Lipschitz regularity and approximate differentiability of the Diperna-Lions flow

Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow

LUIGIAMBROSIO(*) - MYRIAMLECUMBERRY(**) - STEFANIAMANIGLIA(***)

1. Introduction.

In a recent paper [7] Le Bris and Lions studied, among other things, the differentiability properties of the flow X(t;x):[0;T]R^d!R^d as- sociated to a vectorfieldb:(0;T)R^d!R^dhaving 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)

" ! Z(t;x;y)

…1:1†

where the convergence, as "#0, occurs with respect to the local con- vergence in measure inR^d_xR^d_y, and uniformly in time. The mapZ(t;x;y) can be considered, according to this limiting procedure, a kind of ``deri- vative'' of the flowX(t;) atxalong the directiony.

This result raises several questions about the nature of Z 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 inR^d_xR^d_yto the a.e. convergence to 0 as"#0 of the quantities

Z

BR(0)

1^ X(t;x‡"y) X(t;x)

" Z(t;x;y)

dy R>0:

…1:2†

(*) Indirizzo dell'A.: Scuola Normale Superiore, Piazza dei Cavalieri, 56126 Pisa, Italy; e-mail: [email protected]

(**) Indirizzo dell'A.: Max-Planck Institut fuÈr Mathematik, Inselstrasse 22-26, D-04103 Leipzig, Germany; e-mail: [email protected]

(***) Indirizzo dell'A.: Dipartimento di Matematica, UniversitaÁ di Pisa, 56126 Pisa, Italy; e-mail: [email protected]

Notice that elementary Fubini-type arguments show that this passage is possible only for a sequence ("i)#0, but the convergence to 0 of the in- tegrals (1.2)onlyalong 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 (fortfixed) is found, involving only thexvariable, and an example that no Lipschitz property can be derived from this weak differ- entiability property is explicitely given.

Assuming for the sake of simplicity in this introductory discussion that bis autonomous and that bothband its divergence are globally bounded, we are able to answer positively these questions under an assumption slightly stronger thanW_loc^1;1, namely that the local maximal function ofjrbj belongs toL¹_loc(this holds if and only ifjrbjln(2‡ jrbj)2L¹_loc). Under this assumption we show in Theorem 3.3 that Z(t;x;y) is representable as L(t;x)y for suitable linear maps L(t;x):R^d!R^d (see also [4] and Re- mark 3.7); moreover, for any ballB_R(0) and anyd>0 we can find a Borel setABR(0) such that

L^d…BR(0)nA†<d and X(t;)j_A is a Lipschitz map for anyt2 [0;T]:

It turns also out that indeed the mapL(t;x) can be characterizedL^d‡1-a.e.

in [0;T]Aas 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). Fur- thermore, 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 theexplicitformula 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 inR^2d

Y^"(t;x;y):ˆ X(t;x);X(t;x‡"y) X(t;x)

"

associated to the vectorfields

b(x);b(x‡"y) b(x)

"

and on the theory of renormalized solutions for the limit vectorfield (b(x);rb(x)y) (see also [10] for related results in aBVcontext). Our strategy

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 oflnjrX(t;)jcan be controlled byjrbj(X(t;x));

looking for a suitable discrete counterpart of this fact we considered the quantities (here and in the sequelR

denotes the averaged integral)

b~^"_t(x):ˆ

Z

B"(x)

f jX(t;y) X(t;x)j

"

dy;

wheref(s) is of the formln(1‡s^l) for somel0. Their formal limit is Z

B1(0)

f…jrX(t;x)yj†dy;

a quantity comparable tof(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(0)(x)x_B1(0)(y) under the mapY^"(t;) to obtain thatb^"_t satisfy a transport inequality

d

dtb^"_t ‡Dx(bb^"_t)r_t^" with r^"_t(x):ˆl^d

Z

Bl(0)

jb(x‡"y) b(x)j

"jyj w^"_t(x;y)dy;

whose right hand side can be controlled by the maximal function ofjrbj.

Standard representation results for the solutions of transport problems then give estimates from above onb^"_t and then onb~^"_t.

It is not clear whether our argument can be improved, getting Lipschitz properties in the W_loc^1;1 case, or even in the BVloc case considered in [2].

Some extensions of our result, together with some other open problems, are discussed in Remark 3.8.

2. Notation and preliminary results.

Given a mapw(t;x) depending on time and space, we will systematically use the notationwtfor the mapx7!w(t;x), while a derivative with respect to time will be denoted byf_in the case of ODE's and by_dt^d f in the case of PDE's. The least Lipschitz constant of a Lipschitz function f will be de- noted by Lip f.

