Homogenization of linear transport equations. A new approach

Homogenization of linear transport equations. A new approach Tome 7 (2020), p. 479-495.

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HOMOGENIZATION OF

LINEAR TRANSPORT EQUATIONS.

A NEW APPROACH

by Marc Briane

Abstract. — The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fieldsbε(x), the solutions of which uε(t, x)agree att= 0with a bounded sequence ofL^p_loc(R^N)for somep∈(1,∞). Assuming that the sequencebε· ∇w_ε¹ is compact inL^q_loc(R^N)(qconjugate ofp) for some gradient field

∇w¹_εbounded inL^N_loc(R^N)^N, and that there exists a uniformly bounded sequenceσε>0such that σεbε is divergence free if N= 2or is a cross product of (N−1) bounded gradients in L^N_loc(R^N)^N if N>3, we prove that the sequence σεuε converges weakly to a solution to a linear transport equation. It turns out that the compactness ofbε· ∇w¹_εis a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples.

Résumé(Homogénéisation d’équations de transport linéaires. Une nouvelle approche)

Cet article propose une nouvelle approche de l’homogénéisation des équations de transport linéaires induites par une suite uniformément bornée de champs de vecteurs bε(x) et dont les solutionsuε(t, x)coïncident ent= 0avec une suite bornée de L^p_loc(R^N)pour un certain p ∈ (1,∞). En supposant que la suite bε· ∇w¹_ε est compacte dans L^q_loc(R^N) (q exposant conjugué dep) pour un champ de gradients∇w¹_ε borné dansL^N_loc(R^N)^N et qu’il existe une suite uniformément bornéeσε >0 telle que σεbε est à divergence nulle siN= 2ou est un produit vectoriel de (N−1) gradients bornés dans L^N_loc(R^N)^N siN >3, on montre que la suiteσεuε converge faiblement vers une solution d’une équation de transport. Il s’avère que la compacité debε· ∇w¹_ε remplace la condition d’ergodicité du cas périodique bidimensionnel classique et permet de traiter des champs de vecteurs non périodiques en toute dimension. Le résultat d’homogénéisation est illustré par différents exemples généraux.

Contents

1. Introduction. . . 480

2. The main result. . . 483

3. Examples. . . 490

References. . . 494

2010Mathematics Subject Classification. — 35B27, 35F05, 37C10.

Keywords. — Homogenization, transport equation, dynamic flow, rectification.

1. Introduction

In this paper we study the homogenization of the sequence of linear transport equations indexed byε >0,

(1.1)







∂u_ε

∂t −bε(x)· ∇xuε= 0 in (0, T)×R^N, uε(0,·) =u⁰_ε in R^N.

where N > 2, T > 0 and p ∈ [1,∞] with conjugate exponentq. Equation (1.1) is associated with the flowYε(t, x)defined by

(1.2)







∂Y_ε(t, x)

∂t =b_ε(Y_ε(t, x)), t∈R Yε(0, x) =x∈R^N.

Using the DiPerna-Lions transport theory [8, Cor. II.1], if for instance bε is a vec- tor field in L^∞(R^N)^N ∩W_loc^1,q(R^N)^N with bounded divergence and the initial condi- tionu⁰_ε is inL^p(R^N), then there exists a unique solutionuε(t, x)to equation (1.1) in L^∞(0, T;L^p(R^N)).

Tartar [21] showed that the homogenization of first-order hyperbolic equations may lead to nonlocal effective equations with memory effects. In this framework Amirat, Hamdache and Ziani obtained memory effects for the homogenization of the transport equation (1.1), first in dimension two when bε· ∇xuε =bε(t, x2)∂x₁uε [1], and in a more general setting [2, 3] they derived an integral representation of the memory term of the homogenized equation using Young’s measures and a Radon transform.

In the present paper, rather than determining the “closure” set of the homogenized equations of (1.1) as in [1, 2, 3], we provide some sufficient conditions under which the homogenized equation of (1.1) remains a transport equation, namely the nature of equation (1.1) is preserved through homogenization. Assuming thatbε(x) =b(x/ε) is a divergence free periodically oscillating vector field, Brenier [4] showed thanks to the ergodic theorem that the solutionu_εof (1.1) converges. This result was improved in dimension two by Hou and Xin [15] who proved thanks to a two-scale convergence approach that the homogenization of (1.1) leads to the averaged transport equation with R

T²b(y)dy (T^N denotes the N-dimensional torus). Golse [12, Th. 8] extended these results to the locally periodic case b_ε(x) = b(x, x/ε) with div_yb(x,·) = 0, as- suming some ergodic property involving the flow associated with−b(x,·). In [12, 13]

and in the generalizations [11, 10] with an application to Physics [9], Golse et al. also studied the perturbed differential system satisfied by the pair(Iε, Jε)inR^N ×T^N:

(1.3)









 dI_ε

dt =εf(I_ε, J_ε), t∈R dJε

dt =ω(Iε) +εg(Iε, Jε),

with f(I,·),g(I,·)periodic, which after anε-rescaling is associated with a Liouville partial differential equation of type (1.1) but more complicated. Assuming among

others thatω satisfies the Kolmogorov non-degeneracy type ergodic condition (1.4) meas

I∈R^N, |I|6R:|ω(I)·ξ|< α −−−→

α→0 0 uniformly inξ∈S^N−1, or some variant, Golse et al. obtained an error estimate between the solution to system (1.3) and the averaged system with R

T^Nf(I, y)dy, as well as the velocity averaging (homogenization) of the associated Liouville partial differential equation.

