Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach

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Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach

Thomas Blanc, Mihai Bostan, Franck Boyer

To cite this version:

Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2017, 37 (9), pp.4637-4676. �10.3934/dcds.2017200�.

�hal-01242679v3�

Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach

Thomas BLANC ^∗ , Miha¨ı BOSTAN ^† , Franck BOYER ^‡ (January 19, 2017)

Abstract

We perform the asymptotic analysis of parabolic equations with stiff transport terms.

This kind of problem occurs, for example, in collisional gyrokinetic theory for tokamak plasmas, where the velocity diffusion of the collision mechanism is dominated by the velocity advection along the Laplace force corresponding to a strong magnetic field. This work appeal to the filtering techniques. Removing the fast oscillations associated to the singular transport operator, leads to a stable family of profiles. The limit profile comes by averaging with respect to the fast time variable, and still satisfies a parabolic model, whose diffusion matrix is completely characterized in terms of the original diffusion matrix and the stiff transport operator. Introducing first order correctors allows us to obtain strong convergence results, for general initial conditions (not necessarily well prepared).

Keywords: Average operators, Ergodic means, Unitary groups, Multiple scales, Homoge- nization.

AMS classification: 35Q75, 78A35

1 Introduction

General framework. In many applications we deal with partial differential equations with disparate scales. The solutions of the problems in hand fluctuate at very different scales and for the moment, solving numerically for both slow and fast scales seems out of reach.

Depending on the particular regimes we are interested in, it could be worth to solve an averaged problem with respect to the slow variable, after smoothing out the fast oscillations.

In this work we focus on parabolic models perturbed by stiff transport operators (

∂ t u ^ε − div y (D(y)∇ _y u ^ε ) + 1

ε b(y) · ∇ _y u ^ε = 0, (t, y) ∈ R + × R ^m

u ^ε (0, y) = u ⁱⁿ (y), y ∈ R ^m .

(1) Here b : R ^m → R ^m and D : R ^m → M _m (R) are given fields of vectors and symmetric positive definite matrices, and ε > 0 is a small parameter destinated to converge to 0. If the vector

∗

Aix Marseille Universit´ e, CNRS, Centrale Marseille, Institut de Math´ ematiques de Marseille, UMR 7373, Chˆ ateau Gombert 39 rue F. Joliot Curie, 13453 Marseille FRANCE. E-mail : thomas.blanc@univ-amu.fr

†

Aix Marseille Universit´ e, CNRS, Centrale Marseille, Institut de Math´ ematiques de Marseille, UMR 7373, Chˆ ateau Gombert 39 rue F. Joliot Curie, 13453 Marseille FRANCE. E-mail : mihai.bostan@univ-amu.fr

‡

Institut de Math´ ematiques de Toulouse, UMR 5219, Universit´ e de Toulouse, CNRS, UPS IMT, F-31062

Toulouse Cedex 9, France. E-mail : franck.boyer@math.univ-toulouse.fr

field b is divergence free, the energy balance writes 1

2 d dt

Z

R

^m

(u ^ε (t, y)) ² dy + Z

R

^m

D(y)∇ _y u ^ε · ∇ _y u ^ε dy = 0.

Therefore, since the matrices D(y) are positive definite, the L ² norms of the solutions (u ^ε ) _ε>0 decrease in time, and we expect that the limit model still behaves like a parabolic one, whose diffusion matrix field needs to be determined.

Motivating example. This work is motivated by the study of collisional models for the gyrokinetic theory in tokamak plasmas. The fluctuations of the presence density of charged particles are due to the transport in space and velocity (under the action of electro-magnetic fields), but also to the collision mechanisms. In the framework of the magnetic confinement fusion, the external magnetic fields are very large, leading to a stiff velocity advection, due to the magnetic force qv ∧ B ^ε = qv ∧ ^B _ε . Here q stands for the particle charge and B ^ε = ^B _ε represents a strong magnetic field, when ε goes to 0. Using a Fokker-Planck operator for taking into account the collisions between particles, we are led to the Fokker-Planck equation

∂ t f ^ε +v ·∇ _x f ^ε + q m

E + v ∧ B ε

·∇ _v f ^ε = ν div v (Θ∇ _v f ^ε +vf ^ε ), (t, x, v) ∈ R + × R ³ × R ³ (2) where E is the electric field, m is the particle mass, ν is the collision frequency and Θ is the temperature. Notice that in the Fokker-Planck equation, the diffusion occurs only in the velocity space, and therefore (2) is different from (1). Nevertheless, we expect that the main arguments used for the treatment of (1) still apply when investigating the asymptotic behavior of (2), see Section 3.2. The asymptotic analysis of (2), when neglecting the collisions is now well understood [8, 25, 26, 28]. It can be handled by averaging the perturbed model along the characteristic flow associated to the dominant transport operator. Recently, models including collisions have been analyzed formally by using the averaging method [9, 10]. In particular, it was emphasized that, averaging with respect to the fast cyclotronic motion leads to diffusion not only in velocity, but also with respect to the perpendicular space directions, see (32). At least at the formal level, the theoretical study presented in this work allows to perform the asymptotic analysis of the Fokker-Planck equation, under strong magnetic fields, see Section 3.2. The study of the averaged diffusion matrix field is crucial when determining the equilibria of the limit Fokker-Planck equation (2), when ε goes to zero. Numerical results concerning strongly anisotropic elliptic and parabolic problems were obtained in [21, 24, 18].

The advection-diffusion problems with large drift in periodic setting have been extensively studied by many authors [23, 22, 33, 5, 34, 29]. For the homogenization of the associated eigenvalue problems, we refer to [6, 30]. The aim of this paper is to obtain an homogenization result for the problem (1) without periodicity assumptions on the advection field and for initial data not necessarely well prepared. The effective limit problem comes by appealing to an ergodic theory result. It is a parabolic problem, whose diffusion matrix field appears as the average of the original diffusion matrix field, along a flow. We specify sufficient assumptions which guarantee the well definition of such an average matrix field. Moreover, we study the behavior of the solutions of (1), when ε goes to zero, and estimate the convergence rate in L ² , by constructing a suitable corrector. The asymptotic analysis of the system (1) was provided also in [29] in the new framework of weak Σ-convergence along a flow. This approach leads to the same effective limit problem as our, but with a different interpretation based on the theory of ergodic algebra with mean value.

Our paper is organized as follows. The main results are introduced in Section 2. We

indicate the main lines of our arguments by performing formal computations. In Section 3

we present a brief overview on the construction of the average operators for matrix fields through the ergodic theory. Section 4 is devoted to obtaining uniform estimates for solutions of (1), in view of convergence results. In Section 5 we establish two-scale convergence results, in the ergodic setting, which allows us to handle situations with non periodic fast variables.

Up to our knowledge, these results have not been reported yet. The proofs of the main theorems are detailed in Section 6. Some technical arguments are presented in Appendix A.

2 Presentation of the main results and formal approach

The subject matter of this paper concentrates on the asymptotic analysis of (1), when ε becomes small. Obviously, the fast time oscillations come from the large advection field ^b(y) _ε . Indeed, when neglecting the diffusion operator, the problem (1) reduces to a transport model, whose solution writes

u ^ε (t, y) = u ⁱⁿ (Y (−t/ε; y)), (t, y) ∈ R + × R ^m . (3) Here (s, y) ∈ R × R ^m → Y (s; y) ∈ R ^m stands for the characteristic flow of b · ∇ _y

dY

ds = b(Y (s; y)), (s, y) ∈ R × R ^m , Y (0; y) = y, y ∈ R ^m . This flow is well defined under standard smoothness assumptions

( b ∈ W _loc ^1,∞ ( R ^m ),

∃C > 0 such that |b(y)| ≤ C(1 + |y|), y ∈ R ^m . (4) Under the above hypotheses the flow Y is global and smooth, Y ∈ W _loc ^1,∞ (R × R ^m ). Moreover, we assume that

b is divergence free, (5)

which guarantees that the transformation y ∈ R ^m → Y (s; y) ∈ R ^m is measure preserving for any s ∈ R . Motivated by (3), we introduce the new unknowns

v ^ε (t, z) = u ^ε (t, Y (t/ε; z)), (t, z) ∈ R + × R ^m , ε > 0 (6) and we expect to get stability for the family (v ^ε ) ε>0 , when ε goes to 0. In that case we will deduce that, for small ε > 0, u ^ε behaves like v(t, Y (−t/ε; y)), for some profile v = lim ε&0 v ^ε , that is, u ^ε appears as the composition product between a stable profile and the fast oscillating flow Y (−t/ε; y). We prove mainly two strong convergence results for general initial conditions (not necessarily well prepared), whose simplified versions are stated below. For detailed assertions see Theorems 2.2, 2.3.

