Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow

Transcription

1 REND. SEM. MAT. UNIV. PADOVA, Vol ) Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow LUIGI AMBROSIO *) - MYRIAM LECUMBERRY **) - STEFANIA MANIGLIA ***) 1. Introduction. In a recent paper [7] Le Bris and Lions studied, among other things, the differentiability properties of the flow X t; x) : [; T]! associated to a vectorfield b : ; T)! having a Sobolev regularity with respect to the space variable. Under suitable global conditions on b analogous to those considered in [8], where the flow X has been first characterized, they show that X t; x y) X t; x) 1:1! t; x; y) where the convergence, as #, occurs with respect to the local convergence in measure in x Rd y, and uniformly in time. The map t; x; y) can be considered, according to this limiting procedure, a kind of ``derivative'' of the flow X t; ) atx along the direction y. This result raises several questions about the nature of and the convergence of the difference quotients: the main one is whether we can infer some kind of Lipschitz property of the flow from this convergence. This is indeed closely related to the problem of passing from the local convergence in measure in x Rd y to the a.e. convergence to as # of the quantities 1:2 X t; x y) X t; x) 1 ^ t; x; y) dy R > : B R ) *) Indirizzo dell'a.: Scuola Normale Superiore, Piazza dei Cavalieri, Pisa, Italy; l.ambrosio@sns.it **) Indirizzo dell'a.: Max-Planck Institut fuèr Mathematik, Inselstrasse 22-26, D-413 Leipzig, Germany; lecumberry@mis.mpg.de ***) Indirizzo dell'a.: Dipartimento di Matematica, UniversitaÁ di Pisa, Pisa, Italy; maniglia@mail.dm.unipi.it

2 3 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia Notice that elementary Fubini-type arguments show that this passage is possible only for a sequence i )#, but the convergence to of the integrals 1.2) only along some sequence i ) does not seem to lead to any kind of Lipschitz property. Indeed, after the completion of this paper, in a joint work of the first author and Maly [4], a new characterization of the convergence 1.1 for t fixed) is found, involving only the x variable, and an example that no Lipschitz property can be derived from this weak differentiability property is explicitely given. Assuming for the sake of simplicity in this introductory discussion that b is autonomous and that both b and its divergence are globally bounded, we are able to answer positively these questions under an assumption slightly stronger than W 1;1 loc, namely that the local maximal function of jrbj belongs to L 1 loc this holds if and only if jrbj ln 2 jrbj) 2 L1 loc ). Under this assumption we show in Theorem 3.3 that t; x; y) is representable as L t; x)y for suitable linear maps L t; x) :! see also [4] and Remark 3.7); moreover, for any ball B R ) and any d > we can find a Borel set A B R ) such that L d B R ) n A < d and X t; )j A is a Lipschitz map for any t 2 [; T]: It turns also out that indeed the map L t; x) can be characterized L d 1 -a.e. in [; T] A as the classical differential, given by Rademacher theorem, of any Lipschitz extension of X t; )j A see also Section 2.1 for a different characterization in terms of the so-called approximate differential). Furthermore, combining ``forward'' and ``backward'' Lipschitz estimates we obtain in Theorem 3.4 also bi-lipschitz estimates, on large sets depending on time. The countable Lipschitz property immediately implies that several classical identities known to be true under the assumptions of the Cauchy- Lipschitz theorem), as the explicit formula for the density transported by the flow, are still true in this setting, see Corollary 3.5. The strategy in [7] is based on the analysis of the flow in R 2d Y t; x; y) :ˆ X t; x); X t; x y) X t; x) associated to the vectorfields b x y) b x) b x); and on the theory of renormalized solutions for the limit vectorfield b x); rb x)y) see also [1] for related results in a BV context). Our strategy