They also extended this result to a non-periodic framework.

Returning to the periodic case b_ε(x) = b(x/ε), Tassa [22] replaced in dimension two the divergence free ofbby the existence of a periodic positive regular function σ such that

(1.5) div (σb) = 0 in R²,

i.e., σis an invariant measure for b by the Liouville theorem. The main assumption of the periodic framework bε(x) = b(x/ε) of [4, 15, 22] is the ergodicity of the flow associated with b (see, e.g. [20, Lect. 1], or [19, Chap. II, § 5]), namely any periodic invariant function by the flow is constant, or equivalently, for any periodic regular functionv,

(1.6) b· ∇v= 0 inR² =⇒ ∇v= 0 in R²,

together withb6= 0inR². By virtue of the Kolmogorov theorem (see, e.g. [20, Lect. 11]

or [22, §2]) in dimension two with b6= 0, condition (1.6) is equivalent to the ergodic assumption

(1.7) Z

T²

b1(y)σ(y)dy, Z

T²

b2(y)σ(y)dy are rationally independent,

which is equivalent to the irrationality of the rotation number. In the locally periodic caseb_ε(x) =b(x, x/ε)of [12, §8] the ergodicity assumption states that for a.e.x∈R^N, the fluctuation of the vector fieldb(x,·)around its average value is ergodic with respect to the flow associated with−b(x,·).

In the present paper, besides the closure results of [1, 2, 3] and the ergodic ap- proaches of [4, 15, 12, 13, 22], we propose a new approach which holds both in a non-periodic framework and in any dimension, assuming that the vector fieldbε sat- isfies a non-ergodic condition which preserves the nature of equation (1.1) through homogenization. More precisely, the ergodic assumption (1.6) or (1.7) of the peri- odic framework is now replaced by the existence of a sequence w¹_ε in C¹(R^N) and q∈(1,∞)such that

(1.8) 0< bε· ∇w¹_ε −→ θ0>0 strongly inL^q_loc(R^N),

which is equivalent in the periodic case to the existence of a periodic gradient ∇w satisfying

(1.9) b· ∇w= 1 in R^N.

Moreover, the invariant measureσ of the periodic case is replaced by a sequenceσε

satisfying 0 < c⁻¹ < σε < c for some constant c > 1, and (see Remark 2.1 for an

equivalent expression)

(1.10) div (σεbε) = 0 ifN = 2 and σεbε=∇w²_ε× · · · × ∇w^N_ε ifN >3.

The case where σ_εb_ε is only divergence free in dimension N > 3 remains open.

In this way the vector field bε is naturally associated with the vector field Wε :=

(w¹_ε, . . . , w^N_ε) which induces a global rectification of the field bε in the direction e1

(see Remark 2.1). Then, assuming in addition to (1.8), (1.10) that W_ε is uniformly proper (see condition (2.1) below) and converges both in C_loc⁰ (R^N)^N and weakly in W_loc^1,N(R^N)^N, we prove (see Theorem 2.2) that, up to a subsequence, σ_εu_εconverges weakly inL^∞(0, T;L^p(R^N))to a solutionv to the transport equation

(1.11)







∂v

∂t −ξ0· ∇x

v σ0

= 0 in(0, T)×R^N v(0,·) =v⁰ inR^N,

where σ0 is the weak-∗ limit of σε in L^∞(R^N), ξ0 is the weak limit of σεbε in L^N/(N−1)_loc (R^N)^N and v⁰ the weak limit of σεu⁰_ε in L^p(R^N). Moreover, if σε con- verges strongly toσ₀ inL¹_loc(R^N)(see Remark 2.4) oru⁰_εconverges strongly tou⁰in L^p_loc(R^N), then up to a subsequence uε converges weakly inL^∞(0, T;L^p(R^N))to a solutionuto the transport equation

(1.12)







∂u

∂t −ξ₀ σ0

· ∇xu= 0 in (0, T)×R^N u(0,·) =u⁰ in R^N.

The convergence ofuεalso turns out to be strong inL^∞(0, T;L²_loc(R^N))ifu⁰_εconverges strongly tou⁰in L^p_loc(R^N)with p >2(see the second part of Theorem 2.2).

The compactness condition (1.8) is the main assumption of Theorem 2.2. It is equivalent to the compactness of the productσεdet(DWε)which is connected to the vector fieldbεby (1.10). The examples of Section 3 show that this condition may be satisfied in quite general situations.

Section 2 is devoted to the statement of the main result and to its proof. Section 3 deals by three applications of Theorem 2.2. In Section 3.1 we study the case of a diffeo- morphismWεonR²such thatdet(DWε)is compact inL^p_loc(R²)for someq∈(1,∞).

