Homogenization of linear transport equations. A new approach
aa r X i v : . [ m a t h . A P ] M a y Homogenization of linear transport equations.A new approach.
Marc Briane
Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, [email protected]
May 23, 2019
Abstract
The paper is devoted to a new approach of the homogenization of linear transportequations induced by a uniformly bounded sequence of vector fields b ε ( x ), the solutionsof which u ε ( t, x ) agree at t = 0 with a bounded sequence of L p loc ( R N ) for some p ∈ (1 , ∞ ).Assuming that the sequence b ε · ∇ w ε is compact in L q loc ( R N ) ( q conjugate of p ) for somegradient field ∇ w ε bounded in L N loc ( R N ) N , and that there exists a uniformly boundedsequence σ ε > σ ε b ε is divergence free if N = 2 or is a cross product of ( N − L N loc ( R N ) N if N ≥
3, we prove that the sequence σ ε u ε convergesweakly to a solution to a linear transport equation. It turns out that the compactnessof b ε · ∇ w ε is a substitute to the ergodic assumption of the classical two-dimensionalperiodic case, and allows us to deal with non-periodic vector fields in any dimension. Thehomogenization result is illustrated by various and general examples. Keywords: homogenization, transport equation, dynamic flow, rectification
Mathematics Subject Classification:
In this paper we study the homogenization of the sequence of linear transport equations indexedby ε > ∂u ε ∂t − b ε · ∇ x u ε = 0 in (0 , T ) × R N , N ≥ u ε (0 , · ) = u ε in R N . (1.1)where T > p ∈ [1 , ∞ ] with conjugate exponent q . Using the DiPerna-Lions transporttheory [5, Corollary II.1], if for instance b ε is a vector field in L ∞ ( R N ) N ∩ W ,q loc ( R N ) N withbounded divergence and the initial condition u ε is in L p ( R N ), then there exists a unique solution u ε ( t, x ) to equation (1.1) in L ∞ (0 , T ; L p ( R N )).Tartar [14] has showed that the homogenization of first-order hyperbolic equations maylead to nonlocal effective equations with memory effects, and E [6] has also obtained fromthe homogenization of (1.1) effective higher-order hyperbolic equations. Hence, an interestingproblem consists in finding sufficient conditions for which the weak limit of the solution u ε toequation (1.1) is still a solution to a first-order transport equation. This type of homogenization1esult has first been derived in dimension two by Brenier [1] and by Hou, Xin [8], assumingthat b ε ( x ) = b ( x/ε ) where b is a divergence free periodic regular vector field. These works havebeen extended by E [6, Sec. 5] when b ε ( x ) = b ( x, x/ε ) with b ( x, y ) divergence free both in x and y , and by Tassa [15] when there exists a periodic positive regular function σ (which is calledan invariant measure for b ) such thatdiv ( σb ) = 0 in R . (1.2)The main assumption of the periodic framework of [1, 8, 6, 15] is the ergodicity of the flowassociated with b (see, e.g. , [13, Lect. 1], or [12, Chap. II, § v , b · ∇ v = 0 in R ⇒ ∇ v = 0 in R , (1.3)together with b = 0 in R . By virtue of the Kolmogorov theorem (see, e.g. , [13, Lect. 11] or [15,Sec. 2]) in dimension two with b = 0, condition (1.3) is equivalent to h b ih b i / ∈ Q . Here, we present a new approach which holds both in the non-periodic framework and inany dimension with a suitable vector field b ε . The ergodic assumption (1.3) together with b = 0is now replaced by the existence of a sequence w ε in C ( R N ) and q ∈ (1 , ∞ ) such that0 < b ε · ∇ w ε → θ > L q loc ( R N ) , (1.4)which is equivalent in the periodic case to the existence of a periodic gradient ∇ w satisfying b · ∇ w = 1 in R N . (1.5)Moreover, the invariant measure σ of the periodic case is replaced by a sequence σ ε satisfying0 < c − < σ ε < c for some constant c >
