Contractive metrics for scalar conservation laws

Transcription

1 Contractive metrics for scalar conservation laws François Bolley 1, Yann Brenier 2, Grégoire Loeper 34 Abstract We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L 1 - contraction property shown by Kružkov. Existence and uniqueness of solutions to scalar conservation laws in one space dimension have been established by Kružkov in the framework of entropy solutions (see [4] for instance), and among the properties satised by these solutions it is known that the L 1 norm between any two of them is a non-increasing function of time. In this work we shall focus on a class of entropy solutions such that a certain distance between the space derivatives of any two such solutions is also nonincreasing in time. On this class of solutions this result extends the L 1 norm contraction property. More precisely we consider as initial data nondecreasing functions on with limits and 1 at and + respectively. These properties are preserved by the conservation law, and corresponding solutions have been shown in [2] to arise in some models of pressureless gases, obtained as a continuous limit of systems of sticky particles. Noticing that the distributional space derivative of these functions are probability measures, we may consider the Wasserstein distance between the space derivatives of any two such solutions, and we shall prove in this paper that this distance is a nonincreasing function of time, constant in the case of classical solutions. 1 Ecole normale supérieure de Lyon, Umpa, 46, allée d'italie, F Lyon Cedex 7. fbolley@umpa.ens-lyon.fr 2 Laboratoire J.A.Dieudonne, UM CNS 6621, Universite de Nice Sophia-Antipolis, Parc Valrose, 618 Nice Cedex 2. brenier@math.unice.fr 3 Fields Institute & University of Toronto 4 Ecole Polytechnique Fédérale de Lausanne, SB-IMA, 115 Lausanne. gregoire.loeper@epfl.ch 1

2 1 INTODUCTION TO THE ESULTS 2 1 Introduction to the results Given a locally Lipschitz real-valued function f on, called a ux, we consider the scalar conservation law { ut + f(u) x =, t >, x, (1) u(,.) = u, with unknown u = u(t, x) and initial datum u L (), and where the subscripts stand for derivation. We shall consider solutions that are called entropy solutions (see [5] for instance) and are dened as follows: a function u = u(t, x) L ([, + [ ) is said to be an entropy solution of (1) on [, + [ if the entropy inequality (2) E(u) t + F (u) x holds in the sense of distributions for all convex Lipschitz function E on, and with associated ux F dened by (3) F (u) = This means that + u f (v) E (v) dv. ( E(u) ϕt + F (u) ϕ x ) dt dx + E(u (x)) ϕ(, x) dx for all nonnegative ϕ in the space Cc ([, + [ ) of C functions on [, + [ with compact support. We shall also consider classical solutions, that is, functions u = u(t, x) in C 1 (], + [ ) C([, + [ ) satisfying (1) pointwise. In particular any classical solution to (1) satises (2), i.e. is an entropy solution, and conversely any entropy solution satises (1) in the distribution sense. For entropy solutions, the following result is due to Kružkov (see [4]): Theorem 1.1 For every u L (), there exists a unique entropy solution u to (1) in L ([, + [ ) C([, + [, L 1 loc ()). Moreover for classical solutions, we have (see [5] for instance): Theorem 1.2 Given a C 2 ux f and a C 1 bounded initial datum u such that f u is nondecreasing on, the unique entropy solution u to (1) is a classical solution. In this work we shall consider initial data in the subset U of L () dened by

3 1 INTODUCTION TO THE ESULTS 3 Denition 1.3 A function v : belongs to U if it is nondecreasing, right-continuous, and has limits and 1 at and + respectively. The following proposition expresses that this set is preserved by the conservation law (1): Proposition 1.4 Given an initial datum u U, the entropy solution u given by Theorem 1.1 is such that u(t,.) belongs to U for all t. More precisely, given any t, the L () function u(t,.) is a.e. equal to an element of the set U, which on the other hand is characterized by Proposition 1.5 The distributional derivative v x measure on, and for any x, of any v U is a Borel probability v(x) = v x (], x]). Conversely, if µ is a probability measure on, then v dened on as belongs to U, and v x = µ. v(x) = µ(], x]) Consequently the map v v x is one-to-one from U onto the set P of probability measures on (and U can be seen as the set of repartition functions of real-valued random variables). Propositions 1.4 and 1.5 allow us to characterize at any time the distance between two solutions (with initial datum in U) in terms of their space derivatives, in particular by means of the Wasserstein distances: given any real number p 1, the Wasserstein distance of order p is dened on the set of probability measures on by ( ) 1/p W p (µ, µ) = inf x y p dπ(x, y) π 2 where π runs over the set of probability measures on 2 with marginals µ and µ; these distances are considered here in a broad sense with possibly innite values. This paper aims at proving that the Wasserstein distances between the space derivatives of any two such entropy solutions is a nonincreasing function of time: Theorem 1.6 Given a locally Lipschitz real-valued function f on and two initial data u and ũ in U, let u and ũ be the associated entropy solutions to (1). Then, for any t and p 1, we have (with possibly innite values) W p (u x (t,.), ũ x (t,.)) W p (u x, ũ x).

