Uniqueness of Weak Solutions toSystems of Conservation LawsAlberto Bressan(�) and Philippe LeFloch(��)(�) S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy,(��) CMAP & CNRS, Ecole Polytechnique, 91128 Palaiseau, France.Abstract. Consider a strictly hyperbolic n�n system of conservation laws in one space dimension:ut + F (u)x = 0: (�)Relying on the existence of the Standard Riemann Semigroup generated by (�), we establish theuniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumptionon the variation of u along space-like segments.

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1 - Introduction.We are concerned with the uniqueness of solutions to the Cauchy problem for an hyperbolicn � n system of conservation laws in one space dimension:ut + F (u)x = 0; (1:1)u(0; x) = �u(x): (1:2)Let � IRn be an open set containing the origin, and let F : 7! IRn be a smooth map. Assumethat the system (1.1) is strictly hyperbolic and that each characteristic �eld is either linearlydegenerate or genuinely nonlinear [12, 18]. For every initial data �u 2 L1 with small total variation,a well known theorem of Glimm [10] provides the global existence of weak solutions.In subsequent papers [7, 8, 13, 14, 15, 16, 17], several authors have shown that those solutionswhich satisfy various additional regularity and entropy admissibility conditions are necessarilyunique. Apart from the scalar case [11, 20], however, these uniqueness results are applicable onlywithin a class of functions which is strictly smaller than the class where an existence theorem isknown.An alternative approach to the problem of uniqueness and continuous dependence was pro-posed in [1]. Instead of the single Cauchy problem, one considers a whole ow of solutions of(1.1), continuously depending on the initial data. The main results in [1, 5] can be summarized asfollows.Theorem 1. There exists a closed domain D � L1(IR; IRn), constants �0; L, and a continuoussemigroup S : D � [0;1[ 7! D with the properties:(i) Every function �u 2 L1 with Tot.Var.(�u) � �0 lies in D.(ii) For all �u; �v 2 D, t; s � 0 one has St�u� Ss�v L1 � L�jt � sj + k�u� �vkL1�.(iii) If �u 2 D is piecewise constant, then for t > 0 su�ciently small the function u(t; �) = St�ucoincides with the solution of (1.1)-(1.2) obtained by piecing together the standard self-similarsolutions of the corresponding Riemann problems.(iv) Each trajectory t 7! u(t; �) = St�u is a weak, entropy-admissible solution of the correspondingCauchy problem (1.1)-(1.2).(v) Let (u�)��1 be a sequence of approximate solutions of (1.1)-(1.2) generated by a wave-fronttracking algorithm, or by the Glimm scheme with uniformly distributed sampling. Then L1-lim�!1 u�(t; �) = St�u for every t � 0. 1

The positively invariant domain D has the formD = clnu 2 L1(IR; IRn); u is piecewise constant; V (u) + C0 �Q(u) < �0o; (1:3)for some constants C0; �0 > 0. Here V (u) and Q(u) denote the total strength of waves and thewave interaction potential of u, while cl denotes closure. On a given domain D, the semigroup Swith the above properties is unique.Following [1], we say that a map S with the properties (i){(iii) is a Standard Riemann Semi-group (SRS). The existence of such a semigroup was proved in [3] for 2� 2 and recently in [5] forgeneral n�n systems. The uniqueness of the SRS (up to the domain), together with the properties(iv)-(v), was shown in [1]. See [2] for a general survey. At this stage, given the Cauchy problem(1.1)-(1.2), in order to establish the uniqueness of a solution u = u(t; x) it thus su�ces to showthat u coincides with the corresponding semigroup trajectory:u(t; �) = St�u 8t � 0: (1:4)A condition which implies (1.4) was derived in [1], based on a set of local integral estimates.In the present paper we prove that any weak, entropy-admissible solution of (1.1)-(1.2) actuallysatis�es (1.4), provided that the variation of u does not grow too wildly. Roughly speaking, werequire that, if a space-like segment � is contained inside the domain of dependency of anothersegment e�, than the total variation of u along � should be controlled by the variation of u alonge�. This yields a new uniqueness theorem for solutions to hyperbolic systems of conservation laws,valid within the same class of functions where global existence theorems are known.Some basic notations are recalled below. Let A(u) = DF (u) be the Jacobian matrix of F atthe point u. Smooth solutions of (1.1) thus satisfy the equivalent quasilinear systemut +A(u)ux = 0: (1:5)Let �1(u) < � � � < �n(u) be the eigenvalues of A(u) and choose right and left eigenvectors ri(u),li(u), i = 1; : : : ; n, normalized so thatri � �i(u) := limh!0 �i�u+ hri(u)�� �i(u)h � 0; (1:6)jrij � 1; li; rj� = � 1 if i = j,0 if i 6= j. (1:7)We shall be concerned with a solution u de�ned on a strip [0; T ] � IR. Uniqueness will beproved under three assumptions. 2

