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DUKE MATHEMATICAL JOURNALVol. 111, No. 2, c© 2002

UNIQUENESS OF CONTINUOUS SOLUTIONS FORBV VECTOR FIELDS

FERRUCCIO COLOMBINI and NICOLAS LERNER

AbstractWe consider a vector field whose coefficients are functions of bounded variation,with a bounded divergence. We prove the uniqueness of continuous solutions for theCauchy problem.

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3572. Statement of the results. . . . . . . . . . . . . . . . . . . . . . . . . . 360

2.1. A local result for bounded vector fields. . . . . . . . . . . . . . . 3602.2. A global result for transport equations. . . . . . . . . . . . . . . 361

3. Lemmas on nonnegative solutions. . . . . . . . . . . . . . . . . . . . 3643.1. Local results for bounded vector fields. . . . . . . . . . . . . . . 3643.2. Global results for transport equations with integrable coefficients. . 367

4. Commutation lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 3704.1. The DiPerna-Lions commutation argument. . . . . . . . . . . . . 3704.2. Commutation for a BV vector field. . . . . . . . . . . . . . . . . 372

5. Uniqueness results. . . . . . . . . . . . . . . . . . . . . . . . . . . .3745.1. Local results. . . . . . . . . . . . . . . . . . . . . . . . . . . .3745.2. Global results for transport equations. . . . . . . . . . . . . . . . 375

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379A.1. On the divergence of a vector field. . . . . . . . . . . . . . . . . 379A.2. A Log-Lipschitz function is not inW1,1 . . . . . . . . . . . . . . 380A.3. A weakened version of condition (2.11). . . . . . . . . . . . . . 382

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .383

1. IntroductionThe study of transport equations with irregular coefficients has been flourishing inthe last decade, mainly following the paper by R. DiPerna and P. Lions [DL]. The

DUKE MATHEMATICAL JOURNALVol. 111, No. 2, c© 2002Received 30 August 2000. Revision received 27 January 2001.2000Mathematics Subject Classification. Primary 35F05, 34A12, 26A45.

357

358 COLOMBINI and LERNER

Eulerian approach, in which the partial differential equation (PDE)

∂t +

∑1≤ j ≤d

a j (t, x)∂ j (1.1)

is under consideration, was developed in [DL], under the assumption ofW1,1-regular-ity for the a j (t, ·) (and bounded divergence), yielding uniqueness forL∞-solutions.In fact, L p′

-solutions are proven unique fora j in W1,p (and bounded divergence)for p ∈ [1,∞]. It should be pointed out here that this result was new even for Lip-schitz coefficients for which the classical method of Carleman estimates (see, e.g.,[H1, Chapter 28]) would give uniqueness only forL2

loc-solutions, whereas [DL] givesuniqueness forL1

loc-solutions as well. Other important results with applications tofluid mechanics were recently given by B. Desjardins [De1], [De2], [De3], F. Bouchutand L. Desvillettes [BD], and Bouchut and F. James [BJ]. The papers by G. Petrovaand B. Popov [PP] and by F. Poupaud and M. Rascle [PR], as well as the recent noteby Lions [Li ], raise the question of uniqueness for BV vector fields, that is, for thosewhose coefficients areL1 with derivative measure. In the present paper, we give anaffirmative answer to the question of uniqueness for continuous solutions of BV trans-port equations with bounded divergence. In particular, we prove the following localtheorem.

THEOREM 1.1Let X be a vector field with coefficients inBV ∩ L∞, and let c be a Radon measure.Let S be a C1-oriented hypersurface, noncharacteristic for X. Let u be a continuousfunction such that

Xu = cu, suppu ⊂ S+,

where S+ is the half-space above the oriented S. Then if the positive part of thedivergence(div X)+ belongs to L∞ as well as c+, the function u vanishes in a neigh-borhood of S.

This theorem is a consequence of Theorem 2.1. We also provide in Section 2 a spe-cific statement for transport equations with BV coefficients and nonfinite speed ofpropagation. Our proof is divided into two steps. The first step, described in Section3, is a proof of uniqueness for nonnegative solutions. It turns out that it is quite easyto prove that nonnegative solutions are unique. In fact, ifX is a bounded vector field,with L1

loc-divergence and(div X)+ ∈ L∞, the assumptions

0 ≤ w ∈ L∞, Xw = 0

are sufficient to imply uniqueness through a noncharacteristic hypersurface. (In fact,the aboveXw = 0 can be weakened toXw ≤ 0.) Of course, the strong assumption

VECTOR FIELDS OF BOUNDED VARIATION 359

here is thatw is supposed to be nonnegative. However, almost nothing is then requiredfrom the vector field, which should only be bounded, with some natural requirementson its divergence. The second step, developed in Section 5, is devoted to proving that

Xu = 0 implies X(u2) = 0.

To get this we use an approximation argument; in Section 4 we review the DiPerna-Lions commutation argument. Of course, we do not prove convergence inL1 of thekey commutator. We point out that the weak convergence (vague topology of mea-sures) is enough for our argument to work. Once this is proved, we need only applythe first step on nonnegative solutions. In fact, more generally we show that, assumingonly some boundedness or integrability property of the coefficients and the positivepart of the divergence, it is easy to prove uniqueness for nonnegative solutions, pro-vided the vector field is noncharacteristic with respect to the initial hypersurface. Thenif we know thatu satisfiesXu = 0 and thatu vanishes on the initial surface, we pickup a nonnegative functionα(u), for example,u2 if u is bounded (oru2/1 + u2 if uis in L p for a finite p), and we try to prove thatX(α(u)) is zero as well. For this weneed some approximation argument. It would also be interesting to study the linksbetween the Eulerian and the Lagrangian approach, the latter being concerned withthe properties of the ordinary differential equation (ODE)

x(t) = a(t, x(t)

), x(0) = x0. (1.2)

This problem has a long history, going back to the mid-nineteenth century. The mainwidely known result is the Cauchy-Lipschitz theorem, providing existence, unique-ness, and stability for (1.2) under an assumption of Lipschitz continuity fora with re-spect to the variablex (and integrability in the time variablet). In [ChL], the authorsproved that a hypothesis somewhat weaker than Lipschitz continuity could replacethe classical assumptions without altering the main conclusions. The requirementsin that paper were the existence of a positive nondecreasing continuous modulus ofcontinuityω andR0 > 0 such that, for allR ∈ (0, R0),∫ R

0

dr

ω(r )= +∞,

and the condition ona, for ‖x1 − x2‖ < R0, that∥∥a(t, x1)− a(t, x2)∥∥ ≤ α(t)ω

(‖x1 − x2‖

), α ∈ L1

loc. (1.3)

Let t 7→ ψ(t, x) stand for a continuous solution of

ψ(t, x) = x +

∫ t

0a(s, ψ(s, x)

)ds.


Settingν(r ) =∫ r

R0ds/(ω(s)), we see that the functionν is of classC1, negative for

0 < r < R0, and such thatν′(r ) = 1/ω(r ) > 0. One can prove, for‖x1 − x2‖ smallenough, that

∥∥ψ(t, x1)− ψ(t, x2)∥∥ ≤ ν−1

(ν(‖x1 − x2‖

)+

∫ t

0α(s)ds

). (1.4)

In the classical Lipschitz case,ω(r ) = r , ν(r ) = ln r , so that we get the familiar∥∥ψ(t, x1)− ψ(t, x2)∥∥ ≤ ‖x1 − x2‖ e

∫ t0 α(s)ds.