Theorem We denote by (u ^ε ) _ε>0 the variational solutions of (1) and by (v ^ε ) _ε>0 the functions v ^ε (t, z) = u ^ε (t, Y (t/ε; z)), (t, z) ∈ R + × R ^m , ε > 0.

1. Under suitable hypotheses on the vector field b, the matrix field D and the initial con- dition u ⁱⁿ , the family (v ^ε ) _ε>0 converges strongly in L ^∞ _loc ( R + ; L ² ( R ^m )) to the unique variational solution v ∈ L ^∞ (R + ; L ² (R ^m )) of (26), whose diffusion matrix field hDi comes by averaging the matrix field D along the flow of the vector field b (cf. Theorem 2.1), that is

hDi = lim

S→+∞

1 S

Z S 0

∂Y (−s; Y (s; ·))D(Y (s; ·)) ^t ∂Y (−s; Y (s; ·)) ds.

2. Under additional regularity hypotheses, we have the error estimate sup

t∈[0,T ]

ku ^ε (t, ·) − v(t, Y (−t/ε; ·))k _L

2

( R

^m

) ≤ C T ε.

The problem satisfied by v ^ε is obtained by performing the change of variable y = Y (t/ε; z) in (1). A straightforward computation based on the chain rule leads to

∂ _t v ^ε (t, z) = ∂ _t u ^ε (t, Y (t/ε; z)) + 1

ε b(Y (t/ε; z)) · (∇ _y u ^ε )(t, Y (t/ε; z)).

To deal with the diffusion term, we need to compute the action of the operator div y (D∇ _y ·) when applied to a function of the form v ◦ Y (−s, ·) for some s ∈ R . A tedious but straight- forward computation shows that we can write, for any integrable function z 7→ w(z), any matrix field y 7→ C(y), and any s ∈ R,

div _y {C∇ _y (w ◦ Y (−s, ·))} = (div _z (G(s)C ∇ _z w)) ◦ Y (−s, ·) in D ⁰ ( R ^m ), (7) where the family of matrix fields (G(s)C) _s is defined by

(G(s)C)(z) :=∂Y ⁻¹ (s; z)C(Y (s; z)) ^t ∂Y ⁻¹ (s; z)

=∂Y (−s; Y (s; z)) C(Y (s; z)) ^t ∂Y (−s; Y (s; z)) , (s, z) ∈ R × R ^m , (8) Using those notations, the change of variable y = Y (t/ε; z) in (1) leads to

∂ _t v ^ε − div _z ((G(t/ε)D)∇ _z v ^ε ) = 0, (t, z) ∈ R + × R ^m

v ^ε (0, z) = u ^ε (0, z) = u ⁱⁿ (z), z ∈ R ^m , ε > 0 (9) The new diffusion problem (9) seems simpler than the original problem (1), because the singular term ¹ _ε b · ∇ _y has disappeared. Nevertheless, the new model depends on a fast time variable s = t/ε, through the diffusion matrix field G(t/ε)D, and a slow time variable t.

We deal with a two-scale problem in time. As often in asymptotic analysis of multiple scale problems, a way to understand the behavior of the solutions (v ^ε ) ε>0 when ε goes to 0 and to identify the limit problem is to use a formal development whose terms depend both on the slow and fast time variables

v ^ε (t, z) = v(t, t/ε, z) + εv ¹ (t, t/ε, z) + .... (10) This method is used in many frameworks such as periodic homogenization for elliptic and parabolic systems [1, 32], transport equations [13, 19] or kinetic equations [11]. Plugging the Ansatz (10) in (9) and identifying the terms of the same order with respect to ε, lead to the hierarchy of equations

∂ _s v = 0 (11)

∂ t v − div z (G(s)D∇ _z v) + ∂ s v ¹ = 0 (12) .. .

Equation (11) says that the first profile v does not depend on the fast time variable s, that

is v = v(t, z). We expect that v is the limit of the family (v ^ε ) _ε>0 , when ε goes to 0. The

slow time evolution of v is given by (12), but we need to eliminate the second profile v ¹ .

Actually v ¹ appears as a Lagrange multiplier which guarantees that at any time t, the profile

v satisfies the constraint ∂ _s v = 0. In the periodic case, we eliminate v ¹ by taking the average

over one period. However, without periodicity assumptions, we need to use ergodic averaging techniques, that is to write







∂ _t v − div _z

S→+∞ lim 1 S

Z S 0

G(s)D ds

∇ _z v

= 0, (t, z) ∈ R + × R ^m

v(0, z) = u ⁱⁿ (z), z ∈ R ^m .

(13)

The key point is that (G(s)) s∈ R is a C ⁰ -group of unitary operators (on some Hilbert space to be determined), and thanks to von Neumann’s ergodic mean theorem [35], the limit hDi = lim S→+∞ 1

S

R S

0 G(s)D ds makes sense. The Hilbert space which realizes (G(s)) s∈ R as a C ⁰ - group of unitary operators appears as a L ² weighted space, with respect to some field of symmetric positive definite matrices. We assume that there is a matrix field P such that

t P = P, P (y)ξ · ξ > 0, ξ ∈ R ^m \ {0}, y ∈ R ^m , P ⁻¹ , P ∈ L ² _loc (R ^m ) (14)

[b, P ] = 0, in D ⁰ (R ^m ), (15)

where the bracket between a vector field c and a matrix field A is a matrix field formally defined by

[c, A] := (c · ∇ _y )A − ∂ _y cA − A ^t ∂ _y c. (16) The condition (15) naturally appears in our problem because of the following proposition.

Proposition 2.1 Consider a vector field c ∈ W _loc ^1,∞ ( R ^m ) (not necessarily divergence free) with at most linear growth at infinity and A(y) ∈ L ¹ _loc (R ^m ) a matrix field.

1. We can express the commutator between the advection operator c · ∇ _y and the diffusion operator div y (A(y)∇ _y ) as follows

c · ∇ _y , div _y (A(y)∇ _y )

= div _y ([c, A]∇ _y ) + ^t A(y)∇ _y div _y c

· ∇ _y . (17) In particular, if div _y c = 0, this commutator is also a diffusion operator with respect to the matrix field [c, A].

2. The following assertions are equivalent (a) We have [c, A] = 0 in D ⁰ ( R ^m ) (b) For any s ∈ R , y ∈ R ^m , we have

A(Y (s; y)) = ∂Y (s; y)A(y) ^t ∂Y (s; y).

where Y stands for the flow associated with the vector field c.

(c) For any s ∈ R , we have

G(s)A = A, where G is defined in (8).

Proof.

1. A straightforward computation first shows that the commutator between c · ∇ _y and div _y is given by

[c · ∇ _y , div _y ]ξ = ξ · ∇ _y div _y c − div _y (∂ _y c ξ), ξ ∈ (C ² ( R ^m )) ^m .

Using the above formula with ξ = A(y)∇ _y u, for a smooth u, one gets c · ∇ _y (div y (A(y)∇ _y u)) − div y (c · ∇ _y (A(y)∇ _y u))

= A(y)∇ _y u · ∇ _y div y c − div y (∂ y c A(y)∇ _y u). (18) Taking into account that

c · ∇ _y (A(y)∇ _y u) = (c · ∇ _y A)∇ _y u + A(y)(∂ ² u)c(y)

= (c · ∇ _y A)∇ _y u + A(y)∇ _y (c · ∇ _y u) − A(y) ^t ∂ _y c∇ _y u we deduce by (18)

c · ∇ _y (div y (A(y)∇ _y u)) − div y (A(y)∇ _y (c · ∇ _y u)) = A(y)∇ _y u · ∇ _y div y c + div y ((c · ∇ _y A − ∂ y cA(y) − A(y) ^t ∂ y c)∇ _y u), which is the claimed formula.