We denote byL^dthe Lebesgue measure inR^dand byvdthe Lebesgue measure of the unit ball ofR^d. Recall that a sequence of Borel maps (f_h) is said to be locally convergent in measure tof if

h!1lim L^d…fx2B_R(0): jf_h(x) f(x)j>dg† ˆ0 8R>0; d>0:

Equivalently, one can say that 1^ jf_h fj !0 inL¹_loc(R^d).

2.1 ±Approximate differentiability.

We start by recalling the classical definition of approximate differ- entiability: a Borel mapX:R^d!R^m is said to beapproximately differ- entiableatx2R^dif there exists a linear mapL:R^d!R^m such that the difference quotients

y7!X(x‡"y) X(x)

"

locally converge in measure as"#0 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 mapX, differentiable in the classical sense at~ x, such thatX(x)~ ˆX(x) and the coincidence set fy: X(y)ˆX(y)g~ has density 1 atx. The latter for- mulation can be used, in conjunction with Rademacher theorem, to show that ifXj_Ais a Lipschitz map for some setAR^d, thenXis approximately differentiable atL^d-almost any point ofA: it suffices to find a Lipschitz extensionX~ to the whole of R^d of Xj_A (see for instance 2.10.43 of [9]) to obtain the approximate differentiability property at any point of density 1 ofAwhereX~ is classically differentiable. It is worth to mention also (see 3.1.8 of [9]) a converse statement: approximate differentiability at any point of a Borel setAimplies that we can coverAby an increasing family of Borel setsA_hsuch that the restriction ofXj_Ahis a Lipschitz map for anyh.

In connection with Sobolev (or evenBV) functions, the following clas- sical result holds (see for instance [1], Lemma 3.81 and Theorem 3.83):

THEOREM2.1 [Approximate differentiability of Sobolev functions]. Let VR^dbe an open set and let f 2W_loc^1;1(V;R^m). Then we have

limr#0

Z

Br(x)

jf(y) f(x) rf(x)(y x)j

jy xj dyˆ0 for L^d-a:e:x2V:

…2:1†

Furthermore Z

Br(x)

jf(y) f(x)j jy xj dy

Z¹

0

Z

Btr(x)

jrfj(y)dydt for any ball Br(x)V:

…2:2†

In the following theorem we state a basic criterion for approximate differentiability: basically it says that if the asymptotic L¹ norm of trun-

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 withf(t)ˆln(1‡t^l) withlsufficiently large.

THEOREM 2.2. Let f_i:[0;‡1)![0;‡1) be subadditive and non- decreasing functions such thatsup_isupfiˆ ‡1, and let X:R^d!R^mbe a Borel map. Assume that

lim sup

i!1 lim sup

r#0

Z

Br(x)

f_i jX(y) X(x)j r

dy<‡1 8x2A

for some Borel set AR^d. Then X is approximately differentiable atL^d- a.e. x2A.

PROOF. We denote byc(x) the double limsup appearing in the state- ment and we assume with no loss of generality thatL^d(A)<‡1. Since c(x) is finite for anyx2A, for any" >0 we can find a compact setKA andM2Rsuch thatL^d(AnK)< "andcM 1 onK,jXj MonK.

Furthermore, by applying Egorov theorem to the family of functions gk(x):ˆsup

ik lim sup

r#0

Z

Br(x)

fi jX(y) X(x)j r

dy x2K

we can find a compact setK⁰KsatisfyingL^d(KnK⁰)< "such that lim sup

r#0

Z

Br(x)

f_i jX(y) X(x)j r

dy<M 8x2K⁰

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 chooseiin such a way that

fi(lM)> 2Mvd

c_d for some lM0

and we apply in an analogous way Egorov theorem again to find a compact setK⁰⁰K⁰such thatL^d(K⁰nK⁰⁰)< "and

Z

Br(x)

f_i jX(y) X(x)j r

dyM 8x2K⁰⁰ …2:3†

for r<r0, with r0>0 independent of x. Notice that by construction L^d(AnK⁰⁰)<3".