In the so-called Log-Lipschitz case, we have

ω(r ) = r ln(1

r

), ν(r ) = − ln

(ln

(1

r

));

we get ∥∥ψ(t, x1)− ψ(t, x2)∥∥ ≤ ‖x1 − x2‖

e−∫ t0 α(s)ds

.

The paper [ChL] provides as well the existence of solutions under assumption (1.3).These results are true in infinite dimension, that is, fora valued in a Banach space.However, in finite dimension, with the help of Peano’s existence theorem, this re-sult goes back to Osgood in 1904 [Fl]. The paper by H. Bahouri and J.-Y. Chemin[BC] studies for the transport equation the particular case of (1.3) in whichω(r ) =

r ln(1/r ); at any rate, their paper is an interesting excursion away fromW1,1 terri-tory†. On the other hand, the existence of a flow for the ODE (1.2), guaranteed byassumption (1.3), does not seem to imply trivially a uniqueness result for the associ-ated PDE. Although the generalization of the results of [BC] seems very likely, with(1.3) replacing theLL regularity assumption, it would probably require some signifi-cant effort. This may be related to [CoL], where a wave equation withLL coefficientswas studied and in which the energy method had to be substantially modified.

2. Statement of the results

2.1. A local result for bounded vector fieldsLet� be an open set ofRn. Let

X =

∑1≤ j ≤n

a j (x)∂

∂x j, a j ∈ L∞

loc(�) ∩ BVloc(�), (2.1)

†See the appendix for an example of a Log-Lipschitz function which is not inW1,1.


be a real-valued‡ vector field. Note that this implies divX ∈ M (�) = D ′(0)(�), theRadon measures on�. Foru ∈ C0(�), wedefine

Xu =

∑1≤ j ≤n

∂

∂x j(a j u)− u div X. (2.2)

The productu div X makes sense sinceu is continuous and divX is a Radon measure.We note that foru ∈ C1(�) we have the usualXu =

∑1≤ j ≤n a j ∂ j u. Let Sbe aC1-

oriented hypersurface of�. For allx0 ∈ S, there exists a neighborhoodV0 of x0 suchthat S ∩ V0 = {x ∈ V0, ϕ(x) = 0} with ϕ ∈ C1(V0), such thatdϕ 6= 0 on V0;moreover,S+ ∩ V0 is defined as{x ∈ V0, ϕ(x) ≥ 0}. (We say then thatϕ is a definingfunction for S.) We assume thatS is noncharacteristic forX; that is, there exists aneighborhoodK0 of x0 such that, with an essential infimum,

infK0

X(ϕ) > 0. (2.3)

Note that this property does not depend on the choice of the defining function for theoriented hypersurfaceS. In fact, if ϕ is anotherC1-defining function forS, we haveϕ = eϕ with e ∈ C0 and positive, so thatdϕ − edϕ is continuous and vanishes atx0,since

ϕ(x0 + h)− ϕ(x0) = ϕ(x0 + h)e(x0 + h)−

=0︷︸︸︷ϕ(x0)e(x0)

=(ϕ′(x0)h + o(h)

)e(x0 + h) = e(x0)ϕ

′(x0)h + o(h).

Consequently, ifK is a neighborhood ofx0, K ⊂ K0, whereK0 is given in (2.3),

infK

X(ϕ) ≥ (infK0

e)(infK0

X(ϕ))− ‖X‖L∞(K0)

‖dϕ − edϕ‖L∞(K ) > 0

since one may shrinkK aroundx0 ∈ S ad libitum. It is important also to notice thatassumption (2.3) is stable underC1-perturbation ofϕ. Let us assume that (2.3) issatisfied nearx0 ∈ S, and letφ be aC1-function defined onV0. We get from (2.3)

infK0

X(φ) ≥ infK0

X(ϕ)− ‖X‖L∞(K0)

∥∥d(ϕ − φ)∥∥

L∞(K0)> 0

for ‖d(ϕ − φ)‖L∞(K0)small enough.

‡A distribution a is said to be real valued whenevera = a, where the complex conjugatea is definedby duality:

⟨a, ψ

⟩=

⟨a, ψ

⟩. Complex vector fields are very complicated objects whose study is beyond the

scope of this paper. Even with polynomial coefficients, a nonzero complex vector field could fail to belocally solvable, as shown by the Hans Lewy counterexample. On the other hand, the uniqueness for thenoncharacteristic Cauchy problem could fail for first-order operators withC∞ complex-valued coefficients.


THEOREM 2.1Let� be an open set inRn, and let X be a bounded real-valued vector field satisfying(2.1). Let S be a C1-oriented hypersurface, noncharacteristic for X as in (2.3). Let ube a continuous function such that, for some Radon measure c,

Xu = cu, suppu ⊂ S+.

Then if the positive part(2c+ div X)+ ∈ L∞

loc, the function u vanishes in a neighbor-hood of S.

Remark.It would be natural to consider real vector fields of type (2.1) but satisfyingthe extra condition divX ∈ L1 (and(2c + div X)+ ∈ L∞). For u ∈ L∞, we candefineXu =

∑1≤ j ≤n ∂(a j u)/∂x j − u div X. For anL1-functionc, the uniqueness

through a noncharacteristic hypersurface ofL∞-solutions ofXu = cu is certainly anatural question that we are unfortunately unable to answer.

2.2. A global result for transport equationsLet us first introduce some notation. We use the familiarD ′(m)(Rd) to denote thedistributions of orderm onRd, the dual space ofCm

c (Rd). Below we denote byM =

D ′(0)(Rd) the Radon measure onRd. Its subspaceMb denotes the Banach space ofmeasures with finite total mass; this is the dual space ofC0

(0)(Rd), the continuous

functions tending to zero at infinity. Fora ∈ M , with K a compact subset ofRd, wedefine

NK (a) = sup06≡ϕ∈C0

c (K )

∣∣〈a, ϕ〉M ,C0c

∣∣/ sup|ϕ|, (2.4)

whereC0c(K ) stands for the functions ofC0

c(Rd) supported inK . We note that, fora ∈ Mb, and all compactK , we have

NK (a) ≤ ‖a‖Mb.

The Banach space BV(Rd) is defined as

BV(Rd) ={u ∈ L1(Rd),∇u ∈ Mb

}.

A modern treatment of most of the main properties of these functions can be found inthe book by W. Ziemer [Z, Chapter 5] (see also the classic book by Federer [Fe]). Wedefine also

BV loc(Rd) ={u ∈ L1

loc(Rd),∇u ∈ M

}.

The set of continuous bounded functions onRd is denoted byC0b(R

d). Let us considera vector field

X = ∂t +

∑1≤ j ≤d

a j (t, x)∂ j (2.5)


defined on(0, T0) × Rd whereT0 > 0. We assume that the coefficientsa j are realvalued and such that

a j ∈ L1((0, T0); BV loc(Rd)

); (2.6)

that is, for allψ ∈ C∞c (Rd),ψa j ∈ L1((0, T0); BV(Rd)). This implies of course that

the divergence divX =∑

1≤ j ≤d ∂ j (a j ) is real valued and such that

div X ∈ L1((0, T0); M);

that is,∀ψ ∈ C∞

c (Rd), ψ div X ∈ L1((0, T0); Mb

).