1, and (see Remark 2.1 for an equivalent expression)div ( σ ε b ε ) = 0 if N = 2 and σ ε b ε = ∇ w ε × · · · × ∇ w Nε if N ≥ . (1.6)The case where σ ε b ε is only divergence free in dimension N ≥ b ε is naturally associated with the vector field W ε := ( w ε , . . . , w Nε ) which induces aglobal rectification of the field b ε in the direction e (see Remark 2.1). Then, assuming in addi-tion to (1.4), (1.6) that W ε is uniformly proper (see condition (2.1) below) and converges bothin C ( R N ) N and weakly in W ,N loc ( R N ) N , we prove (see Theorem 2.2) that up to a subsequence σ ε u ε converges weakly in L ∞ (0 , T ; L p ( R N )) to a solution v to the transport equation ∂v∂t − ξ · ∇ x (cid:18) vσ (cid:19) = 0 in (0 , T ) × R N v (0 , · ) = v in R N , (1.7)where σ is the weak- ∗ limit of σ ε in L ∞ ( R N ), ξ is the weak limit of σ ε b ε in L N ′ loc ( R N ) N and v the weak limit of σ ε u ε in L p ( R N ). Moreover, if σ ε converges strongly to σ in L ( R N ) (seeRemark 2.4) or u ε converges strongly to u in L p loc ( R N ), then up to a subsequence u ε convergesweakly in L ∞ (0 , T ; L p ( R N )) to a solution u to the transport equation ∂u∂t − ξ σ · ∇ x u = 0 in (0 , T ) × R N u (0 , · ) = u in R N . (1.8)2he convergence of u ε also turns out to be strong in L ∞ (0 , T ; L ( R N )) if u ε converges stronglyto u in L p loc ( R N ) with p > σ ε det( DW ε ) which is connected to the vector field b ε by (1.6).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 dealsby three applications of Theorem 2.2. In Section 3.1 we study the case of a diffeomorphism W ε on R such that det( DW ε ) is compact in L p loc ( R ) for some q ∈ (1 , ∞ ). In Section 3.2 weextend the periodic case of [1, 8, 6, 15] with b ε ( x ) = b ( x/ε ) and the periodic case of [2, Sec. 4]on the asymptotic of the flow associated with b , in the light of Theorem 2.2 with a periodicallyoscillating function σ ε ( x ) = σ ( x/ε ) (see Proposition 3.1). In Section 3.3 we consider the case ofa diffeomorphism W ε which agrees at a fixed time t to a flow X ε ( t, · ) associated with a suitablevector field a ε (see Proposition 3.2). In this general setting assumption (1.4) holds simply whendiv a ε is compact in L q loc ( R N ) for some q ∈ (1 , ∞ ). Notations • ( e , . . . , e N ) denotes the canonical basis of R N . • · denotes the scalar product in R N and | · | the associated norm. • I N is the unit matrix of R N × N , and R ⊥ is the clockwise 90 ◦ rotation matrix in R × . • For M ∈ R N × N , M T denotes the transpose of M . • Y N := [0 , N , and h f i denotes the average-value of a function f ∈ L ( Y N ). • For any open set Ω of R N and k ∈ N ∪ {∞} , C kc (Ω), respectively C kb (Ω), denotes the spaceof the C k functions with compact support in Ω, respectively bounded in Ω. • For k ∈ N ∪ {∞} and p ∈ [1 , ∞ ], C k♯ ( Y N ) denotes the space of the Y N -periodic functionsin C k ( R N ), and L p♯ ( Y N ) denotes the space of the Y N -periodic functions in L p loc ( R N ) ( i.e. in L p ( K ) for any compact set K of R N ). • For u ∈ L ( R N ) and U = ( U j ) ≤ j ≤ d ∈ L ( R N ) N . ∇ x u := ( ∂ x , . . . , ∂ x N ) and DU := (cid:2) ∂ x i U j (cid:3) ≤ i,j ≤ d . • For ξ , . . . , ξ N in R N , the cross product ξ × · · · × ξ N is defined by ξ · ( ξ × · · · × ξ N ) = det ( ξ , ξ , . . . , ξ N ) for ξ ∈ R N , (1.9)where det is the determinant with respect to the canonical basis ( e , . . . , e N ). • o ε denotes a term which tends to zero as ε → • C denotes a constant which may vary from line to line.3 The main result
Let W ε = ( w ε , . . . , w Nε ), ε >
0, be a sequence of vector fields in C ( R N ) N which is uniformlyproper , i.e. for any compact set K of R N there exists a compact set K ′ of R N satisfying W − ε ( K ) ⊂ K ′ for any small enough ε > , (2.1)and let W ∈ C ( R N ) N be such that W ε → W in C ( R N ) N and W ε ⇀ W in W ,N loc ( R N ) N . (2.2)Let b ε be a vector field in C b ( R N ) N ∩ W ,q loc ( R N ) N with bounded divergence and let σ ε be apositive function in C ( R N ) ∩ W ,q loc ( R N ) satisfying for some constant c > c − ≤ σ ε ≤ c and σ ε b ε = ( R ⊥ ∇ w ε if N = 2 ∇ w ε × · · · × ∇ w Nε if N ≥ , in R N . (2.3)Also assume that for p ∈ (1 , ∞ ) with conjugate exponent q , there exists a positive function θ in C ( R N ) such that θ ε := b ε · ∇ w ε > R N and θ ε → θ > L q loc ( R N ) . (2.4)Finally, assume: • either that there exists a constant B > | div b ε | ≤ B a.e. in R N , (2.5) • or the regularity condition b ε ∈ C b ( R N ) N , σ ε ∈ C ( R N ) and u ε ∈ C ( R N ) . (2.6) Remark 2.1.