(A1) (Conservation Equations) The function u = u(t; x) is a weak solution of the Cauchyproblem (1.1)-(1.2), taking values within the domain of the Standard Riemann Semigroup S.More precisely, u is a measurable function from [0; T ]� IR into � IRn, with u(t; �) 2 D forevery t. The initial condition (1.2) holds, together withZ Z �u't + f(u)'x� dxdt = 0 (1:8)for every C1 function ' with compact support contained inside the open strip ]0; T [�IR.(A2) (Entropy Condition) Let u have an approximate jump discontinuity at some point (�; �) 2]0; T [�IR. More precisely, assume that there exists states u�; u+ 2 and a speed � 2 IR suchthat, calling U(t; x) := �u� if x < � + �(t� �),u+ if x > � + �(t� �), (1:9)there holds lim�!0+ 1�2 Z �+���� Z �+���� ���u(t; x)� U(t; x)��� dxdt = 0: (1:10)Then, for some i 2 f1; : : : ; ng, one has the entropy inequality [12]:�i(u�) � � � �i(u+): (1:11)A precise statement of our regularity assumption requires some preliminary de�nitions. Fixa constant �̂, strictly greater than all wave speeds. By a space-like segment we mean a set of theform � := n(t; x); x 2 [a; b]; t = �(x) = �x + �o;for some �; � with j�j < 1=�̂. In particular, a horizontal segment is a set of the forme� := �(�; x); x 2 [c; d];for some �xed � . We say that the above segment � lies inside the domain of dependency of e� if[a; b] � [c; d]; 0 � �(a)� � � a� c�̂ ; 0 � �(b)� � � d� b�̂ :The total variation of the function u along the segment � is written Tot.Var.(u; �). The followingassumption implies that the variation of u does not grow too wildly, as time increases.(A3) (Tame Variation Assumption) There exists a constant K such thatTot.Var.(u; �) � K � Tot.Var.(u; e�) (1:12)3

for every horizontal segment e� and every space-like segment � in the domain of dependencyof e�.By standard interaction estimates, it is well known that every solution of (1.1) constructedby wave-front tracking or by the Glimm scheme satis�es the above assumption. More generally,we expect that any approximation scheme allowing control of the total variation will producesolutions which satisfy (A3). It is conceivable that (A3) may be deduced from (A1)-(A2), forarbitrary solutions with bounded variation. Based on the theory of generalized characteristics [6],the detailed analysis in [19] shows that the property (A3) actually holds for every countably regularBV solution of a strictly hyperbolic, genuinely nonlinear 2� 2 system of conservation laws.The main result proved in this paper is the following.Theorem 2. Let the system (1.1) generate the Standard Riemann Semigroup S : D� [0;1[ 7! D.Then, for every �u 2 D, T > 0, the Cauchy problem (1.1)-(1.2) has a unique weak solution u :[0; T ] 7! D satisfying the assumptions (A1){(A3).A proof of Theorem 2 will be given in Section 3.2 - Preliminary results.Lemma 1. Let u : [0; T ]�IR 7! IRn be any function which satis�es the Tame Variation Assumption(A3), with Tot.Var.�u(0; �)� <1. Then the map t 7! u(t; �) is Lipschitz continuous, i.e. u(t; �)� u(s; �) L1 � L0jt� sj (2:1)for some constant L0 and all t; s 2 [0; T ].Proof. The condition (1.12) impliesTot.Var.�u(�; �)� � K � Tot.Var.�u(0; �)� 8� 2 [0; T ]: (2:2)Consider any interval [�; � + h] � [0; T ]. For each x 2 IR, we can apply (1.12) choosing e� to bethe segment with endpoints (�; x� h�̂), (�; x+ h�̂), and � the segment with endpoints (� + h; x),(�; x+ h�̂). This yields��u(� + h; x)� u(�; x)�� � ��u(� + h; x)� u(�; x+ h�̂)��+ ��u(�; x+ h�̂)� u(�; x)��� (K + 1) � Tot.Var.�u(�); [x� h�̂; x+ h�̂]�: (2:3)Introduce the function �(x) := Tot.Var.�u(�); ]�1; x]�:4