2. This characterization is proved in [12, Prop. 3.8].

Let us now introduce some useful function spaces. We recall that for any two matrices in M _m ( R ), the notation A : B stands for tr( ^t AB).

Definition 2.1 For any matrix field M : R ^m → M _m (R) ∈ L ² _loc (R ^m ) made of positive definite symmetric matrices, we introduce the weighted L ² space

H _M = n

A : R ^m → M _m ( R ) measurable : M ^1/2 AM ^1/2 ∈ L ² o , which is a Hilbert space for the natural scalar product

(A, B) M :=

Z

R

^m

(M ^1/2 AM ^1/2 ) : (M ^1/2 BM ^1/2 ) dy = Z

R

^m

M A : BM dy, ∀A, B ∈ H M . The associated norm is denoted by |A| _M .

Similarly we introduce the Banach space H _M ^∞ = n

A : R ^m → M _m ( R ) measurable : M ^1/2 AM ^1/2 ∈ L ^∞ o , equipped with the norm

|A| _H

^∞

M

:= |M ^1/2 AM ^1/2 | _L

^∞

.

Assume that there is a continuous function ψ, which is left invariant by the flow of b, and goes to infinity when |y| goes to infinity

ψ ∈ C( R ^m ), ψ ◦ Y (s; ·) = ψ for any s ∈ R , lim

|y|→+∞ ψ(y) = +∞. (19) Since the sets {ψ ≤ k}, for k ∈ N are compact sets invariant by the flow of b, we will be able to perform our analysis in the local spaces

H _M,loc =

A : R ^m → M _m ( R ) measurable : 1 _{ψ≤k} A ∈ H _M for any k ∈ N .

We say that a family (A i ) i ⊂ H M,loc converges in H M,loc toward some A ∈ H M,loc iff for any k ∈ N , the family (1 _{ψ≤k} A i ) i converges in H M toward 1 _{ψ≤k} A.

We define similar spaces for vector fields.

Definition 2.2 With the same assumption as in the previous definition, we define X M := {c : R ^m → R ^m measurable : M ^1/2 c ∈ L ² }

which is a Hilbert space for the scalar product

(c, d) _M :=

Z

R

^m

M(y) : c(y) ⊗ d(y) dy, ∀c, d ∈ X _M ,

and the Banach space

X _M ^∞ = {c : R ^m → R ^m measurable : M ^1/2 c ∈ L ^∞ ( R ^m )}, equipped with the norm |c| _X

_m^∞

:= |M ^1/2 c| _L

^∞

.

The following properties are straightforward.

Proposition 2.2 We consider the same assumptions as in the Definitions above.

• We have the continuous embedding C _c ⁰ ( R ^m , M _m ( R )) ⊂ H _M

• If M ⁻¹ ∈ L ² _loc ( R ^m ) then H M ⊂ L ¹ _loc ( R ^m ; M _m ( R )).

Moreover the space H _M

⁻¹

can be identified to the dual of H M through the L ² pairing that can be writing as

hA, Bi _M,M

⁻¹

:=

Z

R

^m

A(y) : B(y) dy = Z

R

^m

M ^1/2 AM ^1/2 : M ^−1/2 BM ^−1/2 dy

≤|A| _M |B| _M

⁻¹

, ∀A ∈ H M , ∀B ∈ H _M

⁻¹

.

Finally, the maps A ∈ H _M → M AM ∈ H _M

⁻¹

and B ∈ H _M

⁻¹

→ M ⁻¹ BM ⁻¹ ∈ H _M are linear isometries that are reciprocal one from each other.

We can now come back to the problem of giving a sense to the average process in (13).

Given a matrix field P satisfying (14), (15), we set Q = P ⁻¹ and we shall prove that the family of applications G(s) : H _Q → H _Q , s ∈ R , is a C ⁰ -group of unitary operators on H _Q (see Proposition 3.1). Thanks to the von Neumann’s ergodic theorem (see Theorem 3.1 and [35]

for more details), we find that the average of a matrix field hAi := lim S→+∞ 1 S

R S

0 G(s)A ds is well defined and coincides with the orthogonal projection on the invariant subspace {B ∈ H _Q : G(s)B = B for any s ∈ R } see also [12, 13]. Under the assumption (19), the group (G(s)) s∈ R also acts on H Q,loc . In particular, any matrix field of H _Q ^∞ ⊂ H Q,loc possesses an average in H _Q,loc , still denoted by h·i, as for matrix fields in H _Q .

Theorem 2.1 Assume that (4), (5), (14), (15), (19) hold true. We denote by L the in- finitesimal generator of the group (G(s)) s∈ R .

1. For any matrix field A ∈ H Q we have the strong convergence in H Q

hAi := lim

S→+∞

1 S

Z r+S r

∂Y (−s; Y (s; ·))A(Y (s; ·)) ^t ∂Y (−s; Y (s; ·))

| {z }

=(G(s)A)(y)

ds = Proj _{ker L} A

uniformly with respect to r ∈ R .

2. If A ∈ H Q is a field of symmetric positive semi-definite matrices, then so is hAi.

3. Let S ⊂ R ^m be an invariant set of the flow of b, that is Y (s; S) = S for any s ∈ R . If A ∈ H _Q is such that

Q ^1/2 (y)A(y)Q ^1/2 (y) ≥ αI m , y ∈ S , for some α > 0, then we have

Q ^1/2 (y) hAi (y)Q ^1/2 (y) ≥ αI _m , y ∈ S and in particular, hAi (y) is positive definite for y ∈ S.

4. If A ∈ H Q ∩ H _Q ^∞ , then hAi ∈ H Q ∩ H _Q ^∞ and

| hAi | _Q ≤ |A| _Q , | hAi | _H

^∞

Q

≤ |A| _H

^∞

Q

. 5. For any matrix field A ∈ H _Q,loc , the family

1 S

Z S 0

∂Y (−s; Y (s; ·))A(Y (s; ·)) ^t ∂Y (−s; Y (s; ·)) ds

S>0

converges in H _Q,loc , when S goes to infinity. Its limit, denoted by hAi, satisfies 1 {ψ≤k} hAi =

1 {ψ≤k} A

, for any k ∈ N

where the symbol h·i in the right hand side stands for the average operator on H Q . In particular, any matrix field A ∈ H _Q ^∞ has an average in H _Q,loc and | hAi | _H

^∞

Q

≤ |A| _H

^∞

Q

. If A ∈ H _Q,loc is such that

Q ^1/2 (y)A(y)Q ^1/2 (y) ≥ αI m , y ∈ R ^m , for some α > 0, then we have

Q ^1/2 (y) hAi (y)Q ^1/2 (y) ≥ αI _m , y ∈ R ^m .

The construction presented above of the average matrix field relies on the hypotheses (4), (14), (15). However, the asymptotic analysis of the family (u ^ε ) _ε>0 requires to estimate the derivatives of (u ^ε ) _ε>0 , up to the order three, see Propositions 4.3, 4.4. To obtain those estimates, it is very convenient to use derivatives along vector fields which commute with the derivatives along b, such vector fields are said to be in involution with respect to b. In this case, the uniform regularity of (u ^ε ) _ε>0 comes easily by taking the derivatives of (1) along these vector fields in involution with respect to b. More precisely, we shall assume that there is a matrix field R(y) such that

( det R(y) 6= 0, y ∈ R ^m , R ∈ L ¹ _loc ( R ^m )

(b · ∇ _y )R + R∂ _y b = 0 in D ⁰ ( R ^m ). (20) The equation satisfied by R in (20) is equivalent to R(Y (s; y))∂Y (s; y) = R(y), (s, y) ∈ R × R ^m , which also writes

∂Y (s; y)R ⁻¹ (y) = R ⁻¹ (Y (s; y)), (s, y) ∈ R × R ^m , (21)

and is finally equivalent to (b · ∇ _y )R ⁻¹ = ∂ _y bR ⁻¹ , saying that the columns of R ⁻¹ are vector

fields in involution with b, see [12] Proposition 3.4. for more details. The vector fields in the

columns of R ⁻¹ are denoted b _i , 1 ≤ i ≤ m. At any point y ∈ R ^m they form a basis for R ^m cf.