3 Lipschitz regularity and approximate differentiability, etc. 31 still uses the same difference quotients, but does not require this extension of the theory. Our starting point has been the observation that, in a smooth setting, the time derivative of ln jrx t; )j can be controlled by jrbj X t; x)); looking for a suitable discrete counterpart of this fact we considered the quantities here and in the sequel enotes the averaged integral) ~b t x) :ˆ jx t; y) X t; x)j f dy; B x) where f s) is of the form ln 1 s ^ l) for some l. Their formal limit is f jrx t; x)yj dy; B 1 ) a quantity comparable to f jrx t; x)j). Then we consider the push-forward b t of b ~ t under the map X t; ) and the push forward w t x; y) of x BR ) x)x B1 ) y) under the map Y t; ) to obtain that b t satisfy a transport inequality d dt b t D x bb t ) r t with r t x) :ˆ ld B l ) jb x y) b x)j w t x; y) dy; jyj whose right hand side can be controlled by the maximal function of jrbj. Standard representation results for the solutions of transport problems then give estimates from above on b t and then on b ~ t. It is not clear whether our argument can be improved, getting Lipschitz properties in the W 1;1 loc case, or even in the BV loc case considered in [2]. Some extensions of our result, together with some other open problems, are discussed in Remark Notation and preliminary results. Given a map w t; x) depending on time and space, we will systematically use the notation w t for the map x 7! w t; x), while a derivative with respect to time will be denoted by f _ in the case of ODE's and by d dt f in the case of PDE's. The least Lipschitz constant of a Lipschitz function f will be denoted by Lip f. We denote by L d the Lebesgue measure in and by v d the Lebesgue measure of the unit ball of. Recall that a sequence of Borel maps f h )is said to be locally convergent in measure to f if lim L d fx 2 B R ) : jf h x) f x)j > dg ˆ 8R > ; d > : h!1 Equivalently, one can say that 1 ^jf h f j!inl 1 loc Rd ).

4 32 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia 2.1 ± Approximate differentiability. We start by recalling the classical definition of approximate differentiability: a Borel map X :! R m is said to be approximately differentiable at x 2 if there exists a linear map L :! R m such that the difference quotients X x y) X x) y 7! locally converge in measure as #toly. This is obviously a local property and we still denote by rx x) the approximate differential whenever no ambiguity arises. The approximate differentiability condition can also be stated in a seemingly stronger but equivalent way, by saying that there is a map X, ~ differentiable in the classical sense at x, such that X x) ~ ˆ X x) and the coincidence set fy : X y) ˆ ~X y)g has density 1 at x. The latter formulation can be used, in conjunction with Rademacher theorem, to show that if Xj A is a Lipschitz map for some set A, then X is approximately differentiable at L d -almost any point of A: it suffices to find a Lipschitz extension X ~ to the whole of of Xj A see for instance of [9]) to obtain the approximate differentiability property at any point of density 1 of A where X ~ is classically differentiable. It is worth to mention also see of [9]) a converse statement: approximate differentiability at any point of a Borel set A implies that we can cover A by an increasing family of Borel sets A h such that the restriction of Xj Ah is a Lipschitz map for any h. In connection with Sobolev or even BV) functions, the following classical result holds see for instance [1], Lemma 3.81 and Theorem 3.83): THEOREM 2.1 [Approximate differentiability of Sobolev functions]. V be an open set and let f 2 W 1;1 loc V; Rm ). Then we have 2:1 dy ˆ for L d -a:e: x 2 V: lim r# Furthermore B r x) jf y) f x) rf x) y x)j jy xj Let 2:2 B r x) jf y) f x)j jy xj 1 dy B tr x) jrf j y)dydt for any ball B r x) V: In the following theorem we state a basic criterion for approximate differentiability: basically it says that if the asymptotic L 1 norm of trun-

5 Lipschitz regularity and approximate differentiability, etc. 33 cated difference quotients can be bounded independently of the truncation level, then the map is approximately differentiable. More precisely, in order to study the Lipschitz properties of the flow, we are going to apply Remark 2.3 with f t) ˆ ln 1 t ^ l) with l sufficiently large. THEOREM 2.2. Let f i : [; 1)! [; 1) be subadditive and nondecreasing functions such that sup i sup f i ˆ 1, and let X :! R m be a Borel map. Assume that jx y) X x)j lim sup lim sup f i dy < 1 8x 2 A i!1 r# r B r x) for some Borel set A. Then X is approximately differentiable at L d - a.e. x 2 A. PROOF. We denote by c x) the double limsup appearing in the statement and we assume with no loss of generality that L d A) < 1. Since c x) is finite for any x 2 A, for any > we can find a compact set K A and M 2 R such that L d A n K) <and c M 1onK, jxj M on K. Furthermore, by applying Egorov theorem to the family of functions jx y) X x)j g k x) :ˆ sup lim sup f i dy x 2 K ik r# r B r x) we can find a compact set K K satisfying L d K n K ) <such that jx y) X x)j lim sup f i dy < M 8x 2 K r# r B r x) for i sufficiently large independent of x. Denoting by c d the Lebesgue measure of the intersection of two open balls with radius 1 whose distance between the centers is 1, we choose i in such a way that f i l M ) > 2Mv d for some l M c d and we apply in an analogous way Egorov theorem again to find a compact set K K such that L d K n K ) <and jx y) X x)j 2:3 f i dy M 8x 2 K r B r x)

