Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations
aa r X i v : . [ m a t h . A P ] J a n Asymptotics of ODE’s flows everywhere or almost-everywhere in thetorus: from rotation sets to homogenization of transport equations
Marc Briane & Lo¨ıc Herv´e
Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, [email protected] & [email protected]
Contents D b
73 A NSC for homogenization of the transport equation 84 Comparison between the seven conditions 13
Abstract
In this paper, we study various aspects of the ODE’s flow X solution to the equation ∂ t X ( t, x ) = b ( X ( t, x )), X (0 , x ) = x in the d -dimensional torus Y d , where b is a regular Z d -periodic vector field from R d in R d . We present an original and complete picture inany dimension of all logical connections between the following seven conditions involvingthe field b : • the everywhere asymptotics of the flow X , • the almost-everywhere asymptotics of the flow X , • the global rectification of the vector field b in Y d , • the ergodicity of the flow related to an invariant probability measure which is abso-lutely continuous with respect to Lebesgue’s measure, • the unit set condition for Herman’s rotation set C b composed of the means of b related to the invariant probability measures, • the unit set condition for the subset D b of C b composed of the means of b related tothe invariant probability measures which are absolutely continuous with respect toLebesgue’s measure, • the homogenization of the linear transport equation with oscillating data and theoscillating velocity b ( x/ε ) when b is divergence free.The main and surprising result of the paper is that the almost-everywhere asymptotics ofthe flow X and the unit set condition for D b are equivalent when D b is assumed to be nonempty, and that the two conditions turn to be equivalent to the homogenization of thetransport equation when b is divergence free. In contrast, using an elementary approachbased on classical tools of PDE’s analysis, we extend the two-dimensional results of Oxtobyand Marchetto to any d -dimensional Stepanoff flow: this shows that the ergodicity of theflow may hold without satisfying the everywhere asymptotics of the flow. eywords: ODE’s flow, transport equation, asymptotics of the flow, rectification of fields,invariant measure, rotation set, ergodicity, homogenization
Mathematics Subject Classification:
In this paper we study various aspects of the ODE’s flow X in the torus Y d ∂X∂t ( t, x ) = b ( X ( t, x )) , t ∈ R X (0 , x ) = x ∈ R d , (1.1)where b is a Z d -periodic vector field in C ( R d ) d (denoted by b ∈ C ♯ ( Y d ) d ), which completelydetermines the flow X .First, we are interested in the asymptotics of the flow X depending on whether it can holdalmost-everywhere (a.e.), or everywhere (e.) in Y d , namely ∃ lim t →∞ X ( t, x ) t µ -a.e. x ∈ Y d or ∃ lim t →∞ X ( t, x ) t ∀ x ∈ Y d , for some probability measure µ on Y d . If the flow X is ergodic with respect to some invariantprobability measure µ , i.e. that µ agrees with its image measure µ X ( t, · ) for any t ∈ R (seeSection 1.2 below), then Birkhoff’s theorem (see, e.g. , [9, Theorem 1, Section 2, Chapter 1])ensures thatlim t →∞ X ( t, x ) t = lim t →∞ (cid:18) t Z t b ( X ( s, x )) ds (cid:19) = Z Y d b ( y ) dµ ( y ) µ -a.e. x ∈ Y d . The non empty set I b composed of the invariant probability measures for the flow X plays afondamental role in ergodic theory. Associated with the set I b , the rotations sets of [18] arestrongly connected to the asymptotic behavior of the flow. In particular, the compact convexHerman rotation set [13] defined by C b := (cid:26)Z Y d b ( y ) dµ ( y ) : µ ∈ I b (cid:27) (1.2)characterizes the everywhere asymptotics of the flow, since by [8, Proposition 2.1] we have forany ζ ∈ R d , C b = { ζ } ⇔ ∀ x ∈ Y d , lim t →∞ X ( t, x ) t = ζ . (1.3)We also consider the subset D b of C b defined by D b := (cid:26)Z Y b ( y ) σ ( y ) dy : σ ∈ L ♯ ( Y d ) and σ ( y ) dy ∈ I b (cid:27) , (1.4)which is a priori less interesting than Herman’s rotation set, since it may be empty and itis not compact in general. But surprisingly, the set D b characterizes the almost-everywhereasymptotics of the flow, which is the first result of our paper. More precisely, assuming theexistence of an a.e. positive invariant density function with respect to Lebesgue’s measure, weprove that for any ζ ∈ R d (see Theorem 2.1), D b = { ζ } ⇔ lim t →∞ X ( t, x ) t = ζ a.e. x ∈ Y d . (1.5)2n the other hand, as a natural association with flow (1.1), we consider the linear transportequation with oscillating data ∂u ε ∂t ( t, x ) − b ( x/ε ) · ∇ x u ε ( t, x ) = f ( t, x, x/ε ) in (0 , T ) × R d u ε (0 , x ) = u ( x, x/ε ) for x ∈ R d , (1.6)where f ( t, x, y ) and u ( x, y ) are suitably regular and Z d -periodic functions with respect tovariable y . In their famous paper [10] DiPerna and Lions showed the strong proximity be-tween ODE’s flows (1.1) and transport equations, in particular when the velocity has a gooddivergence. In the context of homogenization, the linear transport equation with oscillatingdata (1.6) as ε → b is divergence free and the associatedflow X is ergodic, Brenier [4] first obtained the weak convergence of the solution u ε to thetransport equation. Following this seminal work, the homogenization of the transport equationwas obtained for instance in [11, 12, 15, 24] with various conditions, but which are all based onthe ergodicity of the flow. Extending the result of [4] with ergodicity arguments, Peirone [21]proved the convergence of the solution to the two-dimensional transport equation (1.6) with f ( t, x, y ) = 0 and u ( x, y ) independent of y , under the sole assumption that b is a non vanishingfield in C ♯ ( Y ) . More recently, the homogenization of the transport equation with f ( t, x, y ) = 0and u ( x, y ) independent of y , was derived in [6] (see [7] for a non periodic framework) underthe global rectification of the vector field b , which is not an ergodic condition, i.e. the existenceof a C -diffeomorphism Ψ on Y d and of a vector ξ ∈ R d such that ∀ y ∈ Y d , ∇ Ψ( y ) b ( y ) = ξ. (1.7)This result was extended in [8] replacing the classical ergodic condition by the unit rotation setcondition C b = 1, or, equivalently, the everywhere asymptotics (1.3) of the flow.In the present paper, we prove (see Theorem 3.2) that the homogenization of transportequation (1.6) with a divergence free velocity field, holds if, and only if, one of the equivalentconditions of (1.5) is satisfied. It is a quite new result beyond all the former results basedon the sufficient conditions induced either by the ergodic condition, or by the unit Herman’srotation set condition. The proof of this result which is partly based on two-scale convergence[19, 1], clearly shows (see Remark 3.1) the difference between the ergodic approach of [15], andthe present approach through the unit set condition (1.5) which turns out to be optimal.Therefore, we establish strong connections between the three following a priori foreignnotions: the oscillations in the transport equation (1.6), the means of b only related to theinvariant measures for the flow X which are absolutely continuous with respect to Lebesgue’smeasure, and finally the almost-everywhere asymptotics of X . More generally, owing to thisnew material we do build the complete array of all logical connections between the followingseven conditions (see Theorem 4.1 and Figure 1 below): • the global rectification (1.7) of the vector field b , • the ergodicity of the flow X (1.1) related to an invariant probability measure which isabsolutely