Let us consider a real-valued measurec such that

c ∈ L1((0, T0); M). (2.7)

We examine weak solutions of{∂tu +

∑1≤ j ≤d a j ∂ j u = cu + f in (0, T0)× Rd,

u|t=0 = u0 in Rd.(2.8)

This means that for allϕ ∈ C∞c ([0, T0)× Rd), using∗ the notationϕ0(x) = ϕ(0, x),∫ T0

0

[ ∫Rd

u(−X(ϕ)−ϕ div X −cϕ

)dx

]dt =

∫ T0

0

[ ∫Rd

f ϕ dx]

dt +∫

Rdu0ϕ0 dx.

(2.9)In particular, (2.9) makes sense whenc(t, ·) and(div X)(t, ·) are Radon measures andu(t, ·) is a continuous function. The integrals onRd in the formula (2.9) should thenbe written as brackets of duality; more precisely, foru ∈ L∞((0, T); C0(Rd)) theleft-hand side of (2.9) is defined as∫ T0

0

∫Rd

u(t, x)(−∂tϕ(t, x)−

∑1≤ j ≤d

a j (t, x)∂ jϕ(t, x))

dx dt

−

∫ T0

0

⟨(div X + c)(t, ·), ϕ(t, ·)u(t, ·)

⟩M ,C0

cdt.

Let us check that the last term makes sense in the above expression. Sinceϕ ∈

C∞c ([0, T0) × Rd), there existsK0, a compact subset ofRd such that, for allt ,

∗It is of course important to notice thatϕ does not necessarily vanish att = 0. In fact, (2.9) is a compli-cated but standard way of writing∂t u +

∑j ∂ j (a j u)− udivX = cu + f + δ ⊗ u0, where v stands for the

extension ofv by 0 to {t < 0}.


suppϕ(t, ·) ⊂ K0. Letχ0 ∈ C∞c (Rd), identically 1 onK0. Then we have∣∣⟨(div X+c)(t, ·), ϕ(t, ·)u(t, ·)

⟩M ,C0

c

∣∣ =∣∣⟨χ0(div X+c)(t, ·), ϕ(t, ·)u(t, ·)

⟩Mb,C0

(0)

∣∣≤

∥∥χ0(div X + c)(t, ·)∥∥

Mbsup

t∈(0,T0)

∥∥(ϕu)(t, ·)∥∥

L∞(Rd)∈ L1(0, T0).

THEOREM 2.2Let X be the vector field (2.5), and let assumptions (2.6) and (2.7) be satisfied. More-over, we assume that

(2c + div X)+ ∈ L1((0, T0); L∞(Rd)). (2.10)

We also assume∗ that

limλ→+∞

∫ T0

0λ−1

∫λ≤|x|≤2λ

∣∣a j (t, x)∣∣ dx dt = 0. (2.11)

Let u be a function in L∞((0, T0); C0b(R

d)) such that, in the sense of (2.9),{∂tu +

∑1≤ j ≤d a j ∂ j u = cu in (0, T0)× Rd,

u|t=0 = 0 in Rd.(2.12)

Then u vanishes in(0, T0)× Rd.

3. Lemmas on nonnegative solutions

3.1. Local results for bounded vector fieldsLet � be an open set ofRn, and let p ∈ [1,+∞]. We denote byp′ the conjugateexponent ofp such that 1= 1/p + 1/p′. Let

X =

∑1≤ j ≤n

a j (x)∂

∂x j, a j ∈ L∞

loc(�), div X ∈ L ploc, (3.1)

be anL∞

loc real-valued vector field withL ploc-divergence. Foru ∈ L p′

loc(�), we define

Xu =

∑1≤ j ≤n

∂

∂x j(a j u)− u div X. (3.2)

∗We note that this assumption is satisfied whena j (t, x)/(1 + |x|) ∈ L1((0, T0) × Rd). We give in theappendix a weaker condition than (2.11) which is satisfied in particular whenever

a j (t, x)

(1 + |x|) ln(1 + |x|)∈ L1((0, T0)× Rd)

.


Note that assumptions (3.1) are invariant through aC2-diffeomorphism.† Let S be aC1-oriented hypersurface of� (see Section 2 for a precise definition).

LEMMA 3.1Let� be an open set inRn, let p ∈ [1,+∞], and let X be a bounded real-valuedvector field with divergence in Lploc. Let S be a C1-oriented hypersurface, nonchar-

acteristic for X as in (2.3). Let u be a Lp′

loc-function such that, for some functionc ∈ L p

loc,Xu ≤ cu, suppu ⊂ S+, and u≥ 0. (3.3)

Then if(c + div X)+ ∈ L∞

loc, (3.4)

the function u vanishes in a neighborhood of S.

ProofLet us consider a pointx0 ∈ Sandϕ a defining function forS in a neighborhood ofx0.We know that, on an open neighborhoodV0 of x0, with u0 = u|V0 ≥ 0, u0 ∈ L p′

(V0),we have

Xu0 ≤ cu0, suppu0 ⊂ {ϕ ≥ 0}, Xϕ ≥ ρ0 > 0. (3.5)

Let us consider the followingC1-function defined onV0:

ψ(x) = ϕ(x)+ |x − x0|2, θ

(ψ(x)

)=

1

2

((α2

− ψ(x))+

)2,

whereα is a positive parameter such that the closed ballB(x0, α) with centerx0 andradiusα is included inV0. We have

supp(θ(ψ)

)⊂ {ψ ≤ α2

}

andsupp

(u0θ(ψ)

)⊂ {ϕ ≥ 0} ∩ {ψ ≤ α2

} = Kα 3 x0,

which is a compact subset ofV0 (as a closed subset ofB(x0, α)). Let χ ∈ C∞c (V0;

[0,1]), χ = 1 on a neighborhood ofKα. Sinceψ andθ(ψ) areC1-functions, andX(χ) = 0 on a neighborhood of suppu0θ(ψ), we have

∑1≤ j ≤n

a j u0∂ j(θ(ψ)χ

)= u0χθ

′(ψ)X(ψ)+

=0︷︸︸︷u0θ(ψ)X(χ) .

†The assumption divX ∈ L ploc, div X ∈ Lq

loc is invariant by aC2-change of coordinates. Also, this assump-tion is indeed local since with8 ∈ C∞

c (�) the vector field8X still satisfies the same assumption. Thisis not the case for the assumptionX ∈ W1,1 and divX ∈ L∞, which is not local.


We calculate, withdm standing for the Lebesgue measure,∫cu0θ(ψ)χ dm ≥

⟨Xu0, θ(ψ)χ︸︷︷︸

≥0

⟩D ′(1)(V0),C1

c (V0)

= −

∑1≤ j ≤n

∫a j u0∂ j

(θ(ψ)χ

)dm−

∫u0θ(ψ)χ div X dm

=

∑1≤ j ≤n

∫χu0a j ∂ j (ψ)(α

2− ψ)+ dm−

∫u0θ(ψ)χ div X dm.