Integrating (2.3) and recalling (2.2) one obtainsZ 1�1 ��u(� + h; x)� u(�; x)�� dx � (K + 1) Z 1�1 ��(x+ h�̂)� �(x� h�̂)� dx= 2h�̂(K + 1) � Tot.Var.�u(�)�� 2h�̂K(K + 1) �Tot.Var.�u(0)�:This yields (2.1) with Lipschitz constant L0 = 2�̂K(K + 1) � Tot.Var.�u(0)�:Lemma 2. Let w : ]a; b[ 7! IRn be a function such thatsupa<�1<�2<b ���� Z �2�1 w(x) dx���� � C2(b� a) � V 2; (2:4)Tot.Var.�w; ]a; b[� < 2KV (2:5)for some constants C2; K; V , with V < 1=4. ThenZ ba ��w(x)��dx � �C2 + 4K�(b� a)V 3=2: (2:6)Proof. It su�ces to prove (2.6) assuming that w is a smooth function. By a density argument,the same inequality must then hold for every BV function. Let x; h be given, with0 < h < (b� a)=2; 0 < x < (a+ b)=2:Integrating the identity w(x) = w(y)� Z yx wx(z) dzfor y 2 [x; x+ h] , we obtainh jw(x)j � ���� Z x+hx w(y) dy����+ Z x+hx Z yx ��wx(z)�� dzdy:Dividing by h and integrating once more we arrive atZ (a+b)=2a ��w(x)��dx � 1h Z (a+b)=2a ���� Z x+hx w(y) dy����dx+ 1h Z (a+b)=2a Z x+hx Z yx ��wx(z)�� dzdydx:= E1 + E2: (2:7)The �rst term on the right hand side of (2.7) is bounded byE1 � b� a2h � supa<�1<�2<b ���� Z �2�1 w(x) dx���� � C2 (b� a)22h � V 2 : (2:8)5

Introducing the (nondecreasing) function�(x) := Z xa ��wx(z)��dz = Tot.Var.�w; [a; x]�;the second term can be estimated byE2 = 1h Z (a+b)=2a Z x+hx ��(y)� �(x)� dydx� Z (a+b)=2a 1h Z x+hx ��(x+ h)� �(x)� dydx= Z (a+b)=2+h(a+b)=2 � Z a+ha !�(x) dx� h � ��a+ b2 + h�� h � Tot.Var.�w; ]a; b[�� h � 2KV : (2:9)Choosing h := (b� a)V 1=2 < b� a2 ; (2:10)from the estimates (2.7)-(2.9) we deduceZ (a+b)=2a ��w(x)�� dx � �C22 + 2K�(b� a)V 3=2:An entirely similar argument shows that this estimate is also valid for the subinterval �(a+b)=2; b�,completing the proof.Lemma 3. Assume that the function u satis�es the Tame Variation Assumption (A3). Let (�; �)be a point of discontinuity for the function x 7! u(�; x) (with � �xed) and denote by u(�; ��) itsleft and right limits at x = �. Then we have��u(t; x)� u(�; �+)�� � (K + 1) � Tot.Var.�u; ]�; 2x� �]� if x > � + �̂(t � �); t � �; (2:11)��u(t; x)� u(�; ��)�� � (K + 1) � Tot.Var.�u; [2x� �; �[� if x > � � �̂(t � �); t � �: (2:12)In particular if u satis�es the conditions (1.9)-(1.10) in (A2) for some u+, u� and �, then u� =u(�; ��).Proof. Consider for instance a point (t; x) with x � � + �̂(t � �). The estimate (2.11) is thenobtained from (1.14), choosing e� to be the segment with endpoints ��; x � �̂(t � �)�, (�; 2x� �)and � the segment with endpoints (t; x), (�; 2x� �).6