(20) and are supposed smooth and sublinear

( b i ∈ W _loc ^1,∞ (R ^m ), div y b i ∈ L ^∞ (R ^m ), 1 ≤ i ≤ m,

∀i ∈ {1, ..., m}, ∃C _i > 0 such that |b _i (y)| ≤ C _i (1 + |y|), y ∈ R ^m , (22) which guarantees the existence of the global flows Y i (s; y) ∈ W _loc ^1,∞ (R × R ^m ), i ∈ {1, ..., m}.

We claim that the hypotheses (20) imply (14), (15). Clearly R ⁻¹ ∈ L ^∞ _loc ( R ^m ), since b i , which are the columns of R ⁻¹ , are supposed to be locally bounded on R ^m . Since y → R ⁻¹ (y) is continuous, the function y → det R ⁻¹ (y) remains away from 0 on any compact set of R ^m , implying that R = (R ⁻¹ ) ⁻¹ ∈ L ^∞ _loc ( R ^m ). In particular ^t RR, ( ^t RR) ⁻¹ are locally bounded, and therefore locally square integrable on R ^m . We define Q = ^t RR, P = Q ⁻¹ = R ^{−1 t} R ⁻¹ and observe that (14), (15) are satisfied. Indeed, P (y) is symmetric, positive definite, locally square integrable, together with its inverse Q = P ⁻¹ and, thanks to (21), we have

P(Y (s; y)) = R ⁻¹ (Y (s; y)) ^t R ⁻¹ (Y (s; y))

= ∂Y (s; y)R ⁻¹ (y) ^t R ⁻¹ (y) ^t ∂Y (s; y)

= ∂Y (s; y)P (y) ^t ∂Y (s; y)

saying that [b, P ] = 0 in D ⁰ ( R ^m ) cf. Proposition 2.1. Under the hypotheses (22), the spaces H _Q , H _Q ^∞ satisfy

H _Q =

A : R ^m → M _m ( R ) measurable : R A ^t R ∈ L ² ( R ^m ) , H _Q ^∞ = {A : R ^m → M _m ( R ) measurable : R A ^t R ∈ L ^∞ ( R ^m )}.

Given the family (b _i ) 1≤i≤m of vector fields in involution with respect to b, we construct the following Sobolev-type space on R ^m

H _R ¹ :=

m

\

i=1

dom(b _i · ∇ _y )

={u ∈ L ² ( R ^m ) : b i · ∇ _y u ∈ L ² ( R ^m ), ∀i ∈ {1, ..., m}}

(23)

which is a Hilbert space with the scalar product (u, v) _R =

Z

R

^m

u(y)v(y) dy +

m

X

i=1

Z

R

^m

(b _i · ∇ _y u)(b _i · ∇ _y v) dy, u, v ∈ H _R ¹ .

The associated norm is denoted by | · | _R . The operators b i · ∇ _y are the infinitesimal generators of the C ⁰ -groups of linear transformations on L ² ( R ^m ) given by

τ _i (s)u := u ◦ Y _i (s; ·), u ∈ L ² ( R ^m ), s ∈ R , i ∈ {1, ..., m}.

The hypothesis div y b i ∈ L ^∞ ( R ^m ) plays a crucial role when looking for a bound for the Jacobian determinant of ∂Y _i .

Remark 2.1 Notice that every element of H _R ¹ has a gradient in the distribution sense that belongs to L ² _loc ( R ^m ). More precisely, for any u ∈ H _R ¹ , we have

∇ _y u = ^t R ^t b 1 · ∇ _y u, ..., b m · ∇ _y u , and

|u| ² _R = kuk ² _L

2

( R

^m

) + |∇u| ² _P . (24)

It will be convenient in the sequel to make use of the following differential operators

∇ ^R _y := ^t R ⁻¹ ∇ _y = ^t (b ₁ · ∇ _y , ..., b _m · ∇ _y ), as well as its higher order versions

(∇ ^R _y ⊗ ∇ ^R _y ) ij := b i · ∇ _y (b j · ∇ _y ), i, j ∈ {1, ..., m},

(∇ ^R _y ⊗ ∇ ^R _y ⊗ ∇ ^R _y ) _ijk := b _i · ∇ _y (b _j · ∇ _y (b _k · ∇ _y )), i, j, k ∈ {1, ..., m}.

The results in Section 3 hold true under the hypotheses (4), (14), (15) but in Sections 4, 6 and Appendix A we shall need the stronger hypotheses (20) instead of (14), (15).

Moreover we assume that ( _t

D = D, D ∈ H _Q ^∞ , b ∈ X _Q ^∞

∃ α > 0 such that Q ^1/2 (y)D(y)Q ^1/2 (y) ≥ αI m , y ∈ R ^m

(25) where Q = ^t RR and the columns of R ⁻¹ are given by the vector fields b ₁ , ..., b _m .

In view of Theorem 2.1, the formal limit problem (13) of the family of systems (9) becomes ∂ _t v − div _z (hDi ∇ _z v) = 0, (t, z) ∈ R + × R ^m

v(0, z) = u ⁱⁿ (z), z ∈ R ^m . (26)

Under some regularity assumptions (see Section 6), we will obtain a strong convergence result for the family (v ^ε ) ε>0 in L ^∞ _loc ( R + ; L ² ( R ^m )), toward the solution v of the problem (26). Coming back to the family (u ^ε ) _ε>0 , through the change of variable in (6), and thanks to the fact that for any s ∈ R, Y (s; ·) is measure preserving, we justify that at any time t ∈ R + , ε > 0, u ^ε (t, ·) behaves (in L ² ( R ^m )) like the composition product between v(t, ·) and Y (−t/ε; ·), that is

ε&0 lim ku ^ε (t, .) − v(t, Y (−t/ε; ·))k _L

2

= lim

ε&0 kv ^ε (t, .) − v(t, .)k _L

2

= 0, uniformly with respect to t ∈ [0, T ], for any T ∈ R + .

Passing to the limit in (9), when ε & 0, requires a two-scale analysis. This can be achieved by standard homogenization arguments, in the setting of periodic fast variables. The aim of this paper is to obtain results without periodicity assumptions, a framework where not so much results are available. The key point is to appeal to ergodic means along C ⁰ -groups of unitary operators. More precisely, the main difficulty is passing to the limit in the integral

term Z +∞

0

Z

R

^m

G(t/ε)D∇ _z v ^ε · ∇ _z Φ dzdt

that appears in the variational formulation of (9), for any smooth test function Φ. The argument combines the weak convergence of (∇ _z v ^ε ) _ε>0 as ε & 0 in L ² ([0, T ]; X _P ), T ∈ R + , and the strong convergence of

1 S

R S

0 G(s)D ds

S>0 , as S → +∞, in H Q,loc . More exactly, we prove that for any family of functions (w ^β ) β>0 , admiting the same modulus of continuity in C([0, T ]; X _P ) and converging weakly in L ² ([0, T ]; X _P ), as β & 0, toward some w ⁰ , we have

(β,ε)→(0,0) lim Z T

0

Z

R

^m

θ(t, y) ⊗ w ^β (t, y) : G(t/ε)D dydt = Z T

0

Z

R

^m

θ(t, y) ⊗ w ⁰ (t, y) : hDi dydt for any θ ∈ L ² ([0, T ]; X P ), see Propositions 5.1, 5.2. The strong convergences of (∇ _z v ^ε ) ε>0

in L ² ([0, T ]; X P ) and (v ^ε ) ε>0 in L ^∞ ([0, T ]; L ² ( R ^m )) come by energy balances. This is a

consequence of a general result, see Proposition 5.3. Under the coercivity condition in (25),

we prove that the weak convergence of (w ^β ) _β>0 in L ² ([0, T ]; X _P ) toward w ⁰ together with the inequality

lim sup

(β,ε)→(0,0)

Z T 0

Z

R

^m

w ^β (t, y) ⊗ w ^β (t, y) : G(t/ε)D dydt ≤ Z T

0

Z

R

^m

w ⁰ (t, y) ⊗ w ⁰ (t, y) : hDi dydt imply the strong convergence of (w ^β ) _β>0 in L ² ([0, T ]; X _P ) toward w ⁰ .