6 34 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia for r < r, with r > independent of x. Notice that by construction L d A n K ) < 3. We now claim that the restriction of X to K is a Lipschitz map. Indeed, for any pair of points x; y 2 K we can estimate jx x) X y)j with 2Mjx yj=r if jx yj r.ifr :ˆ jx yj < r we apply 2.3) twice and the subadditivity of f i to obtain 1 v d r d B r x) B r x)\b r y) f i f i jx x) X y)j r jx z) X y)j dz r dz B r y) Since L d B r x) \ B r y)) ˆ c d r d we obtain jx x) X y)j f i 2v dm ; r c d jx z) X x)j dz 2M: r f i so that our choice of l M and the monotonicity of f i give jx x) X y)j l M jx yj: p REMARK 2.3. Let f : [; 1)! [; 1) be a subadditive and nondecreasing function. The argument used in the proof of Theorem 2.1 shows that the conditions jx y) X x)j sup f dy M and jxj M 1 on A r r2 ;r ) B r x) for some M, M 1, r > imply that Lip Xj A ) max 2M 1 ; l provided f l) > 2Mv d =c d. r 2.2 ± Maximal functions. Let f 2 L 1 loc Rd ) be a nonnegative function. The local maximal function f ] is defined by f ] x) :ˆ sup f y) dy: t2 ;1) B t x)

7 Lipschitz regularity and approximate differentiability, etc. 35 It is well known see for instance [12]) that the weak L 1 estimate L d fx 2 B R ) : f ] x) > lg C d) f y) dy 8l > l B R 1 )\ff >lg gives that f ] is finite L d -a.e., and that 2:4 f ]p dx C d)p22p p 1 B R ) B R 1 ) j f j p dx 8p 2 1; 1): In the critical case p ˆ 1 we have 2:5 f ] dx v d C d) B R ) B R 1 ) f ln 2 f ) dx: 2.3 ± Flow associated to a vectorfield. In this section we consider a vectorfield B t; z) ˆ B t z) satisfying the following conditions: [P1] B 2 L 1 [; T]; W 1;p loc Rm ; R m ) ; jbj [P2] 1 jzj 2 L1 [; T]; L 1 R m ) L 1 [; T]; L 1 R m ) ; [P3] [div B t ] 2 L 1 [; T]; L 1 R m ). We denote by L the constant 2:6 L :ˆ e R T k[div B t] k dt 1 : If also [P4] [div B t ] 2 L 1 [; T]; L 1 R m ) holds, we set 2:7 R T ~L :ˆ e k[div B t] k dt 1 : The following definition of flow is a variant of the one adopted in [8], as it does not involve the semigroup property. Basically in this definition the flow is considered as a measurable map x 7! X ; x) with values in the space of continuous maps, while in [8] it is considered as a continuous map t 7! X t; ), with a suitable metric in the space of measurable maps in that induces the convergence in measure. See Remark 6.7 of [2] for the proof of the equivalence between the two definitions, at least under the assumptions [P1], [P2], [P3].

8 36 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia DEFINITION 2.4 [Flow]. We say that Y t; z) : [; T] R m! R m is a flow starting at time ) relative to a vectorfield B t; z) if the following two conditions are satisfied: a) for L m -a.e. z 2 R m the map t7!y t; z) is an absolutely continuous integral solution of the ODE _g ˆ B t; g) in [; T], with g ) ˆ z; b) Y t; ) # L m CL m for some constant C independent of t. An important consequence of condition b), that we will use later on, is the property 2:8 L 1 ft 2 ; T) : t; Y t; x)) 2 Ng dx ˆ R m for any L m 1 -negligible set N ; T) R m.indeed,denotingbyn t the t-sections of N, we have that the integral above equals T L m fx 2 R m : Y t; x) 2 N t g dt ˆ T Y t; ) # L m N t ) dt ˆ ; where in the last equality we used the fact that N t is L m -negligible for L 1 -a.e. t 2 ; T). THEOREM 2.5. Under assumptions [P1], [P2], [P3] there exists a flow, uniquely determined in [; T] R m up to sets with L m -negligible projection on R m. Moreover, property b) holds with C ˆ L, the constant defined in 2:6 : The flow has also the following additional properties: a) there exist vectorfields B h, smooth with respect to the space variable, such that 2:9 k[div B ht ] k 1 k[div B t ] k 1 ; jb h j 1 jzj 2 L1 [; T]; L 1 R m ) ; B h 2 L 1 [; T]; W 1;1 loc Rm ; R m ) and such that the classical flows Y h associated to B h satisfy lim max jy h t; z) Y t; z)j^1dz ˆ 8R > : h!1 t2[;t] B R ) b) If [P4] holds, then Y t; ) # L m ~ L 1 L m,with ~ Lasin 2:7.