In Section 3.2 we extend the periodic case of [4, 15, 22] withb_ε(x) =b(x/ε)and the periodic case of [5, §4] on the asymptotic of the flow associated withb, in the light of Theorem 2.2 with a periodically oscillating functionσε(x) =σ(x/ε)(see Proposi- tion 3.1). In Section 3.3 we consider the case of a diffeomorphismW_εwhich agrees at a fixed timetto a flowX_ε(t,·)associated with a suitable vector fielda_ε(see Proposi- tion 3.2). In this general setting assumption (1.8) holds simply whendivaεis compact in L^q_loc(R^N)for someq∈(1,∞).

Notations

– (e1, . . . , eN)denotes the canonical basis ofR^N.

– ·denotes the scalar product inR^N and| · |the associated norm.

– IN is the unit matrix ofR^N×N, andR_⊥ is the clockwise90^◦ rotation matrix in R^2×2.

– ForM ∈R^N×N,M^T denotes the transpose ofM.

– Y_N := [0,1)^N, andhfidenotes the average-value of a functionf ∈L¹(Y_N).

– For any open setΩofR^N andk∈N∪ {∞},C_c^k(Ω), respectivelyC_b^k(Ω), denotes the space of theC^k functions with compact support inΩ, respectively bounded inΩ.

– Fork∈N∪ {∞} andp∈[1,∞],C_]^k(Y_N) denotes the space of theY_N-periodic functions in Ck(R^N), andL^p_](YN)denotes the space of the YN-periodic functions in L^p_loc(R^N)(i.e., inL^p(K)for any compact setK ofR^N).

– Foru∈L¹_loc(R^N)andU = (Uj)16j6d∈L¹_loc(R^N)^N.

∇xu:= (∂x₁u, . . . , ∂x_Nu) and DU:=

∂x_iUj

16i,j6d.

The matrix-valued functionDU stands for the transposition of the Jacobian matrix of the vector fieldU.

– Forξ¹, . . . , ξ^N in R^N, the cross productξ²× · · · ×ξ^N is defined by (1.13) ξ¹·(ξ²× · · · ×ξ^N) = det (ξ¹, ξ², . . . , ξ^N) forξ¹∈R^N, wheredetis the determinant with respect to the canonical basis(e1, . . . , eN).

– oε denotes a term which tends to zero asε→0.

– C denotes a constant which may vary from line to line.

Acknowledgements. — The author is very grateful to the unknown referees for sev- eral comments which have improved the presentation of the paper and enlarged the bibliography.

2. The main result

LetW_ε= (w_ε¹, . . . , w^N_ε), ε >0, be a sequence of vector fields in C¹(R^N)^N which isuniformly proper, i.e., for any compact set Kof R^N there exists a compact subset K⁰ ofR^N satisfying

(2.1) W_ε⁻¹(K)⊂K⁰ for any small enoughε >0, and letW ∈C¹(R^N)^N be such that

(2.2) W_ε−→W inC_loc⁰ (R^N)^N and W_ε* W inW_loc^1,N(R^N)^N.

Letb_εbe a vector field inC_b⁰(R^N)^N∩W_loc^1,q(R^N)^N with bounded divergence and letσ_ε be a positive function inC⁰(R^N)∩W_loc^1,q(R^N)satisfying for some constantc >1, (2.3) c⁻¹6σ_ε6c and σ_εb_ε=







R_⊥∇w²_ε ifN = 2

∇w²_ε× · · · × ∇w^N_ε ifN >3,

inR^N. Also assume that for p∈ (1,∞) with conjugate exponent q, there exists a positive functionθ0in C⁰(R^N)such that

(2.4) θε:=bε· ∇w¹_ε>0 inR^N and θε−→θ0>0 strongly inL^q_loc(R^N).

Finally, assume:

– either that there exists a constantB >0such that

(2.5) |divb_ε|6B a.e. inR^N,

– or the regularity condition

(2.6) bε∈C_b¹(R^N)^N, σε∈C¹(R^N) and u⁰_ε∈C¹(R^N).

Remark2.1. — The definition (2.3) ofbεcan be also written for any dimensionN >2 as the existence of(N−1) gradients∇w_ε², . . . ,∇w^N_ε satisfying

(2.7) ∀ξ∈R^N, σ_εb_ε·ξ= det ξ,∇w²_ε, . . . ,∇w_ε^N .

In dimensionN >3this is exactly the definition of the cross product∇w²_ε×· · ·×∇w^N_ε (see (1.13)). In dimension N = 2 this means exactly thatσεbε =R_⊥∇w_ε², which is equivalent to

(2.8) div (σ_εb_ε) = 0 in R².

However, in dimension N>3 condition (2.3) is stronger thanσεbεdivergence free.

The definition (2.3) ofbεand the definition (2.4) ofθεare equivalent to the global rectification of the fieldbεby the diffeomorphismWε

(2.9) DW_ε^Tb_ε=θ_εe₁ inR^N,

in the direction e1with the compact rangeθε.

Then, we have the following homogenization result.