Theorem 2.2 Assume that the hypotheses (4), (5) (19), (22), (25), (35) hold true together with all the regularity conditions in Proposition 4.4. We suppose that u ⁱⁿ ∈ H _R ² and we denote by (u ^ε ) ε>0 the variational solutions of (1) and by (v ^ε ) ε>0 the functions

v ^ε (t, z) = u ^ε (t, Y (t/ε; z)), (t, z) ∈ R + × R ^m , ε > 0.

Then the family (v ^ε ) ε>0 converges strongly in L ^∞ _loc (R + ; L ² (R ^m )) to the unique variational solution v ∈ L ^∞ ( R + ; L ² ( R ^m )) of (26). The function v has the regularity

∂ t v, ∇ ^R _z v, ∇ ^R _z ⊗ ∇ ^R _z v ∈ L ^∞ _loc (R + ; L ² (R ^m )), ∂ t ∇ ^R _z v ∈ L ² _loc (R + ; L ² (R ^m ))

and (∇ _z v ^ε ) _ε>0 converges toward ∇ _z v in L ² _loc ( R + ; X _P ) when ε goes to 0. Moreover, the strong convergence of (v ^ε ) ε>0 in L ^∞ _loc (R + ; L ² (R ^m )), when ε goes to 0, holds true for any initial condition u ⁱⁿ ∈ L ² ( R ^m ).

It is easily seen that (u ^ε ) _ε>0 converges weakly in L ^∞ ( R + ; L ² ( R ^m )), as ε & 0, toward hvi, see also [12]. We deduce that (u ^ε ) ε>0 converges strongly in L ^∞ _loc (R + ; L ² (R ^m )), as ε & 0, toward v iff v = hvi, or equivalently, iff the initial condition is well prepared, i.e., u ⁱⁿ ∈ ker T . Under additional hypotheses we can justify that v ^ε = v + O(ε) in L ^∞ _loc ( R + ; L ² ( R ^m )), as suggested by the formal Ansatz (10).

Theorem 2.3 Assume that the hypotheses (4), (5), (19), (22), (25) hold true. Moreover, we assume that the solution v of the limit model (26) is smooth enough, that is

∇ ^R _z v ∈ L ^∞ _loc (R + ; L ² (R ^m )), ∇ ^R _z ⊗ ∇ ^R _z v ∈ L ^∞ _loc (R + ; L ² (R ^m ))

∇ ^R _z ∂ _t v ∈ L ¹ _loc ( R + ; L ² ( R ^m )), ∇ ^R _z ⊗ ∇ ^R _z ∂ _t v ∈ L ¹ _loc ( R + ; L ² ( R ^m ))

∇ ^R _z ⊗ ∇ ^R _z ⊗ ∇ ^R _z v ∈ L ² _loc ( R + ; L ² ( R ^m )), ∇ ^R _z ⊗ ∇ ^R _z ⊗ ∇ ^R _z ⊗ ∇ ^R _z v ∈ L ¹ _loc ( R + ; L ² ( R ^m )) and that there is a smooth matrix field C ∈ H _Q ^∞ , that is

div _y (RC ), b _k · ∇ _y div _y (RC ), b _l · ∇ _y (b _k · ∇ _y div _y (RC )) ∈ L ^∞ ( R ^m ), k, l ∈ {1, ..., m}

RC ^t R, b _k · ∇ _y (RC ^t R), b _l · ∇ _y (b _k · ∇ _y (RC ^t R)) ∈ L ^∞ ( R ^m ), k, l ∈ {1, ..., m}

such that the following decomposition holds true

D = hDi + L(C), hCi = 0.

We denote by (u ^ε ) _ε>0 the variational solutions of (1). Then for any T ∈ R + , there is a constant C T such that

sup

t∈[0,T ]

ku ^ε (t, ·) − v(t, Y (−t/ε; ·))k _L

2

( R

^m

) ≤ C _T ε

Z T 0

|∇ _y u ^ε (t, ·) − ∇ _y v(t, Y (−t/ε; ·))| ² _P dt 1/2

≤ C T ε.

3 The average of a matrix field

3.1 Definition and properties

Motivated by the computations leading to (8), we consider the family of linear transformations (G(s)) s∈ R , acting on matrix fields. It happens that (G(s)) s∈ R is a C ⁰ -group of unitary operators on H Q . For any function f = f (y), y ∈ R ^m , the notation f s = f s (z) stands for the composition product f _s = f ◦ Y (s; ·).

Proposition 3.1 Assume that the hypotheses (4), (5), (14), (15), (19) hold true.

1. The family of applications

A → G(s)A := ∂Y ⁻¹ (s; ·)A _s ^t ∂Y ⁻¹ (s; ·) = ∂Y (−s; Y (s; ·))A _s ^t ∂Y (−s; Y (s; ·)) is a C ⁰ -group of unitary operators on H _Q .

2. If A is a field of symmetric matrices, then so is G(s)A, for any s ∈ R .

3. If A is a field of positive semi-definite matrices, then so is G(s)A, for any s ∈ R . 4. Let S ⊂ R ^m be an invariant set of the flow of b, that is Y (s; S ) = S, for any s ∈ R . If

there is α > 0 such that Q ^1/2 (y)A(y)Q ^1/2 (y) ≥ αI _m , y ∈ S, then for any s ∈ R we have Q ^1/2 (y)(G(s)A)(y)Q ^1/2 (y) ≥ αI m , y ∈ S.

5. The family of applications (G(s)) s∈ R acts on H _Q,loc , that is, if A ∈ H _Q,loc , then G(s)A ∈ H _Q,loc for any s ∈ R . Moreover, we have

1 _{ψ≤k} G(s)A = G(s)(1 _{ψ≤k} A), A ∈ H _Q,loc , s ∈ R , k ∈ N . Proof.

1. Thanks to the characterization in Proposition 2.1 we know that

P s = ∂Y (s; ·)P ^t ∂Y (s; ·), s ∈ R . (27) For any s ∈ R we consider the matrix field O(s; ·) = Q ^1/2 s ∂Y (s; ·)Q ^−1/2 . Observe that O(s; ·) is a field of orthogonal matrices, for any s ∈ R . Indeed we have, thanks to (27)

t O(s; ·)O(s; ·) = Q ^{−1/2 t} ∂Y (s; ·)Q ^1/2 _s Q ^1/2 _s ∂Y (s; ·)Q ^−1/2

= Q ^−1/2 ∂Y ⁻¹ (s; ·)P _s ^t ∂Y ⁻¹ (s; ·) −1

Q ^−1/2

= Q ^−1/2 P ⁻¹ Q ^−1/2

= I m

implying that for any matrix field A we have

Q ^1/2 G(s)AQ ^1/2 = Q ^1/2 ∂Y ⁻¹ (s; ·)A _s ^t ∂Y ⁻¹ (s; ·)Q ^1/2 = ^t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·). (28) It is easily seen that if A ∈ H _Q , then for any s ∈ R

|G(s)A| ² _Q = Z

R

^m

Q ^1/2 G(s)AQ ^1/2 : Q ^1/2 G(s)AQ ^1/2 dy

= Z

R

^m

t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·) : ^t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·) dy

= Z

R

^m

Q ^1/2 _s A _s Q ^1/2 _s : Q ^1/2 _s A _s Q ^1/2 _s dy

= Z

R

^m

Q ^1/2 AQ ^1/2 : Q ^1/2 AQ ^1/2 dy = |A| ² _Q

proving that G(s) is a unitary transformation for any s ∈ R . The group property of the family (G(s)) s∈ R follows easily from the group property of the flow (Y (s; ·)) s∈ R

G(s)G(t)A = ∂Y ⁻¹ (s; ·)(G(t)A) _s ^t ∂Y ⁻¹ (s; ·)

= ∂Y ⁻¹ (s; ·)∂Y ⁻¹ (t; Y (s; ·))(A _t ) _s ^t ∂Y ⁻¹ (t; Y (s; ·)) ^t ∂Y ⁻¹ (s; ·)

= ∂Y ⁻¹ (t + s; ·)A _t+s ^t ∂Y ⁻¹ (t + s; ·) = G(t + s)A, A ∈ H Q .