We now claim that the restriction ofXtoK⁰⁰is a Lipschitz map. Indeed, for any pair of points x;y2K⁰⁰ we can estimate jX(x) X(y)j with 2Mjx yj=r₀ifjx yj r₀. Ifr:ˆ jx yj<r₀we apply (2.3) twice and the subadditivity offito obtain

1 v_dr^d

Z

Br(x)\Br(y)

f_i jX(x) X(y)j r

dz

Z

Br(x)

f_i jX(z) X(y)j r

dz‡ Z

Br(y)

f_i jX(z) X(x)j r

dz2M:

SinceL^d(Br(x)\Br(y))ˆc_dr^dwe obtain f_i jX(x) X(y)j

r

2v_dM cd ; so that our choice ofl_Mand the monotonicity off_igive

jX(x) X(y)j l_Mjx yj: p REMARK 2.3. Let f :[0;‡1)![0;‡1) be a subadditive and non- decreasing function. The argument used in the proof of Theorem 2.1 shows that the conditions

r2(0;rsup0)

Z

Br(x)

f jX(y) X(x)j r

dyM and jXj M₁ on A

for someM0,M10,r0>0 imply that Lip(Xj_A)max 2M₁

r0 ;l

providedf(l)>2Mv_d=c_d.

2.2 ±Maximal functions.

Letf 2L¹_loc(R^d) be a nonnegative function. Thelocal maximal function f^]is defined by

f^](x):ˆ sup

t2(0;1)

Z

Bt(x)

f(y)dy:

It is well known (see for instance [12]) that the weakL¹estimate L^dfx2BR(0): f^](x)>lg

C(d) l

Z

BR‡1(0)\ff>lg

f(y)dy 8l>0

gives thatf^] is finiteL^d-a.e., and that Z

BR(0)

f^]pdxC(d)p2^2p p 1

Z

BR‡1(0)

jfj^pdx 8p2(1;1):

…2:4†

In the critical casepˆ1 we have Z

BR(0)

f^]dxv_dR^d‡C(d) Z

BR‡1(0)

fln(2‡f)dx:

…2:5†

2.3 ±Flow associated to a vectorfield.

In this section we consider a vectorfield B(t;z)ˆB_t(z) satisfying the following conditions:

[P1]B2L¹[0;T];W_loc^1;p(R^m;R^m)

; [P2]_1‡jzj^jBj 2L¹ [0;T];L¹(R^m)

‡L¹…[0;T];L¹(R^m)†;

[P3] [divB_t] 2L¹…[0;T];L¹(R^m)†.

We denote byLthe constant L:ˆeRT

0 k[divBt] k₁dt: …2:6†

If also

[P4] [divBt]^‡2L¹…[0;T];L¹(R^m)† holds, we set

L~ :ˆeRT

0 k[divBt]^‡k1dt: …2:7†

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 mapx7!X(;x) with values in the space of continuous maps, while in [8] it is considered as a continuous map t7!X(t;), with a suitable metric in the space of measurable maps in R^d 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].

DEFINITION2.4 [Flow]. We say that Y(t;z):[0;T]R^m!R^mis a flow (starting at time 0) relative to a vectorfield B(t;z) if the following two conditions are satisfied:

(a)forL^m-a.e. z2R^mthe map t7!Y(t;z)is an absolutely continuous integral solution of the ODEg_ˆB(t;g)in[0;T], withg(0)ˆz;

(b)Y(t;)_#L^mCL^m for some constant C independent of t.

An important consequence of condition (b), that we will use later on, is the property

Z

R^m

L¹…ft2(0;T): (t;Y(t;x))2Ng†dxˆ0 …2:8†

for anyL^m‡1-negligible setN(0;T)R^m. Indeed, denoting byN_tthe t-sections ofN, we have that the integral above equals

Z^T

0

L^m…fx2R^m: Y(t;x)2Ntg†dtˆ Z^T

0

Y(t;)_#L^m(Nt)dtˆ0;

where in the last equality we used the fact thatNt isL^m-negligible for L¹-a.e.t2(0;T).

THEOREM2.5. Under assumptions[P1], [P2], [P3]there exists a flow, uniquely determined in[0;T]R^mup to sets withL^m-negligible projec- tion onR^m. Moreover, property (b) holds with CˆL, 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

k[divBht]k₁ k[divBt]k₁; …2:9†

jB_hj

1‡ jzj2L¹…[0;T];L¹(R^m)†; Bh2L¹[0;T];W_loc^1;1(R^m;R^m) and such that the classical flows Y_hassociated to B_hsatisfy

h!1lim Z

BR(0)

t2[0;T]maxjY_h(t;z) Y(t;z)j ^1dzˆ0 8R>0:

(b)If[P4]holds, then Y(t;)_#L^mL~ ¹L^m, withL as in~ …2:7†.

PROOF. The existence of the flow is proved in [8], together with its uni- queness according to the definition of flow adopted therein. Uniqueness ac- cording 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] (see also [3]) by taking asB_hthe standard mollifications ofBw.r.t. the space variable. Statement (b) can be easily proved by approximation, using the explicit expression for the densities ofYh(t;)_#L^d, namely

1

detrY_ht;[Y_h(t;)] ¹(x): SinceD_h(t;x):ˆdetrY_h(t;x) solves the ODE

[D⁰_h(t;x)ˆ(divB_ht(Y_h(t;x)))D_h(t;x)];

taking (2.9) into account with the positive parts we obtain a uniform upper bound onD_hand therefore a uniform lower bound on the densities. p LEMMA 2.6[Logarithmic sup estimate]. Let Y be a flow relative to B.