Observing that the total variation in the right hand side of (2.11) converges to zero uniformlyas (x; t)! (�; �), we deducelim�!0+ 1�2 Z �+�=�̂� Z �+��+�̂(t��) ��u(t; x)� u(�; �+)�� dxdt � lim�!0+ K + 1�̂ � Tot.Var.�u; ]�; � + 2�[� = 0:(2:13)Comparing (2.13) with (1.10), we conclude that u+ = u(�; �+). Similarly, u� = u(�; ��).Lemma 4. Let u be a weak solution of (1.1). Let the conditions (1.9)-(1.10) hold at some point(�; �). Then the states u�; u+ and the speed � satisfy the Rankine-Hugoniot equations�(u+ � u�) = f(u+)� f(u�): (2:14)Together with the entropy condition in (A2), Lemma 4 shows that the states u�; u+ areconnected either by an admissible shock or by a contact discontinuity, propagating with the speed�.Proof of Lemma 4. To derive (2.14), we integrate the conservation laws (1.1) on a family ofparallelograms with a �xed center (�; �):Ds := n(x; t) : jt� � j � s; ��x � � � �(t� �)�� � so (s > 0):From the basic equations (1.1) we deduceZ �+s+�s��s+�s u(� + s; x) dx� Z �+s��s��s��s u(� � s; x) dx+ Z �+s��s �f(u)� �u�(t; � + s+ �(t� �)) dt� Z �+s��s �f(u)� �u�(t; � � s + �(t� �)) dt = 0:Integrating once more for s 2]0; �] and dividing by �2 we obtain1�2 Z �0 Z �+s+�s��s+�s u(� + s; x) dxds� 1�2 Z �0 Z �+s��s��s��s u(� � s; x) dxds+ 1�2 Z �0 Z �+s��s �f(u)� �u�(t; � + s + �(t� �)) dtds� 1�2 Z �0 Z �+s��s �f(u)� �u�(t; � � s + �(t� �)) dtds= 0: (2:15)By (1.9)-(1.10), each of the four terms on the left hand side of (2.15) has a limit when �! 0. Forinstance 1�2 Z �0 Z �+s+�s��s+�s u(� + s; x) dxds ! u� + u+27

1�2 Z �0 Z �+s��s �f(u)� �u�(t; � + s + �(t� �))dtds ! f(u+)� �u+:We thus obtain u� + u+2 � u� + u+2 + �f(u+)� �u+� � �f(u�)� �u�� = 0;proving (2.14).We conclude this section by recalling two local integral estimates that characterize the trajec-tories of a Standard Riemann Semigroup.Two types of local approximate solutions for (1.1) will be considered. One is derived from theself-similar solution of a Riemann problem, the other is obtained by \freezing" the coe�cients ofthe corresponding quasilinear hyperbolic system in a neighborhood of a given point.Let w : IR 7! IRn be any BV function and �x any point � 2 IR where w has a jump. Call! = !(t; x) the unique self-similar entropy solution of the Riemann problemut + F (u)x = 0; u(0; x) = �w(��) if x < 0,w(�+) if x > 0. (2:16)As usual, let �̂ be an upper bound for all wave speeds. For t � � , de�neU ](t; x) := �!(t; x� �) if jx� �j � �̂t,w(x) if jx� �j > �̂t. (2:17)Observe that the function t 7! U ](t; �) is Lipschitz continuous w.r.t. the L1 distance, andapproaches w as t! 0+ .Next, call eA := DF�w(�)� the Jacobian matrix of F computed at the point w(�). For t � 0,de�ne U [(t; x) as the solution of the linear hyperbolic Cauchy problem with constant coe�cientsU [t + eAU [x = 0; U [(0) = w: (2:18)Lemma 5. Let S be a Standard Riemann Semigroup generated by the system (1.1), with domainD as in (1.3). Then, there exists a constant C1 such that the following holds. For every functionw 2 D, every � 2 IR and h; � > 0, with the above de�nitions one has1h Z �+��h�̂���+h�̂ ����Shw�(x)� U ](h; x)��� dx � C1 � Tot.Var.�w; ]� � �; �[ [ ]�; � + �[ �; (2:19)1h Z �+��h�̂���+h�̂ ����Shw�(x)� U [(h; x)��� dx � C1 � �Tot.Var.�w; ]� � �; � + �[ ��2: (2:20)8