We obtain

0 ≥

∫χu0(α

2− ψ)+

[X(ψ)−

1

2(α2

− ψ)+(div X + c)]

dm. (3.6)

Now on the set{x ∈ V0, ϕ(x)+ |x − x0|

2= ψ(x) ≤ α2}

∩{x, ϕ(x) ≥ 0

}we have, using now (3.4)–(3.5),

X(ψ)−1

2(α2

− ψ)+(div X + c) ≥ ρ0 − 2‖X‖L∞(B(x0,α))α

−1

2α2

∥∥(div X + c)+∥∥

L∞(B(x0,α))≥ ρ0/2

if α is chosen small enough with respect toρ0 and‖X‖L∞(V0). On the other hand, the

term ∫χu0

((α2

− ψ)+)2(div X + c)− dm

makes sense and is nonnegative. This yields

0 ≥

∫χu0(α

2− ψ)+ dm,

and since the integrand is nonnegative, we getχu0(α2−ψ)+ = 0. Since on a neigh-

borhood ofx0 we haveχ = 1 andα2− ψ > 0, we indeed obtain thatu0 vanishes

nearx0. The proof of Lemma 3.1 is complete.

Remark.Lemma 3.1 provides a local uniqueness result for noncharacteristic boundedvector fields with bounded positive part of the divergence; the assumptions are invari-ant∗ by C2-change of coordinates. The local problem makes sense since the speed ofpropagation is finite.

∗The assumptionsX ∈ L∞

loc, div X ∈ L ploc, c ∈ L p

loc, (div X+c)+ ∈ L∞

loc are indeed invariant byC2-changeof coordinates and local in the sense of the footnote on page364.


The proof of the following lemma is identical to the proof of Lemma 3.1, exceptfor replacing integrals by brackets of duality. Note also that (3.2) makes sense forucontinuous,a j , div X ∈ D ′(0)(�).

LEMMA 3.2Let � be an open set inRn, and let X be a bounded real-valued vector field withdivergence inD ′(0)(�). Let S be a C1-oriented hypersurface, noncharacteristic for Xas in (2.3). Let u be a continuous function such that, for some measure c∈ D ′(0)(�),

Xu ≤ cu, suppu ⊂ S+, and u≥ 0.

Then if(c + div X)+ ∈ L∞

loc,

the function u vanishes in a neighborhood of S.

The proof of this lemma is obtained by copying the proof of Lemma 3.1 and properlyreplacing integrals by brackets of duality; inequality (3.6) should be replaced by

0 ≥

∫χu0X(ψ)(α2

− ψ)+ dm−1

2

⟨div X + c

((α2

− ψ)+)2χu0

⟩D ′(0)(�),C0

c (�)

≥

∫χu0X(ψ)(α2

− ψ)+ dm−1

2

∫(div X + c)

+︸︷︷︸∈L∞

((α2

− ψ)+)2χu0 dm.

The end of the proof is identical.

3.2. Global results for transport equations with integrable coefficientsLet us consider a vector field

X = ∂t +

∑1≤ j ≤d

a j (t, x)∂ j (3.7)

defined on(0, T0) × Rd whereT0 > 0. We assume that the coefficientsa j are realvalued and such that

a j ∈ L1((0, T0); L1loc(R

d)). (3.8)

We assume moreover that the divergence divX =∑

1≤ j ≤d ∂ j (a j ) is such that

div X ∈ L1((0, T0); L1loc(R

d)). (3.9)

Let us consider a real-valued functionc such that

c ∈ L1((0, T0); L1loc(R

d)). (3.10)


We examine the weak solutions of{∂tu +

∑1≤ j ≤d a j ∂ j u = cu + f in (0, T0)× Rd,

u|t=0 = u0 in Rd.(3.11)

This means that for allϕ ∈ C∞c ([0, T0)× Rd), using the notationϕ0(x) = ϕ(0, x),∫ T0

0

[ ∫Rd

u(−X(ϕ)−ϕ div X −cϕ

)dx

]dt =

∫ T0

0

[ ∫Rd

f ϕ dx]

dt+∫

Rdu0ϕ0 dx.

(3.12)In particular, (3.11) makes sense when thea j , c, and divX satisfy (3.8)–(3.10) andubelongs toL∞((0, T0), L∞

loc(Rd)). We assume forj = 1, . . . ,d the global condition

(2.11). This condition is satisfied in particular whenever the functionsa j (t, x)/(1 +

|x|) belong toL1((0, T0)× Rd).

LEMMA 3.3Let X be a transport equation as in (3.7). We assume that the coefficients aj and csatisfy (3.8)–(3.10). Moreover, we assume (2.11) and

(c + div X)+ ∈ L1((0, T0); L∞(Rd)). (3.13)

Let u be a nonnegative L∞((0, T0)× Rd)-function such that{∂tu +

∑1≤ j ≤d a j ∂ j u ≤ cu in (0, T0)× Rd,

u|t=0 = 0 in Rd.(3.14)

Then u vanishes in(0, T0)× Rd.

ProofRelationship (3.14) implies that, for any nonnegativeϕ ∈ C1

c([0, T0)× Rd),

∫ T0

0

∫Rd

u∂tϕ dx dt+∫ T0

0

∑j

∫Rd

uaj ∂ jϕ dx dt

+

∫ T0

0

∫Rd

uϕ (div X + c)dx dt ≥ 0. (3.15)

Sinceu andϕ are nonnegative, using (3.13) we get, with 0≤ ν0 ∈ L1(0, T0),∫ T0

0

∫Rd

u∂tϕ dx dt+∫ T0

0

∑j

∫Rd

uaj ∂ jϕ dx dt+∫ T0

0

∫Rd

uϕν0(t)dx dt ≥ 0.

(3.16)


We can chooseϕ(t, x) = τ(t)σ (|x|2λ−2), where 0≤ τ ∈ C1

c([0, T0)), λ > 0, and

C1(R+; [0,1])

3 σ =

{1 on [0,1],

0 on[4,∞),with −1/2 ≤ σ ′

≤ 0.

Plugging this into (3.16), we obtain∫∫(0,T0)×Rd

u(t, x)σ(|x|

2λ−2)[− τ (t)− ν0(t)τ (t)

]dt dx

≤ ‖u‖L∞

∑j

∫∫(0,T0)×{λ≤|x|≤2λ}

τ(t)∣∣a j (t, x)x j

∣∣λ−2 dt dx. (3.17)

We want now to choose the functionτ by solving the ODE

τ (t)+ ν0(t)τ (t) = −(T1 − t)+, whereT1 is given in(0, T0). (3.18)

We have

τ(t) =

∫ T1

texp

(−

∫ t

sν0(r )dr

)(T1 − s)+ ds (3.19)

and in particularτ(t) = 0 for t ≥ T1. The function (3.19) is nonnegative and con-tinuous with a derivative inL1, so that, approximatingτ by a smooth function, weobtain∗ that (3.17) is satisfied forτ given in (3.19). This gives∫ ∫

(0,T1)×Rdu(t, x)σ

(|x|

2λ−2)(T1 − t)+ dt dx ≤ ‖u‖L∞

·

∑j

∫∫(0,T1)×{λ≤|x|≤2λ}

1

2(T1 − t)2

∣∣a j (t, x)x j∣∣λ−2 dt dxexp

∫ T1

0ν0(r )dr . (3.20)

Using the Beppo Levi monotone convergence theorem (note thatσ ′ takes non-positive values), the left-hand side of (3.20) tends, forλ → +∞, to∫∫

(0,T1)×Rdu(t, x)(T1 − t)+ dt dx, (3.21)

whereas the lim sup of the right-hand side is bounded above by

‖u‖L∞ T21

∑j

lim supλ→+∞

1

λ

∫∫(0,T1)×{λ≤|x|≤2λ}

∣∣a j (t, x)∣∣ dt dxexp

∫ T1

0ν0(r )dr ,

which is zero by hypothesis (2.11). As a consequence, sinceu is nonnegative, weget thatu vanishes on(0, T1) × Rd for all T1 < T0. The proof of Lemma 3.3 iscomplete.