4 1 INTODUCTION TO THE ESULTS 4 We shall see in Section 2 that for p = 1 the distance W 1 satises W 1 (v x, ṽ x ) = v ṽ L 1 () for all v, ṽ U. Hence Theorem 1.6 reads in the case p = 1: u(t,.) ũ(t,.) L 1 () u ũ L 1 (). Thus, for initial proles in U, we recover the L 1 -contraction property given by Kružkov. In the case of classical solutions, the result of Theorem 1.6 is improved, since the Wasserstein distance between two solutions is conserved: Theorem 1.7 Given a C 1 real-valued function f on, let u and ũ in U be two initial data such that the associated entropy solutions u and ũ to (1) are classical solutions, increasing in x for all t. Then for any t and p 1 we have (with possibly innite values) W p (u x (t,.), ũ x (t,.)) = W p (u x, ũ x). From these general results can be induced some corollaries in the case of initial data in the subsets U p of U dened as: Denition 1.8 Let p 1. A function v in U belongs to U p if its distributional derivative v x has nite moment of order p, that is, if x p dv x (x) is nite. As in Proposition 1.5 the map v v x is one-to-one from U p onto the set P p of probability measures on with nite moment of order p. But we shall note in Section 2 that the map W p on P p P p denes a distance on P p. Then the real-valued map d p dened on U p U p by d p (v, ṽ) = W p (v x, ṽ x ) induces a distance on U p, and for the associated topology we have Corollary 1.9 Given a locally Lipschitz function f on, p 1 and u U p, the entropy solution u to (1) belongs to C([, + [, U p ). In particular for p = 1 and the previous result can be precised by d 1 (v, ṽ) = W 1 (v x, ṽ x ) = v ṽ L 1 (), Corollary 1.1 Given a locally Lipschitz function f on and u solution u to (1) is such that U 1, the entropy u(t,.) u(s,.) L 1 () t s f L ([,1]).

5 2 WASSESTEIN DISTANCES 5 This known result holds under weaker assumptions (for u with bounded variation, see [5]), but in our case it will be recovered in a straightforward way. Finally Theorem 1.7 can be precised in the U p framework in the following way: Corollary 1.11 Given a C 2 convex ux f and two C 1 increasing initial data u and ũ in U p for some p 1, the following three properties hold: 1. the associated entropy solutions u and ũ are classical solutions; 2. u(t,.) and ũ(t,.) belong to U p and are increasing for all t ; 3. for all t, we have (with nite values) W p (u x (t,.), ũ x (t,.)) = W p (u x, ũ x). The paper is organized as follows. The denition and some properties of Wasserstein distances are discussed in greater detail in Section 2. In Section 3 we consider the case of classical solutions, proving Theorem 1.7 and Corollary Then the general case of entropy solutions is studied in Sections 4 and 5: more precisely in Section 4 we introduce a time-discretized scheme, show the W p contraction property for this discretized evolution and prove the convergence of the corresponding approximate solution toward the entropy solution; Theorem 1.6 and its corollaries follow from this in Section 5. In Section 6 we shall nally see how such results extend to viscous conservation laws. 2 Wasserstein distances In this section p is a real number with p 1, P (resp. P p ) stands for the set of probability measures on (resp. with nite moment of order p) and dx for the Lebesgue measure on. The Wasserstein distance of order p, valued in {+ }, is dened on P P by (4) W p (µ, µ) = inf π ( 2 x y p dπ(x, y) where π runs over the set of probability measures on 2 with marginals µ and µ. It is equivalently dened by (5) ( 1 W p (µ, µ) = inf X µ,x µ ) 1/p ) 1/p X µ (w) X µ (w) p dw where the inmum is taken over all random variables X µ and X µ on the probability space (], 1[, dw) with respective laws µ and µ. It takes nite values on P p P p and indeed denes a distance on P p. For complete references about the Wasserstein distances and related topics the reader can refer to [6]. We only mention that both inma in (4) and (5) are achieved, and for the second denition we shall precise some random variables that achieve the inmum. For this purpose we introduce the notion of generalized inverse:

6 2 WASSESTEIN DISTANCES 6 Denition 2.1 Let v belong to U. Then its generalized inverse is the function v 1 dened on ], 1[ by v 1 (w) = inf{x ; v(x) > w}. Then v 1 is a nondecreasing random variable on (], 1[, dw) by denition, with law v x since 1 1 ( ) f(v 1 (w)) dw = f (s)1 {s v 1 (w)} ds dw ( 1 ) = f (s)1 {v(s) w} dw ds = f (s)(1 v(s)) ds = f(s) dv x (s) for all f in C 1 c (). In particular its repartition function is v. Moreover this generalized inverse achieves the inmum in (5): Proposition 2.2 Let v and ṽ in U. Then we have (with possibly innite values) W p (v x, ṽ x ) = ( 1 for all p 1. In particular for p = 1 we also have ) 1/p v 1 (w) ṽ 1 (w) p dw W 1 (v x, ṽ x ) = v ṽ L 1 (). Proof. The general result is proved in [6]. The result specic to the case p = 1 follows by introducing, for a given v U, the map dened on ], 1[ by { 1 if v(x) > w jv(x, w) = if v(x) w, for which we have for almost every w ], 1[, and v 1 ṽ 1 (w) = 1 jv jṽ (x, w) dx jv jṽ (x, w) dw = v ṽ (x) for almost every x. Integrating the rst equality on w in ], 1[ and the second one on x in, we deduce 1 v 1 ṽ 1 (w) dw = v ṽ (x) dx.

7 3 THE CASE OF CLASSICAL SOLUTIONS: THEOEM 1.7 AND COOLLAY 7 Given v U, its generalized inverse v 1 is actually the a.e. unique nondecreasing random variable on (], 1[, dw) with law v x. Given any other random variable X on (], 1[, dw) with law v x, v 1 is called the (a.e. unique) nondecreasing rearrangement of X (see [6]). We conclude this section recalling a result relative to the convergence of probability measures. A sequence (µ n ) of probability measures on is said to converge weakly toward a probability measure µ if, as n goes to +, ϕ dµ n tends to ϕ dµ for all bounded continuous real-valued functions ϕ on (or equivalently for all C functions ϕ with compact support, that is, if µ n converges to µ in the distribution sense). Given p 1 this convergence is metrized on P p by the distance W p as shown by the following proposition (see [6]): Proposition 2.3 Let p 1, (µ n ) a sequence of probability measures in P p and µ P. Then the following statements are equivalent: i) (W p (µ n, µ)) converges to ; ii) (µ n ) converges weakly to µ and sup n x x p dµ n (x) tends to as goes to innity. In this proposition we do not a priori assume that µ belongs to P p, but it can be noted that this property is actually induced by any of both hypotheses i) and ii). For measures in P we have the weaker result: Proposition 2.4 Let p 1, (µ n ) and (ν n ) two sequences in P converging weakly to µ and ν in P respectively. Then (with possibly innite values) W p (µ, ν) lim inf n + W p(µ n, ν n ). 3 The case of classical solutions: Theorem 1.7 and corollary 3.1 Proof of Theorem 1.7 We consider two classical solutions u and ũ to (1) such that u(t,.) and ũ(t,.) belong to U and are increasing for all t, and we shall prove that as a consequence of Proposition 2.2. W p (u x (t,.), ũ x (t,.)) = W p (u x, ũ x)