Then Z

BR(0)

max[0;T]ln 1‡ jY(t;z)j 1‡R

dz kBk 8R>0;

…2:10†

wherekBkdenotes the infimum of all sums Lk B¹

1‡ jzjk_L1(L¹)‡v_mR^mk B²

1‡ jzjk_L1(L¹);

among all decompositions ofjBj=(1‡ jzj) andLis defined in (2.6).

PROOF. Letwtbe the density ofY(t;)_#x_BR(0)L^mw.r.t.L^mand notice that kwtk₁ˆvmR^m and kwtk₁ L, by property (b) of the flow. Using property (a) of the flow we get

Z

BR(0)

max[0;T] ln 1‡ jY(t;z)j 1‡R

dz Z

BR(0)

Z^T

0

jY(t;_ z)j 1‡ jY(t;z)jdtdz

ˆ Z^T

0

Z

BR(0)

jBt(Y(t;z))j 1‡ jY(t;z)jdzdt

Z^T

0

Z

R^m

jBtjwt

1‡ jzjdzdt:

Splitting jBj=(1‡jzj) in the sum of a function in L¹(L¹) and a function inL¹(L¹) and minimizing among all possible decompositions we obtain

(2.10). p

3. Approximate differentiability of the flow.

LEMMA 3.1. Assume that b:(0;T)R^d!R^d fulfils [P1], [P2], [P3]

and let X(t;x)be the flow associated to b. Let" >0and let

Y^"(t;x;y):ˆ X(t;x);X(t;x‡"y) X(t;x)

"

:

Then Y_"is the flow relative to the vectorfield B^"(t;x;y)ˆB^"_t(x;y)inR^2d defined by

B^"(t;x;y):ˆ b_t(x);bt(x‡"y) bt(x)

"

: …3:1†

In particular condition (b) is fulfilled with CˆL², where L:ˆeRT

0 k[divbt] k₁dt: …3:2†

PROOF. It is immediate to check that the conditionX(t;_ x)ˆb_t(X(t;x)) L¹-a.e. in [0;T] for L^d-a.e.ximplies thatY_^"(t;x;y)ˆB^"_t(Y^"(t;x;y))L¹- a.e. in [0;T] forL^2d-a.e. (x;y) (precisely, forL^d-a.e.x, the property holds for anyy). In order to check thatY^"(t;)_#L^2dL²L^2d(withLas in (3.2)) we write the inequality in an integral form

Z

R^d

W X(t;x);X(t;x‡"y) X(t;x)

"

dxL² Z

R^dR^d

W(x;y)dxdy

for any nonnegativeW2Cc(R^dR^d) and we notice that the property is trivially true if bt2C¹ (indeed, in this case the divergence of B^"_t(x;y) is

divb_t(x)‡divb_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 appli-

cation of Theorem 2.5(a) to the vectorfieldb. p

LEMMA 3.2. Assume that b:(0;T)R^d!R^d fulfils [P1], [P2], [P3]

and let X(t;x)be the flow associated to b. Letbbe a bounded nonnegative function satisfying

d

dtb‡D_x(bb)ˆr2L¹_loc[0;T)R^d …3:3†

in the sense of distributions, and assume that

(i)t7!b_tis w-continuous between[0;T)and L¹(R^d);

(ii) sup

[0;T]AjX(;x)j<‡1for some Borel set AR^d. Then we have

t2Q\[0;T)sup b_t(X(t;x))Lb₀(x)‡L Z^T

0

jrj_s(X(s;x))ds

forL^d-a.e. x2A.

PROOF. We first extend the PDE to negative times (as done im- mediately before Theorem 4.1 in [2]), settingbtˆ0 andrtˆ0 fort<0, and b_tˆb₀fort<0. Accordingly, we setX(t;x)ˆxfort<0. Then we mollify w.r.t. the space variable both sides to obtain smooth functionsb^d_t ˆb_tr_d such that

d

dtb^d‡D_x(bb^d) jrj^d in ( 1;T) R^d withjrj^d! jrjinL¹_loc ( 1;T)R^d

asd#0 (by the commutator estimate in [8]). Finally we mollify again w.r.t. the time variable both sides to obtain smooth functionsb^d;hˆ(b^d)r_hsuch that

d

dtb^d;h‡Dx(bb^d;h)c^d;h in ( 1;T h) R^d with

c^d;h:ˆ jrj^dr_h‡ j(D_x(bb^d))r_h D_x(b(b^dr_h))j:

Notice that the smoothness ofb^d w.r.t. the spatial variables immediately gives, expanding the spatial divergences, that c^d;h! jrj^d in L¹_loc ( 1;T)R^d

as h#0. The smoothness ofb^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)R^d)

d

dtb^d;h‡b rb^d;h divbtb^d;h‡c^d;h immediately gives

d dt

heRt

1divbt(X(t;x))dtb^d;h_t (X(t;x))i

eRt

1divbt(X(t;x))dtc^d;h_t (X(t;x)) 8t2( 1;T h) for a.e.x2A. Precisely, this happens for thosex2Asuch that the trans-

port inequality above fails atX(t;x) for a set of timestwith strictly positive 1-dimensional Lebesgue measure; this set isL^d-negligible by (2.8). Hence, ifh2(0;1) we get

b^d;h_t (X(t;x))Lb^d;h₁(x)‡L Z^t

1

c^d;h_s (X(s;x))ds …3:4†

ˆLb₀r_d(x)‡L Z^t

1

c^d;h_s (X(s;x))ds:

For anyt2[0;T) we can use thew-continuity property oft7!b_t(ensuring the strong convergence ofb^d;h_t tob^d_t ash#0) to pass to the limit ash#0 in (3.4), using also condition (b) in Definition 2.4, to obtain

b^d_t(X(t;x))Lb₀r_d(x)‡L Z^t

1

jrj^d_s(X(s;x))dsˆ

ˆLb₀r_d(x)‡L Z^t

0

jrj^d_s(X(s;x))ds

forL^d-a.e.x2A. Passing now to the limit asd#0 and using again condition (b) in Definition 2.4, we eventually obtain

b_t(X(t;x))Lb₀(x)‡L Z^t

0

jrj_s(X(s;x))ds forL^d-a.e.x2 A

for anyt2[0;T). By lettingtvary in the countable setQ\[0;T), we obtain

t2Q\[0;T)sup b_t(X(t;x))Lb₀(x)‡L Z^T

0

jrj_s(X(s;x))ds …3:5†

forL^d-a.e.x2A. p

THEOREM3.3 [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(0)and anyd>0we can find a Borel set AB_R(0)such that L^d(B_R(0)nA)<dand the restriction of X(t;)to A is a Lipschitz map for any t2[0;T].

In particular X(t;)is approximately differentiableL^d-a.e. inR^dfor any t2[0;T].

PROOF. We consider the flowY^"and the associated vectorfieldB^"as in Lemma 3.1 and a ballBR(0). By Lemma 2.6 we obtain

L^d fx2BR(0): max

t2[0;T]jX(t;x)j>Mg

ln 1‡M 1‡R

₁

kbk for any M>R, hence we can find a constant M₁>R and a Borel set A1BR(0) such thatL^d(BR(0)nA1)<d=2 and

max[0;T]jX(t;x)j M1 8x2A1: …3:6†

We define

N:ˆ

Z

B1(0)

ln(1‡ jyj)dy:

Since Z

A1

Z^T

0

jrbsj(X(s;x))ds dxL Z^T

0

Z

BM1(0)

jrbsj(y)dyds<‡1;

we can findM₂such that

L^d(A1nA)<d=2 with A:ˆ

(

x2A1: Z^T

0

jrbsj(X(s;x))ds<M2

) : Eventually we defineM:ˆNL‡vdL²M2and chooselsufficiently large, such thatln(1‡l)>2ML=c~ d, where

L~ :ˆeRT

0 k[divbt]^‡k1dt: Notice that by constructionL^d(BR(0)nA)<d.

Step 1. We fix the initial measure mˆx_A(x)x_B1(0)(y)L^2d to obtain, by Lemma 3.1, thatY^"(t;)_#mL²L^2d, withLdefined in (3.2). We denote by

w^"_t(x;y) the density ofY^"(t;)_#mw.r.t.L^2dand notice thatkw^"k₁L²and

thatw^"solve the following Cauchy problem for the continuity equation:

d

dtw^"‡D_x;y(B^"w^")ˆ0; w^"(0;x;y)ˆx_A(x)x_B1(0)(y):

…3:7†

Moreover, for any nonnegativeW2C_c(R^d) and anyt2[0;T] we have Z

R^d

W(x) Z

R^d

w^"_t(x;y)dydxˆ

Z

AB1(0)