Theorem 2.2. — Let T > 0, let p ∈ (1,∞) and let u⁰_ε be a bounded sequence in L^p(R^N), N > 2. Assume that conditions (2.1) to (2.4) together with (2.5) or (2.6) hold true. Let uε be the solution to the transport equation (1.1) and set vε:=σεuε. Then, up to a subsequence v_ε converges weakly inL^∞(0, T;L^p(R^N)) to a solution v to the transport equation

(2.10)







∂v

∂t −ξ0· ∇x

v σ₀

= 0 in(0, T)×R^N v(0,·) =v⁰ inR^N, where (Cof denotes the cofactors matrix)

ξ0= Cof (DW)e1∈C⁰(R^N)^N, (2.11)











σ_εb_ε* ξ₀ in L^N/(N_loc ⁻¹⁾(R^N)^N, σε* σ0 inL^∞(R^N)∗,

σεu⁰_ε* v⁰ inL^p(R^N).

(2.12)

Moreover, if in addition b_ε ∈ W_loc^1,p/(p−2)(R^N)^N with p > 2 and the sequence u⁰_ε converges strongly to u⁰∈L^p_loc(R^N)with

σ0∈W^1,∞(R) and ξ0∈L^∞(R^N)^N ∩W_loc^1,p/(p−2)(R^N)^N,

then uε converges strongly in L^∞(0, T;L²_loc(R^N)) to the solution uto the transport equation

(2.13)







∂u

∂t − ξ₀ σ0

· ∇xu= 0 in(0, T)×R^N u(0,·) =u⁰ inR^N.

Remark 2.3. — If in Theorem 2.2 we assume in addition that σ0 is in W^1,∞(R^N) andξ₀belongs toL^∞(R^N)^N∩W_loc^1,q(R^N)^N, then by virtue of [8, Cor. II.1] there exists a unique solutionvto the transport equation (2.10).

Remark2.4. — In addition to the conditions (2.1) to (2.4) assume thatσ_εconverges strongly in L¹_loc(R^N) to σ0 ∈ W_loc^1,q(R^N). Then, we have v = σ0u and v⁰ = σ0u⁰ where u⁰ is the weak limit of u⁰_ε in L^p(R^N), which implies that equation (2.10) is equivalent to equation (2.13). Therefore,uεconverges weakly inL^∞(0, T;L^p(R^N))to a solutionuto the transport equation (2.13).

To prove Theorem 2.2 we need the followingL^p-estimate.

Lemma2.5. — Letbε∈L^∞(R^N)^N∩W_loc^1,q(R^N)^N with bounded divergence be such that – either estimate (2.5)holds true,

– or both conditions (2.3)and (2.6)hold true.

Then, there exists a constant C >0 such that for any u⁰_ε∈L^p(R^N) with p∈[1,∞), the solutionuε to equation (1.1)satisfies the estimate

(2.14) kuε(t,·)k_Lp(R^N)6Cku⁰_εk_Lp(R^N) for a.e.t∈(0, T), Proof of Theorem2.2. — First of all, note that by (2.3) and (2.4) we have (2.15) det(DWε) =σεθε>0 in R^N.

This combined with property (2.1) and Hadamard-Caccioppoli’s theorem [6] (or Hada- mard-Lévy’s theorem) implies thatWεis aC¹-diffeomorphism onR^N. Moreover, since by (2.15) det(DWε) is positive and by (2.2) Wε converges weakly in W_loc^1,N(R^N)^N, by virtue of Müller’s theorem [16] det(DW_ε) weakly converges to det(DW) in L¹_loc(R^N). Hence, passing to the limit in (2.15) together with the strong conver- gence (2.4) ofθε, the weak convergence (2.12) ofσεand the boundedness (2.3) of σε

we get that

(2.16) det(DW) =σ₀θ₀>c⁻¹θ₀>0 a.e. inR^N,

which taking into account the continuity of DW and θ0 implies that det(DW)>0 in R^N. Moreover, again by the uniform character of (2.1) W is a proper mapping.

Therefore,W is also a C¹-diffeomorphism onR^N. The weak formulation of equation (1.1) is that for any functionφ∈C_c¹([0, T)×R^N),

(2.17)

Z T 0

Z

R^N

uε

∂φ

∂t dx dt+ Z

R^N

u⁰_ε(x)φ(0, x)dx= Z T

0

Z

R^N

uε div (φ bε)dx dt.

Using a density argument with σε ∈W_loc^1,q(R^N), we can replace the test function φ byσεϕfor anyϕ∈C_c¹([0, T)×R^N). This combined with the divergence free ofσεbε

leads us to the new formulation (2.18)

Z T 0

Z

R^N

σ_εu_ε∂ϕ

∂t dx dt+ Z

R^N

σ_ε(x)u⁰_ε(x)ϕ(0, x)dx

= Z T

0

Z

R^N

uεσεbε· ∇xϕ dx dt.

We pass easily to the limit in the left hand-side of (2.18). The delicate point comes from the right-hand side of (2.18).

By theL^p-estimate (2.14) of Lemma 2.5 combined with the uniform boundedness of σε in (2.3) there exists a subsequence, still denoted by ε, such that vε = σεuε

converges weakly to some functionvin L^∞(0, T;L^p(R^N)).

Let ψ∈ C_c¹([0, T)×R^N) the support of which is contained in some compact set [t₀, t₁]×K of[0, T)×R^N, and define

(2.19) ϕε(t, x) :=ψ(t, Wε(x)) for(t, x)∈(0, T)×R^N,

so that∇xϕε(t, x) =DWε(x)∇yψ(t, y). Hence, making the change of variables y = W_ε(x)and using (2.9) we deduce that

(2.20) Z T

0

Z

R^N

vε(t, x)bε(x)· ∇xϕε(t, x)dx dt

= Z T

0

Z

W_ε⁻¹(K)

vε(t, x)bε(x)· ∇xϕε(t, x)dx dt

= Z T

0

Z

K

vε(t, W_ε⁻¹(y))θε(W_ε⁻¹(y))e1· ∇yψ(t, y) det(DW_ε⁻¹)(y)dy dt.