8 3 THE CASE OF CLASSICAL SOLUTIONS: THEOEM 1.7 AND COOLLAY 8 by The map u is increasing from to 1, so has a (true) inverse X(,.) dened on ], 1[ u (X(, w)) = w. Then, given w ], 1[, we consider a characteristic curve t X(t, w) solution of (6) X t (t, w) = f ( u(t, X(t, w)) ) for t, and taking value X(, w) at t =. Since f is C 1 and u is bounded there exists a (non necessarily unique) solution X(., w) to (6) by Peano Theorem (see [3] for instance); moreover by a classical computation from (1) it is known to satisfy (7) u(t, X(t, w)) = w for all t, from which it follows that ( ) X t (t, w) = f (u(t, X(t, w))) = f (w) and hence (8) X(t, w) = X(, w) + tf (w). In particular there exists a unique solution X(., w) to (6). Now given t, X(t,.) is the (true) inverse of the increasing function u(t,.) (by (7)), and Proposition 2.2 writes But from (8) we obtain W p (u x (t,.), ũ x (t,.)) = ( 1 1/p. X(t, w) X(t, w) dw) p (9) X(t, w) X(t, w) = X(, w) X(, w). This result ensures in particular that W p (u x (t,.), ũ x (t,.)) remains constant in time, may its initial value be nite or not; note however that (9) is actually much stronger that Theorem Proof of Corollary 1.11 We assume that f is a C 2 convex function on, and u is a C 1 increasing initial prole in U p. First of all we note that the associated entropy solutions u is a classical solution in view of Theorem 1.2: this result is proved in [5] for instance, and its proof also ensures that u(t,.) is increasing for all t.

9 4 TIME DISCETIZATION OF THE CONSEVATION LAW 9 Then we check that the moment property is preserved by the conservation law, that is, that u(t,.) also belongs to U p for any t. Indeed, given t, we have by the change of variable w = [u(t,.)](x): x p u x (t, x) dx = 1 1 X(t, w) p dw = X(, w) + tf (w) p dw [ 1 ] 2 p 1 X(, w) p dw + t p f p L (],1[) which is nite since 1 X(, w) p dw = x p u x(x) dx is nite by assumption. This ends the proof of Corollary Time discretization of the conservation law In the previous section we have seen that the classical solutions are obtained through the method of characteristics, that we now summarize in our case: given an initial prole u in U such that the corresponding solution u is C 1 and increasing in x for all t, let X(,.) be its inverse, dened by u (X(, w)) = w for all w ], 1[. Let then X(, w) evolve into (1) X(t, w) = X(, w) + tf (w) for all t and w ], 1[ (see (8)). The solution u(t,.) is then the inverse of the increasing map X(t,.), that is, is the unique solution of X(t, u(t, x)) = x. In the general case, dening X(,.) in some similar way, there is no hope for the function X(t,.) dened by (1) to be increasing for t > ; inverting it would thus lead to a multivalued function, and no more to the entropy solution of the conservation law, as in the particular case discussed above. However, averaging (or "collapsing") this multivalued function into a single-valued function, Y. Brenier showed in [1] how to build an approximate solution to the conservation law. We now precisely describe this so-called Transport-Collapse method in our case.

10 4 TIME DISCETIZATION OF THE CONSEVATION LAW Denition and W p contraction property of the discretized solution Let u U be some xed initial prole, with generalized inverse X(,.) given as in Denition 2.1 by X(, w) = inf{x ; u (x) > w} for all w ], 1[. X(,.) can be seen as a random variable on the probability space ], 1[ equipped with the Lebesgue measure dw; its law is u x, as pointed out after Denition 2.1. We let then X(,.) evolve according to the method of characteristics, denoting X(h, w) = X(, w) + hf (w) for all h and almost every w ], 1[. Again, given h, X(h,.) can be seen as a random variable on ], 1[; let then T h u be its repartition function, that is, the function belonging to U and dened at any x as the Lebesgue measure of the set {w ], 1[; X(h, w) x}. It is given by T h u (x) = 1 1 {X(h,w) x} (w) dw. We summarize this construction in the following denition: Denition 4.1 Let v U with generalized inverse X(,.) dened on ], 1[ by X(, w) = inf{x ; v(x) > w}. Then, given h, and letting X(h, w) = X(, w) + hf (w) for almost every w ], 1[, we dene the U function T h v on by T h v(x) = 1 1 {X(h,w) x} (w) dw. In the case of Section 3 (see (8)), it turns out that X(h,.) is the (true) inverse of T h u, and (h, x) T h u (x) is exactly the entropy solution to equation (1) with initial datum u in U. This does not hold anymore in the general case, but will allow us to build an approximate solution S h u by iterating the operator T h. Let us rst give two important properties of T h : Proposition 4.2 Let h, T h dened as above and p 1. Then i) T h v belongs to U p if so does v. ii) For any v and ṽ in U we have (with possibly innite values unless v and ṽ U p ) W p ([T h v] x, [T h ṽ] x ) W p (v x, ṽ x ).