The definition (2.3) of b ε can be also written for any dimension N ≥ N −
1) gradients ∇ w ε , . . . , ∇ w Nε satisfying ∀ ξ ∈ R N , σ ε b ε · ξ = det (cid:0) ξ, ∇ w ε , . . . , ∇ w Nε (cid:1) . (2.7)In dimension N ≥ ∇ w ε × · · · × ∇ w Nε (see (1.9)). In dimension N = 2 this means exactly that σ ε b ε = R ⊥ ∇ w ε , which is equivalent todiv ( σ ε b ε ) = 0 in R . (2.8)However, in dimension N ≥ σ ε b ε divergence free.The definition (2.3) of b ε and the definition (2.4) of θ ε are equivalent to the global rectifica-tion of the field b ε by the diffeomorphism W ε DW Tε b ε = θ ε e in R N , (2.9)in the direction e with the compact range θ ε .Then, we have the following homogenization result.4 heorem 2.2. Let
T > , let p ∈ (1 , ∞ ) and let u ε be a bounded sequence in L p ( R N ) . Assumethat conditions (2.1) to (2.4) together with (2.5) or (2.6) hold true. Let u ε be the solution tothe transport equation (1.1) and set v ε := σ ε u ε . Then, up to a subsequence v ε converges weaklyin L ∞ (0 , T ; L p ( R N )) to a solution v to the transport equation ∂v∂t − ξ · ∇ x (cid:18) vσ (cid:19) = 0 in (0 , T ) × R N v (0 , · ) = v in R N , (2.10) where ( Cof denotes the cofactors matrix) ξ = Cof ( DW ) e ∈ C ( R N ) N , (2.11) σ ε b ε ⇀ ξ in L N ′ loc ( R N ) N , σ ε ⇀ σ in L ∞ ( R N ) ∗ , σ ε u ε ⇀ v in L p ( R N ) . (2.12) Moreover, if in addition b ε ∈ W ,p/ ( p − ( R N ) N with p > and the sequence u ε converges stronglyto u ∈ L p loc ( R N ) with σ ∈ W , ∞ ( R ) and ξ ∈ L ∞ ( R N ) N ∩ W ,p/ ( p − ( R N ) N , then u ε convergesstrongly in L ∞ (0 , T ; L ( R N )) to the solution u to the transport equation ∂u∂t − ξ σ · ∇ x u = 0 in (0 , T ) × R N u (0 , · ) = u in R N . (2.13) Remark 2.3.
If in Theorem 2.2 we assume in addition that σ is in W , ∞ ( R N ) and ξ belongsto L ∞ ( R N ) N ∩ W ,q loc ( R N ) N , then by virtue of [5, Corollary II.1] there exists a unique solution v to the transport equation (2.10). Remark 2.4.
In addition to the conditions (2.1) to (2.4) assume that σ ε converges strongly in L ( R N ) to σ ∈ W ,q loc ( R N ). Then, we have v = σ u and v = σ u where u is the weak limitof u ε in L p ( R N ), which implies that equation (2.10) is equivalent to equation (2.13). Therefore, u ε converges weakly in L ∞ (0 , T ; L p ( R N )) to a solution u to the transport equation (2.13).To prove Theorem 2.2 we need the following L p -estimate. Lemma 2.5.
Let b ε ∈ L ∞ ( R N ) N ∩ W ,q loc ( 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 > such that for any u ε ∈ L p ( R N ) with p ∈ [1 , ∞ ) , the solution u ε to equation (1.1) satisfies the estimate k u ε ( t, · ) k L p ( R N ) ≤ C k u ε k L p ( R N ) for a.e. t ∈ (0 , T ) , (2.14) Proof of Theorem 2.2.
