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

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).

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 field b is divergence free, the energy balance writes 1 2 d dt R m (u ε (t, y)) 2 dy + 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 (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 [7,24,25,27]. 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 [8,9]. 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 [20,23,17].
The advection-diffusion problems with large drift in periodic setting have been extensively studied by many authors [22,21,32,4,33]. For the homogenization of the associated eigenvalue problems, we refer to [5,29]. 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 [28] 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 a detailed version of the paper available in [6].
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 . ( 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 .
Under the above hypotheses the flow Y is global and smooth, Y ∈ W 1,∞ loc (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 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 2.4, 2.5.
1. Under suitable hypotheses on the vector field b, the matrix field D and the initial condition 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 D comes by averaging the matrix field D along the flow of the vector field b (cf. Theorem 2.3), that is 2. Under additional regularity hypotheses, we have the error estimate 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 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 straightforward computation shows that we can write, for any integrable function z → w(z), any matrix field y → C(y), and any s ∈ R, where the family of matrix fields (G(s)C) s is defined by Using those notations, the change of variable y = Y (t/ε; z) in (1) leads to 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 This method is used in many frameworks such as periodic homogenization for elliptic and parabolic systems [1,31], transport equations [12,18] or kinetic equations [10]. Plugging the Ansatz (10) in (9) and identifying the terms of the same order with respect to ε, lead to the hierarchy of equations 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 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 [34], the limit D = lim S→+∞ 1 S 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 where the bracket between a vector field c and a matrix field A is a matrix field formally defined by The condition (15) naturally appears in our problem because of the following proposition.
Proposition 1. Consider a vector field c ∈ W 1,∞ loc (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 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 where Y stands for the flow associated with the vector field c. (c) For any s ∈ R, we have where G is defined in (8).
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)
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).
which is a Hilbert space for the natural scalar product The associated norm is denoted by |A| M .
Similarly we introduce the Banach space 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) = +∞.
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 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 ( We define similar spaces for vector fields.
equipped with the norm |c| X ∞ m := |M 1/2 c| L ∞ . The following properties are straightforward.
Proposition 2. We consider the same assumptions as in the Definitions above.
• We have the continuous embedding C 0 ). 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 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). Thanks to the von Neumann's ergodic theorem (see Theorem 3.1 and [34] for more details), we find that the average of a matrix field A := lim S→+∞ 1 S 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 [11,12]. 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 · , as for matrix fields in H Q . Theorem 2.3. Assume that (4), (5), (14), (15), (19) hold true. We denote by L the infinitesimal generator of the group (G(s)) s∈R . uniformly with respect to r ∈ R. 2. If A ∈ H Q is a field of symmetric positive semi-definite matrices, then so is A . 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

For any matrix field
for some α > 0, then we have and in particular, A (y) is positive definite for y ∈ S.

THOMAS BLANC, MIHAI BOSTAN AND FRANCK BOYER
5. For any matrix field A ∈ H Q,loc , the family converges in H Q,loc , when S goes to infinity. Its limit, denoted by A , satisfies where the symbol · 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 for some α > 0, then we have 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 7, 8. 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) = 0, y ∈ R m , R ∈ L 1 loc (R m ) The equation satisfied by R in (20) is equivalent to R(Y (s; y))∂Y (s; y) = R(y), for (s, y) ∈ R × R m , which also writes 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 [11] 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 which guarantees the existence of the global flows Y i (s; y) ∈ W 1,∞ loc (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 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 which is a Hilbert space with the scalar product 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 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 1.
Notice that every element of H 1 R has a gradient in the distribution sense that belongs to L 2 loc (R m ). More precisely, for any u ∈ H 1 R , we have It will be convenient in the sequel to make use of the following differential operators , as well as its higher order versions .., m}. The results in Section 3 hold true under the hypotheses (4), (14), (15) but in Sections 4 and 6 we shall need the stronger hypotheses (20) instead of (14), (15).
Moreover we assume that where Q = t RR and the columns of R −1 are given by the vector fields b 1 , ..., b m . In view of Theorem 2.3, the formal limit problem (13) of the family of systems (9) becomes 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 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 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 , 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 ) come by energy balances. This is a consequence of a general result, see Proposition 13. Under the coercivity condition in (25), we prove that the weak convergence of (w β ) β>0 in Theorem 2.4. Assume that the hypotheses (4), (5) (19), (22), (25), (35) hold true together with all the regularity conditions in Proposition 8. We suppose that u in ∈ H 2 R and we denote by (u ε ) ε>0 the variational solutions of (1) and by 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 , when ε goes to 0, holds true for any initial condition u in ∈ L 2 (R m ).
Theorem 2.5. Assume that the hypotheses (4), (5), (19), (22), (25) hold true. Moreover, we assume that the solution v of the limit model (26) .., m} such that the following decomposition holds true 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 3. The average of a matrix field.