First, using successively the Hölder inequality combined with theL^p-estimate (2.14), the inclusion (2.1) and theL^q-strong convergence (2.4) ofθε, we have

Z T 0

Z

K

vε(t, W_ε⁻¹(y)) (θε−θ0)(W_ε⁻¹(y))e1· ∇yψ(t, y) det(DW_ε⁻¹)(y)dy dt 6Cψ

Z T 0

Z

K

vε(t, W_ε⁻¹(y))

pdet(DW_ε⁻¹)(y)dy 1/p

× Z

K

(θε−θ0)(W_ε⁻¹(y))

qdet(DW_ε⁻¹)(y)dy 1/q

dt

6Cψ

Z T 0

kvε(t,·)kL^p(K⁰)kθε−θ0kL^q(K⁰)dt=oε, which implies that

Z T 0

Z

K

vε(t, W_ε⁻¹(y))θε(W_ε⁻¹(y))e1· ∇yψ(t, y) det(DW_ε⁻¹)(y)dy dt

= Z T

0

Z

K

vε(t, W_ε⁻¹(y))θ0(W_ε⁻¹(y))e1· ∇yψ(t, y) det(DW_ε⁻¹)(y)dy dt+oε.

Next, by the uniform convergence (2.2)

∇yψ(t, Wε(x))−→ ∇yψ(t, W(x)) inC_loc⁰ ([0, T]×R^N).

Then, making the inverse change of variables x= W_ε⁻¹(y) together with (2.1) and using the weak convergence ofvε tov inL^∞(0, T;L^p(R^N)), we have

Z T 0

Z

K

v_ε(t, W_ε⁻¹(y))θ₀(W_ε⁻¹(y))e₁· ∇_yψ(t, y) det(DW_ε⁻¹)(y)dy dt

= Z T

0

Z

K⁰

vε(t, x)θ0(x)e1· ∇yψ(t, Wε(x))dx dt

= Z T

0

Z

K⁰

v(t, x)θ0(x)e1· ∇yψ(t, W(x))dx dt+oε.

Letϕ∈C_c¹([0, T)×R^N)and define similarly to (2.19)

ϕ(t, x) :=ψ(t, W(x)) for(t, x)∈[0, T)×R^N,

so that ∇xϕ(t, x) = DW(x)∇yψ(t, y). Therefore, passing to the limit in (2.20) we obtain that

(2.21) Z T

0

Z

R^N

v_ε(t, x)b_ε(x)· ∇_xϕ_ε(t, x)dx dt

= Z T

0

Z

R^N

v(t, x)θ0(x) DW(x)^T−1

e1· ∇xϕ(t, x)dx dt+oε.

On the other hand, using (2.9), (2.3) and the Murat-Tartar div-curl lemma in L^N/(N−1)-L^N (see, e.g. [17, Th. 2]) with convergences (2.2), (2.4), (2.12) we get that (2.22) DW_ε^T(σ_εb_ε) =σ_εθ_εe₁* DW^Tξ₀=σ₀θ₀e₁ weakly inL¹_loc(R^N).

This combined with (2.16) yields equality (2.11). Convergences (2.21) and (2.22) imply that

Z T 0

Z

R^N

v_εb_ε· ∇xϕ_εdx dt −−−→

ε→0

Z T 0

Z

R^N

v σ0

ξ₀· ∇xϕ dx dt.

Finally, passing to the limit in formula (2.18) with ϕ_ε, it follows that for any ϕ ∈ C_c¹([0, T)×R^N),

Z T 0

Z

R^N

v∂ϕ

∂t dx dt+ Z

R^N

v⁰(x)ϕ(0, x)dx= Z T

0

Z

R^N

v

σ₀ξ0· ∇xϕ dx dt,

which taking into account thatξ₀is divergence free yields the weak formulation of the desired limit equation (2.10). This concludes the proof of the first part of Theorem 2.2.

Now, assume in addition thatbε∈W_loc^1,p/(p−2)(R^N)^N with p >2 andu⁰_ε converges strongly tou⁰in L^p(R^N)with

σ0∈W^1,∞(R^N) and ξ0∈L^∞(R^N)^N ∩W_loc^1,p/(p−2)(R^N)^N.

By [8, Th. II.3 and Corollary II.1]u²_εis the unique solution to the equation (1.1) with initial condition(u⁰_ε)², or equivalently, for any φ∈C_c¹([0, T)×R^N),

Z T 0

Z

R^N

u²_ε∂φ

∂t dx dt+ Z

R^N

(u⁰_ε)²(x)φ(0, x)dx= Z T

0

Z

R^N

u²_εdiv (φ bε)dx dt, Replacinguεbyu²_ε in the first part of Theorem 2.2 and using the strong convergence ofu⁰_ε we get that the sequence σ_εu²_ε converges weakly inL^∞(0, T;L^p/2(R^N))to the solutionwto the transport equation

(2.23)







∂w

∂t −ξ0· ∇x

w σ0

=∂w

∂t − ξ0

σ0

· ∇xw+ξ0· ∇σ0

σ²₀ w= 0 in (0, T)×R^N

w(0,·) =σ₀(u⁰)² in R^N.