First of all, note that by (2.3) and (2.4) we havedet( DW ε ) = σ ε θ ε > R N . (2.15)This combined with property (2.1) and Hadamard-Caccioppoli’s theorem [3] (or Hadamard-L´evy’s theorem) implies that W ε is a C -diffeomorphism on R N . Moreover, since by (2.15)det( DW ε ) is positive and by (2.2) W ε converges weakly in W ,N loc ( R N ) N , by virtue of M¨uller’stheorem [9] det( DW ε ) weakly converges to det( DW ) in L ( R N ). Hence, passing to the limit5n (2.15) together with the strong convergence (2.4) of θ ε , the weak convergence (2.12) of σ ε and the boundedness (2.3) of σ ε we get thatdet( DW ) = σ θ ≥ c − θ > R N , (2.16)which taking into account the continuity of DW and θ implies that det( DW ) > R N .Moreover, again by the uniform character of (2.1) W is a proper mapping. Therefore, W isalso a C -diffeomorphism on R N .The weak formulation of equation (1.1) is that for any function φ ∈ C c ([0 , T ) × R N ), ˆ T ˆ R N u ε ∂φ∂t dx dt + ˆ R N u ε ( x ) φ (0 , x ) dx = ˆ T ˆ R N u ε div ( φ b ε ) dx dt. (2.17)Using a density argument with σ ε ∈ W ,q loc ( R N ), we can replace the test function φ by σ ε ϕ forany ϕ ∈ C c ([0 , T ) × R N ). This combined with the divergence free of σ ε b ε leads us to the newformulation ˆ T ˆ R N σ ε u ε ∂ϕ∂t dx dt + ˆ R N σ ε ( x ) u ε ( x ) ϕ (0 , x ) dx = ˆ T ˆ R N u ε σ ε b ε · ∇ x ϕ dx dt. (2.18)We pass easily to the limit in the left hand-side of (2.18). The delicate point comes from theright-hand side of (2.18).By the L p -estimate (2.14) of Lemma 2.5 combined with the uniform boundedness of σ ε in(2.3) there exists a subsequence, still denoted by ε , such that v ε = σ ε u ε converges weakly tosome function v in L ∞ (0 , T ; L p ( R N )).Let ψ ∈ C c ([0 , T ) × R N ) the support of which is contained in some compact set [ t , t ] × K of [0 , T ) × R N , and define ϕ ε ( t, x ) := ψ ( t, W ε ( x )) for ( t, x ) ∈ (0 , T ) × R N , (2.19)so that ∇ x ϕ ε ( t, x ) := DW ε ( x ) ∇ y ψ ( t, y ). Hence, making the change of variables y = W ε ( x ) andusing (2.9) we deduce that ˆ T ˆ R N v ε ( t, x ) b ε ( x ) · ∇ x ϕ ε ( t, x ) dx dt = ˆ T ˆ W − ε ( K ) v ε ( t, x ) b ε ( x ) · ∇ x ϕ ε ( t, x ) dx dt = ˆ T ˆ K v ε ( t, W − ε ( y )) θ ε ( W − ε ( y )) e · ∇ y ψ ( t, y ) det( DW − ε )( y ) dy dt. (2.20)First, using successively the H¨older inequality combined with the L p -estimate (2.14), the inclu-sion (2.1) and the L q -strong convergence (2.4) of θ ε , we have (cid:12)(cid:12)(cid:12)(cid:12) ˆ T ˆ K v ε ( t, W − ε ( y )) ( θ ε − θ )( W − ε ( y )) e · ∇ y ψ ( t, y ) det( DW − ε )( y ) dy dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ψ ˆ T (cid:18) ˆ K (cid:12)(cid:12) v ε ( t, W − ε ( y )) (cid:12)(cid:12) p det( DW − ε )( y ) dy (cid:19) p (cid:18) ˆ K (cid:12)(cid:12) ( θ ε − θ )( W − ε ( y )) (cid:12)(cid:12) q det( DW − ε )( y ) dy (cid:19) p dt ≤ C ψ ˆ T k v ε ( t, · ) k L p ( K ′ ) k θ ε − θ k L q ( K ′ ) dt = o ε , which implies that ˆ T ˆ K v ε ( t, W − ε ( y )) θ ε ( W − ε ( y )) e · ∇ y ψ ( t, y ) det( DW − ε )( y ) dy dt ˆ T ˆ K v ε ( t, W − ε ( y )) θ ( W − ε ( y )) e · ∇ y ψ ( t, y ) det( DW − ε )( y ) dy dt + o ε . ∇ y ψ ( t, W ε ( x )) → ∇ y ψ ( t, W ( x )) in C ([0 , T ] × R N ) . Then, making the inverse change of variables x = W − ε ( y ) together with (2.1) and using theweak convergence of v ε to v in L ∞ (0 , T ; L p ( R N )), we have ˆ T ˆ K v ε ( t, W − ε ( y )) θ ( W − ε ( y )) e · ∇ y ψ ( t, y ) det( DW − ε )( y ) dy dt = ˆ T ˆ K ′ v ε ( t, x ) θ ( x ) e · ∇ y ψ ( t, W ε ( x )) dx dt = ˆ T ˆ K ′ v ( t, x ) θ ( x ) e · ∇ y ψ ( t, W ( x )) dx dt + o ε . Let ϕ ∈ C c ([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 ∇ x ϕ ( t, x ) := DW ( x ) ∇ y ψ ( t, y ). Therefore, passing to the limit in (2.20) we obtain that ˆ T ˆ R N v ε ( t, x ) b ε ( x ) · ∇ x ϕ ε ( t, x ) dx dt = ˆ T ˆ R N v ( t, x ) θ ( x ) (cid:0) DW ( x ) T (cid:1) − e · ∇ x ϕ ( t, x ) dx dt + o ε . (2.21)On the other hand, using (2.9), (2.3) and the Murat-Tartar div-curl lemma in L N ′ - L N (see, e.g. , [10, Th´eor`eme 2]) with convergences (2.2), (2.4), (2.12) we get that DW Tε ( σ ε b ε ) = σ ε θ ε e ⇀ DW T ξ = σ θ e weakly in L ( R N ) . (2.22)This combined with (2.16) yields equality (2.11). Convergences (2.21) and (2.22) imply that ˆ T ˆ R N v ε b ε · ∇ x ϕ ε dx dt −→ ε → ˆ T ˆ R N vσ ξ · ∇ x ϕ dx dt. Finally, passing to the limit in formula (2.18) with ϕ ε , it follows that for any ϕ ∈ C c ([0 , T ) × R N ), ˆ T ˆ R N v ∂ϕ∂t dx dt + ˆ R N v ( x ) ϕ (0 , x ) dx = ˆ T ˆ R N vσ ξ · ∇ x ϕ dx dt, which taking into account that ξ is divergence free yields the weak formulation of the desiredlimit equation (2.10). This concludes the proof of the first part of Theorem 2.2.Now, assume in addition that b ε ∈ W ,p/ ( p − ( R N ) N with p > u ε converges strongly to u in L p ( R N ) with σ ∈ W , ∞ ( R N ) and ξ ∈ L ∞ ( R N ) N ∩ W ,p/ ( p − ( R N ) N . By [5, Theorem II.3and Corollary II.1] u ε is the unique solution to the equation (1.1) with initial condition ( u ε ) ,or equivalently, for any φ ∈ C c ([0 , T ) × R N ), ˆ T ˆ R N u ε ∂φ∂t dx dt + ˆ R N ( u ε ) ( x ) φ (0 , x ) dx = ˆ T ˆ R N u ε div ( φ b ε ) dx dt, Replacing u ε by u ε in the first part of Theorem 2.2 and using the strong convergence of u ε weget that the sequence σ ε u ε converges weakly in L ∞ (0 , T ; L p/ ( R N )) to the solution w to thetransport equation ∂w∂t − ξ · ∇ x (cid:18) wσ (cid:19) = ∂w∂t − ξ σ · ∇ x w + ξ · ∇ σ σ w = 0 in (0 , T ) × R N w (0 , · ) = σ ( u ) in R N . (2.23)7ote that by virtue of [5, Corollary II.1] the solution w to equation (2.23) is unique due to theregularities σ ∈ W , ∞ ( R N ), ξ ∈ L ∞ ( R N ) N ∩ W ,p/ ( p − ( R N ) N with divergence free. Moreover,again by [5, Theorem II.3 and Corollary II.1] v is the unique solution to the equation inducedby (2.10) ∂ ( v ) ∂t − ξ σ · ∇ x ( v ) + 2 ξ · ∇ σ σ v = 0 in (0 , T ) × R N v (0 , · ) = ( σ u ) in R N , or equivalently, for any φ ∈ C c ([0 , T ) × R N ), ˆ T ˆ R N v ∂φ∂t dx dt + ˆ R N ( σ u ) ( x ) φ (0 , x ) dx = ˆ T ˆ R N v div (cid:18) φ ξ σ (cid:19) dx dt + ˆ T ˆ R N v ξ · ∇ σ σ φ dx dt. Replacing the test function φ by ϕ/σ by a density argument, it follows that for any function ϕ ∈ C c ([0 , T ) × R N ), ˆ T ˆ R N v σ ∂ϕ∂t dx dt + ˆ R N σ ( x ) ( u ) ( x ) ϕ (0 , x ) dx = ˆ T ˆ R N v div (cid:18) ϕ ξ σ (cid:19) dx dt + ˆ T ˆ R N v ξ · ∇ σ σ ϕ dx dt = ˆ T ˆ R N v σ div (cid:18) ϕ ξ σ (cid:19) dx dt + ˆ T ˆ R N v σ ξ · ∇ σ σ ϕ dx dt, which shows that v /σ is also a solution to equation (2.23). By uniqueness we thus get that w = v /σ . Similarly, the solution u to equation (2.13) agrees with v/σ . Finally, using thesetwo equalities we have for any compact set K of R N , ˆ T ˆ K σ ε ( u ε − u ) dx dt = ˆ T ˆ K ( σ ε u ε − σ ε u ε u + σ ε u ) dx dt −→ ε → ˆ T ˆ K ( w − v u + σ u ) dx dt = 0 , which concludes the proof of Theorem 2.2. (cid:3) Proof of Lemma 2.5.