9 Lipschitz regularity and approximate differentiability, etc. 37 PROOF. The existence of the flow is proved in [8], together with its uniqueness according to the definition of flow adopted therein. Uniqueness according to Definition 2.4 a priori a weaker one) is proved in [2] for the case of bounded vectorfields and in [3] in the general case. Statement a) is proved in [8] seealso[3])bytakingasb h the standard mollifications of B w.r.t. the space variable. Statement b) can be easily proved by approximation, using the explicit expression for the densities of Y h t; ) # L d, namely 1 : detry h t; [Y h t; )] 1 x) Since D h t; x) :ˆ det ry h t; x) solves the ODE [D h t; x) ˆ div B ht Y h t; x))) D h t; x)]; taking 2.9) into account with the positive parts we obtain a uniform upper bound on D h and therefore a uniform lower bound on the densities. p LEMMA 2.6 [Logarithmic sup estimate]. Let Y be a flow relative to B. Then 1 jy t; z)j 2:1 max ln dz kbk 8R > ; [;T] 1 R B R ) where kbk denotes the infimum of all sums Lk B1 1 jzj k L 1 L 1 ) v mr m k B2 1 jzj k L 1 L 1 ) ; among all decompositions of jbj= 1 jzj) and L is defined in 2.6). PROOF. Let w t be the density of Y t; ) # x BR )L m w.r.t. L m and notice that kw t k 1 ˆ v m R m and kw t k 1 L, by property b) of the flow. Using property a) of the flow we get 1 jy t; z)j T j Y t; max ln dz _ z)j [;T] 1 R 1 jy t; z)j dtdz B R ) B R ) T ˆ B R ) jb t Y t; z))j 1 jy t; z)j dzdt T R m jb t jw t 1 jzj dzdt: Splitting jbj= 1 jzj) in the sum of a function in L 1 L 1 ) and a function in L 1 L 1 ) and minimizing among all possible decompositions we obtain 2.1). p

10 38 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia 3. Approximate differentiability of the flow. LEMMA 3.1. Assume that b : ; T)! fulfils [P1], [P2], [P3] and let X t; x) be the flow associated to b. Let >and let Y X t; x y) X t; x) t; x; y) :ˆ X t; x); : Then Y is the flow relative to the vectorfield B t; x; y) ˆ B t x; y) in R2d defined by B t; x; y) :ˆ b t x); b t x y) b t x) 3:1 : In particular condition b) is fulfilled with C ˆ L 2, where 3:2 L :ˆ e R T k[div b t] k dt 1 : PROOF. It is immediate to check that the condition X t; _ x) ˆ b t X t; x)) L 1 -a.e. in [; T] for L d -a.e. x implies that Y _ t; x; y) ˆ B t Y t; x; y)) L 1 - a.e. in [; T] for L 2d -a.e. x; y) precisely, for L d -a.e. x, the property holds for any y). In order to check that Y t; ) # L 2d L 2 L 2d with L as in 3.2)) we write the inequality in an integral form X t; x y) X t; x) W X t; x); dx L 2 W x; y) dxdy for any nonnegative W 2 C c ) and we notice that the property is trivially true if b t 2 C 1 indeed, in this case the divergence of B t x; y) is div b t x) div b t x y)). The general case can be immediately achieved using the integral form and the stability property of the flows with respect to approximations by locally Lipschitz vectorfields, ensured by an application of Theorem 2.5 a) to the vectorfield b. p LEMMA 3.2. Assume that b : ; T)! fulfils [P1], [P2], [P3] and let X t; x) be the flow associated to b. Let b be a bounded nonnegative function satisfying d 3:3 dt b D x bb) ˆ r 2 L 1 loc [; T) in the sense of distributions, and assume that i) t 7! b t is w -continuous between [; T) and L 1 );