∗In fact, if τε is a standard regularization ofτ , we keepτε ≥ 0 and τε(t) = 0 for t ≥ T1 + ε in such away that τε − τ and (τε − τ)ν0 both converge to 0 inL1(0, T0).


The following lemma is similar to Lemma 3.3.

LEMMA 3.4Let X be a transport equation as in (3.7). We assume that aj ,divX, c belong toL1((0, T0); M ). Moreover, we assume (3.13) and

lim supλ→+∞

1

λ

∫ T0

0N{λ≤|x|≤2λ}

(a j (t, ·)

)dt = 0 (3.22)

(see (2.4) for the definition of N). Let u be a nonnegative L∞((0, T0); C0b(R

d))-function such that (3.14) is satisfied. Then u vanishes in(0, T0)× Rd.

ProofIt is easy to check that (3.15) is satisfied, with the integral

∫Rd uaj ∂ jϕ dx dt re-

placed by the bracket of duality〈a j (t, ·),u(t, ·)∂ jϕ(t, ·)〉M ,C0c

and the integral∫Rd uϕ (div X + c) replaced by〈(div X + c)(t, ·),u(t, ·)ϕ(t, ·)〉M ,C0

c. Assumption

(3.13) implies (3.16) as well. The sequel of the proof is identical, with the right-handside of (3.17) replaced by

‖u‖L∞ λ−1∑

j

∫ T0

0τ(t)N{λ≤|x|≤2λ}

(a j (t, ·)

)dt.

Remark. It seems possible to relax assumption (2.11) and, in particular, to assumeonly that the lim sup is finite (see also the footnote on page363 and the appendixfor a different kind of weakening of (2.11)). However, it is not clear that this gener-alization is worthwhile, since the proof gets much more complicated. On the otherhand, the fact that the vector field here is a priori unbounded forces somehow a globalassumption, which should replace a standard convexification procedure workable inthe bounded case. On the other hand, in Lemmas 3.3 and 3.4, we could have requiredglobal integrability conditions for the coefficients of the vector field; this would haveled to slightly different results.

4. Commutation lemmas

4.1. The DiPerna-Lions commutation argumentSince we want to use this now classical argument in a slightly different context in-volving BV functions, we start with a quick review of some identities attached tocommutation of vector fields with a standard mollifier. Let us consider a vector field


X defined inRn,

X =

∑1≤ j ≤n

a j (x)∂

∂x j, (4.1)

so that, for somep ∈ [1,+∞], the coefficientsa j belong toW1,ploc (R

n). For u ∈

L p′

loc(Rn), we define

Xu =

∑j

∂ j (a j u)− u div X. (4.2)

Let ρ be aC∞c nonnegative function supported in the unit ball ofRn such that

∫ρ =

1. We set, forε > 0, ρε(·) = ε−nρ(·/ε). We consider the linear operator defined on

L p′

loc(Rn) by

Rεu = X(u ∗ ρε)− (Xu) ∗ ρε . (4.3)

We define also the translation operator

(τzu)(x) = u(x − z). (4.4)

The following identity holds:

(Rεu)(x) =

∫ {(τεz div X)(x)ρ(z)+

1

ε

∑j

[(Id − τεz)a j

](x)∂ j ρ(z)

}·[(τεz − Id)u

](x)dz. (4.5)

From this identity,∗ we get immediately, forK0 compact inRn, K1 = K0 + unit ball,0< ε ≤ 1, 1/q + 1/q′

= 1,

‖Rεu‖L1(K0)≤ ‖div X‖Lq(K1)

∫ ∥∥(τεz − Id)u∥∥

Lq′(K0)

ρ(z)dz

+

∑j

∥∥∇a j∥∥

Lq(K1)

∫ ∥∥(τεz − Id)u∥∥

Lq′(K0)

∣∣∂ j ρ(z)‖z∣∣ dz. (4.6)

∗The sum in (4.5) can be written as∑1≤ j,k≤n

∫∫ 1

0(∂ka j )(x − εθz)zk(∂ j ρ)(z)

(u(x − εz)− u(x)

)dz dθ,

so that if ρ(z) = ρ0(|z|2), this term is in fact∑1≤ j,k≤n

∫∫ 1

0(∂ka j )(x − εθz)2zkz j ρ

′

0(|z|2)

(u(x − εz)− u(x)

)dz dθ.

As pointed out in [De3], we thus have only to deal with∑1≤ j,k≤n

(∂ka j )(x − εθz)2zkz j =

∑1≤ j,k≤n

zkz j[(∂ka j )(x − εθz)+ (∂ j ak)(x − εθz)

].

This means that choosing a radial mollifierρ allows us to control only the symmetric part of the gradient,∂ j ak + ∂ka j .


LEMMA 4.1Let p be in[1,∞], and let X be a vector field in W1,ploc (R

n). Then with Rε defined in

(4.3), we have, for u∈ L p′

loc (1/p + 1/p′= 1),

limε→0

Rεu = 0 in L1loc.

ProofIf p′ < ∞, estimate (4.6) withq = p suffices to get the result. Let us assume nowu ∈ L∞

loc, and letY be aC∞c vector field. DenotingRε in (4.3) by Rε,X , we get from

(4.6), for Rε,Y with anyq′ < ∞,

limε→0

∥∥Rε,Yu∥∥

L1(K0)= 0.

This yields from (4.6), forX − Y with q = 1, with C0 depending only onρ,

lim supε→0

∥∥Rε,Xu∥∥

L1(K0)≤ lim sup

ε→0

∥∥Rε,X−Yu∥∥

L1(K0)

≤ C0 ‖X − Y‖W1,1(K1)‖u‖L∞(K1)

,

and the result by density ofC∞c in W1,1.

4.2. Commutation for aBV vector fieldAs was noticed in a paper by Bouchut (see [Bo, Remark 3.2]), the convergence tozero inL1 of Rεu doesnot follow from the previous commutation lemma. We provebelow a weak convergence result that is enough for our uniqueness theorem.

LEMMA 4.2Let X be a vector field inBV loc(Rn), that is, with coefficients in L1loc with first deriva-tives inD ′(0). Then with Rε defined in (4.3), we have, for u∈ C0(Rn),

limε→0

Rεu = 0 weakly inD ′(0),

that is,∀ψ ∈ C0

c(Rn), lim

ε→0〈Rεu, ψ〉D ′(0),C0

c= 0.


ProofGoing back to (4.5) and using notation (4.4), we get∗

⟨Rεu, ψ

⟩=

∫ [⟨τεz div X, ψ(τεz − Id)u

⟩ρ(z)

+1

ε

∑j

⟨(Id − τεz)a j , ψ(τεz − Id)u

⟩∂ j ρ(z)

]dz

=

∫ ⟨τεz div X, ψ(τεzu − u)

⟩ρ(z)dz

+

∑1≤ j,k≤n

∫∫ 1

0

⟨τεθz∂ka j , ψ(τεzu − u)

⟩zk∂ j ρ(z)dz dθ

=

∫ ⟨τεz div X, ψ(τεzu − u)

⟩ρ(z)dz

+

∑1≤ j,k≤n

∫∫ 1

0

⟨∂ka j ,

(τ−εθzψ

)(τεz−εθz − τ−εθz)u

⟩zk∂ j ρ(z)dz dθ.