If the uniform boundedness (2.5) of div b ε is satisfied, then using theestimate (17) of [5, Proposition II.1] for the solution to the regularized equation of (1.1) andthe lower semi-continuity of the L p -norm ( p < ∞ ) we get estimate (2.14).Otherwise, assume that conditions (2.3) and (2.6) hold true. Using the regularity of thedata the proof is based on an explicit expression of the solution to equation (1.1) from the flow Y ε associated with the vector field b ε by ∂Y ε ( t, x ) ∂t = b ε ( Y ε ( t, x )) , t ∈ R Y ε (0 , x ) = x ∈ R d . (2.24)Let u ε be a function in C ( R N ) N ∩ L p ( R N ). It is classical that the regular solution u ε to thetransport equation (1.1) is given by u ε ( t, x ) = u ε ( Y ε ( t, x )) for ( t, x ) ∈ [0 , T ] × R N . (2.25)8et t ∈ [0 , T ]. Making the change of variables combined with the semi-group property of theflow y = Y ε ( t, x ) ⇔ x = Y ε ( − t, y ) , we get that ˆ R N (cid:12)(cid:12) u ε ( Y ε ( t, x )) (cid:12)(cid:12) p dx = ˆ R N (cid:12)(cid:12) u ε ( y ) (cid:12)(cid:12) p (cid:12)(cid:12) det( D y Y ε ( − t, y )) (cid:12)(cid:12) dy. (2.26)Moreover, by (2.24) and the Liouville formula we have for any ( τ, y ) ∈ R × R N ,det( D y Y ε ( τ, y )) = exp (cid:18) ˆ τ (div b ε )( Y ε ( s, y )) ds (cid:19) . However, since by (2.3) σ ε b ε is divergence free, we have ˆ τ (div b ε )( Y ε ( s, y )) ds = − ˆ τ (cid:18) ∇ σ ε · b ε σ ε (cid:19) ( Y ε ( s, y )) ds = − ˆ τ ∂∂s (cid:0) ln σ ε ( Y ε ( s, y )) (cid:1) ds = ln (cid:18) σ ε ( y ) σ ε ( Y ε ( τ, y )) (cid:19) . This combined with the boundedness of σ ε in condition (2.3) implies that ∀ ( τ, y ) ∈ R × R N , < det( D y Y ε ( τ, y )) = σ ε ( y ) σ ε ( Y ε ( τ, y )) ≤ c . Hence, we deduce from (2.26) that ˆ R N | u ε ( x ) | p dx = ˆ R N (cid:12)(cid:12) u ε ( Y ε ( t, x )) (cid:12)(cid:12) p dx ≤ c ˆ R N (cid:12)(cid:12) u ε ( y ) (cid:12)(cid:12) p dy, which yields the desired estimate (2.14). This concludes the proof of Lemma 2.5. (cid:3) The purpose of this section is to illustrate the homogenization of the transport equation (1.1)by various oscillating fields b ε which satisfy the assumptions of Theorem 2.2. It means givingexamples of diffeomorphism W ε on R N satisfying the rectification (2.9) of the vector field b ε where the sequence θ ε > L q loc ( R N ) for some q ∈ (1 , ∞ ). Let α ε , α ∈ C ( R ) be such that for some constant c > α ε → α in C ( R ) , α ′ ε ≥ c in R , α ′ ε → α ′ in L ( R ) , (3.1)and let β ε , β ∈ C ( R ) be such that for some constant C > β ε → β in C ( R ) , | β ε | ≤ C in R , β ′ ε is bounded in L ∞ loc ( R ) , (3.2)Consider the vector field W ε ∈ C ( R N ) N defined by W ε ( x ) := (cid:0) α ε ( x ) exp (cid:8) β ε ( α ε ( x ) α ε ( x )) (cid:9) , α ε ( x ) exp (cid:8) − β ε ( α ε ( x ) α ε ( x )) (cid:9)(cid:1) , x ∈ R , (3.3)9hich is based on the characterization of the holomorphic mappings on C with constantJacobian [11]. The gradient of W ε is given by ∇ w ε ( x ) = exp (cid:8) β ε ( α ε ( x ) α ε ( x )) (cid:9) α ′ ε ( x ) (cid:0) α ε ( x ) α ε ( x ) β ′ ε ( α ε ( x ) α ε ( x )) (cid:1) α ′ ε ( x ) α ε ( x ) β ′ ε ( α ε ( x ) α ε ( x ) (cid:1) ! ∇ w ε ( x ) = exp (cid:8) − β ε ( α ε ( x ) α ε ( x )) (cid:9) − α ′ ε ( x ) α ε ( x ) β ′ ε ( α ε ( x ) α ε ( x )) α ′ ε ( x ) (cid:0) − α ε ( x ) α ε ( x ) β ′ ε ( α ε ( x ) α ε ( x )) (cid:1)! . Also define b ε := R ⊥ ∇ w ε and σ ε := 1, so that conditions (2.3) and (2.5) are fulfilled.By (3.1) and (3.2) we have W ε ( x ) → W ( x ) := (cid:0) α ( x ) exp (cid:8) β ( α ( x ) α ( x )) (cid:9) , α ( x ) exp (cid:8) − β ( α ( x ) α ( x )) (cid:9)(cid:1) in C ( R ) ,W ε ⇀ W in H ( R ) , so that conditions (2.2) is satisfied, and b ε · ∇ w ε ( x ) = det( DW ε )( x ) = α ′ ε ( x ) α ′ ε ( x ) → α ′ ( x ) α ′ ( x ) in L ( R ) , (3.4)so that condition (2.4) is satisfied with p = 2. Moreover, since by (3.1) ∀ t ∈ R , | α ε ( t ) − α ε (0) | ≥ c | t | , the sequence α ε (0) converges, and β ε is uniformly bounded in R , condition (2.1) holds for W ε .Note that the oscillations of the drift b ε in equation (1.1) are only due to the oscillations ofthe sequence β ′ ε which does not appear in the convergence (3.4) of the Jacobian. This section extends the periodic framework of [1, 8, 6, 15] and [2, Corollary 4.4].Let W = ( w , . . . , w N ) be a vector field in C ( R N ) N , and let M be a matrix in R N × N suchthat (cid:0) x W ( x ) − M x (cid:1) ∈ C ♯ ( Y N ) N and σ := det( DW ) > R N . (3.5)Consider the periodic vector field b ∈ C ♯ ( Y N ) N defined by σ b := ( R ⊥ ∇ w if N = 2 ∇ w × · · · × ∇ w N if N ≥ . (3.6)We have the following result. Proposition 3.1.