The continuity of the group, i.e., lim s→0 G(s)A = A strongly in H _Q , is left to the reader.

2. Notice that G(s) commutes with transposition

t (G(s)A) = ^t ∂Y ⁻¹ (s; ·)A _s ^t ∂Y ⁻¹ (s; ·)

= ∂Y ⁻¹ (s; ·) ^t A _s ^t ∂Y ⁻¹ (s; ·)

= G(s) ^t A.

In particular, if ^t A = A, then ^t (G(s)A) = G(s)A.

3. We use the formula (28). For any ξ, η ∈ R ^m , the notation ξ ⊗ η stands for the matrix whose (i, j) entry is ξ i η j . For any ξ ∈ R ^m we have

G(s)A : Q ^1/2 ξ ⊗ Q ^1/2 ξ = Q ^1/2 G(s)AQ ^1/2 : ξ ⊗ ξ

= ^t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·) : ξ ⊗ ξ

= Q ^1/2 _s A _s Q ^1/2 _s : O(s; ·)(ξ ⊗ ξ) ^t O(s; ·)

= Q ^1/2 _s A s Q ^1/2 _s : (O(s; ·)ξ) ⊗ (O(s; ·)ξ)

= A s : (Q ^1/2 _s O(s; ·)ξ) ⊗ (Q ^1/2 _s O(s; ·)ξ).

As A is a field of positive semi-definite matrices, therefore G(s)A is a field of positive semi- definite matrices as well.

4. Assume that there is α > 0 such that Q ^1/2 AQ ^1/2 ≥ αI m on S. As before we write for any ξ ∈ R ^m , y ∈ S

Q ^1/2 G(s)AQ ^1/2 : ξ ⊗ ξ = (Q ^1/2 AQ ^1/2 ) _s : (O(s; ·)ξ) ⊗ (O(s; ·)ξ) ≥ α|O(s; ·)ξ| ² = α|ξ| ² saying that Q ^1/2 G(s)AQ ^1/2 ≥ αI _m on S.

5. Here G(s) stands for the application A → ∂Y (−s; Y (s; ·))A(Y (s; ·)) ^t ∂Y (−s; Y (s; ·)) independently of A being in H _Q or in H _Q,loc . As ψ is left invariant by the flow of b, so is 1 _{ψ≤k} , for any k ∈ N . If A belongs to H _Q,loc , we have

1 {ψ≤k} G(s)A = G(s)(1 {ψ≤k} A) ∈ H Q , k ∈ N, s ∈ R

saying that G(s)A ∈ H Q,loc , s ∈ R . Moreover, the applications (G(s)) s∈ R preserve locally the norm of H _Q

1 _{ψ≤k} G(s)A Q =

G(s)(1 _{ψ≤k} A) Q =

1 _{ψ≤k} A

Q , k ∈ N , s ∈ R .

We denote by L the infinitesimal generator of the group G in H _Q L : dom(L) ⊂ H _Q → H _Q , domL =

A ∈ H _Q : ∃ lim

s→0

G(s)A − A

s in H _Q

and we set L(A) = lim s→0 G(s)A−A

s for any A ∈ dom(L). Notice that C _c ¹ ( R ^m ) ⊂ dom(L)

and L(A) = (b · ∇ _y )A − ∂ y bA − A ^t ∂ y b, as soon as A ∈ C _c ¹ (R ^m ) (use the hypothesis Q ∈

L ² _loc ( R ^m ) and the dominated convergence theorem). The main properties of the operator L

are summarized below (see [12, Prop. 3.13] for details)

Proposition 3.2 Assume that the hypotheses (4), (14), (15) hold true.

1. The domain of L is dense in H Q and L is closed.

2. The matrix field A ∈ H _Q belongs to dom(L) iff there is a constant C > 0 such that

|G(s)A − A| _Q ≤ C|s|, s ∈ R .

3. The operator L is skew-adjoint in H _Q and we have the orthogonal decomposition H _Q = ker L ⊕ ^⊥ Range L.

Remark 3.1 When working on H _Q,loc , the generator of (G(s)) s∈ R , which is still denoted by L, is defined by

A ∈ dom(L) iff ∃ lim

s→0

G(s)(1 _{ψ≤k} A) − 1 _{ψ≤k} A

s ∈ H Q , k ∈ N

and

1 _{ψ≤k} L(A) = lim

s→0

G(s)(1 _{ψ≤k} A) − 1 _{ψ≤k} A

s , k ∈ N .

The transformations (G(s)) s∈ R also behave nicely in weighted L ^∞ spaces. More precisely, for any s ∈ R, and any A ∈ H _Q ^∞ , we have G(s)A ∈ H _Q ^∞ and |G(s)A| _H

^∞

Q

= |A| _H

^∞

Q

. Indeed, thanks to (28) and to the orthogonality of O(s; ·), we can write

Q ^1/2 G(s)AQ ^1/2 : Q ^1/2 G(s)AQ ^1/2 = ^t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·) : ^t O(s; ·)Q ^1/2 _s A _s Q ^1/2 _s O(s; ·)

= (Q ^1/2 A Q ^1/2 : Q ^1/2 A Q ^1/2 ) s .

We are now in position to apply the von Neumann’s ergodic mean theorem.

Theorem 3.1 (von Neumann’s ergodic mean theorem, see [35]) Let (G(s)) _{s∈ R} be a C ⁰ -group of unitary operators on an Hilbert space (H, (·, ·)) and L be its infinitesimal gener- ator. Then for any x ∈ H, we have the strong convergence in H

S→+∞ lim 1 S

Z r+S r

G(s)x ds = Proj _KerL x, uniformly with respect to r ∈ R .

The proof of Theorem 2.1 comes immediately, by applying Theorem 3.1 to the group in Proposition 3.1.

Proof of Theorem 2.1. The first and second statements are obvious.

3. For any ξ ∈ R ^m , ψ ∈ C _c ⁰ (S), ψ ≥ 0 we have ψ(·)P ^1/2 ξ ⊗ P ^1/2 ξ ∈ H Q and we can write, thanks to (28)

(G(s)A, ψ(·)P ^1/2 ξ ⊗ P ^1/2 ξ) _Q = Z

R

^m

ψ(y)Q ^1/2 G(s)AQ ^1/2 : ξ ⊗ ξ dy

= Z

R

^m

ψ(y) ^t O(s; y)Q ^1/2 _s A _s Q ^1/2 _s O(s; y)ξ · ξ dy

= Z

R

^m

ψ(y)Q ^1/2 _s A _s Q ^1/2 _s : O(s; y)ξ ⊗ O(s; y)ξ dy

≥ α Z

R

^m

|O(s; y)ξ| ² ψ(y) dy

= α|ξ| ² Z

R

^m

ψ(y) dy.

Taking the average over [0, S ] and letting S → +∞ yield Z

R

^m

ψ(y)Q ^1/2 hAi Q ^1/2 : ξ ⊗ ξ dy = (hAi , ψP ^1/2 ξ ⊗ P ^1/2 ξ) _Q ≥ Z

R

^m

α|ξ| ² ψ(y) dy

implying that

Q ^1/2 (y) hAi (y)Q ^1/2 (y) ≥ αI m , y ∈ S.

4. Obviously, for any A ∈ H _Q , we have by the properties of the orthogonal projection on ker L that | hAi | _Q = |Proj _{ker L} A| _Q ≤ |A| _Q . For the last inequality, consider M ∈ M _m ( R ) a fixed matrix, ψ ∈ C _c ⁰ (R ^m ), ψ ≥ 0 and, as before, observe that ψP ^1/2 M P ^1/2 ∈ H Q , which allows us to write

(G(s)A,ψP ^1/2 M P ^1/2 ) _Q = Z

R

^m

Q ^1/2 G(s)AQ ^1/2 : ψM dy

= Z

R

^m

t O(s; y)Q ^1/2 _s A _s Q ^1/2 _s O(s; y) : ψM dy

= Z

R

^m

Q ^1/2 _s A _s Q ^1/2 _s : O(s; y)M ^t O(s; y) dy

≤ Z

R

^m

q

Q ^1/2 s A _s Q ^1/2 s : Q ^1/2 s A _s Q ^1/2 s

p O(s; y)M ^t O(s; y) : O(s; y)M ^t O(s; y)ψ dy

≤ |A| _H

^∞

Q

(M : M ) ^1/2 Z

R

^m

ψ(y) dy.