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 Proposition 3. Assume that the hypotheses (4), (5), (14), (15), (19) hold true.

The family of applications
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 then G(s)A ∈ H Q,loc for any s ∈ R. Moreover, we have Proof. 1. Thanks to the characterization in Proposition 1 we know that For any s ∈ R we consider the matrix field O(s; is a field of orthogonal matrices, for any s ∈ R. Indeed we have, thanks to (27) implying that for any matrix field A we have It is easily seen that if A ∈ H Q , then for any s ∈ R 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 The continuity of the group, i.e., lim s→0 G(s)A = A strongly in H Q , is left to the reader.
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 . 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 We denote by L the infinitesimal generator of the group G in H Q loc (R m ) and the dominated convergence theorem). The main properties of the operator L are summarized below (see [11,Prop. 3.13] for details) Proposition 4. Assume that the hypotheses (4), (14), (15) The operator L is skew-adjoint in H Q and we have the orthogonal decomposi- Remark 2. When working on H Q,loc , the generator of (G(s)) s∈R , which is still denoted by L, is defined by The transformations (G(s)) s∈R also behave nicely in weighted L ∞ spaces. More precisely, for any s ∈ R, and any Indeed, thanks to (28) and to the orthogonality of O(s; ·), we can write We are now in position to apply the von Neumann's ergodic mean theorem.
Theorem 3.1 (von Neumann's ergodic mean theorem, see [34]). Let (G(s)) s∈R be a C 0 -group of unitary operators on an Hilbert space (H, (·, ·)) and L be its infinitesimal generator. Then for any x ∈ H, we have the strong convergence in H lim S→+∞ 1 S r+S r G(s)x ds = Proj KerL x, uniformly with respect to r ∈ R.
The proof of Theorem 2.3 comes immediately, by applying Theorem 3.1 to the group in Proposition 3.
Proof of Theorem 2.3. The first and second statements are obvious.
3. For any ξ ∈ R m , ψ ∈ C 0 c (S), ψ ≥ 0 we have ψ(·)P 1/2 ξ ⊗ P 1/2 ξ ∈ H Q and we can write, thanks to (28) Taking the average over [0, S] and letting S → +∞ yield implying that 4. Obviously, for any A ∈ H Q , we have by the properties of the orthogonal projection on ker L that | A | Q = |Proj ker L A| Q ≤ |A| Q . For the last inequality, consider M ∈ M m (R) a fixed matrix, ψ ∈ C 0 c (R m ), ψ ≥ 0 and, as before, observe that ψP 1/2 M P 1/2 ∈ H Q , which allows us to write Taking the average over [0, S] and letting S → +∞, lead to We deduce that

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 It is easily seen that for any k, l ∈ N we have almost everywhere on {ψ ≤ min(k, l)}, and thus, there is a matrix field denoted by A , 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 The inclusion H ∞ Q ⊂ H Q,loc follows by the compactness of {ψ ≤ k}, k ∈ N. By the fourth statement we have 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 By the third statement we deduce that for any k ∈ N

3.2.
Examples. In this section we explicitly compute the average matrix field in three cases. We deal with periodic and almost-periodic flows.

The Fokker-Planck equation.
We inquire now about the Fokker-Planck equation. For simplicity, we assume that the magnetic 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 In this case, the flow to be considered corresponds to the vector field We introduce the notation c( It is easily seen that and R(θ) stands for the rotation in R 2 of angle θ ∈ R. The Jacobian matrix of the flow writes 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 appeal to the weak formulation of (29), written for the test function (t, y) → ϕ(t, Y (−t/ε; y)), y = ( . Denoting by f in the initial density, we obtain 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 Motivated by Theorems 2.4, 2.5, we expect that (g ε ) ε converges in L ∞ loc (R + ; L 2 (R 6 )) to the solution of the problem where the matrix field D and the vector fields C, V are given by ergodic averages Observe that D is the average of the diffusion matrix D entering the Fokker-Planck equation (29) cf. Theorem 2.3 After direct computations we obtain (thanks to the periodicity of the flow) 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 [14,15,16,26,35]. It remains to clarify the existence of the ergodic average for vector fields. This comes from the fact that the family of maps c ∈ X Q → ∂ y Y (−s; Y (s; ·))c(Y (s; ·)) ∈ X Q , s ∈ R forms a C 0 -group of unitary operators in X Q , see also Proposition 3. Consequently, thanks to the von Neumann's theorem, we deduce that the vector fields C and V introduced in (30), (31) are well defined (see Theorem 2.3). 4. Well posedness for the perturbed problem and uniform estimates. For solving (1), we appeal to variational methods. We use the continuous embedding . Proposition 5. Assume that b satisfies (4), (5), (19), that R and R −1 satisfy (20) and (22), and that D satisfies (25).
1. For any ε > 0, the bilinear form a ε : is well defined, continuous and coercive on H 1 R with respect to L 2 (R m ).