continuous with respect to Lebesgue’s measure, • the everywhere asymptotics of the flow X in (1.3), • the almost-everywhere asymptotics of the flow X in (1.5),3 the unit set condition for Herman’s rotation set C b (1.2), • the unit set condition for D b (1.4), • the homogenization of the transport equation (1.6) when b is divergence free in R d .In addition, the following pairs of conditions cannot be compared in general:- the global rectification of b and the ergodicity of X ,- the ergodicity of X and the everywhere asymptotics of X ,- the ergodicity of X and the unit set condition for C b .The proof of the three last items involves the Stepanoff flow [23] (see Example 4.1) in whichthe vector field b has a non empty finite zero set, and is parallel to a fixed direction ξ ∈ R d with incommensurable coordinates. Using a purely ergodic approach, Oxtoby [20] and laterMarchetto [17] proved that any two-dimensional flow homeomorphic to a Stepanoff flow admitsa unique invariant probability measure µ for the flow which does not load the zero set of b ,that µ is absolutely continuous with respect to Lebesgue’s measure on Y , and finally that theflow is ergodic with respect to µ . Moreover, the set D b is a unit set, but the rotation set C b is a closed line set of R , possibly not reduced to a unit set. We extend (see Proposition 4.1)the two-dimensional results of [20, 17] on the Stepanoff flow to any dimension d ≥
2, thanks toa new and elementary approach based on classical tools of PDE’s analysis. Finally, owing toanother two-dimensional flow (see Example 4.2 and Proposition 4.4) we obtain that the set D b may be either empty or a singleton, while the rotation set C b is a closed line set of R possiblynot reduced to a singleton. • ( e , . . . , e d ) denotes the canonical basis of R d . • “ · ” denotes the scalar product and | · | the euclidian norm in R d . • Y d , d ≥
1, denotes the d -dimensional torus R d / Z d , which is identified to the unit cube[0 , d in R d . • C kc ( R d ), k ∈ N ∪ {∞} , denotes the space of the real-valued functions in C k ( R d ) withcompact support in R d . • C k♯ ( Y d ), k ∈ N ∪ {∞} , denotes the space of the real-valued functions f ∈ C k ( R d ) whichare Z d -periodic, i.e. ∀ k ∈ Z d , ∀ x ∈ R d , f ( x + k ) = f ( x ) . (1.8) • The abbreviations “a.e.” for almost everywhere, and “e.” for everywhere will be usedthroughout the paper. The simple mention “a.e.” refers to the Lebesgue measure on R d . • dx or dy denotes the Lebesgue measure on R d . • For a Borel measure µ on Y d , extended by Z d -periodicity to a Borel measure ˜ µ on R d (seedefinition (1.21) below), a ˜ µ -measurable function f : R d → R is said to be Z d -periodic˜ µ -a.e. in R d , if ∀ k ∈ Z d , f ( · + k ) = f ( · ) ˜ µ -a.e. on R d . (1.9) • For a Borel measure µ on Y d , L p♯ ( Y d , µ ), p ≥
1, denotes the space of the µ -measurablefunctions f : Y d → R such that R Y d | f ( x ) | p dµ ( x ) < ∞ .4 L p♯ ( Y d ), p ≥
1, simply denotes the space of the Lebesgue measurable functions f in L p loc ( R d ), which are Z d -periodic dx -a.e. in R d . • M loc ( R d ) denotes the space of the non negative Borel measures on R d , which are finiteon any compact set of R d . • M ♯ ( Y d ) denotes the space of the non negative Radon measures on Y d , and M p ( Y d ) denotesthe space of the probability measures on Y d . • D ′ ( R d ) denotes the space of the distributions on R d . • For a Borel measure µ on Y d and for f ∈ L ♯ ( Y d , µ ), f µ denotes the µ -mean of f on Y d f µ := Z Y d f ( y ) dµ ( y ) , (1.10)which is simply denoted by f when µ is Lebesgue’s measure. The same notation is usedfor a vector-valued function in L ♯ ( Y d , µ ) d . • The notation I b in (1.18) will be used throughout the paper. Let b : R d → R d be a vector-valued function in C ♯ ( Y d ) d . Consider the dynamical system ∂X∂t ( t, x ) = b ( X ( t, x )) , t ∈ R X (0 , x ) = x ∈ R d . (1.11)The solution X ( · , x ) to (1.11) which is known to be unique (see, e.g. , [14, Section 17.4]) inducesthe dynamic flow X associated with the vector field b , defined by X : R × R d → R d ( t, x ) X ( t, x ) , (1.12)which satisfies the semi-group property ∀ s, t ∈ R , ∀ x ∈ R d , X ( s + t, x ) = X ( s, X ( t, x )) . (1.13)The flow X is actually well defined in the torus Y d , since ∀ t ∈ R , ∀ x ∈ R d , ∀ k ∈ Z d , X ( t, x + k ) = X ( t, x ) + k. (1.14)Property (1.14) follows immediately from the uniqueness of the solution X ( · , x ) to (1.11) com-bined with the Z d -periodicity of b .A possibly signed Borel measure µ on Y d is said to be invariant for the flow X if ∀ t ∈ R , ∀ ψ ∈ C ♯ ( Y d ) , Z Y d ψ (cid:0) X ( t, y ) (cid:1) dµ ( y ) = Z Y d ψ ( y ) dµ ( y ) . (1.15)For a non negative Borel measure µ on Y d , a function f ∈ L ♯ ( Y d , µ ) is said to be invariant forthe flow X with respect to µ , if ∀ t ∈ R , f ◦ X ( t, · ) = f ( · ) µ -a.e. in Y d . (1.16)5he flow X is said to be ergodic with respect to some invariant probability measure µ , if ∀ f ∈ L ♯ ( Y d , µ ) , invariant for X w.r.t. µ, f = f µ µ -a.e. in Y d . (1.17)Then, define the set I b := (cid:8) µ ∈ M p ( Y d ) : µ invariant for the flow X (cid:9) , (1.18)where M p ( Y d ) is the set of probability measures on Y d . From the set of invariant probabilitymeasures we define the so-called Herman [13] rotation set C b := (cid:26) b µ = Z Y b ( y ) dµ ( y ) : µ ∈ I b (cid:27) , (1.19)and its subset D b := (cid:26) σ b = Z Y b ( y ) σ ( y ) dy : σ ∈ L ♯ ( Y d ) and σ ( y ) dy ∈ I b (cid:27) (1.20)which is restricted to the invariant probability measures which are absolutely continuous withrespect to Lebesgue’s measure. If there is no such invariant measure, then the set D b is empty(see Remark 4.1).We have the following characterization of an invariant measure known as Liouville’s theorem,which can also be regarded as a divergence-curl result with measures (see [8, Proposition 2.2]and [8, Remark 2.2] for further details). Proposition 1.1 (Liouville’s theorem)
Let b ∈ C ♯ ( Y d ) d , and let µ ∈ M ♯ ( Y d ) . We definethe Borel measure ˜ µ ∈ M loc ( R d ) on R d by Z R d ϕ ( x ) d ˜ µ ( x ) = Z Y d ϕ ♯ ( y ) dµ ( y ) , where ϕ ♯ ( · ) := X k ∈ Z d ϕ ( · + k ) for ϕ ∈ C c ( R d ) . (1.21) Then, the three following assertions are equivalent: ( i ) µ is invariant for the flow X , i.e. (1.15) holds, ( ii ) ˜ µ b is divergence free in the space R d , i.e.div (˜ µ b ) = 0 in D ′ ( R d ) , (1.22)( iii ) µ b is divergence free in the torus Y d , i.e. ∀ ψ ∈ C ♯ ( Y d ) , Z Y d b ( y ) · ∇ ψ ( y ) dµ ( y ) = 0 . (1.23) Remark 1.1
Since any function ψ ∈ C ∞ ♯ ( Y d ) can be represented as a function ϕ ♯ for a suitablefunction ϕ ∈ C ∞ c ( R d ) (see [5, Lemma 3.5]), we deduce that the mapping M ♯ ( Y d ) → (cid:26) ν ∈ M loc ( R d ) : ∀ ϕ ∈ C c ( R d ) , ϕ ♯ = 0 ⇒ Z R d ϕ ( x ) dν ( x ) = 0 (cid:27) µ ˜ µ is bijective. Therefore, the measure ˜ µ of (1.21) completely characterizes the measure µ . By virtue of [8, Proposition 2.1] (see also [18]) Herman’s set C b satisfies the following result. Proposition 1.2 ([8, 18])
Let b ∈ C ♯ ( Y d ) d . Then, we have for any ζ ∈ R d , C b = { ζ } ⇔ ∀ x ∈ Y d , lim t →∞ X ( t, x ) t = ζ . (1.24)6 The rotation subset D b We have the following characterization of the singleton condition satisfied by D b , which has tobe compared to the one satisfied by C b in Proposition 1.2 above. Theorem 2.1
Let b ∈ C ♯ ( Y b ) d be such that there exists an a.e. positive function σ ∈ L ♯ ( Y d ) with σ = 1 , satisfying div( σ b ) = 0 in R d . Then, the flow X associated with b satisfies for any ζ ∈ R d , D b = { ζ } ⇔ lim t →∞ X ( t, x ) t = ζ , a.e. x ∈ Y d . (2.1) Proof.