This implies, withK0 = suppψ , K1 = K0 + unit ball, 0< ε ≤ 1, χ1 ∈ C0c(Rn)

identically 1 nearK1,∣∣〈Rεu, ψ〉∣∣ ≤

∫‖τ−εzu − u‖L∞(K1)

ρ(z)dz‖χ1 div X‖Mb‖ψ‖L∞ (4.7)

+

∑1≤ j,k≤n

∫∫ 1

0‖τεz−εθzu − τ−εθzu‖L∞(K1)

∣∣zk∂ j ρ(z)∣∣ dz dθ

×∥∥χ1∂ka j

∥∥Mb

‖ψ‖L∞ ,

which gives the result of the lemma sinceu is continuous and the integration in thevariablesz, θ takes place on compact sets.

Remark.In fact, if div X ∈ L1loc, for χ ∈ C0

c(Rn), there existsχ ∈ C0c(Rn) such that,

for all u ∈ C0(Rn),‖χRεu‖L1 ≤ ‖χ∇a‖Mb

‖χu‖L∞ .

This implies that the sequenceχRεu is bounded inL1 and thus that one can extracta convergent subsequence for the weak∗ topology onMb, that is, theσ(Mb,C0

(0))

topology. On the other hand, the convergence ofRεu to zero inD ′(1)(Rn) is obvious.

∗The computation is more transparent using integrals instead of brackets of duality. Also, operator (4.4)acts on distributions, and forw ∈ D ′(Rn), z ∈ Rn,

τzw = w −

∑j

z j

∫ 1

0τθz(∂ jw)dθ.


5. Uniqueness results

5.1. Local results

Proof of Theorem 2.1Let� be an open set ofRn, and let

X =

∑1≤ j ≤n

a j (x)∂

∂x j

be a vector field whose coefficientsa j are locally bounded with derivatives inD ′(0).As a consequence, the divergence

∑j ∂ j (a j ) belongs toD ′(0). Let c be inD ′(0). Let

Sbe aC1-oriented hypersurface, noncharacteristic forX as in (2.3). We assume alsothat

(2c + div X)+ ∈ L∞

loc. (5.1)

Let u be a continuous function such that

suppu ⊂ S+, Xu = cu. (5.2)

We want to prove that the functionu actually vanishes in a neighborhood ofS. Weprove thatX(u2) = 2cu2 and apply Lemma 3.2 to get the answer. Let us calculate forϕ ∈ C1

c(�) the bracket⟨X(u2), ϕ

⟩D ′(1)(�),C1

c (�)= −

∫u2X(ϕ)dm− 〈div X,u2ϕ〉D ′(0)(�),C0

c (�)

= −

∫χ2u2X(ϕ)dm− 〈ϕ div X, χ2u2

〉D ′(0),C0c,

whereχ ∈ C1c(�) is identically 1 on the support ofϕ. We obtain, using the notation

of Section 4,⟨X(u2), ϕ

⟩D ′(1),C1

c

= − limε→0+

[ ∫χu(χu ∗ ρε)X(ϕ)dm+

⟨ϕ div X, χu(χu ∗ ρε)

⟩D ′(0),C0

c

].

Since the functionχu ∗ ρε is C1, we get⟨X(u2), ϕ

⟩D ′(1),C1

c= limε→0+

⟨χuX(χu ∗ ρε)+ (χu ∗ ρε)X(χu), ϕ

⟩D ′(1),C1

c.

We have, sinceχ ∈ C1c(�),

X(χu) = χX(u)+ uX(χ) = χcu + uX(χ), (5.3)


and sinceϕ∇χ = 0 andχϕ = ϕ, we obtain⟨X(u2), ϕ

⟩D ′(1),C1

c= limε→0+

⟨χuX(χu ∗ ρε)+ (χu ∗ ρε)cu, ϕ

⟩D ′(1),C1

c. (5.4)

Now we write, using the notation of Section 4 and (5.3),

X(χu ∗ ρε) = Rε(χu)+ X(χu) ∗ ρε = Rε(χu)+ (χcu) ∗ ρε +(uX(χ)

)∗ ρε,

yielding⟨X(u2), ϕ

⟩D ′(1),C1

c= limε→0+

⟨χuRε(χu)+ χu(χcu ∗ ρε)+ (χu ∗ ρε)cu, ϕ

⟩D ′(1),C1

c

= limε→0+

⟨χuRε(χu), ϕ

⟩D ′(1),C1

c+ 2〈c,u2ϕ〉D ′(0),C0

c

= limε→0+

⟨Rε(χu), χuϕ

⟩D ′(0),C0

c+ 2〈c,u2ϕ〉D ′(0),C0

c.

From Lemma 4.2, we get, sinceχuϕ is continuous,

limε→0+

⟨Rε(χu), χuϕ

⟩D ′(0),C0

c= 0,

which givesX(u2) = 2cu2.

Sinceu2 is nonnegative and supported inS+, we can apply Lemma 3.2, provided that

(2c + div X)+ ∈ L∞

loc.

The proof of Theorem 2.1 is complete.

Remark. If X is a W1,1∩ L∞ vector field,c ∈ L1, the L∞-solutions ofXu = cu

are such thatXu2= 2cu2 (a consequence of Lemma 4.1). On the other hand, for

1 < p ≤ ∞, if X is aW1,p∩ L∞ vector field,c ∈ L p, theL p′

-solutions ofXu =

cu are such thatX(α(u)) = cα′(u)u with α(u) = u2/(1 + u2) (a consequence ofLemma 4.1). Lemma 3.1 yields uniqueness of these solutions, provided that(div X)+andc+ are bounded.

5.2. Global results for transport equations

Proof of Theorem 2.2We consider now a transport equation

X = ∂t +

∑1≤ j ≤d

a j (t, x)∂ j


defined on(0, T0) × Rd whereT0 > 0. We assume that the coefficientsa j (t, x) arereal valued and such that

a j (t, x) ∈ L1((0, T0); BV loc(Rd)). (5.5)

We assume also that the divergence divX =∑

j ∂ j (a j ) is such that

div X ∈ L1((0, T0); L1loc(R

d)). (5.6)

Let us consider a real-valued functionc such that

c ∈ L1((0, T0); L1loc(R

d)). (5.7)

We assume also that global condition (2.11) is satisfied and that

(2c + div X)+ ∈ L1((0, T0); L∞

loc(Rd)

).

Let us check a function

u(t, x) ∈ L∞((0, T0); C0

b(Rd)

)(5.8)

such thatXu = cu andu|t=0 = 0. This means in fact that

∂tv + Y(t)v = dv on (−T0, T0)× Rd, (5.9)

wherev is the extension of the functionu by zero on{t < 0}, Y(t) the extension ofthe vector field

∑j a j (t, x)∂ j by zero on{t < 0}, andd the extension of the function

c by zero on{t < 0}. We donot assume here thatu is nonnegative, and we want toprove thatu ≡ 0. It is enough to prove that(∂t +Y(t))(v2) = 2dv2 on(−T0, T0)×Rd,since Lemma 3.3 could then be applied to the nonnegative bounded functionv2. Wewrite, for8 ∈ C1

c((−T0, T0)× Rd),

E(v,8) =⟨∂t (v

2)+ Y(t)(v2),8⟩D ′(1)(R1+d),C1

c (R1+d)

= −

∫∫v28dx dt+

∫∫v2Y∗(8)dx dt.