11 Lipschitz regularity and approximate differentiability, etc. 39 ii) sup [;T]A jx ; x)j < 1 for some Borel set A. Then we have for L d -a.e. x 2 A. sup b t X t; x)) Lb x) L t2q\[;t) T jrj s X s; x)) ds PROOF. We first extend the PDE to negative times as done immediately before Theorem 4.1 in [2]), setting b t ˆ and r t ˆ for t <, and b t ˆ b for t <. Accordingly, we set X t; x) ˆ x for t <. Then we mollify w.r.t. the space variable both sides to obtain smooth functions b d t ˆ b t r d such that d dt bd D x bb d ) jrj d in 1; T) with jrj d!jrj in L 1 loc 1; T) R d as d# by the commutator estimate in [8]). Finally we mollify again w.r.t. the time variable both sides to obtain smooth functions b d;h ˆ b d ) r h such that with d dt bd;h D x bb d;h ) c d;h in 1; T h ) c d;h :ˆjrj d r h j D x bb d )) r h D x b b d r h ))j: Notice that the smoothness of b d w.r.t. the spatial variables immediately gives, expanding the spatial divergences, that c d;h!jrj d in L 1 loc 1; T) R d as h#. The smoothness of b d;h and the fact that it solves the transport inequality not only in the distributions sense, but also a.e. in 1; T h) ) immediately gives d h e dt R t 1 div b t X t;x)) dt b d;h t d dt bd;h b rb d;h div b t b d;h c d;h i X t; x)) R t e div b t X t;x)) dt 1 c d;h t X t; x)) 8t 2 1; T h) for a.e. x 2 A. Precisely, this happens for those x 2 A such that the trans-

12 4 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia port inequality above fails at X t; x) for a set of times t with strictly positive 1-dimensional Lebesgue measure; this set is L d -negligible by 2.8). Hence, if h 2 ; 1) we get 3:4 t b d;h t X t; x)) Lb d;h 1 x) L 1 ˆ Lb r d x) L c d;h s X s; x)) ds t 1 c d;h s X s; x)) ds: For any t 2 [; T) we can use the w -continuity property of t 7! b t ensuring the strong convergence of b d;h t to b d t as h#) to pass to the limit as h# in 3.4), using also condition b) in Definition 2.4, to obtain b d t X t; x)) Lb r d x) L t 1 t ˆ Lb r d x) L jrj d s X s; x)) ds ˆ jrj d s X s; x)) ds for L d -a.e. x 2 A. Passing now to the limit as d# and using again condition b) in Definition 2.4, we eventually obtain t b t X t; x)) Lb x) L jrj s X s; x)) ds for L d -a.e. x 2 A for any t 2 [; T). By letting t vary in the countable set Q \ [; T), we obtain 3:5 for L d -a.e. x 2 A. sup b t X t; x)) Lb x) L t2q\[;t) T jrj s X s; x)) ds p THEOREM 3.3 [Lipschitz estimate]. Assume that b fulfils [P1] for some p > 1, [P2], [P3], [P4] and let X t; x) be the flow associated to b. Then, for any ball B R ) and any d > we can find a Borel set A B R ) such that L d B R ) n A) < d and the restriction of X t; ) to A is a Lipschitz map for any t 2 [; T]. In particular X t; ) is approximately differentiable L d -a.e. in for any t 2 [; T].

13 Lipschitz regularity and approximate differentiability, etc. 41 PROOF. We consider the flow Y and the associated vectorfield B as in Lemma 3.1 and a ball B R ). By Lemma 2.6 we obtain L d fx 2 B R ) : max jx t; x)j > Mg ln 1 M 1 kbk t2[;t] 1 R for any M > R, hence we can find a constant M 1 > R and a Borel set A 1 B R ) such that L d B R ) n A 1 ) < d=2 and 3:6 We define max jx t; x)j M 1 8x 2 A 1 : [;T] N :ˆ ln 1 jyj) dy: B 1 ) Since T T jrb s j X s; x)) ds dx L jrb s j y) dyds < 1; A 1 B M1 ) we can find M 2 such that L d A 1 n A) < d=2 with A :ˆ x 2 A 1 : T jrb s j X s; x)) ds < M 2 ): Eventually we define M :ˆ NL v d L 2 M 2 and choose l sufficiently large, such that ln 1 l) > 2M L=c ~ d, where ~L :ˆ e k[div b t] k dt 1 : Notice that by construction L d B R ) n A) < d. R T Step 1. We fix the initial measure m ˆ x A x)x B1 ) y)l 2d to obtain, by Lemma 3.1, that Y t; ) # m L 2 L 2d, with L defined in 3.2). We denote by w t x; y) the density of Y t; ) # m w.r.t. L 2d and notice that kw k 1 L 2 and that w solve the following Cauchy problem for the continuity equation: d 3:7 dt w D x;y B w ) ˆ ; w ; x; y) ˆ x A x)x B1 ) y): Moreover, for any nonnegative W 2 C c ) and any t 2 [; T] we have W x) w x; y) dydx ˆ W X t; x)) dxdy Lv d W x) dx t AB 1 )