11 4 TIME DISCETIZATION OF THE CONSEVATION LAW 11 Proof. It is really similar to what has been done in Section 3 as for Corollary i) T h v belongs to U as a repartition function of a random variable, and we have x p d[t h v] x (x) = = 1 1 X(h, w) p dw X(, w) + hf (w) p dw 1 2 p 1 X(, w) p + hf (w) p dw [ ] 2 p 1 x p dv x (x) + h p f p L (],1[), which ensures that [T h v] x has nite moment of order p if so does v x. ii) On one hand the generalized inverses X(,.) and X(,.) of v and ṽ respectively satisfy (11) W p (v x, ṽ x ) = ( 1 X(, w) X(, ) 1/p w) p dw by Proposition 2.2 (with nite values if both v and ṽ belong to U p, and possibly innite otherwise). On the other hand X(h,.) and X(h,.) have respective law [T h v] x and [T h ṽ] x, so (12) W p ([T h u ] x, [T h ũ ] x ) ( 1 by denition of the Wasserstein distance. But X(h, w) X(h, ) 1/p w) p dw X(h, w) X(h, w) = X(, w) X(, w) for almost every w ], 1[ by denition, which concludes the argument by (11) and (12). Note again that (12) holds only as an inequality since X(h,.) and X(h,.) are not necessarily nondecreasing, which was the case in the example discussed in Section 3. We now use the operator T h dened above to build an approximate solution S h u to the conservation law (1): Denition 4.3 Let h be some positive number and v U. For any t decomposed as t = (N + s)h with N N and s < 1, we let where T h N+1 v = v and Th v = T h (Th Nv). S h v(t,.) = (1 s) Th N v(.) + s T N+1 h v(.)

12 4 TIME DISCETIZATION OF THE CONSEVATION LAW 12 These iterations make sense because T h v U if v U, S h v(t,.) U (resp. U p ) for any h, t and v U (resp. U p ). We now prove two contractions properties on these approximate solutions. We rst have the L 1 () contraction property: Proposition 4.4 Let h be some xed positive number and S h dened as above. Then, for any v U and s, t we have S h v(t,.) S h v(s,.) L 1 () t s f L ([,1]). Proof. As in [1] we rst observe that T h V V L 1 () h f L (],1[) for any V U, then let t = (M + µ)h and s = (N + ν)h with M, N N and µ, ν < 1, and, for instance assuming that M > N, prove the proposition by applying this rst bound to V = S h v(kh,.) = Th k v for k = N + 1,..., M 1. Then we have the W p contraction property: Proposition 4.5 Let h be some xed positive number and S h dened as above. Then, given v and ṽ in U, we have for any t : W p ([S h v] x (t,.), [S h ṽ] x (t,.)) W p (v x, ṽ x ). Proof. It follows from Proposition 4.2 (about T h ) and to the convexity of the W p distance to the power p, in the sense that W p p (αµ 1 + (1 α)µ 2, αν 1 + (1 α)ν 2 ) αw p p (µ 1, ν 1 ) + (1 α)w p p (µ 2, ν 2 ) for all real number α [, 1] and probability measures µ 1, µ 2, ν 1 and ν 2 (see [6] for instance). We shall now recall the convergence of the scheme toward the entropy solution of the conservation law. 4.2 Convergence of the scheme in the L 1 loc () sense In this section we consider the space C([, + [, L 1 loc ()) equipped with the topology dened by the semi-norms m q nm (f) = sup f(x) dx t [,n] m for any integers n and m and f C([, + [, L 1 loc ()). Then we have Proposition 4.6 Let u U. Then, as h goes to, the function S h u converges in C([, + [, L 1 loc ()) to the entropy solution of (1) with initial datum u.