Denoting byvε(t)(x) = [v(t, ·)∗ρε](x) thex-regularization of the functionv, we get,usingv ∈ L∞(C0

b(Rd)), thatvε is bounded uniformly inε and converges for almost

all t to v(t),

E(v,8) = −

∫∫v28dx dt+ lim

ε→0

∫∫v2ε (t)Y

∗(t)(8(t)

)dx dt,

and thus

E(v,8) = −

∫∫v28dx dt+ lim

ε→0

∫∫2Y(t)

(vε(t)

)vε(t)8(t)dt. (5.10)


From the equality

[Y(t)v(t)

]∗ ρε =

∑j

∈L1loc(R

d)︷︸︸︷[a j (t)v(t)] ∗∂ j ρε −

∈L1loc(R

d)︷︸︸︷[v(t) div Y(t)] ∗ρε (5.11)

and assumptions (5.5)–(5.8), we get[Y(t)v(t)] ∗ ρε ∈ L1loc(R

1+d) and thus, usingequation (5.9),

v(t) ∗ ρε =[−Y(t)v(t)+ d(t)v(t)

]∗ ρε ∈ L1

loc(R1+d). (5.12)

On the other hand, we have⟨Y(t)

(vε(t)

)vε(t)8(t)

⟩D ′(0)(Rd),C0

c (Rd)=

⟨Rε,Y(t)

(v(t)

)vε(t)8(t)

⟩D ′(0),C0

c

+⟨[

Y(t)v(t)]∗ ρεvε(t)8(t)

⟩D ′(0),C0

c.

(5.13)

From Lemma 4.2 and estimate (4.7), we get (sincevε(t)8(t) → v(t)8(t) in C0c(Rd))∫ T0

0

⟨Rε,Y(t)

(v(t)

)vε(t)8(t)

⟩D ′(0),C0

cdt → 0 with ε. (5.14)

We are left with∫ T0

0

⟨[Y(t)v(t)

]∗ ρεvε(t)8(t)

⟩D ′(0),C0

cdt

=

∫ T0

0

⟨[−v(t)+ d(t)v(t)

]∗ ρεvε(t)8(t)

⟩D ′(0),C0

cdt. (5.15)

We get then from (5.10) and (5.13)–(5.15),

E(v,8) =

∫int2dv28dx dt−

∫∫v28dx dt− lim

ε→0

∫∫2vεvε8dx dt. (5.16)

On the other hand, sincevε ∈ L∞

loc(R1+d) and∂tvε ∈ L1

loc(R1+d), we have∂t (v

2ε ) =

2∂t (vε)vε ; as a matter of fact, ifw ∈ L∞

loc(Rn), ∂x1w ∈ L1

loc(Rn), one checks∗ with a

standard mollifier that∂x1(w2) = 2w∂x1(w). Applying this to our situation, we get∫∫

2vεvε8dx dt = 〈2vε∂tvε,8〉D ′(1)(R1+d),C1c (R1+d)

=⟨∂t (v

2ε ),8

⟩D ′(1),C1

c= −

∫∫v2ε∂t8dx dt. (5.17)

∗With⟨D ′(0),C0

c

⟩brackets of duality, one has⟨

∂x1(w2), ϕ

⟩= −

∫w2∂x1ϕ dx = lim

ε

⟨∂x1(wwε), ϕ

⟩= lim

ε

[〈w∂x1wε , ϕ〉 + 〈wε∂x1w, ϕ〉

].

The last term tends to∫ϕw∂x1w, whereas the other is equal to

∫(∂x1w)εwϕ and goes to

∫wϕ∂x1w, so

that eventually∂x1(w2) = 2w∂x1w.


Eventually, taking the limit whenε goes to zero, we get

E(v,8) =

∫∫2dv2 dx dt, (5.18)

which is the desired result. Assuming (5.5) and (5.8) and replacing (5.6)–(5.7) by

c, div X ∈ L1((0, T0); M), (5.19)

let us check that the previous proof gives the result for Theorem 2.2, using Lemma3.4 instead of Lemma 3.3. In fact, equality (5.10) is unchanged and (5.11) gives[

Y(t)v(t)]∗ ρε ∈ L1

loc(Rt ,M ),

so that equality (5.12) givesvε(t) ∈ L1loc(Rt ,M ). Equality (5.13) is unchanged, and

(5.14)–(5.15) still hold, whereas (5.16) should be replaced by

E(v,8) =

∫2⟨d(t), v2(t)8(t)

⟩M ,C0

cdt −

∫∫v28dx dt (5.20)

− limε→0

∫2⟨vε(t), vε(t)8(t)

⟩M ,C0

cdt.

To get the result, we need to prove that ifw ∈ L∞

loc(Rt ,C0(Rdx)) and ∂tw ∈

L1loc(Rt ,M ), then

∂t (w2) = 2w∂tw,

that is,

−

∫∫w2∂t8dx dt = 2

∫ ⟨w(t), w(t)8(t)

⟩M ,C0

cdt. (5.21)

In fact, we have⟨∂t (w

2), ψ⟩D ′(1)(R1+d),C1

c (R1+d)= −

∫∫w2∂t (ψ)dx dt

= − limθ→0

∫∫w(w ∗ ρθ )∂tψ dx dt, (5.22)

since

‖wwθ∂tψ − w2∂tψ‖L1(R1+d) ≤ ‖w∂tψ‖L∞ ‖w − wθ‖L1(suppψ) → 0 with θ.(5.23)

Consequently, we have⟨∂t (w

2), ψ⟩D ′(1)(R1+d),C1

c (R1+d)

= limθ→0

(〈∂tw,wθψ〉D ′(1)(R1+d),C1

c (R1+d) +

∫∫wψ∂twθ dx dt

). (5.24)


But ∂twθ → ∂tw in L1loc(R,M ), so the second term in (5.24) converges to∫ ⟨

∂tw(t), w(t)ψ(t)⟩M ,C0

cdt.

On the other hand, we have

〈∂tw,wθψ〉D ′(1)(R1+d),C1c (R1+d) =

∫∫w(ψ∂tw)θ dx dt

→

∫ ⟨ψ(t)∂tw(t), w(t)

⟩M ,C0

cdt, (5.25)

yielding (5.21). Consequently, from (5.20),

E(v,8) =

∫2⟨v2(t)d(t),8(t)

⟩M ,C0

cdt −

∫∫v28dx dt+ lim

ε→0

∫∫v2ε 8dx dt,

implying the result(∂t + Y(t))v2= 2dv2. The proof of Theorem 2.2 is complete.

Appendix

A.1. On the divergence of a vector fieldLet (M, ω) be a smooth oriented manifold, and letX be a smooth vector field onM .Using the flow of the vector field, it is possible to define the Lie derivativeLX(ω), andthe fundamental formula of the calculus of variations gives, sinceω is a nondegeneraten-form,

LX(ω) = d(ωcX)+ dωcX = d(ωcX) = ω div X, (A.1)

wherec stands for the interior product. The last equality makes sense without anyregularity assumption onX, for example, for a smoothω and a vector field withdistribution coefficients. Note that Green’s formula can be written as∫

Mω div X =

∫M

d(ωcX) =

∫∂MωcX.

