extended vector fieldF ε , i.e. DF ε f = 0 withF or equivalently as the stationary solution of so that J∇ 2 U + ∇ 2 U J = (∆U )J and where g then denotes a function of the 2 variables (s, Y ). We may now resume the derivation of the equations (i) and (ii) of Theorem 4.1, by first splittingF ε intoF ε = 1 εǦ +Ǩ withǦ It is clear thatǦ now generates a 2π-periodic flow and the corresponding Fourier modes are all vanishing except the modes 1, −1 and 0 (the additional one w.r.t. the case of a constant field): According to Theorem 4.1, we thus have Omitting the argument x in E, U and b, and denoting simply ∇ for ∇ x , we have Besides, we have Finally, at first order in ε, we have Therefore the transport equations on h are at first order in ε: and with the initial condition h(0, 0, y) = f 0 (y). Similarly the transport equations on S are and with the initial condition S(0, 0, y) = 0. Now we make an important comment on these transport equations. First, the transport equation (6.3) coincides with the gyro-kinetic model derived by Frénod & Sonnendrücker in [19]. However, our approach provides more information through the phase S and the dependence in τ . These informations are necessary to correctly reconstruct the full original distribution function f (and not only the averaged model) at first order in ε. This reconstruction may be performed through the relation f (t, x, v) = h(t, τ (t, x, v), x, v) where τ (t, x, v) is a solution to ετ = S(t, τ, x, v). Up to our knowledge, no such construction can be found in the literature.

Magnetic field with varying intensity and varying direction
We now consider the transport kinetic equation in its general form (1.4). This means that now we allow variations of B in both amplitude and direction. Our aim in this part is to extend our previous approach to this more general case.
We first immerse the model (1.4) in an augmented problem in the unkown g(s, t, x, v), as follows with the initial condition g(0, t, x, v) = f (t, x, v). The main interest of this form is that the oscillatory part in the variable s is now driven by the vector field B(x) |B(x)| × v, which, as we shall see, generates a periodic flow with a constant period 2π. In particular the period does not depend on the trajectory, although the unitary vector B(x) |B(x)| depends on this trajectory . Indeed let e 0 a unit vector and let (e 0 , e 1 , e 2 ) an orthonormal basis such that e 0 × e 1 = e 2 and e 1 × e 2 = e 0 . The matrix representing the skew-symmetric linear map L e 0 : v → v × e 0 in the basis (e 0 , e 1 , e 2 ), is simply J = 0 0 0 J . Since exp(tJ ) is 2π-periodic, the flow exp(tL e 0 ) is 2π-periodic. We now apply our methodology on model (6.7). Here the vector field F ε = 1 ε G + K is given by We introduce the following notations Using Theorem 3.5, the vector field K τ can be easily computed from The following elementary identities We then deduce the expression of the Jacobian matrix ∂ t,x,v Φ τ = (∂ t Φ τ , ∂ x Φ τ , ∂ v Φ τ ): where R = (1 − cos τ )∂ x P e(x) v + (sin τ )∂ x L e(x) v = α 0 + αe iτ + αe −iτ , Q τ = (cos τ )I + (1 − cos τ )P e(x) + (sin τ )L e(x) = a 0 + ae iτ + ae −iτ , , and a 0 = P e(x) , α 0 = ∂ x P e(x) v , 2a = I − P e(x) − iL e(x) , 2α = −∂ x P e(x) v + iL e(x) v .

Now we also have
One can easily see that the Fourier expansion of K τ (in the periodic variable τ ) only contains modes k ∈ Z with |k| ≤ 3. Note that we can recover the previous case (B(x) with constant direction and (x, v) ∈ R 2 × R 2 ) by taking P e(x) v = 0, L e(x) ≡ J and α = 0, which means that R = 0 and Q τ = e τ J .
Although all the Fourier coefficients of K τ can be derived from this expression, we just give for simplicity the zero-th mode: with (K 0 ) 3 (x, v) = a 0 E(x) − (a 0 b 0 a 0 + a 0 ba + a 0 ba + ab 0 a + aba 0 + ab 0 a + aba 0 )v = P e(x) E(x) − 4P e(x) ∂ x P e(x) v P e(x) + 1 2 P e(x) ∂ x L e(x) v L e(x) We then deduce the vector field G ε at the 0 th order in ε: The averaged model at the 0 th order in ε can now be written in terms of h(t, τ, x, v) and S(t, τ, x, v). We have |B(x)| · ∇ v S = 0, with the initial conditions: h(0, 0, x, v) = f 0 (x, v) and S(0, 0, x, v) = 0. Note that in the particular case where B(x) has a constant direction B(x) = (0, 0, b(x)) T , we get The averaged equations at the first order in ε can also be derived in the case of a magnetic field B(x) with constant direction B(x) = (0, 0, b(x)) T , with b(x) > 0. In this case we have R = 0, e(x) is the constant unit-vector e z , |B(x)| = b(x), and therefore The computation of K ε at first order in ε can then be derived from Theorem 3.5 and provides, up to a rescaling in time, an asymptotic model which is similar to the one recently derived by Degond & Filbet in [17].