Taking the average over [0, S ] and letting S → +∞, lead to Z

R

^m

Q ^1/2 hAi Q ^1/2 : M ψ(y) dy = (hAi , ψP ^1/2 M P ^1/2 ) Q ≤ |A| _H

^∞

Q

(M : M ) ^1/2 Z

R

^m

ψ(y) dy.

We deduce that

Q ^1/2 (y) hAi (y)Q ^1/2 (y) : M ≤ |A| _H

^∞

Q

(M : M ) ^1/2 , y ∈ R ^m , M ∈ M _m (R) saying that

| hAi | _H

^∞

Q

= ess sup _{y∈ R}

m

q

Q ^1/2 (y) hAi (y)Q ^1/2 (y) : Q ^1/2 (y) hAi (y)Q ^1/2 (y) ≤ |A| _H

^∞

Q

. 5. Let A be a matrix field in H _Q,loc . For any k ∈ N , 1 _{ψ≤k} A belongs to H Q , and by the first statement we know that

S→+∞ lim 1 S

Z S 0

G(s)(1 _{ψ≤k} A) ds =

1 _{ψ≤k} A

∈ H Q , k ∈ N .

It is easily seen that for any k, l ∈ N we have

S→+∞ lim 1 S

Z S 0

G(s)(1 _{ψ≤k} A) ds = lim

S→+∞

1 S

Z S 0

G(s)(1 _{ψ≤l} A) ds

almost everywhere on {ψ ≤ min(k, l)}, and thus, there is a matrix field denoted by hAi, whose restriction on {ψ ≤ k} coincides with

1 _{ψ≤k} A

for any k ∈ N . Notice also that for any k ∈ N we have

1 {ψ≤k} A

= 0 almost everywhere on {ψ > k} and thus we obtain 1 _{ψ≤k} hAi =

1 _{ψ≤k} A

, k ∈ N .

Observe that for any k ∈ N , we have the convergence in H _Q

S→+∞ lim 1 _{ψ≤k} 1 S

Z S 0

G(s)(A) ds = lim

S→+∞

1 S

Z S 0

G(s)(1 _{ψ≤k} A) ds =

1 _{ψ≤k} A

= 1 _{ψ≤k} hAi saying that lim S→+∞ 1

S

R S

0 G(s)A ds = hAi in H Q,loc . The inclusion H _Q ^∞ ⊂ H Q,loc follows by the compactness of {ψ ≤ k}, k ∈ N . By the fourth statement we have

| hAi | _H

^∞

Q

= sup

k∈ N

|1 _{ψ≤k} hAi | _H

^∞

Q

= sup

k∈ N

|

1 {ψ≤k} A

| _H

^∞

Q

≤ sup

k∈ N

|1 _{ψ≤k} A| _H

^∞

Q

= |A| _H

^∞

Q

. Let A be a matrix field of H _Q,loc , such that Q ^1/2 (y)A(y)Q ^1/2 (y) ≥ αI m , y ∈ R ^m , for some α > 0. For any k ∈ N we have 1 _{ψ≤k} A ∈ H _Q and

Q ^1/2 (y)1 _{ψ≤k} A(y)Q ^1/2 (y) ≥ αI m , y ∈ {ψ ≤ k}.

By the third statement we deduce that for any k ∈ N Q ^1/2 (y)1 _{ψ≤k} hAi (y)Q ^1/2 (y) = Q ^1/2 (y)

1 _{ψ≤k} A

(y)Q ^1/2 (y) ≥ αI _m , y ∈ {ψ ≤ k}

saying that Q ^1/2 (y) hAi (y)Q ^1/2 (y) ≥ αI _m , y ∈ R ^m . 3.2 Examples

In this section we explicitly compute the average matrix field in three cases. We deal with periodic and almost-periodic flows.

3.2.1 Periodic flow

Consider the vector field b(y) = (γy 2 , −βy ₁ ), for any y = (y 1 , y 2 ) ∈ R ² , with β, γ ∈ R ^? + . The function ψ(y) = βy ₁ ² + γy ₂ ² , y ∈ R ² , is a coercive invariant associated to the field b. We denote by Y (s; y) the flow of the vector field b. We intend to determine the average along the flow Y of the matrix field

D(y) =

λ ₁ (y) 0 0 λ 2 (y)

, y ∈ R ²

where λ 1 , λ 2 are two given functions. It is easily seen that the flow is 2π/ √

βγ-periodic and writes Y (s; y) = R(−s; β, γ)y, (s, y) ∈ R × R ² , with

R(s; β, γ) =





cos( √

βγ s) − q _γ

β sin( √ βγ s) q β

γ sin( √

βγ s) cos( √ βγ s)



 .

By Theorem 2.1 we deduce that hDi =

√ βγ 2π

Z 2π/ √ βγ 0

∂Y (−s; Y (s; ·))D(Y (s; ·)) ^t ∂Y (−s; Y (s; ·)) ds

=

√ βγ 2π

Z 2π/ √ βγ 0

R(s; β, γ)D(Y (s; ·))R(−s; γ, β) ds

=

hDi ₁₁ hDi ₁₂ hDi ₂₁ hDi ₂₂

where

hDi ₁₁ = 1 2

D

λ 1 [1 + cos(2 p βγ ·)] E

+ γ 2β

D

λ 2 [1 − cos(2 p

βγ ·)] E

hDi ₁₂ = hDi ₂₁ =

√ β 2 √

γ D

λ ₁ sin(2 p βγ ·) E

−

√ γ 2 √

β D

λ ₂ sin(2 p βγ ·) E

hDi ₂₂ = β 2γ

D

λ 1 [1 − cos(2 p βγ ·)] E

+ 1 2

D

λ 2 [1 − cos(2 p βγ ·)] E

and for any function h the notation hλ _i h(·)i stands for

√ βγ 2π

R _2π/

√ βγ

0 λ i (Y (s; y))h(s) ds, i ∈ {1, 2}. Notice that when λ ₁ , λ ₂ are constant functions along the flow Y (s; ·) (that is, when λ 1 , λ 2 depend only on β(y 1 ) ² + γ(y 2 ) ² ), the expression for hDi reduces to

hDi =

1

2 λ ₁ + _2β ^γ λ ₂ 0 0 _2γ ^β λ ₁ + ¹ ₂ λ ₂

! .

Observe that even if λ 2 = 0 everywhere (in which case D(y) is nowhere definite positive), then the averaged diffusion matrix hDi can still be definite positive everywhere.

3.2.2 Almost-periodic flow

Consider the vector field b(y) = (y ₂ , −ω ₁ ² y ₁ , y ₄ , −ω ₂ ² y ₃ ), y = (y ₁ , y ₂ , y ₃ , y ₄ ) ∈ R ⁴ , with ω ₁ , ω ₂ ∈ R ^? incommensurable, i.e ω 1 /ω 2 ∈ / Q. The function ψ(y) = ω ² ₁ y ² ₁ +y ₂ ² +ω ₂ ² y ₃ ² +y ₄ ² , with y ∈ R ⁴ , is a coercive invariant associated to the field b. We denote by Y (s; y) the flow of the vector field b. We consider the matrix field

D(y) =









λ ₁ (y) 0 0 0

0 λ 2 (y) 0 0

0 0 λ ₃ (y) 0

0 0 0 λ ₄ (y)









, y ∈ R ⁴

where λ 1 , λ 2 , λ 3 , λ 4 are four given functions, which are constant along the flow Y (s; ·). The flow writes Y (s; y) = R(−s; ω ₁ , ω ₂ )y, (s, y) ∈ R × R ⁴ , with

R(s; ω ₁ , ω ₂ ) =









cos(sω ₁ ) − _ω ¹

1

sin(sω ₁ ) 0 0

ω 1 sin(sω 1 ) cos(sω 1 ) 0 0

0 0 cos(sω ₂ ) − _ω ¹

2

sin(sω ₂ )

0 0 ω 2 sin(sω 2 ) cos(sω 2 )







 .