In particular, working with coordinates inRn and with a nonvanishingν,

ω = ν(x)dx1 ∧ · · · ∧ dxn, X =

∑1≤ j ≤n

a j (x)∂

∂x j,

we obtaindiv X = X

(ln |ν|

)+

∑1≤ j ≤n

∂ j (a j ). (A.2)


As a matter of fact,∗⟨LX(ω),Y1 ∧ · · · ∧ Yn

⟩= X

(〈ω,Y1 ∧ · · · ∧ Yn〉

)−

∑1≤ j ≤n

⟨ω,Y1 ∧ · · · ∧ [X,Yj ] ∧ · · · ∧ Yn

⟩.

If we chooseYj = ej the constantj th canonical vector, we get

ν div X = X(ν)+ ν∑

j

∂ j (a j ),

which is (A.2). A similar point of view is related to taking adjoints. We have

X∗= −X − div X. (A.3)

In fact, withϕ,ψ ∈ C∞c (M),∫

X(ϕ)ψω =

∫M

LX(ϕψω)−

∫MϕLX(ψω)

=

∫M

d(ϕψωcX)︸︷︷︸=0

−

∫MϕX(ψ)ω −

∫Mϕψω div X,

yielding (A.3).

A.2. A Log-Lipschitz function is not in W1,1

The so-called Log-Lipschitz functions satisfying (1.3) forω1(r ) = r ln(1/r ) fail tobeW1,1, as shown by the following one-dimensional example. The function definedonR by

f (x) = x+e−1/x sin(e1/x), x+ =

{x for x > 0,

0 for x ≤ 0,(A.4)

does not belong toW1,1loc but satisfies with some constantC (and for|x1|, |x2| smaller

than 1) ∣∣ f (x1)− f (x2)∣∣ ≤ C|x1 − x2|

∣∣ln |x1 − x2|∣∣. (A.5)

∗The Lie derivative of a tensorK is defined as

LX(K ) =d

dt(8t

X)∗(K )|t=0.

It is a derivation for the tensor product, it commutes with the contraction, and for a functionf and avector field Y we haveLX( f ) = X f,LX(Y) = [X,Y]. Note that if ω is a one-form,X,Y vector fields,we have

〈dω, X ∧ Y〉 = 〈dωcX,Y〉 = LX(ω)cY −⟨d(ωcX),Y

⟩= X

(〈ω,Y〉

)−

⟨ω, [X,Y]

⟩− Y

(〈ω, X〉

).


In fact, forx > 0,

f ′(x) = e−1/x sin(e1/x)+ x−1e−1/x sin(e1/x)− x−1 cos(e1/x)

= −x−1 cos(e1/x)+ L∞,

and more precisely, since forx > 0, (x−1+1)e−1/x

≤ 1, | f ′(x)| ≤ |x−1 cos(e1/x)|+

1. It is elementary to check thatf ′ /∈ L1loc since

+∞ =

∫+∞

e

| cosu|

u ln udu =

∫ 1

0x−1

∣∣cos(e1/x)∣∣ dx.

We consider now 0≤ x0 < x1 such thatx1 − x0 < 1/e, and we define forθ ∈ (0,1),xθ = x0 + θ(x1 − x0). We need to check that

Q(x1, x0) =| f (x1)− f (x0)|

|x1 − x0| |ln(x1 − x0)|≤

∣∣∣∣ cos(e1/xθ )

xθ |ln(x1 − x0)|

∣∣∣∣ + 1

for someθ ∈ (0,1). So we get thatQ is bounded if|xθ ln(x1 − x0)| ≥ 1. However,denotingx1 − x0 = k−1, x0 = x, we havek ≥ e and∣∣xθ ln(x1 − x0)

∣∣ = (x + θk−1) ln k ≥ x ln k.

If x ln k ≥ 1, we obtainQ(x1, x0) ≤ 2. If x < 1/ ln k, we get, using the fact that thefunctionx 7→ xe−1/x is nondecreasing onR+,

Q = k| f (x + k−1)− f (x)|

ln k

≤2k

ln k

[(x + k−1)e−1/(x+k−1)

]≤

2k

ln k

[((ln k)−1

+ k−1)e−1/[(ln k)−1+k−1

]]

≤2

ln k+

2k

ln2 kexp−

1

k−1 + (ln k)−1

=2

ln k+

2

ln2 kexp

ln2 k

k + ln k→ 0 if k → +∞.

Remarks.The function f defined in (A.4) also provides an example of a function thathas “exactly” the Log-Lipschitz regularity; that is, it is such that

0< lim inf|x1−x2|→0|x1|,|x2|≤1

| f (x1)− f (x2)|

ω1(|x1 − x2|). (A.6)

We say that a functionu is Log{k}-Lipschitz if, for each compact setK , there exists apositive constantC such that forx1, x2 ∈ K , |x1 − x2| ≤ 1/C, we have∣∣u(x1)− u(x2)

∣∣ ≤ C|x1 − x2| (ln ◦ · · · ◦ ln)︸︷︷︸k terms

( 1

|x1 − x2|

). (A.7)


One can prove that the set of Log{k}-Lipschitz functions, although decreasing whenkincreases, is not included inW1,1

loc . (On the other hand, it is trivial to check thatW1,1loc 6⊂

Log-Lipschitz.) The proof is similar to the previous discussion on the functionfgiven by (A.4). Let us simply provide a formula; for an integerk ≥ 1, the followingfunction fk is Log{k}-Lipschitz but is not inW1,1

loc (nor in Log{k+1}-Lipschitz). Wedefine fk(x) = 0 for x ≤ 0, and forx positive we set

fk(x) =x sin[exp{k}(1/x)]∏

1≤ j ≤k exp{ j }(1/x)with exp{k}

= exp◦ · · · ◦ exp︸︷︷︸k terms

.

A.3. A weakened version of condition (2.11)Let β be a positiveL1

loc-function defined on[1,+∞) such that∫+∞

1β(r )dr = +∞. (A.8)

Condition (2.11) in Theorem 2.2 can be replaced by the more general integral condi-tion ∫ T0

0

∫|a j (t, x)|β(|x| + 1)dx dt< ∞. (A.9)

Note that (A.8) and (A.9) are satisfied, for instance, when

a j (t, x)

(1 + |x|) ln(1 + |x|)∈ L1((0, T0)× Rd)

.

Let us check the proof of Lemma 3.3, with assumptions (A.8) and (A.9) replacing(2.11). After (3.16), we chooseϕ(t, x) = τ(t)σ (λ−1B(|x| + 1)), where the functionB is defined forr ≥ 1 by

B(r ) = exp∫ r

1β(ρ)dρ.

In the right-hand side of (3.17),|a j (t, x)x j |λ−2 should be replaced by

1

2

∣∣a j (t, x)∣∣λ−1β

(|x| + 1

)B

(|x| + 1

)and the domain of integration by(0, T0) × {λ ≤ B(1 + |x|) ≤ 4λ}. We are then leftwith ∫ T0

0

∫{λ≤B(1+|x|)≤4λ}

2∣∣a j (t, x)

∣∣β(|x| + 1

)dx dt,

which goes to zero withλ−1 from (A.9), and the fact that the inverse functionB−1

goes to infinity withλ, thanks to (A.8).


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