The bilinear form
is well defined, continuous and coercive on H 1 R with respect to L 2 (R m ). Proof. 1. For any u, v ∈ H 1 R we have |D(y)∇ y u · ∇ y v| = |Q 1/2 (y)D(y)Q 1/2 (y) : ≤ |b| X ∞ Q |P 1/2 (y)∇ y u| |v(y)|, y ∈ R m . Therefore, it is easily seen, thanks to (24), that showing that the bilinear application a ε (·, ·) is well defined and continuous. We inquire now about the coercivity of a ε on H 1 R , with respect to L 2 (R m ). For any u ∈ H 1 R we have, thanks to the anti-symmetry of b · ∇ y a ε (u, u) + α u 2 We emphasize the following inequality, which will be used several times in the sequel 2. We follow the same lines as before since Theorem 2.3 gives that Assume that b satisfies (4), (5), (19), that R and R −1 satisfy (20) and (22), and that D satisfies (25).

ASYMPTOTIC ANALYSIS OF PARABOLIC EQUATIONS 4657
Proof. By [19, Theorems 1 and 2, p. 513] (see also [30]) we deduce that for any u in ∈ L 2 (R m ), there is a unique variational solution u ε for the problem (1), that is Similarly, there is a unique variational solution v for the limit model (26), that is The above estimates come immediately by the energy balance In particular we deduce u ε (t) L 2 (R m ) ≤ u in L 2 (R m ) , for any t ∈ R + , ε > 0, and , for any ε > 0. The estimates for v follow similarly.
Indeed, performing the change of variable z → y = Y (t/ε; z), which is measure preserving, one gets 2α .
Using twice the formula .., m} as soon as those norms are finite.
Up to now, we have considered solutions with initial condition u in ∈ L 2 (R m ). In order to study the stability of the family (v ε ) ε>0 when ε goes to 0, we need more regularity. This will be the object of the next propositions, in which we analyze how the regularity of the initial condition propagates in time. The idea is to take the directional derivative b i · ∇ y of (1), leading to (34) Notice that the key point was to take advantage of the involution between b i and b, for any i ∈ {1, ..., m}, which guarantees that there is no commutator between the first order operators b i · ∇ y and b · ∇ y . More generally, if we apply the directional derivative c · ∇ y in (1), the right hand side of the corresponding equation in (34) will contain the extra term 1 ε [b · ∇ y , c · ∇ y ]u ε , which is clearly unstable, when ε goes to 0, if b and c are not in involution. The estimate for b i · ∇ y u ε follows by using the energy balance of (34), observing that, thanks to the anti-symmetry of b · ∇ y , we get rid of the term of order 1/ε. We assume that for any i, j ∈ {1, ..., m}, the coordinates of the Poisson bracket [b i , b j ] in the basis (b k ) 1≤k≤m are bounded The following next four results collect some uniform estimates on the solutions of our two-scale problem and its weak limit. We do not give here the proofs that are technical and quite standard; they are however given in details in [6].
Proposition 7. Assume that the hypotheses (4), (5), (22), (25), (35) hold true. Moreover we assume that for any i, j ∈ {1, ..., m} If the initial condition belongs to H 1 R , then we have for any The goal is to obtain a uniform bound for (∂ t ∇ R z v ε ) ε>0 in L 2 loc (R + ; L 2 (R m )) which will be necessary for applying Proposition 12. This can be achieved for any initial condition u in ∈ H 2 R .
Proposition 8. Assume that the hypotheses (4), (5), (22), (25), (35) hold true. Moreover we assume that for any i, j, k ∈ {1, ..., m} . If the initial condition u in belongs to H 2 R , then for any T ∈ R + we have sup Here the notation ∇ R ⊗ ∇ R ⊗ ∇ R w stands for the tensor whose entry Finally, we can also obtain estimates for the solution of the limit model (26).