For a proof, see [1, p.217]. Observe that any solution U [ of the linear system (2.18) will cer-tainly satisfy the Tame Variation Assumption (A3), possibly with a larger constant K, dependingonly on the �rst order derivatives of F . In particular, for every � 2 IR, � > 0 we can assumeTot.Var.�U [(�=�̂); ]� � �; � + �[� � K � Tot.Var.�U [(0); ]� � 2�; � + 2�[ �: (2:21)3 - Proof of Theorem 2.It su�ces to show that u(T; �) = ST �u. Indeed, the same assumptions (A1){(A3) hold on anyinterval [0; T 0] � [0; T ].By (A3) and Lemma 1, u is a BV function on the strip ]0; T [�IR, in the sense that thedistributional derivatives Dtu;Dxu are Radon measures. By a well known structure theorem [9,20], there exists a set eN � ]0; T [�IR of 1-dimensional Hausdor� measure zero such that, at everypoint (�; �) =2 eN , u either is continuous or has an approximate jump discontinuity. We claim that,at every such point of discontinuity, the jump cannot have a horizontal tangent. In other words,one can never have (1.10) in the caseU(t; x) = �u� if t < � ,u+ if t > � , u+ 6= u�: (3:1)Indeed, for every positive constant c (arbitrarily large), from (3.1), (1.9) and (2.1) we deducecju+�u�j = lim�!0+ 1�2 Z �+�� Z �+c�� ju+�u�j dxdt = lim�!0+ 1�2 Z �+�� Z �+c�� ��u(t; x)�u(�t; x)�� dxdt� lim�!0+ 1�2 Z �+�� L0 � 2t dt = L0:Hence u+ = u�, against the second assumption in (3.1).Taking the projection of eN on the t-axis, we conclude that there exists a set N � [0; T ] ofmeasure zero (containing the endpoints 0 and T ) such that, at every point (�; �) 2 [0; T ]� IR with� =2 N , the following property holds.(P) Either u is continuous at (�; �), or it has an approximate jump discontinuity, in the sense that(1.9)-(1.10) hold for some states u�; u+ and some � 2 IR. In this second case, by Lemma3, one has u� = u(�; ��), u+ = u(�; �+). Moreover, by Lemma 4 the Rankine-Hugoniotequations (2.14) hold. Hence, by (A2) the entropy condition (1.11) holds as well.9

In the next section of the proof we establish the basic relationlim infh!0+ 1h u(� + h)� Shu(�) L1 = 0 for all � =2 N : (3:2)Let any � 2 [0; T [ be given. Fix any � 2 IR and � > 0. De�ne U [ to be the solution of the linearhyperbolic problem with constant coe�cientsU [t + eAU [x = 0; U [(�) = u(�); (3:3)where eA := DF�u(�; �)�. Call ~�1; : : : ~�n the eigenvalues of eA and let ~li, ~ri be the correspondingleft and right eigenvectors, normalized as in (1.7). Consider the time step h = �=�̂, where �̂ is thegiven upper bound for all wave speeds. By the assumption (A3), for every (t; x) in the interior ofthe trapezoidal regionD� := n(t; x); t 2 [�; � + h]; jx� �j � 2�� �̂(t� �)o;the same argument used in (2.3) yields the estimate��u(t; x)� u(�; �)�� � (K + 1) � Tot.Var.�u(�); ]� � 2�; � + 2�[ �: (3:4)For every i = 1; : : : ; n and every choice of �1; �2, with � � � < �1 < �2 < � + �, we now estimatethe quantity Ei := Z �2�1 ~li; u(� + h; x)� U [(� + h; x)� dx: (3:5)Observing that (3.3) yields~li; U [(� + h; x)� = ~li; U [(�; x� h~�i)� = ~li; u(�; x� h~�i)�;and integrating (1.1) over the domainDi := n(t; x); t 2 [�; � + h]; �1 + (t � � � h)~�i � x � �2 + (t � � � h)~�io;we obtain Ei = Z �+h� D~li; �F (u) � ~�iu�(t; �1 + (t � � � h)~�i)E dt� Z �+h� D~li; �F (u)� ~�iu�(t; �2 + (t� � � h)~�i)Edt: (3:6)To estimate the quantity in (3.6), consider the statesu0 := u(t; �1 + (t � � � h)~�i); u00 := u(t; �2 + (t� � � h)~�i); ~u := u(�; �):10