W(X(t;x))dxdyLvd

Z

R^d

W(x)dx

and therefore Z

R^d

w^"_t(x;y)dyLv_d forL^d-a.e.x; for anyt2 [0;T]:

…3:8†

We are going to apply Remark 2.3 with the function f(t):ˆln(1‡

‡t^l). To this aim we define

b^"_t(x):ˆ

Z

R^d

f(jyj)w^"_t(x;y)dy:

Step 2.(estimates onb^") Letc2C_c¹( 2;2) withc1 on [ 1;1] and let c_R(y)ˆc(y=R). Using the test functionW(x)(fc_R)(jyj), withW2C_c¹(R^d), in (3.7) gives

d dt

Z

R^d

W(x) Z

R^d

(fc_R)(jyj)w^"_t(x;y)dydxˆ …3:9†

ˆ Z

R^d

hrW(x);b_t(x)i Z

R^d

(fc_R)(jyj)w^"_t(x;y)dydx‡

‡ Z

R^d

W(x) Z

Bl(0)

hbt(x‡"y) bt(x);yi

"jyj(1‡ jyj) c_R(jyj)w^"_t(x;y)dydx‡

‡ Z

R^d

W(x) Z

R^d

c⁰_R(jyj)hb_t(x‡"y) b_t(x);_jyj^yi

" f(jyj)w^"_t(x;y)dydx

in the distribution sense in (0;T). Using (3.8), the contribution ofb(x) in the last integral in (3.9) can be estimated by

ln(1‡l)kc⁰k₁ R"

Z

R^d

jW(x)kb(x)j Z

R^d

w^"_t(x;y)dydx

Lv_dln(1‡l)kc⁰k₁ R"

Z

R^d

jW(x)kb(x)jdx:

Using the inequality 1‡ jx‡"yj C‡2"R for x2suppW and y2suppc_R, and writingb=(1‡ jzj) asA‡A⁰withjAj 2L¹ [0;T];L¹(R^d) and jA⁰j 2L¹ [0;T];L¹(R^d)

we can also estimate the contribution of

b(x‡"y) in the last integral of (3.9) as follows:

(C‡2R")ln(1‡l)kc⁰k₁ R"

Z

R^d

jW(x)j Z

fjyjRg

L²jA_tj(x‡"y)‡kA⁰_tk₁w^"_t(x;y)dy dx:

Hence, passing to the limit asR! 1in (3.9), the dominated convergence theorem gives

d dt

Z

R^d

W(x)b^"_t(x)dx ˆ

ˆ Z

R^d

hrW(x);bt(x)ib^"_t(x)dx‡ Z

R^d

W(x) Z

Bl(0)

hbt(x‡"y) bt(x);yi

"jyj(1‡ jyj) w^"_t(x;y)dydx:

SinceWis arbitrary this proves that d

dtb^"‡Dx(bb^")r^"; b^"(0;x)ˆx_A(x)

Z

B1(0)

f(jyj)dy …3:10†

with

r^"(t;x):ˆ

Z

Bl(0)

jhbt(x‡"y) bt(x);yij

"jyj² w^"_t(x;y)dy

L² Z

Bl(0)

jhb_t(x‡"y) b_t(x);yij

"jyj² dy:

By (2.2) in Theorem 2.1 we get

r^"(t;x)L²v_dl^djrb_tj^](x) for l"1:

…3:11†

Notice thatjrb_tj^]2L¹_loc [0;T]R^d

because of the maximal estimate (2.4).

Moreover, by (2.1) and (3.8) we infer lim sup

"#0 r_t^"(x)Lv_djrbtj(x)

…3:12†

forL^d‡1-a.e. (t;x)2(0;T)R^d.

Therefore, by applying Lemma 3.2 and using the definition of Aand

(3.6), Fatou Lemma gives lim sup

"#0 sup

t2Q\[0;T)b^"_t(X(t;x))lim sup

"#0 Lb(0;x)‡L

Z^T

0

jr^"j_s(X(s;x))ds

LN‡v_dL² Z^T

0

jrb_sj(X(s;x))ds<M

for L^d-a.e. x2A. Notice that the bound (3.11) and the integrability of jrb_tj^]are used to ensure that Fatou lemma is applicable.