Proposition 9.
Assume that all the hypotheses of Proposition 7 hold true, together with (19). Then we have for any ). Proposition 10. Assume that all the hypotheses of Proposition 8 hold true, together with (19). Then for any T ∈ R + , we have 5. Two-scale analysis. We intend to investigate the asymptotic behavior of (1), or equivalently (9). For any smooth, compactly supported function ψ(t, z) we have to pass to the limit, when ε 0, in the formulation Clearly, the main difficulty comes from the last integral, which presents two time scales : a slow time variable t and also a fast time variable s = t/ε (not necessarily periodic). We detail here a general two-scale convergence result, based on ergodic averages. We use the notations introduced in Definition 2.1 and Proposition 2. Performing the change of variable s = t ε in the above integral, leads to We split the interval [0, T [ into a finite number of intervals of size greater or equal than T δ,ε . For example let k δ,ε be T T δ,ε . If T /T δ,ε is an integer, that is T /T δ,ε = k δ,ε , we consider the intervals and if T /T δ,ε is not an integer, we take the intervals Notice that in both cases we have k δ,ε intervals, whose sizes are between T δ,ε and 2T δ,ε . We denote by (t k,δ,ε ) 0≤k≤k δ,ε , or simply (t k ) 0≤k≤k δ,ε , the end points of these intervals. The last point is always t k δ,ε = T . Therefore we can write for any The estimate for Σ 2 comes by using (36) Thanks to (37), (38), (39) we deduce Let η be a positive real number and δ > 0 small enough such that δ B L 1 ([0,T ];H P ) < η/2 uniformly with respect to B ∈ B ω (which is possible since B ω is bounded in L 1 ([0, T ]; H P )). Observing that lim ε 0 T δ,ε = lim ε 0 εS δ = 0, and lim ε 0 ω(2T δ,ε ) = 0, we deduce that there is ε = ε(η) such that for any 0 < ε < ε(η) Finally we obtain for any 0 < ε < ε(η), uniformly with respect to B ∈ B ω . In the sequel, we present some consequences of Proposition 11 which will be used when justifying the main result in Theorem 2.4. Everytime that y → A(y) : B(y) belongs to L 1 (R m ), by the notation A, B P,Q we understand with ω non decreasing and lim λ 0 ω(λ) = 0. Then for any family (w β ) β>0 ⊂ W ω which converges weakly in L 2 ([0, T ]; X P ) toward w 0 when β 0, we have Proof. Notice that for any θ, w ∈ L 2 ([0, T ]; X P ) we have (P 1/2 (y)θ(t, y)) ⊗ (P 1/2 (y)w(t, y)) : Q 1/2 (y)G(t/ε)DQ 1/2 (y) dydt and similarly, by using As the family (w β ) β>0 is bounded in L 2 ([0, T ]; X P ), it is enough to check (40) for any θ in a dense subset of L 2 ([0, T ]; X P ), for example for any θ such that P 1/2 θ ∈ C 0 c ([0, T ]×R m ). Take k ∈ N large enough, such that k ≥ ψ(y), y ∈ supp (P 1/2 θ). As D ∈ H ∞ Q belongs to H Q,loc , the matrix field D k = 1 {ψ≤k} D belongs to H Q ∩ H ∞ Q . We appeal to Proposition 11 with C(s) = G(s)D k , C = D k = 1 {ψ≤k} D , cf. Theorem 2.3, and B = {θ ⊗ w : w ∈ W ω }. By Proposition 3 we know that (G(s)) s∈R is a C 0 -group of unitary operators on H Q , implying that C ∈ L ∞ (R; H Q ). By Theorem 2.3 we deduce that lim S→+∞ 1 S s0+S s0 C(s) ds = C, uniformly with respect to s 0 ∈ R.