First of all, by virtue of the Birkhoff theorem applied with the invariant measure σ ( x ) dx with the a.e. positive function σ ∈ L ♯ ( Y d ), combined with the uniform boundednessof X ( t, x ) /t for t ∈ R and x ∈ Y d , there exists a function ξ ∈ L ∞ ♯ ( Y d ) which is invariant for theflow X with respect to Lebesgue’s measure, such thatlim t →∞ X ( t, x ) t = ξ ( x ) a.e. x ∈ Y d . Hence, by Lebesgue’s theorem we get that for any invariant measure σ ( x ) dx with σ ∈ L ♯ ( Y d ), Z Y d b ( x ) σ ( x ) dx = lim t →∞ t Z t (cid:18)Z Y d b ( X ( s, x )) σ ( x ) dx (cid:19) ds = Z Y d lim t →∞ (cid:18) X ( t, x ) − xt (cid:19) σ ( x ) dx = Z Y d ξ ( x ) σ ( x ) dx. (2.2)( ⇒ ) Assume that D b = { ζ } for some ζ ∈ R d . Then, we have for any invariant measure σ ( x ) dx with σ ∈ L ♯ ( Y d ), Z Y d b ( x ) σ ( x ) dx = ζ Z Y d σ ( x ) dx, which by (2.2) implies that Z Y d ( ξ ( x ) − ζ ) σ ( x ) dx = 0 . (2.3)On the other hand, since the non negative and the non positive parts ( ξ − ζ ) ± of ξ − ζ arealso invariant functions for the flow X with respect to Lebesgue’s measure, by Lemma 2.1below the measures ( ξ ( x ) − ζ ) ± σ ( x ) dx are invariant for X . Therefore, putting the measures σ ( x ) dx = ( ξ ( x ) − ζ ) ± σ ( x ) dx in equality (2.3) we get that Z Y d ( ξ ( x ) − ζ ) ( ξ ( x ) − ζ ) ± σ ( x ) dx = ± Z Y d (cid:2) ( ξ ( x ) − ζ ) ± (cid:3) σ ( x ) dx = 0 , which due to the a.e. positivity of σ , implies the right hand-side of (2.1).( ⇐ ) Conversely, we deduce immediately from (2.2) that for any invariant measure σ ( x ) dx with σ ∈ L ♯ ( Y d ), Z Y d b ( x ) σ ( x ) dx = ζ Z Y d σ ( x ) dx, which yields D b = { ζ } . (cid:3) emma 2.1 Let b ∈ C ♯ ( Y d ) d be a vector field in R d such that there exists an a.e. positivefunction σ ∈ L ♯ ( Y d ) with σ = 1 , satisfying div( σ b ) = 0 in R d . Then, a function f in L ∞ ♯ ( Y d ) is invariant for the flow X with respect to Lebesgue’s measure if, and only if, the signed measure f ( x ) σ ( x ) dx is invariant for X .Proof. First of all, for any t ∈ R , X ( t, · ) is a C -diffeomorphism on R d with reciprocal X ( − t, · ),as a consequence of the semi-group property (1.13) satisfied by the flow X . Moreover, by virtueof Liouville’s theorem the jacobian determinant of X ( t, · ) is given by ∀ t ∈ R , ∀ x ∈ Y d , J ( t, x ) := det (cid:0) ∇ x X ( t, x ) (cid:1) = exp (cid:18)Z t (div b )( X ( s, x )) ds (cid:19) . (2.4)Since by Proposition 1.1 the measure ^ σ ( x ) dx = σ ( x ) dx (due to the Z d -periodicity of σ ) isinvariant for the flow X , we have for any function ϕ ∈ C c ( R d ) and any t ∈ R , ϕ ♯ ( X ( − t, · )) = (cid:0) ϕ ( X ( − t, · )) (cid:1) ♯ by (1.14) , and Z R d ϕ ( x ) σ ( x ) dx = Z Y d ϕ ♯ ( x ) σ ( x ) dx = Z Y d ϕ ♯ ( X ( − t, x )) σ ( x ) dx = Z Y d (cid:0) ϕ ( X ( − t, x )) (cid:1) ♯ σ ( x ) dx = Z R d ϕ ( X ( − t, x )) σ ( x ) dx = |{z} x = X ( t,y ) Z R d ϕ ( y ) σ ( X ( t, y )) J ( t, y ) dy. This implies that the jacobian determinant J ( t, · ) satisfies the relation ∀ t ∈ R , σ ( X ( t, y )) J ( t, y ) = σ ( y ) a.e. y ∈ R d . (2.5)Now, let f ∈ L ∞ ♯ ( Y d ). From (2.5) we deduce that for any function ϕ ∈ C c ( R d ) and any t ∈ R , Z R d ϕ ( X ( − t, x )) f ( x ) σ ( x ) dx = |{z} x = X ( t,y ) Z R d ϕ ( y ) f ( X ( t, y )) σ ( X ( t, y )) J ( t, y ) dy = Z R d ϕ ( y ) f ( X ( t, y )) σ ( y ) dy. By virtue of Remark 1.1 combined with the Z d -periodicity of the function f , the former equalityalso reads as ∀ ψ ∈ C ♯ ( Y d ) , ∀ t ∈ R , Z Y d ψ ( X ( − t, x )) f ( x ) σ ( x ) dx = Z Y d ψ ( x ) f ( X ( t, x )) σ ( x ) dx. (2.6)Therefore, due to the a.e. positivity of σ , the function f ∈ L ∞ ♯ ( Y d ) is invariant for the flow X with respect to Lebesgue’s measure, i.e. f ( X ( · , x )) = f ( x ) a.e. x ∈ Y d , if, and only if, thesigned measure f ( x ) σ ( x ) dx is invariant for the flow X . (cid:3) First of all, recall the definition of the two-scale convergence introduced by Nguetseng [19] andAllaire [1], which is easily extended to the time dependent case.8 efinition 3.1
Let T ∈ (0 , ∞ ) . a ) A sequence u ε ( t, x ) in L ((0 , T ) × R d ) is said to two-scale converge to a function U ( t, x, y ) in L ([0 , T ] × R d ; L ♯ ( Y d )) , if we have for any function ϕ ∈ C c ([0 , T ] × R d ; C ♯ ( Y d )) withcompact support in [0 , T ] × R d × Y d , lim ε → Z (0 ,T ) × R d u ε ( t, x ) ϕ ( t, x, x/ε ) dtdx = Z (0 ,T ) × R d × Y d U ( t, x, y ) ϕ ( t, x, y ) dtdxdy, (3.1) b ) According to [1, Definition 1.4] any function ψ ( t, x, y ) ∈ C c ([0 , T ] × R d ; L ♯ ( Y d )) withcompact support in [0 , T ] × R d × Y d , is said to be an admissible function for two-scaleconvergence , if ( t, x ) ψ ( t, x, x/ε ) is Lebesgue measurable and lim ε → Z (0 ,T ) × R d ψ ( t, x, x/ε ) dtdx = Z (0 ,T ) × R d × Y d ψ ( t, x, y ) dtdxdy. (3.2)Then, we have the following two-scale convergence compactness result. Theorem 3.1 ([1], Theorem 1.2, Remark 1.5)
Any sequence u ε ( t, x ) which is bounded in L ((0 , T ) × R d ) two-scale converges, up to extract a subsequence, to some function U ( t, x, y ) in L ((0 , T ) × R d ; L ♯ ( Y d )) . Moreover, equality (3.1) holds true for any admissible function (3.2) . Let b ( y ) ∈ C ♯ ( Y d ) d be a vector field, let u ( x, y ) ∈ C c ( R d ; L ♯ ( Y d )) be an admissible functionwith compact support in R d × Y d , and let f ( t, x, y ) ∈ C c ([0 , T ] × R d ; L ∞ ♯ ( Y d )) be an admissiblefunction with compact support in [0 , T ] × R d × Y d . Consider the linear transport equation withoscillating data ∂u ε ∂t ( t, x ) − b ( x/ε ) · ∇ x u ε ( t, x ) = f ( t, x, x/ε ) in (0 , T ) × R d u ε (0 , x ) = u ( x, x/ε ) for x ∈ R d , (3.3)which by [10, Proposition II.1, Theorem II.2] has a unique solution in L ∞ ((0 , T ); L ( R d )).We have the following criterion for the homogenization of equation (3.3). Theorem 3.2