We then haveD~li; F (u00)� F (u0)� ~�i(u00 � u0)E = D~li; DF (~u) � (u00 � u0)� ~�i(u00 � u0)E+ D~li; A�(u00 � u0)E;where the matrix A� is de�ned byA� := Z 10 hDF�su00 + (1� s)u0��DF (~u)i ds:Therefore, ����D~li; F (u00)� F (u0)� ~�i(u00 � u0)E���� � C � �maxfju0 � ~uj; ju00 � ~uj�2; (3:7)for some constant C depending only on the second order derivatives of the function F . Applying(3.7) to the integrand in (3.6) and recalling (3.4) we thus obtainjEij � Ch � � sup(t;x)2Di ��u(t; x)� u(�; �)���2� Ch(K + 1)2 � �Tot.Var.�u(�); ]� � 2�; � + 2�[��2:Since the above estimate holds for all i = 1; : : : ; n, we concludesup���<�1<�2<�+� �����Z �2�1 �u(� + h; x)� U [(� + h; x)� dx����� � 2C2� ��Tot.Var.�u(�); ]�� 2�; �+ 2�[��2;(3:8)for some constant C2 and all � 2 IR and � > 0, with h = �=�̂.We now derive an additional estimate which is appropriate at points of discontinuity. Fix any� =2 N . Let �01 < � � � < �0N be any �nite collection of points where u(�; �) is discontinuous. Since� =2 N , for each � = 1; : : : ; N by the property (P) the two states u�� := u(�; �0��) are connected byan entropy-admissible shock, travelling with some speed ��. Moreover, introducing the functionsU�(t; x) = � u�� if x < �0� + ��(t� �),u+� if x > �0� + ��(t� �), (3:9)one has lim�!0+ 1� Z �+���� Z ��+����� ��u(t; x)� U�(t; x)�� dxdt = 0 (� = 1; : : : ; N): (3:10)We claim that there exists a sequence of positive numbers h� decreasing to zero such thatlim�!1 NX�=1 1h� Z �0�+�̂h��0���̂h� ��u(� + h� ; x)� U�(� + h� ; x)��dx = 0: (3:11)11

Indeed, in the opposite case, we would haveNX�=1 1h Z �0�+�̂h�0���̂h ju(� + h; x)� U�(� + h; x)j dx � �;for some � > 0 and all h small enough. Therefore,lim infh!0+ NX�=1 1h2 Z h0 Z �0�+�̂s�0���̂s ��u(� + s; x)� U�(� + s; x)��dxds � � > 0;in contradiction with (3.10).We are now in the position to prove the basic estimate (3.2). Fix any " 2 ]0; 1=4[ . Let�01 < : : : < �0N be the points where the total variation measure of u(�) exceeds the value ", i.e.��u(�; �0�+) � u(�; �0��)�� � ":De�ne the corresponding functions U� as in (3.9). By the previous arguments, there exists asequence h� ! 0+ such thatNX�=1 1h� Z �0�+�̂h��0���̂h� ���u(� + h� ; x)� U�(� + h� ; x)���dx � " (3:12)for every �. >From (2.19) in Lemma 5 it follows that the semigroup trajectory satis�es an entirelysimilar estimate:NX�=1 1h� Z �0�+�̂h��0���̂h� ����Sh�u(�)�(x)� U�(� + h� ; x)���dx� C1 � NX�=1Tot.Var.�u(�); ]�0� � 2�̂h� ; �0�[ [ ]�0�; �0� + 2�̂h� [�� "; (3:13)provided that h� is su�ciently small. Combining (3.12) and (3.13) we deduceNX�=1 1h� Z �0�+�̂h��0���̂h� ���u(t; x)� �Sh�u(�)�(x)���dx � 2 ": (3:14)Next, we consider all the remaining points, where u(�) either is continuous or has a jump of size< ". For each �, de�ne �� := �̂h� . Choose �� large enough so that, for all � > �� , the estimate (3.14)holds together with the properties 4�� < min�=2;���;N j�0� � �0��1j; (3:15)12