Note that by virtue of [8, Cor. II.1] the solutionwto equation (2.23) is unique due to the regularitiesσ0∈W^1,∞(R^N),ξ0∈L^∞(R^N)^N∩W_loc^1,p/(p−2)(R^N)^N with divergence free. Moreover, again by [8, Th. II.3 and Corollary II.1] v² is the unique solution to the equation induced by (2.10)







∂(v²)

∂t − ξ₀ σ0

· ∇x(v²) + 2ξ₀· ∇σ0

σ0

v²= 0 in(0, T)×R^N v²(0,·) = (σ0u⁰)² inR^N, or equivalently, for anyφ∈C_c¹([0, T)×R^N),

Z T 0

Z

R^N

v²∂φ

∂t dx dt+ Z

R^N

(σ0u⁰)²(x)φ(0, x)dx

= Z T

0

Z

R^N

v² div φξ₀

σ0

dx dt+

Z T 0

Z

R^N

2v²ξ₀· ∇σ₀

σ²₀ φ dx dt.

Replacing the test functionφbyϕ/σ0 by a density argument, it follows that for any functionϕ∈C_c¹([0, T)×R^N),

Z T 0

Z

R^N

v² σ₀

∂ϕ

∂t dx dt+ Z

R^N

σ0(x) (u⁰)²(x)ϕ(0, x)dx

= Z T

0

Z

R^N

v² div ϕξ₀

σ²₀

dx dt+ Z T

0

Z

R^N

2v²ξ₀· ∇σ₀ σ³₀ ϕ dx dt

= Z T

0

Z

R^N

v² σ0

div ϕ ξ₀

σ0

dx dt+ Z T

0

Z

R^N

v² σ0

ξ₀· ∇σ₀

σ²₀ ϕ dx dt, which shows thatv²/σ0 is also a solution to equation (2.23). By uniqueness we thus get that w = v²/σ0. Similarly, the solution u to equation (2.13) agrees with v/σ0. Finally, using these two equalities we have for any compact setK ofR^N,

Z T 0

Z

K

σε(uε−u)²dx dt= Z T

0

Z

K

(σεu²_ε−2σεuεu+σεu²)dx dt

−−−→ε→0

Z T 0

Z

K

(w−2v u+σ₀u²)dx dt= 0,

which concludes the proof of Theorem 2.2.

Proof of Lemma2.5. — If the uniform boundedness (2.5) of div bε is satisfied, then using the estimate (17) of [8, Prop. II.1] for the solution to the regularized equation of (1.1) and the lower semi-continuity of theL^p-norm (p <∞)we get estimate (2.14).

Otherwise, assume that conditions (2.3) and (2.6) hold true. Using the regularity of the data the proof is based on an explicit expression of the solution to equation (1.1) from the flowYεassociated with the vector field bεby

(2.24)







∂Y_ε(t, x)

∂t =b_ε(Y_ε(t, x)), t∈R Yε(0, x) =x∈R^N.

Letu⁰_ε be a function inC¹(R^N)∩L^p(R^N). It is classical that the regular solutionuε

to the transport equation (1.1) is given by

(2.25) u_ε(t, x) =u⁰_ε(Y_ε(t, x)) for(t, x)∈[0, T]×R^N.

Lett∈[0, T]. Making the change of variables combined with the semi-group property of the flow

y=Y_ε(t, x) ⇐⇒ x=Y_ε(−t, y), we get that

(2.26)

Z

R^N

u⁰_ε(Yε(t, x))

pdx= Z

R^N

u⁰_ε(y)

p

det(DyYε(−t, y)) dy.

Moreover, by (2.24) and the Liouville formula we have for any(τ, y)∈R×R^N, det(DyYε(τ, y)) = exp

Z τ 0

(divbε)(Yε(s, y))ds

.

However, since by (2.3)σεbε is divergence free, we have Z τ

0

(divb_ε)(Y_ε(s, y))ds=− Z τ

0

∇σ_ε·b_ε σε

(Y_ε(s, y))ds

=− Z τ

0

∂

∂s lnσ_ε(Y_ε(s, y))

ds= ln σ_ε(y) σε(Yε(τ, y))

.

This combined with the boundedness ofσεin condition (2.3) implies that

∀(τ, y)∈R×R^N, 0<det(DyYε(τ, y)) = σε(y)

σε(Yε(τ, y)) 6c². Hence, we deduce from (2.26) that

Z

R^N

|uε(x)|^pdx= Z

R^N

u⁰_ε(Yε(t, x))

pdx6c² Z

R^N

u⁰_ε(y)

pdy,

which yields the desired estimate (2.14). This concludes the proof of Lemma 2.5.

3. Examples

The purpose of this section is to illustrate the homogenization of the transport equation (1.1) by various oscillating fieldsb_ε which satisfy the assumptions of Theo- rem 2.2. It means giving examples of diffeomorphismWεonR^N satisfying the recti- fication (2.9) of the vector fieldbε where the sequenceθε>0is compact inL^q_loc(R^N) for someq∈(1,∞).