Remark 5. The previous arguments show that if
for any w ∈ L 2 ([0, T ]; X P ). Indeed, taking into account that the bilinear application is continuous, it is enough to establish (42) for w in the set {θ ∈ L 2 ([0, T ]; X P ) : . And this is a direct consequence of Remark 4 applied with D k = 1 {ψ≤k} D, k ≥ ψ(y), y ∈ supp (P 1/2 θ), since for any θ ∈ L 2 ([0, T ]; X P ) such that When the matrix field D is positive definite, the behavior of the upper limit with respect to (β, ε) for the quadratic term 1. If the matrix field D is positive semi-definite, then we have 2. If (w β ) β>0 converges strongly in L 2 ([0, T ]; X P ) toward w 0 when β 0 (the existence of a modulus of continuity ω in C([0, T ]; X P ) for the family (w β ) β>0 is not necessary here), then we have 3. If there is α > 0 such that Q 1/2 DQ 1/2 ≥ αI m , and then the family (w β ) β>0 converges strongly in L 2 ([0, T ]; X P ) toward w 0 when β 0.
6. Proofs of the main theorems. We establish two convergence results. In Theorem 2.4 we state strong convergence results for (v ε ) ε>0 in L ∞ loc (R + ; L 2 (R m )) and (∇ z v ε ) ε>0 in L 2 loc (R + ; X P ). In Theorem 2.5 we investigate the rate of the above convergences, by introducing a corrector, that is, we justify the leading term in the developement (10).
Proof of Theorem 2.4. Assume that u in belongs to H 2 R , which is the space defined by As u ε is the variational solution of (1), we have for any Φ ∈ C 1 Actually the above formulation holds true for any compactly supported function in R + ×R m , which belongs to W 1,∞ (R + ×R m ). Pick a test function ψ ∈ C 1 c (R + ×R m ) and let us introduce the function Φ ε (t, y) = ψ(t, Y (−t/ε; y)), (t, y) ∈ R + × R m . Thanks to the hypotheses (4), the function Φ ε is compactly supported in R + × R m , belongs to W 1,∞ (R + × R m ) and thus satisfies (46). We perform the change of variable z → y = Y (t/ε; z). Taking the time and space derivatives of the equalities ψ(t, z) = Φ ε (t, Y (t/ε; z)) and v ε (t, z) = u ε (t, Y (t/ε; z)) gives ) and the weak formulation (46), written with the test function Φ ε (t, y) becomes Therefore v ε is the variational solution of (9). By Propositions 7, 8 we have, for Let us consider a sequence (ε k ) k converging to 0 such that We claim that v 0 is the variational solution of (26). For any η ∈ C 1 c (R + ) and Φ ∈ H 1 R , the variational formulation of (9) yields As η Φ belongs to L 1 (R + ; L 2 (R m )), the weak convergence in L ∞ (R + ; We use now Proposition 12 with T > 0 such that supp η ⊂ [0, T [, and Obviously, W ω is bounded in L 2 ([0, T ]; X P ) and for any k ∈ N, t, t ∈ [0, T ], we can write . Therefore W ω is contained in C([0, T ]; X P ) and admits the continuity modulus Applying Proposition 12 with θ(t, z) = η(t)∇ z Φ(z) ∈ L 2 ([0, T ]; X P ) we deduce that Therefore, passing to the limit, when k → +∞, in the variational formulation of v ε k , implies and thus v 0 is the variational solution of (26) (v 0 = v). By the uniqueness of the solution for the limit model (26), we deduce that the convergences in (47), (48) actually hold for the whole family (v ε ) ε lim ε 0 v ε = v weakly in L ∞ (R + ; L 2 (R m )), lim ε 0 ∇ z v ε = ∇ z v weakly in L ∞ loc (R + ; X P ).
It remains to show that our convergence result still holds for any u in ∈ L 2 (R m ). Indeed, for any δ > 0, by density we can consider u in δ ∈ H 2 R such that u in − u in δ L 2 (R m ) ≤ δ/2. We denote by v ε δ (resp. v δ ) the variational solution of (9) (resp. (26)) with the initial condition u in δ . Thanks to the energy balance we obtain easily . By the first part of the proof, since u in δ ∈ H 2 R , we know that for any δ > 0, lim Remark 6. Under the assumptions of Theorem 2.4, the family (v ε ) ε>0 converges strongly in L ∞ loc (R + ; L 2 (R m )), which is a much better situation compared with the usual homogenization results for parabolic equations [13]. The above convergence in L ∞ loc (R + ; L 2 (R m )) relies on energy balances thanks to Proposition 13. The uniform estimates of (∂ t v ε ) ε>0 , (∂ t ∇ R z v ε ) ε>0 in L 2 loc (R + ; L 2 (R m )) (which are available for initial conditions u in ∈ H 2 R cf. Proposition 8) play a crucial role, in order to apply Propositions 12, 13.
The above considerations show that v ε = v + o(1) in L ∞ loc (R + ; L 2 (R m )), when ε 0. As suggested by (10), we expect a convergence rate in O(ε). This can be achieved assuming that the limit solution v is smooth enough and that there is a smooth matrix field C such that that is, cf. Remark 2 The existence of the matrix field C is essential when constructing the corrector term u 1 , see (57).