By Egorov theorem, possibly passing to a slightly smaller setA still satisfyingL^d(BR(0)nA)<d, we can assume the existence of"0>0 such that

"2(0;"sup0) sup

t2Q\[0;T]b^"_t(X(t;x))M forL^d-a.e.x2 A:

…3:13†

Step 3.(Conclusion) We now claim that, setting

~b^"_t(x):ˆ

Z

B1(0)

f jX(t;x‡"y) X(t;x)j

"

dyˆ Z

B"(x)

f jX(t;y) X(t;x)j

"

dy

the inequality

b~^"_t(x) L~

v_db^"_t(X(t;x)) L^d-a.e. inA; for any t2 [0;T]

…3:14†

holds. Indeed, recalling thatw^"_tis the density ofY^"(t;)_#(x_A(x)x_B1(0)(y)L^2d) w.r.t.L^2d, we have the identity

Z

R^d

W(x)b^"_t(x)dxˆ Z

R^dR^d

W(x)f(jyj)w^"_t(x;y)dxdyˆv_d Z

A

W(X(t;x))b~^"_t(x)dx;

that tells us thatb^"_tL^dˆX(t;)_#(v_db~^"_tx_AL^d). From [P4] and Theorem 2.5 we obtainb~^"_t (L=v~ _d)b^"_tX(t;)L^d-a.e. inA, for anyt2[0;T].

Summing up, from (3.13) and (3.14) we infer

"2(0;"sup0) sup

t2Q\[0;T)

Z

B"(x)

f jX(t;y) X(t;x)j

"

dyLM~ v_d

forL^d-a.e.x2Aand we can use the continuity in time of the left hand side to replace the sup on Q\[0;T) by a sup on the whole interval [0;T].

Thanks to Remark 2.3 this implies that, denoting byBthe Borel subset of Awhere the inequality above is fulfilled, we have

LipX(t;)j_B maxf2M₁="₀;lg:

…3:15† p

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 measurerinR^dand anyd>0, t2[0;T]we can find a set Mtsuch thatr(R^dnMt)<dand X(t;)j_Mtis a bi- Lipschitz map.

PROOF. Givens;t2[0;T] we denote byY(t;s;x) the flow associated to bstarting from times(so that the 1-parameter flow in Definition 2.4 with dˆm,Bˆbcorresponds toY(t;0;x)): for any givensit is characterized by the conditions

Y(s;s;x)ˆx; d

dtY(t;s;x)ˆb t;… Y(t;s;x)†; Y(t;s;)_#L^dCsL^d withCindependent oft. Arguing as in [8] (see also Remark 6.7 in [2]) one can use the characterization of the 1-parameter flows to obtain the semi- group property

Y(t;s;x)ˆY t;… r;Y(r;s;x)† for allt2 [0;T]; forL^d-a.e.x …3:16†

for any s;r2[0;T]. Notice that the L^d-negligible exceptional set Nrs a priori depends onr;s.

Given d>0 we can find a ``forward'' setAsuch thatY(t;0;)j_Ais Lip- schitz and r(R^dnA)<d=2. By reversing the time variable we can find a

``backward'' setA_t such that

Y(t;0;)_#r(R^dnA_t)<d 2

andY(0;t;)j_At is Lipschitz with constantl. Finally we defineM_tso that M_t :ˆA\Y(t;0;) ¹(A_t)nN_t0:

SinceMtAwe need only to show lower bounds onjX(t;x) X(t;y)j. To

this aim, notice that forx2Mtwe have

xˆY(0;0;x)ˆY…0;t;Y(t;0;x)†

because, by definition,M_t\N_t0ˆ ;. Therefore, forx;y2M_t, since both Y(t;0;x) andY(t;0;y) belong toA_twe obtain

jx yj ˆ jY(0;0;x) Y(0;0;y)j ˆjY…0;t;Y(t;0;x)† Y…0;t;Y(t;0;y)†j ljY(t;0;x) Y(t;0;y)j:

As a consequencejX(t;x) X(t;y)j ˆ jY(t;0;x) Y(t;0;y)j jx yj=lfor

x;y2Mt. 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 den- sities, a computation that would be impossible without an explicit re- presentation of the densities themselves.

COROLLARY 3.5 [Explicit representation of X(t;)_#r]. Under the as- sumptions of Theorem 3.3, for any absolutely continuous probability measure rˆfL^d in R^d and anyt2[0; T]we have that the density w_t ofX(t;)_#rw.r.t.L^d is representable as

wt ˆ f

jdetrX(t;x)jhX(t;)j_Sti ₁ …3:17†

for a suitable Borel set StR^d whose complement is L^d-negligible.

Furthermore, for any nonnegative Borel functionsW,c, we have the change of variables formula

Z

R^d

W(w_t)cdyˆ Z

R^d

jdetrX(t;x)jW f jdetrX(t;x)j

c(X(t;x))dx:

…3:18†

PROOF. We recall the area formula for Lipschitz maps (see for in- stance 3.2.3 of [9]): ifw:AR^d!R^dis a Lipschitz map, then

Z

A

h(x)jdetrwjdxˆ Z

R^d

X

x2A\w ¹(y)

h(x)dy …3:19†

for any nonnegative Borel functionh. By the remarks made in Section 2.1, this formula still holds for maps that are approximately differentiable at any point ofA, since we can coverAby a sequence of setsA_hsuch thatwj_Ah is Lipschitz for anyh.