3.1. First example. — Letαε, α∈C¹(R)be such that for some constantc >0, (3.1) αε−→α in C_loc⁰ (R), α⁰_ε>c inR, α⁰_ε−→α⁰ inL²_loc(R), and letβ_ε, β∈C¹(R)be such that for some constantC >0,

(3.2) β_ε−→β inC_loc⁰ (R), |βε|6C inR, β_ε⁰ is bounded inL^∞_loc(R), Consider the vector fieldWε∈C¹(R^N)^N defined by

(3.3) W_ε(x) := αε(x1) exp

βε(αε(x1)αε(x2)) , αε(x2) exp

−βε(αε(x1)αε(x2)) , x∈R², which is based on the characterization of the holomorphic mappings on C² with constant Jacobian [18]. The gradient of Wεis given by



















∇w¹_ε(x) = exp

βε(αε(x1)αε(x2)) α⁰_ε(x1) 1 +αε(x1)αε(x2)β_ε⁰(αε(x1)αε(x2)) α⁰_ε(x2)α²_ε(x1)β_ε⁰(αε(x1)αε(x2)

!

∇w²_ε(x)

= exp

−βε(αε(x1)αε(x2)) −α⁰_ε(x1)α²_ε(x2)β_ε⁰(αε(x1)αε(x2)) α⁰_ε(x2) 1−αε(x1)αε(x2)β_ε⁰(αε(x1)αε(x2))

! .

Also definebε:=R⊥∇w²_ε andσε:= 1, so that conditions (2.3) and (2.5) are fulfilled.

By (3.1) and (3.2) we have Wε(x) ^C

0 loc(R²)

−−−−−→W(x) := α(x1) exp

β(α(x1)α(x2)) , α(x2) exp

−β(α(x1)α(x2)) , Wε* W in H_loc¹ (R²),

so that conditions (2.2) are satisfied, and

(3.4) bε· ∇w¹_ε(x) = det(DWε)(x) =α⁰_ε(x1)α⁰_ε(x2)−→α⁰(x1)α⁰(x2) inL²_loc(R²), so that condition (2.4) is satisfied withp= 2. Moreover, since by (3.1)

∀t∈R, |αε(t)−α_ε(0)|>c|t|,

the sequenceαε(0)converges, andβεis uniformly bounded inR, condition (2.1) holds forW_ε.

Note that the oscillations of the drift bε in equation (1.1) are only due to the oscillations of the sequenceβ_ε⁰ which does not appear in the convergence (3.4) of the Jacobian.

3.2. The periodic case. — This section extends the periodic framework of [4, 15, 22], [12, Th. 8], and [5, Cor. 4.4].

Let W = (w¹, . . . , w^N) be a vector field in C²(R^N)^N, and letM be a matrix in R^N×N such that

(3.5) x7→W(x)−M x

∈C_]¹(YN)^N and σ:= det(DW)>0 in R^N. Consider the periodic vector fieldb∈C_]¹(Y^N)^N defined by

(3.6) σ b:=







R_⊥∇w² ifN= 2

∇w²× · · · × ∇w^N ifN>3.

We have the following result.

Proposition 3.1. — Let u⁰_ε ∈ C¹(R^N) be a bounded sequence in L^p(R^N) with p ∈ (1,∞). Assume that conditions (3.5)and (3.6)hold true. Then, the vector fieldsWε

andb_εdefined by

(3.7) Wε(x) :=ε W(x/ε) and bε(x) :=b(x/ε) forx∈R^N, satisfy the assumptions of Theorem 2.2.

Moreover, for any sequence u⁰_ε in L^p(R^N) such that σ(x/ε)u⁰_ε converges weakly to v⁰ in L^p(R^N), the solution uε to equation (1.1) is such that σ(x/ε)uε converges weakly in L^∞(0, T;L^p(R^N)) to the solution v to the equation (2.10) with σ₀ = hσi andξ₀=hσ bi.

Proof of Proposition3.1. — By the quasi-affinity of the determinant (see, e.g. [7,

§4.3.2]) and by (3.5) we have

det(M) = dethDWi=

det(DW)

>0, and by (3.7) there exists a constantC >0such that

(3.8) ∀x∈R^N, |Wε(x)−M x|6Cε,

which implies condition (2.1). Moreover, estimate (3.8) and the uniform bounded of DW_εimply easily the convergences (2.2) with the limitW(x) :=M x.

On the other hand, the definitions (3.5) ofW,σand the definition (3.6) ofbshow clearly that condition (2.3) and the regularity (2.6) hold true. Moreover, we have

θ:=b· ∇w¹=det(DW)

σ = 1 inR^N, which implies (2.4) sinceθε(x) =θ(x/ε) = 1.

Moreover, let u⁰_ε be a sequence in L^p(R^N)such that σ(x/ε)uε converges weakly to v⁰ in L^p(R^N). By virtue of Theorem 2.2 combined with Remark 2.3 and using the weak limit of a periodically oscillating sequence, the sequenceσ(x/ε)uεconverges weakly inL^p(R^N)to the solutionvto the equation (2.10) withσ0=hσiandξ0=hσ bi.