Let b be a divergence free vector field in C ♯ ( Y d ) d , and let X be the flow (1.11) associated with b . Then, we have the equivalence of the two following assertions: ( i ) There exists ζ ∈ R d such that the flow X satisfies the asymptotics lim t →∞ X ( t, x ) t = ζ , a.e. x ∈ Y d , (3.4) or, equivalently, D b = { ζ } . ( ii ) There exists ζ ∈ R d such that for any admissible functions u ( x, y ) ∈ C c ( R d ; L ♯ ( Y d )) withcompact support in [0 , T ] × R d , and f ( t, x, y ) ∈ C c ([0 , T ] × R d ; L ∞ ♯ ( Y d )) with compactsupport in [0 , T ] × R d × Y d , the solution u ε to (3.3) converges weakly in L ∞ ((0 , T ); L ( R d )) to the solution u ( t, x ) to the transport equation ∂u∂t ( t, x ) − ζ · ∇ x u ( t, x ) = f ( t, x, · ) in (0 , T ) × R d u (0 , x ) = u ( x, · ) for x ∈ R d . (3.5)9 oreover, in both cases we have ζ = b .Proof of Theorem 3.2. ( i ) ⇒ ( ii ) . First of all, note that, since b is divergence free in R d , by Proposition 1.1 Lebesgue’smeasure is an invariant probability measure for the flow X associated with b , which impliesthat b ∈ D b = { ζ } and ζ = b .Now, let u ( x, y ) ∈ C c ( R d ; L ♯ ( Y d )) be an admissible function with compact support in[0 , T ] × R d , and let f ( t, x, y ) ∈ C c ([0 , T ] × R d ; L ∞ ♯ ( Y d )) be an admissible function whose supportis contained in [0 , T ] × K , K being a compact set of R d .Denote b ε ( x ) := b ( x/ε ) which is divergence free in R d , and denote f ε ( t, x ) := f ( t, x, x/ε )which is uniformly bounded in [0 , T ] × R d and is compactly supported in [0 , T ] × K . Formally,multiplying (3.3) by u ε ( t, x ), integrating by parts over R d and using Cauchy-Schwarz inequality,we get that for any t ∈ (0 , T ),12 ddt (cid:18)Z R d u ε ( t, x ) dx (cid:19) = 12 ddt (cid:18)Z R d u ε ( t, x ) dx (cid:19) − Z R d div( b ε )( x ) u ε ( t, x ) dx = Z K f ε ( t, x ) u ε ( t, x ) dx ≤ C f (cid:18)Z R d u ε ( t, x ) dx (cid:19) / , where C f is a non negative constant only depending on f . This can be justified following theproof of [10, Proposition II.1]. Hence, we deduce the estimate k u ε ( t, · ) k L ( R d ) ≤ k u ε (0 , · ) k L ( R d ) + C f T a.e. t ∈ (0 , T ) . (3.6)Therefore, estimate (3.6) combined with (recall that the admissible function ψ ( t, x, y ) = u ( x, y )satisfies (3.2)) lim ε → k u ε (0 , · ) k L ( R d ) = k u ( x, y )) k L ( R d × Y d ) , implies that the sequence u ε is bounded in L ∞ ((0 , T ); L ( R d )). Then, up to a subsequence, u ε ( t, x ) two-scale converges to some function U ( t, x, y ) ∈ L ([0 , T ] × R d ; L ♯ ( Y d )), and u ε ( t, x )converges weakly in L ((0 , T ) × R d ) to the mean u ( t, x ) := U ( t, x, · ) = Z Y d U ( t, x, y ) dy for a.e. ( t, x ) ∈ (0 , T ) × R d . (3.7)Next, we follow the two-scale procedure of the proof of [15, Theorem 2.1]. Putting the testfunction ϕ ( t, x ) ∈ C c ([0 , T ) × R d ) in the weak formulation of (3.3), and integrating by partswe have − Z (0 ,T ) × R d ∂ϕ∂t ( t, x ) u ε ( t, x ) dtdx − Z R d ϕ (0 , x ) u ( x, x/ε ) dx + Z (0 ,T ) × R d b ( x/ε ) · ∇ x ϕ ( t, x ) u ε ( t, x ) dtdx = Z R d ϕ ( t, x ) f ( t, x, x/ε ) dtdx. Then, passing to the two-scale limit and using that u ( x, y ) and f ( t, x, y ) are admissible func-tions for two-scale convergence, we get that − Z (0 ,T ) × R d × Y d ∂ϕ∂t ( t, x ) U ( t, x, y ) dtdxdy − Z R d × Y d ϕ (0 , x ) u ( x, y ) dxdy + Z (0 ,T ) × R d × Y d b ( y ) · ∇ x ϕ ( t, x ) U ( t, x, y ) dtdxdy = Z (0 ,T ) × R d × Y d ϕ ( t, x ) f ( t, x, y ) dtdxdy, − Z (0 ,T ) × R d ∂ϕ∂t ( t, x ) u ( t, x ) dx − Z R d ϕ (0 , x ) u ( x, · ) dx + Z (0 ,T ) × R d U ( t, x, · ) b · ∇ x ϕ ( t, x ) dtdx = Z R d × Y d ϕ ( t, x ) f ( t, x, · ) dtdx. (3.8)Similarly, passing to the two-scale limit with the admissible test function ε ϕ ( t, x ) ψ ( x/ε ) forany ϕ ( t, x ) ∈ C c ([0 , T ) × R d ) and any ψ ∈ C ♯ ( Y d ), we get that Z (0 ,T ) × R d × Y d ϕ ( t, x ) b ( y ) · ∇ y ψ ( y ) U ( t, x, y ) dtdxdy = Z (0 ,T ) × R d ϕ ( t, x ) (cid:18)Z Y d U ( t, x, y ) b ( y ) · ∇ y ψ ( y ) dy (cid:19) dtdx = 0 , which by Proposition 1.1 implies thatdiv y ( U ( t, x, · ) b ) = 0 in D ′ ( R d ) , a.e. ( t, x ) ∈ (0 , T ) × R d . (3.9)Then, applying Lemma 2.1 with σ = 1, for a.e. ( t, x ) ∈ (0 , T ) × R d , the function U ( t, x, · )is an invariant function for the flow X associated with b related to Lebesgue’s measure, andso are the positive and negative parts U ± ( t, x, · ) of U ( t, x, · ). Hence, again by Lemma 2.1 themeasures U ± ( t, x, y ) dy are invariant for X , which by the definition (1.20) of D b = { ζ } , impliesthat U ± ( t, x, · ) b = Z Y d b ( y ) U ± ( t, x, y ) dy = (cid:18)Z Y d U ± ( t, x, y ) dy (cid:19) ζ a.e. ( t, x ) ∈ (0 , T ) × R d . (3.10)From (3.10) and (3.7) we deduce that U ( t, x, · ) b = U ( t, x, · ) ζ = u ( t, x ) ζ a.e. ( t, x ) ∈ (0 , T ) × R d . (3.11)Putting this equality in the weak formulation (3.8) we get that for any ϕ ( t, x ) ∈ C c ([0 , T ) × R d ), − Z (0 ,T ) × R d ∂ϕ∂t ( t, x ) u ( t, x ) dx − Z R d ϕ (0 , x ) u ( x, · ) dx + Z (0 ,T ) × R d u ( t, x ) ζ · ∇ x ϕ ( t, x ) dtdx = Z R d × Y d ϕ ( t, x ) f ( t, x, · ) dtdx, which is the weak formulation of the homogenized transport equation (3.5).( ii ) ⇒ ( i ) . First of all, note that the set D b contains the mean b , since by the free divergenceof b and by Proposition 1.1, Lebesgue’s measure is an invariant probability measure for theflow X associated with b .Now, let us prove that any invariant probability measure σ ( x ) dx with σ ∈ L ♯ ( Y d ), for theflow X satisfies the equality σ b = ζ , which will yield the desired equality D b = { ζ } . To thisend, let us first show this for any invariant probability measure v ( x ) /v dx with v ∈ L ∞ ♯ ( Y d ). Byvirtue of Proposition 1.1 such a function v is solution to the equationdiv( v b ) = b · ∇ v = 0 in D ′ ( R d ) . (3.12)11et θ ∈ C c ( R d ), and define for ε > u ε ∈ C ([0 , T ]; C c ( R d )) by u ε ( t, x ) := θ ( x + t ζ ) v ( x/ε ) for ( t, x ) ∈ [0 , T ] × R d , where ζ is the vector involving in the homogenized equation (3.5). By (3.12) we have ∂u ε ∂t ( t, x ) − b ( x/ε ) · ∇ x u ε ( t, x )= v ( x/ε ) ζ · ∇ x θ ( x + t ζ ) − v ( x/ε ) b ( x/ε ) · ∇ x θ ( x + t ζ ) − /ε θ ( x + t ζ ) ( b · ∇ y v )( x/ε )= (cid:0) v ( x/ε ) ζ − ( v b )( x/ε ) (cid:1) · ∇ x θ ( x + t ζ ) = f ( t, x, x/ε ) , where f ( t, x, y ) := (cid:0) v ( y ) ζ − ( v b )( y ) (cid:1) · ∇ x θ ( x + t ζ ) for ( t, x, y ) ∈ [0 , T ] × R d × Y d , is an admissible function in C c ([0 , T ] × R d ; L ∞ ♯ ( Y d )) with compact support in [0 , T ] × R d × Y d .Moreover, we have u ε (0 , x ) = θ ( x ) v ( x/ε ) for x ∈ R d , where θ ( x ) v ( y ) ∈ C c ( R d ; L ♯ ( Y d )) withcompact support in [0 , T ] × R d is also an admissible function. Hence, by the homogenizationassumption the sequence u ε ( t, x ) converges weakly in L ((0 , T ) × R d ) to u ( t, x ) = θ ( x + t ζ ) v solution to the homogenized equation (3.5), i.e. ∀ ( t, x ) ∈ [0 , T ] × R d , ∂u∂t ( t, x ) − ζ · ∇ x u ( t, x ) = f ( t, x, · ) = (cid:0) v ζ − v b (cid:1) · ∇ x θ ( x + t ζ ) . But directly from the expression u ( t, x ) = θ ( x + t ζ ) v , we also deduce that ∀ ( t, x ) ∈ [0 , T ] × R d , ∂u∂t ( t, x ) − ζ · ∇ x u ( t, x ) = 0 . Equating the two former equations we get that for any θ ∈ C c ( R d ), ∀ ( t, x ) ∈ [0 , T ] × R d , (cid:0) v ζ − v b (cid:1) · ∇ x θ ( x + t ζ ) = 0 , which implies that v b = v ζ . (3.13)Now, let σ be a non negative function in L ♯ ( Y d ) with σ = 1, such that σ ( x ) dx is an invariantmeasure for the flow X , or, equivalently, by Lemma 2.1 applied with σ = 1, the function σ is invariant for X with respect to Lebesgue’s measure. Hence, for any n ∈ N , the truncatedfunction σ ∧ n is also invariant for X . Equality (3.13) applied with v = σ ∧ n ∈ L ∞ ♯ ( Y d ), yields( σ ∧ n ) b = σ ∧ n ζ −→ n →∞ σ b = σ ζ = ζ . Thus, we obtain the desired equality D b = { ζ } = { b } , which owing to Theorem 2.1 concludesthe proof of Theorem 3.2. (cid:3) Remark 3.1