14 42 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia and therefore 3:8 wt x; y) dy Lv d for L d -a.e. x; for any t 2 [; T]: We are going to apply Remark 2.3 with the function f t) :ˆ ln 1 t ^ l). To this aim we define b t x) :ˆ f jyj)w x; y) dy: Step 2. estimates on b ) Let c 2 Cc 1 2; 2) with c 1on[ 1; 1] and let c R y) ˆ c y=r). Using the test function W x) f c R ) jyj), with W 2 Cc 1 Rd ), in 3.7) gives 3:9 d W x) f c dt R ) jyj)wt x; y) dydx ˆ ˆ hrw x); b t x)i W x) B l ) W x) c t f c R ) jyj)w x; y) dydx t hb t x y) b t x); yi c jyj 1 jyj) R jyj)w t x; y) dydx jyj i R jyj) hb t x y) b t x); y f jyj)w t x; y) dydx in the distribution sense in ; T). Using 3.8), the contribution of b x) in the last integral in 3.9) can be estimated by ln 1 l)kc k 1 jw x)kb x)j wt x; y) dydx R Lv d ln 1 l)kc k 1 R jw x)kb x)j dx: Using the inequality 1 jx yj C 2R for x 2 supp W and y 2 supp c R, and writing b= 1 jzj)asa A with jaj 2L 1 [; T]; L 1 ) and ja j2l 1 [; T]; L 1 ) we can also estimate the contribution of

15 Lipschitz regularity and approximate differentiability, etc. 43 b x y) in the last integral of 3.9) as follows: C 2R) ln 1 l)kc k 1 jw x)j R fjyjrg L 2 ja t j x y) ka t k 1 w t x; y) dy dx: Hence, passing to the limit as R!1in 3.9), the dominated convergence theorem gives d W x)bt x) dx ˆ dt ˆ hrw x); b t x)ib t x) dx W x) B l ) hb t x y) b t x); yi w t x; y) dydx: jyj 1 jyj) Since W is arbitrary this proves that 3:1 d dt b D x bb ) r ; b ; x) ˆ x A x) B 1 ) f jyj) dy with r t; x) :ˆ B l ) jhb t x y) b t x); yij jyj 2 w t x; y) dy L 2 B l ) jhb t x y) b t x); yij jyj 2 dy: By 2.2) in Theorem 2.1 we get 3:11 r t; x) L 2 v d l d jrb t j ] x) for l 1: Notice that jrb t j ] 2 L 1 loc [; T] R d because of the maximal estimate 2.4). Moreover, by 2.1) and 3.8) we infer 3:12 lim sup r t x) Lv djrb t j x) # for L d 1 -a.e. t; x) 2 ; T). Therefore, by applying Lemma 3.2 and using the definition of A and

16 44 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia 3.6), Fatou Lemma gives lim sup # sup b t t2q\[;t) X t; x)) lim sup Lb ; x) L # LN v d L 2 T T jr j s X s; x)) ds jrb s j X s; x)) ds < M for L d -a.e. x 2 A. Notice that the bound 3.11) and the integrability of jrb t j ] are used to ensure that Fatou lemma is applicable. By Egorov theorem, possibly passing to a slightly smaller set A still satisfying L d B R ) n A) < d, we can assume the existence of > such that 3:13 sup sup 2 ; ) t2q\[;t] b t X t; x)) M for L d -a.e. x 2 A: Step 3. Conclusion) We now claim that, setting ~b t x) :ˆ f dy ˆ f B 1 ) the inequality jx t; x y) X t; x)j B x) jx t; y) X t; x)j dy ~b t x) L 3:14 ~ b t v X t; x)) L d -a.e. in A; for any t 2 [; T] d holds. Indeed, recalling that w t is the density of Y t; ) # x A x)x B1 ) y)l 2d ) w.r.t. L 2d, we have the identity t W x)b x) dx ˆ W x)f jyj)wt x; y) dxdy ˆ v d A W X t; x)) b ~ t x) dx; that tells us that b t L d ˆ X t; ) # v d b ~ t x AL d ). From [P4] and Theorem 2.5 we obtain b ~ t ~ L=v d )b t X t; ) L d -a.e. in A, for any t 2 [; T]. Summing up, from 3.13) and 3.14) we infer sup sup 2 ; ) t2q\[;t) B x) f jx t; y) X t; x)j dy ~ LM v d for L d -a.e. x 2 A and we can use the continuity in time of the left hand side to replace the sup on Q \ [; T) by a sup on the whole interval [; T].