Let u ε ∈ C ( R N ) be a bounded sequence in L p ( R N ) with p ∈ (1 , ∞ ) . Assumethat conditions (3.5) and (3.6) hold true. Then, the vector fields W ε and b ε defined by W ε ( x ) := ε W (cid:16) xε (cid:17) and b ε ( x ) := b (cid:16) xε (cid:17) for x ∈ R N , (3.7) satisfy the assumptions of Theorem 2.2.Moreover, for any sequence u ε in L p ( R N ) such that σ ( x/ε ) u ε converges weakly to v in L p ( R N ) ,the solution u ε to equation (1.1) is such that σ ( x/ε ) u ε converges weakly in L ∞ (0 , T ; L p ( R N )) to the solution v to the equation (2.10) with σ = h σ i and ξ = h σ b i . roof of Proposition 3.1. By the quasi-affinity of the determinant (see, e.g. , [4, Sec. 4.3.2])and by (3.5) we have det( M ) = det h DW i = (cid:10) det( DW ) (cid:11) > , and by (3.7) there exists a constant C > ∀ x ∈ R N , | W ε ( x ) − M x | ≤
Cε, (3.8)which implies condition (2.1). Moreover, estimate (3.8) and the uniform bounded of DW ε implyeasily the convergences (2.2) with the limit W ( x ) := M x .On the other hand, the definitions (3.5) of W , σ and the definition (3.6) of b show clearlythat condition (2.3) and the regularity (2.6) hold true. Moreover, we have θ := b · ∇ w = det( DW ) σ = 1 in R N , which implies (2.4) since θ ε ( x ) = θ ( x/ε ) = 1.Moreover, let u ε be a sequence in L p ( R N ) such that σ ( x/ε ) u ε converges weakly to v in L p ( R N ). By virtue of Theorem 2.2 combined with Remark 2.3 and using the weak limit ofa periodically oscillating sequence, the sequence σ ( x/ε ) u ε converges weakly in L p ( R N ) to thesolution v to the equation (2.10) with σ = h σ i and ξ = h σ b i . The proof of Proposition 3.1 isnow complete. (cid:3) In this section we construct a sequence W ε from a dynamic flow associated with a suitable butquite general sequence of vector fields a ε .Let a ε , a be vector fields in C ( R N ) N such that a ε → a in C ( R N ) N , a ε ⇀ a in W , ∞ loc ( R N ) N ∗ , (3.9)and for some constant A > | a ε | + | div a ε | ≤ A in R N . (3.10)Also assume that there exists q ∈ (1 , ∞ ) such thatdiv a ε → div a in L q loc ( R N ) . (3.11)Consider the dynamic flow X ε associated with the vector field a ε defined by ∂X ε ( t, x ) ∂t = a ε ( X ε ( t, x )) , t ∈ R X ε (0 , x ) = x ∈ R d , (3.12)and let X be the limit flow associated with the limit vector field a .Then, from any sequence of flows X ε we may derive a general sequence of vector fields b ε inducing the homogenization of the transport equation (1.1). Proposition 3.2.
Let u ε be a bounded sequence in L p ( R N ) with p ∈ (1 , ∞ ) . Assume thatconditions (3.9) , (3.10) , (3.11) hold true. For a fixed t > , define the vector field W ε := X ε ( t, · ) from R N into R N , and the vector field b ε by (2.3) with σ ε = 1 . Then, the sequences W ε and b ε satisfy the assumptions of Theorem 2.2.Moreover, for any sequence u ε converging weakly to u in L p ( R N ) , the solution u ε to equa-tion (1.1) converges weakly in L ∞ (0 , T ; L p ( R N )) to a solution u to the equation (2.13) where σ = 1 and ξ = Cof ( D x X ( t, x )) e . emark 3.3. There is a strong correspondance between the conditions (3.9)-(3.10) and (3.11)satisfied by the vector field a ε , and respectively the conditions (2.2) and (2.4) satisfied by thevector fields W ε and b ε . Proof of Proposition 3.2.