From equation (3.9)
Hou and Xin [15] used the ergodicity of the flow X to deducethat U ( t, x, · ) is constant a.e. ( t, x ) ∈ (0 , T ) × R d . However, this condition is not necessary.Indeed, the less restrictive condition used in the above proof is that D b is reduced to the unitset { ζ } . This combined with Lemma 2.1 on invariant measures and functions leads us to equality (3.11) , and allows us to conclude. Comparison between the seven conditions
In the sequel we denote:
Rec if there exist a C -diffeomorphism Ψ on Y d and ξ ∈ R d such that ∇ Ψ b = ξ in Y d . Erg if the ergodic condition (1.17) holds with an invariant probability measure for X ,which is absolutely continuous with respect to Lebesgue’s measure, Asy-a.e. if there exist ζ ∈ R d such that lim t →∞ X ( t, x ) /t = ζ , a.e. x ∈ Y d . Asy-e. if there exist ζ ∈ R d such that lim t →∞ X ( t, x ) /t = ζ , ∀ x ∈ Y d . C b = 1 if the unit set condition holds for Herman’s set C b . D b = 1 if the unit set condition holds for the set D b . Hom if the homogenized equation (3.5) holds when b is divergence free in R d . Theorem 4.1
Let b ∈ C ♯ ( Y d ) d be a non null but possibly vanishing vector field such that thereexists an invariant probability measure σ ( x ) dx with σ ∈ L ♯ ( Y d ) , for the flow X associatedwith b , or, equivalently, D b = Ø . Then, we have a complete array (see Figure 1 below) of allthe logical connections between the above seven conditions, in which:- A grey square means a tautology.- A square with ⇐ means that the condition of the top line implies the condition of the leftcolumn, but not the converse in general.- A square with ⇑ means that the condition of the left column implies the condition of thetop line, but not the converse in general.- A square with ⇔ or m means that the conditions of the top line and of the left column areequivalent.- A dark square means that the conditions of the top line and the left column cannot becompared in general.- Finally, if a square involves condition Hom, then the other condition must be consideredunder the assumption that b is divergence free in R d . Remark 4.1
We may have both C b = 1 and D b = Ø .To this end, consider a gradient field b = ∇ u with u ∈ C ♯ ( Y d ) , such that ∇ u = 0 a.e. in Y d .On the one hand, by virtue of [8, Proposition 2.4] we have C b = { } . On the other hand,assume that there exists an invariant probability measure σ ( x ) dx with σ ∈ L ♯ ( Y d ) , for the flowassociated with ∇ u . Then, by virtue of Proposition 1.1 we have Z Y d σ ( x ) |∇ u ( x ) | dx = Z Y d σ ( x ) ∇ u ( x ) · ∇ u ( x ) dx = 0 , which implies that σ = 0 a.e. in Y d , a contradiction with σ = 1 . Therefore, we get that D b = Ø .Proof of Theorem 4.1.Condition Rec. By virtue of [6, Corollary 4.1] condition
Rec implies condition C b = 1 whichby Proposition 1.2 is equivalent to condition Asy-e. . Moreover, condition
Asy-e. clearly impliescondition
Asy-a.e. which by Theorem 2.1 is equivalent to condition D b = 1, and by Theo-rem 3.2 is equivalent to condition Hom . Therefore, condition
Rec implies condition
Asy-a.e. ,condition
Asy-e. , condition C b = 1, condition D b = 1, and condition Hom .On the other hand, note that if the vector field b vanishes, then condition Rec cannot holdtrue. Otherwise, in equality ∇ Ψ b = ζ the constant vector ζ is necessarily nul, hence due to the13 ec Erg Asy-a.e. Asy-e. C b = 1 D b = 1 HomRec ⇑ ⇑ ⇑ ⇑ ⇑
Erg ⇑ ⇑ ⇑
Asy-a.e. ⇐ ⇐ ⇐ ⇐ ⇔ ⇔
Asy-e. ⇐ ⇑ ⇔ ⇑ ⇑ C b = 1 ⇐ ⇑ m ⇑ ⇑ D b = 1 ⇐ ⇐ m ⇐ ⇐ ⇔ Hom ⇐ ⇐ m ⇐ ⇐ m
Figure 1: Logical connections between the seven conditionsinvertibility of ∇ Ψ, b is the nul vector field, which yields a contradiction. Therefore, since allother conditions may be satisfied with a vanishing vector field b according to the examples of[8, Section 4] combined with Theorem 2.1 and Theorem 3.2, condition Rec cannot be deducedin general from any of the other six conditions.
Conditions Rec and Erg cannot be compared. [6, Corollary 4.1] provides a two-dimensional anda three-dimensional example in which condition
Rec holds true, but not condition
Erg . Condition Erg.
By virtue of Birkhoff’s theorem condition
Erg implies condition
Asy-a.e. whichis equivalent to condition D b = 1 (by Theorem 2.1) and is equivalent to condition Hom (byTheorem 3.2).
Conditions Erg and C b = 1 cannot be compared. Since condition
Rec implies C b = 1, butcondition Rec does not imply in general condition
Erg (by [6, Corollary 4.1]), by a transitivityargument condition C b = 1 does not imply in general condition Erg .On the other hand, extending the two-dimensional results of Oxtoby [20] and Marchetto [17]to any dimension by a different approach, Example 4.1 and Proposition 4.1 below deal with a d -dimensional Stepanoff flow [23, Section 4] defined by ∂S∂t ( t, x ) = b S ( S ( t, x )) = ρ S ( S ( t, x )) ξ, t ∈ R S (0 , x ) = x ∈ R d , (4.1)14here ρ S is a non negative function in C ♯ ( Y d ) with a finite positive number of roots in Y d and σ S := 1 /ρ S ∈ L ♯ ( Y d ), and where ξ is a constant vector of R d with incommensurable coordinates.Under these conditions σ S ( x ) /σ S dx is the unique invariant probability measure on Y d for theflow S , which does not load the zero set of ρ S , and S is ergodic with respect to the measure σ S ( x ) /σ S dx . Hence, condition Erg holds true with the probability measure σ S ( x ) /σ S dx .Moreover, Proposition 4.1 shows that D b S = { ζ } and C b S = [0 , ζ ] with ζ = (1 /σ S ) ξ = 0.Therefore, condition Erg does not imply in general condition C b = 1, or, equivalently, condition Asy-e. . Conditions C b = 1 and D b = 1 . Since D b is assumed to be non empty, condition C b = 1clearly implies condition D b = 1.In contrast, as above mentioned the Stepanoff flow induces that D b S = { ζ } and C b S = [0 , ζ ]with ζ ∈ R d \{ } . Alternatively, Example 4.2 below provides a different class of two-dimensionalvanishing vector fields b such that D b is a singleton, while C b is a closed line set not reduced toa singleton. Therefore, condition D b = 1 does not imply in general C b = 1. Condition D b = 1 . Since condition C b = 1 implies condition D b = 1, but C b = 1 does notimply in general condition Erg , by a transitivity argument condition D b = 1 does not implyin general condition Erg . Moreover, since condition C b = 1 is equivalent to condition Asy-e. ,but condition D b = 1 does not imply in general C b = 1, condition D b = 1 does not imply ingeneral condition Asy-e. . Condition Hom.