17 Lipschitz regularity and approximate differentiability, etc. 45 Thanks to Remark 2.3 this implies that, denoting by B the Borel subset of A where the inequality above is fulfilled, we have 3:15 Lip X t; )j B maxf2m 1 = ; lg: In order to obtain bi-lipschitz estimates we combine forward and backward Lipschitz estimates and use the semigroup property of the flow, as discussed in [8] and [2]. THEOREM 3.4 [bi-lipschitz estimates]. Assume that b fulfils [P1] for some p > 1, [P2], [P3], [P4] and let X t; x) be the flow associated to b. Then for any absolutely continuous probability measure r in and any d >, t 2 [; T] we can find a set M t such that r n M t ) < d and X t; )j Mt is a bi- Lipschitz map. PROOF. Given s; t 2 [; T] we denote by Y t; s; x) the flow associated to b starting from time s so that the 1-parameter flow in Definition 2.4 with d ˆ m, B ˆ b corresponds to Y t; ; x)): for any given s it is characterized by the conditions Y s; s; x) ˆ x; d Y t; s; x) ˆ bt; Y t; s; x) dt ; Y t; s; ) #L d C s L d with C independent of t. Arguing as in [8] see also Remark 6.7 in [2]) one can use the characterization of the 1-parameter flows to obtain the semigroup property p 3:16 Y t; s; x) ˆ Yt; r; Y r; s; x) for all t 2 [; T]; for L d -a.e. x for any s; r 2 [; T]. Notice that the L d -negligible exceptional set N rs a priori depends on r; s. Given d > we can find a ``forward'' set A such that Y t; ; )j A is Lipschitz and r n A) < d=2. By reversing the time variable we can find a ``backward'' set A t such that Y t; ; ) # r n A t ) < d 2 and Y ; t; )j At is Lipschitz with constant l. Finally we define M t so that M t :ˆ A \ Y t; ; ) 1 A t ) n N t : Since M t A we need only to show lower bounds on jx t; x) X t; y)j. To

18 46 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia this aim, notice that for x 2 M t we have x ˆ Y ; ; x) ˆ Y ; t; Y t; ; x) because, by definition, M t \ N t ˆ;. Therefore, for x; y 2 M t, since both Y t; ; x) and Y t; ; y) belong to A t we obtain jx yj ˆjY ; ; x) Y ; ; y)j ˆjY ; t; Y t; ; x) Y ; t; Y t; ; y) j ljy t; ; x) Y t; ; y)j: As a consequence jx t; x) X t; y)j ˆjY t; ; x) Y t; ; y)j jx yj=l for x; y 2 M t. p In the following corollary we give an explicit representation of the density of the absolutely continuous) measures transported by the flow. This enables to compute also integrals of nonlinear functions of the densities, a computation that would be impossible without an explicit representation of the densities themselves. COROLLARY 3.5 [Explicit representation of X t; ) # r]. Under the assumptions of Theorem 3.3, for any absolutely continuous probability measure r ˆ fl d in and any t 2 [;T] we have that the density w t of X t; ) # r w.r.t. L d is representable as f h i 1 3:17 w t ˆ jdet rx t; x)j X t; )j S t for a suitable Borel set S t whose complement is L d -negligible. Furthermore, for any nonnegative Borel functions W, c, we have the change of variables formula f 3:18 W w t )c dy ˆ jdet rx t; x)jw c X t; x)) dx: jdet rx t; x)j PROOF. We recall the area formula for Lipschitz maps see for instance of [9]): if w : A! is a Lipschitz map, then X 3:19 h x)jdet rwj dx ˆ h x) dy A x2a\w 1 y) for any nonnegative Borel function h. By the remarks made in Section 2.1, this formula still holds for maps that are approximately differentiable at any point of A, since we can cover A by a sequence of sets A h such that wj Ah is Lipschitz for any h.

19 Lipschitz regularity and approximate differentiability, etc. 47 Hence, denoting by S 1 the set of points where X t; ) is approximately differentiable, we can apply 3.19) to any Borel set A S 1 with w ˆ X t; ). By applying the semigroup property 3.16) we obtain a Borel set S 2 such that L d n S 2 ) ˆ and with the notation of Theorem 3.4) x ˆ Y ; ; x) ˆ Y ; t; Y t; ; x ˆ Y ; t; X t; x) 8x 2 S 2 ; so that X t; ) is one to one on S 2. Setting S ˆ S 1 \ S 2 we can apply 3.19) with A ˆ S 1 \ X t; ) 1 E) and h ˆ f x S jdet rx t; )j for any Borel set E to obtain f f x) dx ˆ jdet rx t; x)j X t; )j SŠ 1 y) dy: X t;) 1 E) E Since E is arbitrary, this proves 3.17). To prove 3.18) we just apply 3.19) again with f x) h x) ˆ x S x)c X t; x))w : p jdet rx t; x)j In the following theorem we discuss the relation between the approximate differential rx t; x)y and the derivative t; x; y) of the flow considered in [7]. THEOREM 3.6. Assume that b fulfils [P1] for some p > 1, [P2], [P3], [P4], let X t; x) be the flow associated to b. Then for any t 2 [; T] the difference quotients X t; x y) X t; x) t; x; y) :ˆ locally converge in measure in x Rd y as # to t; x; y) ˆrX t; x)y. PROOF. By the very definition of approximate differential the vector fields t; x; y) locally converge in measure in y as # torx t; x)y for any t; x) where rx t; x) is defined. Therefore, since rx t; x) exists for L d - a.e. x 2, one more integration w.r.t. x gives the result. p REMARK 3.7. Using the theory of renormalized solutions for vectorfields of the form B x; y) ˆ b x); rb x)y), developed in [7], one can also