First of all, conditions (2.3) and (2.5) are straightforward, since σ ε = 1 and b ε is divergence free. Fix T >
0. By (3.10) we have ∀ t ∈ [0 , T ] , ∀ x ∈ R N , | X ε ( t, x ) − x | ≤ A T, (3.13)so that the uniform property (2.1) is satisfied by W ε .Let K be a compact set of R N . Again by (3.13) there exists a compact set K ′ of R N suchthat (cid:8) X ε ( t, x ) , ( t, x ) ∈ [0 , T ] × K (cid:9) ⊂ K ′ . (3.14)Let δ >
0. Since a ε converges uniformly to a in K ′ and a ∈ C ( R N ) is k -Lipschitz in K ′ forsome k >
0, we have for any small enough ε > t ∈ [0 , T ], for any x, y ∈ K , (cid:12)(cid:12) X ε ( t, x ) − X ε ( t, y ) (cid:12)(cid:12) ≤ | x − y | + ˆ t (cid:12)(cid:12) a ε ( X ε ( s, x )) − a ε ( X ε ( s, y )) (cid:12)(cid:12) ds ≤ δ + | x − y | + k ˆ t (cid:12)(cid:12) X ε ( s, x ) − X ε ( s, y ) (cid:12)(cid:12) ds. Hence, by Gronwall’s inequality (see, e.g. , [7, Sec. 17.3]) we get that for any small enough ε > ∀ t ∈ [0 , T ] , ∀ x, y ∈ K, | X ε ( t, x ) − X ε ( t, y ) (cid:12)(cid:12) ≤ ( δ + | x − y | ) e kt , which by (3.10) implies that for any small enough ε > ∀ s, t ∈ [0 , T ] , ∀ x, y ∈ K, | X ε ( s, x ) − X ε ( t, y ) (cid:12)(cid:12) ≤ A | s − t | + ( δ + | x − y | ) e kt , namely X ε is uniformly equicontinuous in the compact set [0 , T ] × K . Therefore, by virtueof Ascoli’s theorem this combined with (3.14) and (3.9) implies that up to a subsequence X ε converges uniformly in [0 , T ] × K to a solution X to ∀ t ∈ [0 , T ] , ∀ x ∈ K, X ( t, x ) = x + ˆ t a ( X ( s, x )) ds, i.e. X is the flow associated with the vector field a . Since a belongs to C b ( R N ), the flow X is uniquely determined (see, e.g. , [7, Sec. 17.4]). Therefore, the whole sequence X ε convergesuniformly to X in [0 , T ] × K . Moreover, by the differentiability of the flow (see, e.g. , [7,Sec. 17.6]) we have ∀ t ∈ [0 , T ] , ∀ x ∈ K, D x X ε ( t, x ) = I N + ˆ t D x X ε ( s, x ) D x a ε ( X ε ( s, x )) ds, (3.15)which using (3.9), (3.14) and Gronwall’s inequality implies that there exists a constant c > ∀ t ∈ [0 , T ] , ∀ x ∈ K, | D x X ε ( t, x ) | ≤ | 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 constant c > ∀ t ∈ [0 , T ] , ∀ x ∈ K, c − ≤ det ( D x X ε ( t, x )) = exp (cid:18) ˆ t (div a ε )( X ε ( s, x )) ds (cid:19) ≤ c, (3.16)12hich implies the existence of a constant C > t ∈ [0 , T ] and x ∈ K , (cid:12)(cid:12) det ( D x X ε ( t, x )) − det ( D x X ( t, x )) (cid:12)(cid:12) ≤ C ˆ T | div a ε − div a | ( X ε ( s, x )) ds + C ˆ T (cid:12)(cid:12) (div a )( X ε ( s, x )) − (div a )( X ( s, x )) (cid:12)(cid:12) ds. Hence, using successively Jensen’s inequality with respect to the integral in s , Fubini’s theoremand the change of variables y = X ε ( s, x ) together with (3.14) and (3.16), it follows that thereexists a constant C > t ∈ [0 , T ], (cid:13)(cid:13) det ( D x X ε ( t, · )) − det ( D x X ( t, · )) (cid:13)(cid:13) L q ( K ) ≤ C k div a ε − div a k L q ( K ′ ) + C sup [0 ,T ] × K (cid:12)(cid:12) (div a )( X ε ) − (div a )( X ) (cid:12)(cid:12) . This combined with convergence (3.11) and the uniform convergence of X ε to X in the compactset [0 , T ] × K implies the convergence (2.4) of θ ε = det( D x X ε ( t, · )).Finally, let u ε be a sequence in L p ( R N ) converging weakly to u in L p ( R N ). By virtue ofTheorem 2.2 combined with Remark 2.4 and recalling that σ ε = 1, the sequence u ε convergesweakly in L p ( R N ) to a solution u to the equation (2.13) where σ = 1 and by (2.11) ξ = Cof ( D x X ( t, · )) e in R N . Proposition 3.2 is thus proved. (cid:3)
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