Tot.Var.�u(�); ]a; b[�� " < 14 whenever b� a � 4 �� ; ]a; b[ \ f�01; : : : �0Ng = ;: (3:16)Such a �� certainly exists, because there are only �nitely many points �0� and the total variation ofu(�) is bounded.For any given � > �� we can now select a countable (but locally �nite) set of points f�k; k � 1gsuch that the intervalsIk := [�k � �� ; �k + �� ]; Jk := [�k � 2�� ; �k + 2�� ]satisfy the properties:(i) Sk Ik � IR nSN�=1[�0� � �� ; �0� + �� ].(ii) �0� =2 Jk for every �; k.(iii) Every point x 2 IR is contained in at most three distinct intervals Jk .For each k, call U [k the solution of (3.3) with eA = DF�u(�; �k)�, and de�neVk := Tot.Var.�u(�); �k � 2�� ; �k + 2�� [ �:By (3.16), Vk < 1=4. >From the estimates (3.8), (1.12) and (2.21) it followssup�k���<�1<�2<�k+�� �����Z �2�1 �u(� + h� ; x)� U [(� + h� ; x)� dx����� � 2C2�� � V 2k ; (3:17)Tot.Var.�u(� + h�)� U [(� + h�); ]�k � �� ; �k + �� [ � � 2KVk: (3:18)Applying Lemma 2 to the function w = u(� + h�)� U [k(� + h�), we obtainZ �k+���k��� ���u(� + h� ; x)� U [k(� + h� ; x)��� dx � C3 � 2��V 3=2k ; (3:19)with C3 = C2 + 4K. On the other hand, applying (2.20) with w = u(�) we deduce1h� Z �k+���k��� ����Sh�u(�)�(x)� U [k(� + h� ; x)��� dx � C2 �Tot.Var.�u(�); ]�k � �� ; �k + �� [ ��2 (3:20)The two inequalities (3.19) and (3.20) together yield1h� Z �k+���k��� ���u(� +h� ; x)� �Sh�u(�)�(x)��� dx � �2�̂C3+C2�Tot.Var.�u(�); ]�k��� ; �k+�� [ � "1=2:(3:21)13

Recalling that each point x can belong to at most three of the intervals Jk , summing over k weobtainXk 1h� Z �k+��=2�k���=2 ���u(� + h� ; x)� �Sh�u(�)�(x)��� dx � 3 �2�̂C3 + C2� "1=2 �Tot.Var.�u(�)�: (3:22)>From (3.14) and (3.22) we �nally obtain1h� Z 1�1 ���u(� + h� ; x)� �Sh�u(�)�(x)���dx � 2"+ 3 �2�̂C1 + C2� �Tot.Var.�u(�)� "1=2; (3:23)for every h� with � > �� . Therefore,lim infh!0+ 1h u(� + h)� Shu(�) L1 � 2"+ 3(2�̂C3 + C2) �Tot.Var.�u(�)�"1=2:Since " > 0 was arbitrary, this establishes (3.2).We can now complete the proof of Theorem 2.Let L; L0 be the Lipschitz constants of the semigroup S, and of the map t 7! u(t; �), so that St�u� Ss�v � L�k�u� �vk+ jt� sj�; (3:24) u(t) � u(s) � L0jt � sj: (3:25)Fix any " > 0. Choose an open set J � IR such thatJ � N [ f0; Tg; meas(J) < ": (3:26)Consider the time� := supnt 2 [0; T ]; ST�tu(t)� ST �u � "Lt + L(L0 + L) �meas�[0; t]\ J�o: (3:27)We claim that � = T . To show this, assume that, on the contrary, � < T . By continuity, we thenhave ST��u(�)� ST �u � "L� + L(L0 + L) �meas�[0; � ]\ J�o: (3:28)Two cases may occur.CASE 1: � 2 J . Since J is an open set, there exists h 2 ]0; T � � ] such that [�; � + h] � J . By(3.24)-(3.25) and (3.28) it follows ST���hu(� + h)� ST �u � ST���hu(� + h)� ST���hu(�) + ST���hu(�)� ST���hShu(�) + ST��u(�)� ST �u � L(L0h) + L(Lh) + "L� + L(L0 + L) �meas�[0; � ]\ J�< "L(� + h) + L(L0 + L) �meas�[0; � + h] \ J�:14