20 48 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia show that X t; x); rx t; x)y is the unique flow associated to B, where ``flow'' is understood in a slightly weaker sense than the one adopted in this paper due to the fact that the vectorfield B fails in this case to satisfy condition [P3]). Furthermore, even case not covered by our results, one can show using the stability results of [7] that that the component t; x; y) of the flow is still representable as L t; x)y for suitable linear maps L t; x) :! precisely L t; x)y is the limit in measure of rx h t; x)y, where X h are the approximating flows). in the W 1;1 loc REMARK 3.8 [Extensions and open problems]. 1) As the proof of Theorem 3.3 clearly shows, the W 1;p loc regularity for some p > 1 can be weakened by requiring [P1] with p ˆ 1, [P2], [P3] and 3:2 T B R ) jrb t j ] x) dxdt < 1 8R > : Equivalently, we may require that T B R ) jrb t x)j ln 2 jrb t x)j) dxdt < 1 8R > : One can also notice, in the same spirit of [5], that the expression of r in 3.11) involves only the symmetric difference quotients of b t, namely those of the form hb t x y) b t x); yi jyj that can be controlled using only the symmetric part of the derivative. Therefore, still keeping [P2] and [P3], [P1] and 3.2) can be replaced by T B R ) j rb t ) rb t ) T j ] x) dxdt < 1 8R > ; requiring only that the symmetric part of the distributional derivative of b t is in L 1 loc ; by the results in [5] the flow is well defined also under these weaker conditions.

21 Lipschitz regularity and approximate differentiability, etc. 49 2) The local integrability of the maximal function of b t plays an essential role in the pointwise estimate of r, necessary in order to apply Lemma 3.2. Therefore, as we said in the introduction, it is not clear whether our results can be extended to the W 1;1 case or even to the BV case considered in [2]. 3) The argument used in the proof of Theorem 3.3 does not lead to an explicit bound of the Lipschitz constant of X t; ) as a function of d. More precisely, assuming for simplicity that jbj is globally bounded, we have clearly that the constant M 1 depends only on R, while M 2 and therefore M can be estimated from above with C R)=d. Hence l e C R)=d can be estimated explictly and gives a bound on the L 1 norm of jrxj on [; T] A. On the other hand, due to the application of Egorov theorem, the global Lipschitz constant of X t; )j A depends also on, as 3.15) shows. More precisely, we have jx t; x) X t; y)j ljx yj 8x; y 2 A with jx yj ; t 2 [; T]: REFERENCES [1] L. AMBROSIO -N. FUSCO -D. PALLARA, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2. [2] L. AMBROSIO, Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae, ), pp [3] L. AMBROSIO, Lecture notes on transport equation and Cauchy problem for BV vector fields and applications. Preprint, 24 available at cvgmt.sns.it). [4] L. AMBROSIO -J.MALY Â, Very weak notions of differentiability. Preprint, 25 available at [5] I. CAPUO DOLCETTA -B. PERTHAME, On some analogy between different approaches to first order PDE's with nonsmooth coefficients. Adv. Math. Sci Appl., ), pp [6] F. COLOMBINI -N.LERNER, Uniqueness of continuous solutions for BV vector fields. Duke Math. J., ), pp [7] C. LE BRIS - P. L. LIONS, Renormalized solutions of some transport equations with partially W 1;1 velocities and applications. Annali di Matematica, ), pp [8] R. J. DI PERNA - P. L. LIONS: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., ), pp [9] H. FEDERER, Geometric Measure Theory. Springer, [1] N. LERNER: Transport equations with partially BV velocities. Preprint, 24. [11] P. L. LIONS, Sur les eâquations diffeârentielles ordinaires et les eâquations de transport. C. R. Acad. Sci. Paris SeÂr. I, ), pp

22 5 Luigi Ambrosio - Myriam Lecumberry - Stefania Maniglia [12] E. M. STEIN: Singular integrals and differentiability properties of functions. Princeton University Press, 197. Manoscritto pervenuto in redazione il 28 febbraio 25