Here, we assume that the vector field b is divergence free in R d .On the one hand, consider the constant vector field b = e in R d , which induces the flow X ( t, x ) = x + t e for ( t, x ) ∈ R × R d . Then, any function f ∈ L ♯ ( Y d ) independent of variable x is invariant for the flow X with respectto any invariant probability measure which is absolutely continuous with respect to Lebesgue’smeasure. Hence, the flow X is not ergodic with respect to such an invariant probability measure.Moreover, we have immediately C b = D b = { e } . Therefore, condition Hom which is equivalentto condition D b = 1 (by Theorem 3.2), does not imply in general condition Erg .On the other hand, the two-dimensional divergence free Oxtoby example [20, Section 2] com-bined with the uniqueness result of [20, Theorem 1] (see Example 4.1) provides a flow whichis ergodic with respect to Lebesgue’s measure, and such that C b is not a unit set (see Proposi-tion 4.1). Therefore, since condition Erg implies condition
Hom (see, e.g. , [15, Theorem 3.2])condition
Hom does not imply in general condition C b = 1. Finally, condition Hom does notimply in general either condition
Erg , or condition C b = 1, or, equivalently, condition Asy-e. . The rest of the implications can be easily deduced from the former arguments. (cid:3)
Exemple 4.1
Oxtoby [20] provided an example of a free divergence analytic two-dimensionalvector field b with (0 ,
0) as unique stationary point in Y , such that the associated flow X isergodic with respect to Lebesgue’s measure, and such that Lebesgue’s measure is the uniqueinvariant measure for the flow X among all the invariant probability measures which do notload the point (0 , ρ S is a non negative function in C ♯ ( Y ) with (0 ,
0) as unique stationary point, and where ξ isa constant vector of R with incommensurable coordinates. Stepanoff [23, Section 4] provedthat Birkhoff’s theorem applies if σ S := 1 /ρ S is in L ♯ ( Y ), which is not incompatible with theanalyticity for ρ S . A suitable candidate for ρ S is then the function (see [8, Example 4.2] foranother application) ρ S ( x ) := (cid:0) sin ( πx ) + sin ( πx ) (cid:1) β for x ∈ Y , with β ∈ (1 / , . (4.2)15ore generally, Oxtoby [20, Theorem 1] proved that any two-dimensional flow homeomorphicto a Stepanoff flow with a unique stationary point x , admits a unique invariant probabilitymeasure µ for the flow S (4.1) satisfying µ ( { x } ) = 0, and that S is ergodic with respectto µ . Twenty five years later, Marchetto [17, Proposition 1.2] extended this result to any flowhomeomorphic to a Stepanoff flow with a finite number of stationary points in Y .In what follows, we extend the two-dimensional results of [20, 17] to any dimension d ≥ Proposition 4.1
Consider a d -dimensional, d ≥ , Stepanoff flow S (4.1) where ρ S ∈ C ♯ ( Y d ) is non negative with a finite positive number of roots (the stationary points for S ) in Y d and σ S := 1 /ρ S ∈ L ♯ ( Y d ) , and where ξ ∈ R d has incommensurable coordinates. Then, the measure σ S ( x ) /σ S dx is the unique invariant probability measure on Y d for the flow S , which does notload the zero set of ρ S . The flow S is also ergodic with respect to the measure σ S ( x ) /σ S dx .Moreover, we have D b S = { ζ } and C b S = [0 , ζ ] , where ζ := 1 /σ S ξ . Remark 4.2
Similarly to [20, 17] the result of Proposition 4.1 actually extends to any flowwhich is homeomorphic to a Stepanoff flow.Indeed, let Ψ be a C -diffeomorphism on Y d (see [8, Remark 2.1]). Define the flow ˆ X obtainedthrough the homeomorphism Ψ from the flow X associated with a vector field b ∈ C ♯ ( Y d ) d , by ˆ X ( t, x ) := Ψ (cid:0) X ( t, Ψ − ( x )) (cid:1) for ( t, x ) ∈ R × Y d . (4.3) According to [8, Remark 2.1] the homeomorphic flow ˆ X is the flow associated with the vectorfield ˆ b ∈ C ♯ ( Y d ) d defined by ˆ b ( x ) = ∇ Ψ(Ψ − ( x )) b (Ψ − ( x )) for x ∈ Y d . (4.4) Now, let µ be a probability mesure on Y d , and let ˆ µ be the image measure of µ by Ψ defined by Z Y d ϕ ( x ) d ˆ µ ( x ) = Z Y d ϕ (Ψ( y )) dµ ( y ) for ϕ ∈ C ♯ ( Y d ) . By (4.3) we have ∀ ϕ ∈ C ♯ ( Y d ) , Z Y d ϕ (cid:0) ˆ X ( t, x ) (cid:1) d ˆ µ ( x ) = Z Y d ϕ (cid:0) Ψ( X ( t, y )) (cid:1) dµ ( y ) , ∀ ρ ∈ C ♯ ( Y d ) , ˆ µ ( { ρ = 0 } ) = µ ( { ρ ◦ Ψ = 0 } ) , ∀ f ∈ L ♯ ( Y d ) , ˆ f := f ◦ Ψ − , ∀ t ∈ R , ˆ f (cid:0) ˆ X ( t, x ) (cid:1) = f (cid:0) X ( t, Ψ − ( x )) (cid:1) a.e. x ∈ Y d . (4.5) Also note that, if µ is invariant for X, so is ˆ µ for ˆ X . Therefore, if the homeomorphic flow ˆ X is aStepanoff flow S satisfying the assumptions of Proposition 4.1, we easily deduce from (4.5) thatProposition 4.1 holds true for the flow X . Namely, there exists a unique invariant probabilitymeasure µ on Y d for the flow X , which does not load the zero set of ρ S ◦ Ψ . Moreover, themeasure µ is absolutely continuous with respect to Lebesgue’s measure with an a.e. positivedensity, and the flow X is ergodic with respect to µ . Remark 4.3
Assuming the uniqueness result of Proposition 4.1, the ergodicity of σ S ( x ) /σ S dx and equality C b S = [0 , ζ ] can also be proved using standard arguments of ergodic theory. Indeed, et E b S be the set of all the ergodic invariant probability measures for the flow S . Recall (see, e.g. ,[9, Theorem 2, Chapter 1]) that I b S = conv ( E b S ) , and that two elements in E b S are either equal,or mutually singular. Now, if µ ∈ E b S satisfies µ ( { x } ) > for some zero of ρ S , then µ = δ x due to δ x ∈ E b S . Next, since σ S ( x ) /σ S dx is the unique invariant probability measure on Y d for the flow S , which does not load the zero set of ρ S , it follows from equality I b S = conv ( E b S ) that the flow S is ergodic with respect to σ S ( x ) /σ S dx . Hence, E b S is the finite set E b S = (cid:8) σ S ( x ) /σ S dx (cid:9) ∪ (cid:8) δ x : ρ S ( x ) = 0 (cid:9) . (4.6) Therefore, σ S b S /σ S = ζ provides the unique non zero contribution in C b S through E b S , whichby convex combination implies that C b S = [0 , ζ ] . Equality I b S = conv ( E b S ) and property (4.6) also give D b S = { ζ } .Proof of Proposition 4.1. Assume that µ is an invariant probability measure for the flow S (4.1),which does not load the zero set of ρ S . Then, by virtue of Proposition 1.1 the Borel measure ˜ µ on R d defined by (1.21) is solution to the equationdiv(˜ µ b S ) = div( ρ S ˜ µ ξ ) = 0 in D ′ ( R d ) . Hence, applying Lemma 4.2 below with the measure ν = ρ S µ which is connected to the measure˜ ν = ρ S ˜ µ by (1.21), there exists a constant c ∈ R such that ρ S ( x ) dµ ( x ) = c dx on Y d , i.e. ∀ ϕ ∈ C ♯ ( Y d ) , Z Y d ϕ ( x ) ρ S ( x ) dµ ( x ) = Z Y d c ϕ ( x ) dx. Then, we get that for any n ≥ ∀ ϕ ∈ C ♯ ( R d ) , Z Y d ϕ ( x ) ρ S ( x ) + 1 /n ρ S ( x ) dµ ( x ) = Z Y d c ϕ ( x ) ρ S ( x ) + 1 /n dx. (4.7)However, since measure µ does not load the finite zero set of ρ S in Y d (at this point thisassumption is crucial), we have ϕ ρ S ρ S + 1 /n −→ n →∞ ϕ dµ ( x )-a.e. in Y d , with (cid:12)(cid:12)(cid:12)(cid:12) ϕ ρ S ρ S + 1 /n (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k ∞ ∈ L ♯ ( Y d , µ ) ϕρ S + 1 /n −→ n →∞ ϕρ S dx -a.e. in Y d , with (cid:12)(cid:12)(cid:12)(cid:12) ϕρ S + 1 /n (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k ∞ ρ S ∈ L ♯ ( Y d ) . Therefore, passing to the limit as n → ∞ owing to Lebesgue’s theorem in (4.7), we get that ∀ ϕ ∈ C ♯ ( Y d ) , Z Y d ϕ ( x ) dµ ( x ) = Z Y d c ϕ ( x ) σ S ( x ) dx. We thus obtain the equality dµ ( x ) = σ S ( x ) /σ S dx , which shows the uniqueness of an invariantprobability measure for the flow S , which does not load the zero set of ρ S . Conversely, σ S b S = ξ is clearly divergence free, which by Proposition 1.1 implies that µ is an invariant probabilitymeasure for the flow S . We have just proved that µ is the unique invariant probability measurefor the flow S , which does not load the zero set of ρ S .Now, let us prove that the flow S is ergodic with respect to the measure dµ ( x ) = σ S ( x ) /σ S dx .To this end, let f ∈ L ♯ ( Y d ) be an invariant function for the flow S with respect to measure µ .Let T ± n be the truncation functions at level n ∈ N , defined by T ± n ( t ) := (cid:0) ( ± t ) ∨ (cid:1) ∧ n for t ∈ R . T ± n ( f ) ∈ L ∞ ♯ ( Y d ) are also invariant functions for the flow S with respect tomeasure µ . We know that σ S ( x ) dx is an invariant measure for the flow S . Hence, by virtueof Lemma 2.1 the Radon measures dµ ± n ( x ) := ( T ± n ( f ) σ S )( x ) dx are invariant for S , which byrelation (1.21) and Proposition 1.1 implies that f µ ± n b S = µ ± n b S = ( T ± n ( f ) σ S b S )( x ) dx = T ± n ( f )( x ) ξ dx are divergence free in R d . Therefore, applying Lemma 4.2 with measures dν ( y ) = T ± n ( f )( y ) dy which satisfy ˜ ν = ν , the functions T ± n ( f ) agree with constants c ± n ∈ R a.e. in Y d . However, sincethe sequences T ± n ( f ) converge strongly in L ♯ ( Y d ) to the non negative and the non positive parts f ± of f , the sequences c ± n converge to some constants c ± in R . Hence, the function f = f + − f − agrees with the constant c + − c − a.e. in Y d . This proves the desired property.Next, since σ S ( x ) /σ S dx is the unique invariant probability measure on Y d for the flow S ,among the invariant probability measures which are absolutely continuous with respect toLebesgue’s measure, we have D b S = (cid:26)Z Y d ρ S ( x ) ξ σ S ( x ) /σ S dx (cid:27) = { /σ S ξ } . Note that the former equality can be alternatively deduced from the ergodicity of the flow S combined with Theorem 4.1.On the other hand, set b n := b S + 1 /n for n ≥
1. Since ξ has incommensurable coordinates, wehave (see [8, Example 4.1]) C b n = { ζ n } where ζ n := (cid:18)Z Y d dxρ S ( x ) + 1 /n dx (cid:19) − ζ . Finally, since the function ρ S vanishes in Y d , by virtue of [8, Theorem 3.1] we obtain that C b S = [0 , ζ ] , where ζ = lim n →∞ ζ n = 1 /σ S ξ. Note that the ergodic approach of Remark 4.3 alternatively shows that C b S = [0 , ζ ]. The proofof Proposition 4.1 is now complete. (cid:3) Lemma 4.2
Let ν ∈ M ♯ ( Y d ) , let ˜ ν ∈ M loc ( R d ) be the Borel measure on R d connected to themeasure ν by relation (1.21) , and let ξ ∈ R d be a vector with incommensurable coordinates.Assume that ˜ ν ξ is divergence free in R d , i.e. ∀ ϕ ∈ C ∞ c ( R d ) , Z R d ξ · ∇ ϕ ( x ) d ˜ ν ( x ) = 0 . (4.8) Then, there exists a constant c ∈ R such that dν ( y ) = c dy on Y d . Remark 4.4
In Lemma 4.2 the incommensurability of ξ ′ s coordinates is also a necessary con-dition to get (4.8) . Indeed, assume that there exists a non nul integer vector k ∈ Z d \ { } such that k · ξ = 0 . Then, for any non constant Z -periodic function θ ∈ C ♯ ( Y ) , the function (cid:0) τ : x θ ( k · x ) (cid:1) belongs to C ♯ ( Y d ) , ˜ τ ( x ) dx = τ ( x ) dx , and ∀ x ∈ R d , div( τ ξ )( x ) = θ ′ ( k · x ) k · ξ = 0 , so that the conclusion of Lemma 4.2 does not hold true. roof of Lemma 4.2. Let ( φ n ) n ∈ N be a sequence of mollifiers in C ∞ c ( R d ) with φ n = 1. Applyingsuccessively Fubini’s theorem twice and (4.8), the convolution φ n ∗ ˜ ν ∈ C ∞ ( R d ) satisfies for any n ∈ N and for any ϕ ∈ C ∞ c ( R d ), Z R d (cid:18)Z R d φ n ( x − y ) d ˜ ν ( y ) (cid:19) ξ · ∇ ϕ ( x ) dx = Z R d (cid:18)Z R d φ n ( x − y ) ξ · ∇ ϕ ( x ) dx (cid:19) d ˜ ν ( y )= Z R d (cid:18)Z R d φ n ( x ) ξ · ∇ x ϕ ( x + y ) dx (cid:19) d ˜ ν ( y ) = Z R d (cid:18)Z R d ξ · ∇ y ϕ ( x + y ) d ˜ ν ( y ) (cid:19) φ n ( x ) dx = 0 , or, equivalently, div (cid:0) ( φ n ∗ ˜ ν ) ξ (cid:1) = ∇ ( φ n ∗ ˜ ν ) · ξ = 0 in R d . (4.9)Now, consider ξ , . . . , ξ d − ( d −
1) vectors in R d such that ( ξ , . . . , ξ d − , ξ ) is an orthogonal basisof R d , and let Λ be the matrix in R ( d − × d whose lines are the vectors ξ , . . . , ξ d − , i.e. its entriesare given by Λ ij = ξ ij for ( i, j ) ∈ { , . . . , d − } × { , . . . , d } . Then, make the linear change ofvariables R d → R d x y = (Λ x, ξ · x ) = ( ξ · x, . . . , ξ d − · x, ξ · x ) . Since (4.9) means that ( φ n ∗ ˜ ν )( x ) is independent of the variable y d = ξ · x , it follows that thereexists a function θ n ∈ C ∞ ( R d − ) such that ∀ x ∈ R d , ( φ n ∗ ˜ ν )( x ) = θ n (Λ x ) . Moreover, due to (1.21) and the Z d -periodicity of ( φ n ) ♯ , we have for any x ∈ R d and k ∈ Z d ,( φ n ∗ ˜ ν )( x + k ) = Z R d φ n ( x + k − y ) d ˜ ν ( y ) = Z Y d ( φ n ) ♯ ( x + k − y ) dν ( y )= Z Y d ( φ n ) ♯ ( x − y ) dν ( y ) = Z R d φ n ( x − y ) d ˜ ν ( y ) = ( φ n ∗ ˜ ν )( x ) , which implies that the function φ n ∗ ˜ ν is also Z d -periodic. As a consequence, the regular function θ n satisfies the periodicity condition ∀ k ∈ Z d , ∀ x ∈ R d − , θ n ( x + Λ k ) = θ n ( x ) . Hence, by virtue of the density Lemma 4.3 below we get that θ n is a constant c n ∈ R , and thus φ n ∗ ˜ ν = c n in R d . Therefore, by Fubini’s theorem we have for any ϕ ∈ C ∞ c ( R d ), Z R d c n ϕ ( x ) dx = Z R d (cid:18)Z R d φ n ( x − y ) d ˜ ν ( y ) (cid:19) ϕ ( x ) dx = Z R d (cid:18)Z R d φ n ( x − y ) ϕ ( x ) dx (cid:19) d ˜ ν ( y ) , where the function (cid:0) y R R d φ n ( x − y ) ϕ ( x ) dx (cid:1) converges uniformly to ϕ on R d as n → ∞ ,whose support is included in a fixed compact set of R d , and which is bounded uniformly by k ϕ k ∞ . Therefore, passing to the limit as n → ∞ thanks to Lebesgue’s theorem with respect tomeasure ˜ ν , we get that the sequence ( c n ) n ∈ N converges to some c ∈ R , and that ∀ ϕ ∈ C ∞ c ( R d ) , Z R d c ϕ ( x ) dx = Z R d ϕ ( y ) d ˜ ν ( y ) . Hence, we deduce the equality d ˜ ν ( x ) = c dx on R d , or, equivalently, dν ( y ) = c dy on Y d byvirtue of Remark 1.1. This concludes the proof of Lemma 4.2. (cid:3) emma 4.3 Let ξ be a vector in R d for d ≥ , with incommensurable coordinates, let ξ , . . . , ξ d − be ( d − vectors in R d such that ( ξ , . . . , ξ d − , ξ ) is an orthogonal basis of R d , and let Λ bethe matrix in R ( d − × d whose lines are the vectors ξ , . . . , ξ d − . Then, the lattice Λ Z d is densein R d − .Proof. Lemma 4.3 follows easily from [3, Proposition 6 & Corollary, Section VII.7] which leadsone to Kronecker’s approximation theorem [3, Proposition 7, Section VII.7]. For the reader’sconvenience we propose a more direct proof.Since matrix Λ has rank ( d −
1) and Ker(Λ) = R ξ , we may assume, up to reorder the vectors,that the vectors Λ e , . . . , Λ e d − are linearly independent and that there exist d real numbers α , . . . , α d − , α satisfying Λ e d = d − X i =1 α i Λ e i and e d − d − X i =1 α i e i = α ξ. (4.10)Replacing the vector e d in the first equality of (4.10) and using that Λ ξ = 0, we get thatΛ Z d = d X i =1 Z Λ e i = d − X i =1 ( Z + α i Z ) Λ e i . Assume that there exists j ∈ { , . . . , d − } such that the set ( Z + α j Z ) is not dense in R , or,equivalently, α j ∈ Q . Taking the j -th and d -th coordinates in the second equality of (4.10),it follows that ξ j + α j ξ d = 0, which contradicts the incommensurability of ξ ’s coordinates.Therefore, the set Λ Z d is dense in R d − , which concludes the proof. (cid:3) Exemple 4.2
Consider a two-dimensional vector field b = ρ R ⊥ ∇ u such that ρ ∈ C ♯ ( Y ) isa.e. positive in Y and does vanish in Y , and such that ∇ u ∈ C ♯ ( Y ) does not vanish in Y and ∇ u has incommensurable coordinates. Also assume that σ := 1 /ρ ∈ L ♯ ( Y ). An exampleof such a function is given by (4.2). Note that, by virtue of Proposition 1.1 the probabilitymeasure σ ( x ) /σ dx is invariant for the flow X associated with b .Now, let σ ∈ L ♯ ( Y ) be a non negative function with σ = 1, such that div( σb ) = 0 in R . ByProposition 1.1 σ ( x ) dx is an invariant probability measure for the flow X . Hence, by Fubini’stheorem we have for any T > Z Y σ ( x ) b ( x ) dx = 1 T Z T (cid:18)Z Y σ ( x ) b ( X ( t, x )) dx (cid:19) dt = Z Y (cid:18) X ( T, x ) − xT (cid:19) σ ( x ) dx. (4.11)On the other hand, since the function ρ does vanish in Y together with ρ > Y ,from [8, Lemma 3.1] applied with the invariant probability measure dµ ( x ) := σ ( x ) /σ dx , wededuce that lim T →∞ X ( T, x ) T = ζ := σ bσ = R ⊥ ∇ uσ = (0 ,
0) a.e. x ∈ Y . Therefore, passing to the limit T → ∞ in equality (4.11) thanks to Lebesgue’s theorem, we getthat for any invariant probability measure σ ( x ) dx with σ ∈ L ♯ ( Y ), Z Y σ ( x ) b ( x ) dx = ζ = (0 , , which thus implies that D b = { ζ } . However, by virtue of [8, Corollary 3.4] we obtain that C b = [0 , ζ ]. Therefore, we have C b = ∞ , while D b = 1.20 emark 4.5 The result of Example 4.2 can be deduced from the Proposition 4.1 combinedwith Remark 4.2, using Kolmogorov’s theorem [16] (see, e.g. , [22, Lecture 11], and see also [24,Theorem 2.1] for an elementary proof when one of the coordinates of the vector field does notvanish). Indeed, since the divergence free field R ⊥ ∇ u of Example 4.2 does not vanish in Y ,by virtue of Kolmogorov’s theorem there exists a C -diffeomorphism on Y which transformsthe flow X associated with the vector field b = ρ R ⊥ ∇ u , to a Stepanoff flow satisfying theassumptions of Proposition 4.1 provided the zero set of ρ is finite. Therefore, Remark 4.2allows us to conclude. We can extend Example 4.2 to the following variant of [8, Corollary 3.4], which provides ageneral framework where the sets C b and D b may differ. Proposition 4.4
Let b = ρ Φ ∈ C ♯ ( Y ) be a vector field, where ρ ∈ C ♯ ( Y ) is a non negativefunction with a positive finite number of roots, and where Φ ∈ C ♯ ( Y ) is a non vanishing vectorfield. Also assume that there exists a function u ∈ C ( Y ) with ∇ u ∈ C ♯ ( Y ) , such that ∇ u has incommensurable coordinates and Φ · ∇ u = 0 in Y . Then, the exists a vector ζ ∈ R suchthat C b = [0 , ζ ] , together with D b = Ø or D b = { ζ } .Proof. First of all, define for n ≥
1, the function ρ n := ρ + 1 /n >
0, and the vector field b n := ρ n Φ. By the equality Φ · ∇ u = 0 in Y , we get that u is an invariant function forthe flow X n associated with the vector field b n , with respect to Lebesgue’s measure. Then,following the proof of [8, Corollary 3.4], from the ergodic case of [21, Theorem 3.1] and theincommensurability of ∇ u ’s coordinates, we deduce that there exists a vector ζ n ∈ R such that C b n = { ζ n } .On the one hand, since the function ρ vanishes in Y , by the second case of [8, Theorem 3.1]it turns out that the sequence ( ζ n ) n ≥ converges to some ζ ∈ R , and that C b = [0 , ζ ].On the other hand, assume that the set D b is non empty. Then, there exists an invariantprobability measure σ ( x ) dx with σ ∈ L ♯ ( Y ), for the flow X associated with the vector field b , i.e. σ ( x ) /σ dx ∈ I b . Following the proof of [8, Corollary 3.3] define the probability measure µ n by dµ n ( x ) := C n ρ ( x ) ρ n ( x ) σ ( x ) dx where C n := (cid:18)Z Y d ρ ( x ) ρ n ( y ) σ ( y ) dy (cid:19) − . Note that C n < ∞ , since ρ σ is non negative and not nul a.e. in Y . Due to σ ( x ) /σ dx ∈ I b ,by Proposition 1.1 we have ∀ ϕ ∈ C ♯ ( Y ) , Z Y d b n ( x ) · ∇ ϕ ( x ) dµ n ( x ) = C n Z Y d b ( x ) · ∇ ϕ ( x ) σ ( x ) dx = 0 , which again by Proposition 1.1 implies that µ n ∈ I b n . This combined with C b n = { ζ n } yields ζ n = Z Y d b n ( x ) dµ n ( x ) = C n Z Y d b ( x ) σ ( x ) dx = C n σ b which is actually independent of σ . Due ρ > Y , by Lebesgue’s theorem we get thatthe sequence ( C n ) n ≥ converges to σ = 1. Hence, we deduce that ζ = lim n →∞ ζ n = σ b which is also independent of σ . Therefore, we obtain that D b = { ζ } , which concludes the proofof Proposition 4.4. (cid:3) eferences [1] G. Allaire: “Homogenization and two-scale convergence”,
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