This contradicts the maximality of � .CASE 2: � =2 J . Since N � J , we thus have � =2 N . By (3.2), there exists h 2 ]0; T � � ] such that u(� + h)� Shu(�) < "h:In turn, this implies ST���hu(� + h)� ST �u � ST���hu(� + h)� ST���hShu(�) + ST��u(�)� ST �u < "Lh + "L� + L(L0 + L) �meas�[0; � ]\ I�� "L(� + h) + L(L0 + L) �meas�[0; � + h] \ J�;again contradicting the maximality of � .Using (3.28) with � = T and then (3.27), we obtain u(T )� ST �u � "LT + L(L0 + L) �meas(J) � "�LT + L(L0 + L)�:Since " > 0 was arbitrary, the theorem is proved.Remarks. The assumption (A2) can be relaxed, requiring (1.11) to hold only for � outside a setN � [0; T ] of measure zero.Another basic assumption in Theorem 2 is the existence of a Standard Riemann SemigroupS : D � [0;1[ 7! D such that u(t; �) 2 D for all t. Due to the �nite propagation speed, this can bereplaced by the following local condition.For every (�; �), there exists � > 0 and a semigroup S such that, for all t 2 [�; � + �=�̂], thedomain D of S contains some function w such thatw(x) = u(t; x) for all x 2 J(t);where J(t) := �� � �+ �̂(t� �); � + �� �̂(t� �)� and �̂ is an upper bound for all wave speeds.The proof is obtained in the same way as before, replacing (3.2) withlim infh!0+ 1h ZJ(�+h) ���u(� + h; x)� �Shw�(x)��� dx = 0;where w 2 D is any function coinciding with u(�) on J(�). Relying on the construction in [4], thepresent uniqueness result can thus be extended to some 2� 2 conservation laws with large data.15

References[1] A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995),205-230.[2] A. Bressan, The semigroup approach to systems of conservation laws, Mathematica Contem-poranea, to appear.[3] A. Bressan and R.M. Colombo, The semigroup generated by 2 � 2 conservation laws, Arch.Rational Mech. Anal. 133 (1995), 1-75.[4] A. Bressan and R.M. Colombo, Unique solutions of 2 � 2 conservation laws with large data,Indiana Univ. Math. J. 44 (1995), 677-725.[5] A. Bressan, G. Crasta, and B. Piccoli, Well posedness of the Cauchy problem for n�n systemsof conservation laws, preprint S.I.S.S.A., Trieste 1996.[6] C. Dafermos, Generalized characteristics in hyperbolic systems hyperbolic systems of conser-vation laws, Arch. Rat. Mech. Anal. 107 (1989), 127-155.[7] C. Dafermos and X. Geng, Generalized characteristics, uniqueness and regularity of solutionsin a hyperbolic system of conservation laws, Ann. Inst. Henri Poincar�e - Nonlinear Analysis8 (1991), 231-269.[8] R. J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math.J. 28 (1979), 137{188.[9] L. C. Evans and R. F. Gariepy,Measure Theory and Fine Properties of Functions, C.R.C. Press,1992.[10] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. PureAppl. Math. 18 (1965), 697-715.[11] S. Kruzkov, First-order quasilinear equations with several space variables, Mat. Sb. 123(1970), 228{255. English transl. in Math. USSR Sb. 10 (1970), 217{273.[12] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957),537-566. 16

[13] P.G. LeFloch and Z.P. Xin, Uniqueness via the adjoint problems for systems of conservationlaws, Comm. Pure Appl. Math. 46 (1993), 1499{1533.[14] T. P. Liu, Uniqueness of weak solutions of the Cauchy problem for general 2� 2 conservationlaws, J. Di�erential Equations 20 (1976), 369{388.[15] O. Oleinik, Discontinuous solutions of nonlinear di�erential equations Usp. Mat. Nauk. 12(1957), 3-73; English transl. in Amer. Math. Soc. Transl. Ser.2, 26, 95-172.[16] O. Oleinik, On the uniqueness of the generalized solution of the Cauchy problem for a nonlinearsystem of equations occurring in mechanics, Usp. Mat. Nauk (N.S.) 12 (1957), 169{176 (inRussian).[17] M. Schatzman, Continuous Glimm functionals and uniqueness of solutions of the Riemannproblem, Indiana Univ. Math. J. 34 (1985), 533-589.[18] J. Smoller, Shock Waves and Reaction-Di�usion Equations, Springer-Verlag, New York, 1983.[19] K. Trivisa, A priori estimates in hyperbolic systems of conservation laws via generalized char-acteristics, preprint, Brown University, Providence 1996.[20] A. I. Volpert, The space BV and quasilinear equations, Math. USSR Sb. 2 (1967), 257{267.

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