The proof of Proposition 3.1 is now complete.

3.3. The dynamic flow case. — In this section we construct a sequenceWε from a dynamic flow associated with a suitable but quite general sequence of vector fieldsaε.

Leta_ε, abe vector fields inC¹(R^N)^N such that

(3.9) aε−→a inC_loc⁰ (R^N)^N, aε* a inW_loc^1,∞(R^N)^N∗, and for some constantA >0,

(3.10) |aε|+|div a_ε|6A in R^N. Also assume that there existsq∈(1,∞)such that (3.11) divaε−→div a in L^q_loc(R^N).

Consider the dynamic flow X_εassociated with the vector fielda_ε defined by (3.12)







∂Xε(t, x)

∂t =aε(Xε(t, x)), t∈R Xε(0, x) =x∈R^d,

and letX be the limit flow associated with the limit vector fielda.

Then, from any sequence of flows Xε we may derive a general sequence of vector fieldsbεinducing the homogenization of the transport equation (1.1).

Proposition3.2. — Letu⁰_εbe a bounded sequence inL^p(R^N)withp∈(1,∞). Assume that conditions (3.9),(3.10),(3.11)hold true. For a fixedt >0, define the vector field W_ε:=X_ε(t,·)from R^N intoR^N, and the vector fieldb_εby (2.3)with σ_ε= 1. Then, the sequences Wε andbε satisfy the assumptions of Theorem 2.2.

Moreover, for any sequenceu⁰_ε converging weakly tou⁰ inL^p(R^N), the solution uε

to equation(1.1)converges weakly inL^∞(0, T;L^p(R^N))to a solutionuto the equation (2.13) whereσ0= 1 andξ0= Cof (DxX(t, x))e1.

Remark3.3. — There is a strong correspondence between the conditions (3.9)-(3.10) and (3.11) satisfied by the vector field aε, and respectively the conditions (2.2) and (2.4) satisfied by the vector fieldsW_εandb_ε.

Proof of Proposition3.2. — First of all, conditions (2.3) and (2.5) are straightforward, sinceσε= 1andbε is divergence free. FixT >0. By (3.10) we have

(3.13) ∀t∈[0, T], ∀x∈R^N, |X_ε(t, x)−x|6A T, so that the uniform property (2.1) is satisfied byWε.

LetK be a compact set ofR^N. Again by (3.13) there exists a compact subset K⁰ ofR^N such that

(3.14)

X_ε(t, x),(t, x)∈[0, T]×K ⊂K⁰.

Let δ > 0. Since aε converges uniformly to ain K⁰ and a∈ C¹(R^N) is k-Lipschitz in K⁰ for somek >0, we have for any small enoughε >0 and for anyt∈[0, T], for

anyx, y∈K,

Xε(t, x)−Xε(t, y)

6|x−y|+ Z t

0

aε(Xε(s, x))−aε(Xε(s, y)) ds 6δ+|x−y|+k

Z t 0

X_ε(s, x)−X_ε(s, y) ds.

Hence, by Gronwall’s inequality (see, e.g. [14, §17.3]) we get that for any small enough ε >0,

∀t∈[0, T], ∀x, y∈K, |Xε(t, x)−Xε(t, y)

6(δ+|x−y|)e^kt, which by (3.10) implies that for any small enoughε >0,

∀s, t∈[0, T], ∀x, y∈K, |Xε(s, x)−X_ε(t, y)

6A|s−t|+ (δ+|x−y|)e^kt, namelyX_εis uniformly equicontinuous in the compact set [0, T]×K. Therefore, by virtue of Ascoli’s theorem this combined with (3.14) and (3.9) implies that up to a subsequence Xεconverges uniformly in[0, T]×K to a solutionX to

∀t∈[0, T], ∀x∈K, X(t, x) =x+ Z t

0

a(X(s, x))ds,

i.e.,X is the flow associated with the vector field a. Sinceabelongs to C_b¹(R^N), the flowX is uniquely determined (see, e.g. [14, §17.4]). Therefore, the whole sequenceXε

converges uniformly toX in[0, T]×K. Moreover, by the differentiability of the flow (see, e.g. [14, §17.6]) we have

(3.15) ∀t∈[0, T], ∀x∈K, DxXε(t, x) =IN + Z t

0

DxXε(s, x)Dxaε(Xε(s, x))ds, which using (3.9), (3.14) and Gronwall’s inequality implies that there exists a constant c >0 such that

∀t∈[0, T], ∀x∈K, |D_xX_ε(t, x)|6|I_N|e^ct. Therefore, convergences (2.2) hold true.

On the other hand, by the Liouville formula associated with equation (3.15) and estimate (3.10) we get that there exists a constantc >1such that

(3.16) ∀t∈[0, T], ∀x∈K,

c⁻¹6det (D_xX_ε(t, x)) = exp Z t

0

(div a_ε)(X_ε(s, x))ds

6c, which implies the existence of a constant C > 0 such that for any t ∈ [0, T] and x∈K,

det (DxXε(t, x))−det (DxX(t, x)) 6C

Z T 0

|div aε−div a|(Xε(s, x))ds +C

Z T 0

(diva)(Xε(s, x))−(div a)(X(s, x)) ds.