The incommensurability condition ensures that the flow is not periodic with respect to the variable s. Nevertheless it is almost periodic with respect to s. By Theorem 2.1 we deduce that

hDi = lim

S→+∞

1 S

Z S 0

∂Y (−s; Y (s; ·))D(Y (s; ·)) ^t ∂Y (−s; Y (s; ·)) ds

= lim

S→+∞

1 S

Z S 0

R(s; ω 1 , ω 2 )D(Y (s; ·)) ^t R(s; ω 1 , ω 2 ) ds

=













1

2 λ ₁ (y) + _2ω ¹

2 1

λ ₂ (y) 0 0 0

0 ^ω ₂

²¹

λ 1 (y) + ¹ ₂ λ 2 (y) 0 0

0 0 ¹ ₂ λ ₃ (y) + _2ω ¹

2

2

λ ₄ (y) 0

0 0 0 ^ω ₂

²²

λ ₃ (y) + ¹ ₂ λ ₄ (y)













.

3.3 The Fokker-Planck equation

We inquire now about the Fokker-Planck equation. For simplicity, we assume that the mag- netic field is uniform B ^ε = (0, 0, B/ε), x ∈ R ³ . In the finite Larmor radius regime (i.e., the typical length in the orthogonal directions is much smaller then the typical length in the parallel direction), the presence density of charged particles f ^ε satisfies

∂ _t f ^ε + 1

ε (v ₁ ∂ _x

₁

+v ₂ ∂ _x

₂

)f ^ε +v ₃ ∂ _x

₃

f ^ε + q

m E·∇ _v f ^ε + qB

mε (v ₂ ∂ _v

₁

−v ₁ ∂ _v

₂

)f ^ε = νdiv _v {Θ∇ _v f ^ε +vf ^ε }.

(29) In this case, the flow

(x, v) ∈ R ³ × R ³ 7→ Y (s; x, v) = (X(s; x, v), V (s; x, v)) ∈ R ³ × R ³ , to be considered corresponds to the vector field

b(x, v) · ∇ _x,v = v ₁ ∂ _x

₁

+ v ₂ ∂ _x

₂

+ ω _c (v ₂ ∂ _v

₁

− v ₁ ∂ _v

₂

), ω _c = qB

m , (x, v) ∈ R ⁶ . We introduce the notation c(x, v) · ∇ _x,v = v ₃ ∂ _x

₃

+ _m ^q E · ∇ _v . It is easily seen that

X(s; x, v) = x +

⊥ v

ω _c − R(−ω _c s) ω _c

⊥ v, X ₃ (s; x ₃ ) = x ₃ , V (s; v) = R(−ω _c s)v, V ₃ (s; v ₃ ) = v ₃ where we have used the notations x = (x 1 , x 2 ), v = (v 1 , v 2 ), ^⊥ v = (v 2 , −v ₁ ) and R(θ) stands for the rotation in R ² of angle θ ∈ R . The Jacobian matrix of the flow writes

∂ _x,v Y (s; x, v) =









I ₂ 0 2×1 I

2

−R(−ω

c

s)

ω

c

E 0 2×1

0 1×2 1 0 1×2 0

0 2×2 0 2×1 R(−ω _c s) 0 2×1

0 1×2 0 0 1×2 1









where 0 m×n stands for the null matrix with m lines and n columns, and E = R(−π/2). We indicate the main lines for the asymptotic analysis of the Fokker-Planck equation (29). We ap- peal to the weak formulation of (29), written for the test function (t, y) → ϕ(t, Y (−t/ε; y)), y = (x, v), Y = (X, V ), ϕ ∈ C _c ¹ (R + × R ⁶ ). Denoting by f in the initial density, we obtain

− Z

R

⁶

f _in (y)ϕ(0, y) dy − Z +∞

0

Z

R

⁶

f ^ε (t, y){∂ _t ϕ(t) − 1

ε b · (∇ _z ϕ)(t)}(Y (−t/ε; y)) dydt

− Z +∞

0

Z

R

⁶

f ^ε (t, y)c(t, y) · ^t ∂ _y Y (−t/ε; y) (∇ _z ϕ)(t, Y (−t/ε; y)) dydt

− Z +∞

0

Z

R

⁶

f ^ε (t, y) b(y)

ε · ^t ∂ y Y (−t/ε; y) (∇ _z ϕ)(t, Y (−t/ε; y)) dydt

= −ν Z +∞

0

Z

R

⁶

(Θ∇ _v f ^ε + vf ^ε ) · ^t ∂ _v Y (−t/ε; y) (∇ _z ϕ)(t, Y (−t/ε; y)) dydt.

We introduce the new densities (g ^ε ) _ε given by

f ^ε (t, y) = g ^ε (t, Y (−t/ε; y)), (t, y) ∈ R + × R ⁶ and we use the identity

b(Y (−t/ε; y)) − ∂ y Y (−t/ε; y) b(y) = 0.

After performing the change of variables z = Y (−t/ε; y), the above formulation reduces to

− Z

R

⁶

f in (z)ϕ(0, z) dz − Z +∞

0

Z

R

⁶

g ^ε (t, z)∂ t ϕ(t, z) dzdt

− Z +∞

0

Z

R

^m

g ^ε ∂ _y Y (−t/ε; Y (t/ε; z))c(t, Y (t/ε; z)) · ∇ _z ϕ(t, z) dzdt

= −ν Z +∞

0

Z

R

⁶

Θ∂ v Y (−t/ε; Y (t/ε; z)) ^t ∂ v Y (−t/ε; Y (t/ε; z))∇ _z g ^ε · ∇ _z ϕ dzdt

− ν Z +∞

0

Z

R

⁶

g ^ε (t, z)∂ _v Y (−t/ε; Y (t/ε; z))V (t/ε; z)g ^ε (t, z) · ∇ _z ϕ dzdt.

Motivated by Theorems 2.2, 2.3, we expect that (g ^ε ) _ε converges in L ^∞ _loc ( R + ; L ² ( R ⁶ )) to the solution of the problem

∂ t g + C(t, z) · ∇ _z g = νdiv z {ΘD(z)∇ _z g + V (z)g}, (t, z) ∈ R + × R ⁶

g(0, z) = f _in (z), z ∈ R ⁶

where the matrix field D and the vector fields C, V are given by ergodic averages D(z) = lim

S→+∞

1 S

Z S 0

∂ v Y (−s; Y (s; z)) ^t ∂ v Y (−s; Y (s; z)) ds

C(t, z) = lim

S→+∞

1 S

Z S 0

∂ _y Y (−s; Y (s; z))c(t, Y (s; z)) ds (30)

V(z) = lim

S→+∞

1 S

Z S 0

∂ _v Y (−s; Y (s; z)) V (s; z) ds (31)

= lim

S→+∞

1 S

Z S 0

∂ y Y (−s; Y (s; z)) ^t (0, V (s; z)) ds.

Observe that D is the average of the diffusion matrix D entering the Fokker-Planck equation (29) cf. Theorem 2.1

D =

3

X

i=1

e _v

_i

⊗ e _v

_i

=

0 3×3 0 3×3

0 3×3 I ₃

.

After direct computations we obtain (thanks to the periodicity of the flow) D(z) = lim

S→+∞

1 S

Z S 0

∂ _y Y (−s; Y (s; z))D ^t ∂ _y Y (−s; Y (s; z)) ds

= ω c

2π

Z 2π/ω

c

0

∂ y Y (−s; Y (s; z))

0 3×3 0 3×3

0 3×3 I 3

t ∂ y Y (−s; Y (s; z)) ds

=











2I

2

ω

_c²

0 2×1 − _ω ^E

c

0 2×1

0 1×2 0 0 1×2 0

E

ω

c

0 2×1 I 2 0 2×1

0 1×2 0 0 1×2 1











. (32)

Notice that the averaged Fokker-Planck kernel D contains diffusion terms not only in velocity

variables (as in initial the Fokker-Planck kernel D) but also in space variables (orthogonal

to the magnetic lines). This was actually observed in gyrokinetic theory and numerical

simulations [15, 16, 17, 27, 36].