13 4 TIME DISCETIZATION OF THE CONSEVATION LAW 13 We briey give the steps of the proof, which follows the one of Brenier in [1], adapted to functions of U instead of L 1 (). We rst prove that the family (S h u ) h is relatively compact in C([, + [, L 1 loc ()) by means of Proposition 4.4, and Helly and Ascoli-Arzela Theorems. Then we check that the limit of any sequence of (S h u ) h converging in C([, + [, L 1 loc ()) is an entropy solution to the conservation law (1) with initial datum u. By the uniqueness of this solution ensured by Theorem 1.1, this concludes the proof of Proposition Convergence of the scheme in W p distance sense We rst prove a uniform equiintegrability result on the approximate solutions: Proposition 4.7 Let S h be dened as above, v U p and T. Then sup sup h T t T tends to as goes to innity. x x p d[s h v] x (t, x) Proof. We again denote M = f L (],1[), and rst consider T h itself, writing x x p d[t h v] x (x) = 1 v 1 (w) + hf (w) p 1 { v 1 (w)+hf (w) } dw ( x + hm) p 1 { x +hm } dv x (x) ( 1 + hm ) p x p dv x (x) hm x hm for > hm. From this computation we deduce by iteration x for > NhM, with Thus N j=1 x p d[t N h v] x (x) ( 1 + x N j=1 ) ( hm 1 + jhm ( ) p hm 1 + x p dv x (x) jhm x NhM ) N ( ) hm NhM exp. NhM NhM ( ) pt M x p d[s h v] x (Nh, x) exp x p dv x (x) T M x T M for any N and h such that Nh T.

14 5 THE GENEAL CASE OF ENTOPY SOLUTIONS: THEOEM 1.6 AND COOLLAIES14 >From this we get for instance ( ) 2pT M x p d[s h v] x (t, x) exp x p dv x (x) x 2T M x 2T M for any t and h smaller than T. This concludes the argument since the last integral tends to as goes to innity. >From this we deduce the convergence of the scheme in W p distance sense: Proposition 4.8 Let u U p and u be the entropy solution to (1) with initial datum u. Then, for any t, W p ([S h u ] x (t,.), u x (t,.)) converges to as h goes to. Given t, S h u (t,.) converges to u(t,.) in L 1 loc () as h goes to (by Proof. Proposition 4.6), so [S h u] x (t,.) converges to the probability measure u x (t,.), rst in the distribution sense, then in the weak sense of probability measures, and nally in W p distance by Propositions 4.7 and 2.3. Note in particular that u x (t,.) has nite moment of order p for any t, that is, u(t,.) belongs to U p. 5 The general case of entropy solutions: Theorem 1.6 and corollaries 5.1 Proof of Theorem 1.6 We let p 1 and consider two initial data u and ũ in U with associated entropy solutions u and ũ. Given t, Proposition 4.6 yields again the convergence of [S h u ] x (t,.) to u x (t,.) in the weak sense of probability measures. Since this holds also for ũ, we obtain W p (u x (t,.), ũ x (t,.)) lim inf h by Proposition 2.4. But, for each h, by Proposition 4.5, so nally This concludes the argument. W p ([S h u ] x (t,.), [S h ũ ] x (t,.)) W p ([S h u ] x (t,.), [S h ũ ] x (t,.)) W p (u x, ũ x) W p (u x (t,.), ũ x (t,.)) W p (u x, ũ x).

15 5 THE GENEAL CASE OF ENTOPY SOLUTIONS: THEOEM 1.6 AND COOLLAIES Proof of Corollary 1.9 We recall that in the introduction we have dened a distance on each U p by letting d p (u, ũ) = W p (u x, ũ x ), and we now prove that, given p 1 and u U p, the entropy solution u to the conservation law (1) belongs to C([, + [, U p ). We rst note, in view of the proof of Proposition 4.8, that u(t,.) indeed belongs to U p for all t. Then, given s, we need to prove that d p (u(t,.), u(s,.)) (= W p (u x (t,.), u x (s,.))) tends to as t goes to s. Indeed, on one hand u(t,.) tends to u(s,.) in L 1 loc () by Theorem 1.1, so u x (t,.) tends to u x (s,.), rst in the distribution sense, then in the weak sense of probability measures. On the other hand, given T > s, we now prove that sup x p du x (t, x) goes to t T x as goes to innity. For this, given ε >, let such that sup sup h T t T x x p d[s h u ] x (t, x) ε by Proposition 4.7. Let then ϕ Cc () such that ϕ 1 and ϕ(x) = if x. On one hand ϕ(x) x p d[s h u ] x (t, x) ϕ(x) x p du x (t, x) as h goes to since ϕ(x) x p Cc () and [S h u ] x (t,.) tends to u x (t,.) in distribution sense. On the other hand ϕ(x) x p d[s h u ] x (t, x) ε for all h, t T. Hence at the limit ϕ(x) x p du x (t, x) ε for all t T, from which it follows that which means that indeed sup t T sup t T x x x p du x (t, x) ε, x p du x (t, x) goes to as goes to innity. From these two results we deduce the continuity result by Proposition 2.3.