Hence, denoting byS1the set of points whereX(t;) is approximately differentiable, we can apply (3.19) to any Borel setAS1withwˆX(t;).

By applying the semigroup property (3.16) we obtain a Borel setS₂ such thatL^d(R^dnS₂)ˆ0 and (with the notation of Theorem 3.4)

xˆY(0;0;x)ˆY…0;t;Y(t;0;x† ˆY…0;t;X(t;x)† 8x2S₂; so thatX(t;) is one to one onS2. SettingSˆS1\S2we can apply (3.19) withAˆS1\X(t;) ¹(E) and

hˆ fx_S jdetrX(t;)j for any Borel setER^d to obtain

Z

X(t;) ¹(E)

f(x)dxˆ Z

E

f

jdetrX(t;x)j‰X(t;)j_SŠ ¹(y)dy:

SinceEis arbitrary, this proves (3.17). To prove (3.18) we just apply (3.19) again with

h(x)ˆx_S(x)c(X(t;x))W f(x) jdetrX(t;x)j

: p

In the following theorem we discuss the relation between the approx- imate differential rX(t;x)y and the derivative Z(t;x;y) of the flow con- sidered 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 t2[0;T]the dif- ference quotients

Z^"(t;x;y):ˆX(t;x‡"y) X(t;x)

"

locally converge in measure inR^d_xR^d_yas"#0to Z(t;x;y)ˆ rX(t;x)y.

PROOF. By the very definition of approximate differential the vector fieldsZ^"(t;x;y) locally converge in measure inR^d_yas"#0 torX(t;x)yfor any (t;x) whererX(t;x) is defined. Therefore, sincerX(t;x) exists forL^d- a.e.x2R^d, one more integration w.r.t.xgives the result. p REMARK 3.7. Using the theory of renormalized solutions for vector- fields of the form B(x;y)ˆ(b(x);rb(x)y), developed in [7], one can also

show that

X(t;x);rX(t;x)y

… †

is theunique flowassociated toB, where ``flow'' is understood in a slightly weaker sense than the one adopted in this paper (due to the fact that the vectorfieldBfails in this case to satisfy condition [P3]). Furthermore, even in theW_loc^1;1case not covered by our results, one can show using the stability results of [7] that that the component Z(t;x;y) of the flow is still re- presentable asL(t;x)yfor suitable linear mapsL(t;x):R^d!R^d(precisely L(t;x)y is the limit in measure of rX_h(t;x)y, where X_h are the approx- imating flows).

REMARK3.8 [Extensions and open problems]. (1) As the proof of The- orem 3.3 clearly shows, the W_loc^1;p regularity for somep>1 can be wea- kened by requiring [P1] withpˆ1, [P2], [P3] and

Z^T

0

Z

BR(0)

jrbtj^](x)dxdt<‡1 8R>0:

…3:20†

Equivalently, we may require that Z^T

0

Z

BR(0)

jrbt(x)jln(2‡ jrbt(x)j)dxdt<‡1 8R>0:

One can also notice, in the same spirit of [5], that the expression ofr^" in (3.11) involves only thesymmetricdifference quotients ofb_t, namely those of the form

hbt(x‡"y) bt(x);yi

"jyj

that can be controlled using only the symmetric part of the derivative.

Therefore, still keeping [P2] and [P3], [P1] and (3.20) can be replaced by Z^T

0

Z

BR(0)

j(rb_t)‡(rb_t)^Tj^](x)dxdt<‡1 8R>0;

requiring only that the symmetric part of the distributional derivative ofb_t is inL¹_loc; by the results in [5] the flow is well defined also under these weaker conditions.

(2) The local integrability of the maximal function of bt plays an es- sential 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 whe- ther our results can be extended to theW^1;1case or even to theBV 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 ofX(t;) as a function ofd. More precisely, assuming for simplicity that jbj is globally bounded, we have clearly that the constantM₁depends only onR, whileM₂and thereforeM can be estimated from above withC(R)=d. Hencele^C(R)=d can be esti- mated explictly and gives a bound on theL¹ norm ofjrXjon [0;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 "0, as (3.15) shows. More precisely, we have

jX(t;x) X(t;y)j ljx yj 8x;y2Awithjx yj "₀; t2[0;T]:

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Manoscritto pervenuto in redazione il 28 febbraio 2005