16 6 EXTENSION TO VISCOUS CONSEVATION LAWS Proof of Corollary 1.1 Given t, S h u (t,.) converges to u(t,.) in L 1 loc () by Proposition 4.6, so for all s, t, n we have But u(t,.) u(s,.) L 1 ([ n,n]) = lim h S h u (t,.) S h u (s,.) L 1 ([ n,n]). S h u (t,.) S h u (s,.) L 1 ([ n,n]) S h u (t,.) S h u (s,.) L 1 () t s f L () for all h by Proposition 4.4, so letting h go to we get u(t,.) u(s,.) L 1 ([ n,n]) t s f L (). Since this holds for all n, we obtain Corollary Extension to viscous conservation laws In this section we let ν be a positive number and consider the viscous conservation law (13) u t + f(u) x = ν u xx t >, x with initial datum u L (). Assuming that f is a locally Lipschitz real-valued function on, and calling solution a function u in L ([, + [ ) such that (13) holds in the sense of distributions, it is known that, given u L (), there exists a unique solution u to (13). If moreover u U, then u(t,.) also belongs to U for all t, and the W p contraction property stated in Theorem 1.6 in the inviscid case ν = still holds: Theorem 6.1 Given a locally Lipschitz real-valued function f on and two initial data u and ũ in U, let u and ũ be the associated solutions to (13). Then, for any t and p 1, we have (with possibly innite values) W p (u x (t,.), ũ x (t,.)) W p (u x, ũ x). We briey mention how this contraction property for the viscous conservation law allows to recover the same property for the inviscid equation, given in Theorem 1.6. Given some initial datum u in U and ν >, let indeed u ν be the corresponding solution to the viscous equation (13). Then it is known (see [5] for instance) that u ν (t,.) converges in L 1 loc () to the solution u(t,.) to the inviscid conservation law (1) with initial datum u. From this the argument already used in Section 5.1 (with S h u (t,.) intead of u ν (t,.)) enables to recover Theorem 1.6.

17 EFEENCES 17 The proof of Theorem 6.1 follows the lines of Sections 4 and 5 and makes use of a timediscretization of equation (13) based on the discretization of the inviscid conservation law previously discussed. More precisely, given a time step h >, we rst map u U to T h u as in section 4.1, and then let T h u evolve along the heat equation on a time interval h, that is, map it to T h u = K h T h u where K h is the heat kernel dened on by K h (z) = 1 4πh e z2 4h. Then, dening an approximate solution S h u by iterating the T h operator as in Denition 4.3, we prove that Propositions 4.2 and 4.5 still hold for the new T h and S h operators (Note that the convolution with the heat kernel is a contraction for the Wasserstein distance of any nite order). Proposition 4.4 only holds assuming that v is twice derivable with v in L 1 (): it more precisely reads S h v(t,.) S h v(s,.) L 1 () t s [ f L (],1[) + v L 1 ()]. As in Section 4.2, this enables to prove that, given u in U, twice derivable with (u ) in L 1 (), the family (S h u ) h converges in C([, + [, L 1 loc ()) to the solution of (13) with initial datum u. With this convergence result in hand we follow the lines of Section 5.1 to prove Theorem 6.1 in the case of twice derivable initial data, with L 1 second derivative, while the general case follows by a density argument. Acknowledgments: This work has been partly supported by the European network Hyke, funded by the EC contract HPN-CT eferences [1] Y. Brenier, ésolution d'équations d'évolution quasilinéaires en dimension N d'espace à l'aide d'équations linéaires en dimension N + 1, J. Di. Eq. 5, 3, p (1983). [2] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 6, p (1998). [3] L. Hörmander, Lectures on nonlinear hyperbolic dierential equations, Mathematics & Applications (26), Springer, Berlin (1997).

18 EFEENCES 18 [4] S. Kružkov, Generalized solutions of the Cauchy problem in the large for rst order nonlinear equations, Dokl. Akad. Nauk. SSS, 187, p (1969). [5] D. Serre, Systems of conservation laws I, Cambridge University Press, Cambridge (1999). [6] C. Villani, Topics in optimal transportation, Grad. Stud. Math (58), American Mathematical Society, Providence (23).