On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws

On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws Stefano Bianchini and Alberto Bressan S.I.S.S.A., Via Beirut 4, Trieste 34014 Italy. E-mail addresses: bianchin@mis.mpg.de, bressan@sissa.it Abstract. We study a non linear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the change of shape of a BV solution to a scalar conservation law. Keywords: Area Functional, Parabolic Estimates, Glimm Functional. S.I.S.S.A. Ref. 123/99/M 0

1 - The area swept by a shortening curve Fix two points A, B in the plane IR 2 and consider the family F AB of all polygonal lines joining. A with B. Given γ F AB, say with vertices A = P 0, P 1,..., P n = B, define v i = Pi P i 1 and consider the functional Qγ). = 1 2 n v i v j, 1.1) i,j=1 i<j where stands for the external product in IR 2. Let γ be obtained from γ by replacing the two segments P l 1 P l and P l P l+1 by one single segment P l 1 P l+1, as in fig. 1. The area of the triangle with vertices P l 1, P l, P l+1 satisfies AreaP l 1 P l P l+1 ) = 1 2 v l+1 v l Qγ) Qγ ). 1.2) Indeed, Qγ) Qγ ) = 1 l 1 ) v i v l + v i v l+1 v i v l + v l+1 ) + 1 2 2 v l v l+1 + 1 2 i=0 n j=l+2 ) v l v j + v l+1 v j v l + v l+1 ) v j 1 2 v l v l+1. Next, assume that γ is obtained from γ by a finite sequence of consecutive cuts fig. 2). In other words, let γ 0, γ 1,..., γ n be a sequence of polygonals with γ 0 = γ, γ n = γ and such that each γ i is obtained from γ i 1 by replacing two adjacent segments with a single one. By 1.2), an inductive argument yields Areaγ, γ ) n Areaγ i, γ i 1 ) Qγ) Qγ ). 1.3) i=1 P l γ P l+1 γ P l 1 γ γ A B A B Figure 1. Figure 2. More generally, instead of polygonals, we can define the functional Q on the family of parametric curves in the plane. Following [2], by a parametric curve we mean a continuous map 1

γ : [a, b] IR 2. Two parametric curves γ 1 and γ 2 are regarded as equivalent if, for every ε > 0, there exists a homeomorphism h ε : [a 1, b 1 ] [a 2, b 2 ] possibly order reversing) such that sup γ 1 x) γ 2 h ε x)) < ɛ. x [a 1,b 1 ] Let F be the set of all parametric curves with finite length, i.e. such that Lγ) =. n sup γx i ) γx i 1 ) < +, 1.4) i=1 where the supremum is taken w.r.t. all partitions a = x 0 <... < x n = b. We recall that 1.4) implies the existence of an absolutely continuous parameterization. In the following, up to a homeomorphism h, we can assume that each γ F is absolutely continuous and parametrized by x [0, 1], and we denote with γ x the derivative of γ w.r.t. x. We recall that γ x L 1 [0, 1]; IR 2 ) and Lγ) = We can introduce a metric d on F defined as 1 0 γ x dx. dγ 1, γ 2 ). = inf γ 1 γ 2 h C 0, 1.5) where the infimum is taken over all homeomorphisms h : [0, 1] [0, 1]. functional Q : F IR by setting Qγ). = 1 2 sup { n = 1 2 1 1 0 x i,j=1 j>i γx i ) γx i 1 ) ) γx j ) γx j 1 ) ) } γ x x) γ x y) dydx 1 2 Lγ)2, We can now define a where, as above, the supremum is taken w.r.t. all partitions 0 = x 0 <... < x n = 1. Observe that the definition 1.6) is the natural extension of 1.1). 1.6) It is well known that the length Lγ) of a curve is lower semi continuous w.r.t. the distance 1.5). A similar result holds for the functional Q. Lemma 1. The functional Q is lower semi continuous in F, d). Namely, if dγ ν, γ) 0, then Qγ) lim inf ν Qγ ν). Proof. For every ε > 0 there exists a ν such that dγ ν, γ) < ε for all ν ν, and hence n γx i ) γx i 1 ) ) γx j ) γx j 1 ) ) i,j=1 i<j n i,j=1 i<j γ ν x i ) γ ν x i 1 ) ) γ ν x j ) γ ν x j 1 ) ) + 2n 2 ε 2 Qγ ν ) + 2n 2 ε 2. 2

Letting ε 0 first and then taking the supremum w.r.t. all partitions 0 = x 0 <... < x n = 1, the conclusion follows. Given γ F, by a cut we mean the replacement of the portion of the curve { γx); x [x 1, x 2 ] [0, 1] } with the segment connecting γx 1 ) to γx 2 ), for some x 1, x 2 [0, 1]. We say that γ follows γ, and write γ γ, if there exists a sequence of curves γ n converging to γ in F, d) such that each γ n is obtained from γ by a finite sequence of consecutive cuts fig. 3). Note that, as a consequence of this definition, γ must have the same endpoints of γ. It is easy to see that defines a partial order relation. Given γ, γ F with γ γ, we consider the closed curve γ γ : [0, 2] IR 2 as γ γ ) x) =. { γx) x 0, 1] γ 2 x) x 1, 2] By the area between γ and γ, denoted by Area[γ, γ ], we mean the area of the regions where the winding number of the curve γ γ is odd. For the definition and basic properties of the winding number of a closed curve in the plane we refer to [8]. We remark that the above definition of area between curves is not affected by a change in the parameterizations. Lemma 2. If γ γ, then the area between the two curves satisfies 1.7) Area[γ, γ ] Qγ) Qγ ). 1.8) Proof. In the case where γ is a polygonal and γ is obtained from γ with a finite sequence of cuts, the result was already proved in 1.3). The general case follows by approximation, using Lemma 1. Remark 1. One can give an equivalent definition of the partial order relation by setting γ γ if the following holds: There exists a sequence of parabolic problems on the plane: ξt ν + λ ν t, x)ξx ν c ν t, x)ξxxt, ν x) = 0, ξ ν 0, x) = γx) x [0, 1], ξ ν t, 0) = γ0), ξ ν t, 1) = γ1), t [0, 1], whose solutions at time t = 1 converge to γ, i.e. lim d ξ ν 1, ), γ ) = 0. ν Here λ ν, c ν are smooth functions from [0, 1] [0, 1] IR, with c ν strictly positive. Note that by a change of variable y = yt, x), where y is the unique positive solution to the uniformly parabolic 3 1.9)

system y t + λ ν t, x)y x c ν t, x)y xx = 0, y0, x) = x x [0, 1], yt, 0) = 0, yt, 1) = 1, t [0, 1], equation 1.9) becomes ξ ν t = c ν t, x) y x t, x) ) 2 ξ ν yy = c νt, y)ξ ν yy, 1.9 ) whit c νt, x) c > 0 for all t, x) [0, 1] [0, 1]. Consider now any cut in the curve γ, say obtained by replacing the portion {γx); x [a, b] [0, 1]} by a single segment. This can be uniformly approximated by solutions of 1.9), letting the viscosity coefficient c ν tend to + inside the interval [a, b] and to 0 outside [a, b]. scheme Conversely, any solution of 1.9 ) can be uniformly approximated by a forward difference In fact, if ξ ν n + 1) t, k x ) ξ ν n t, k x ) = t x 2 c ) [ ν n t, k x ξ ν n t, k + 1) x ) 2ξ ν n t, k x ) + ξ ν n t, k 1) x )]. t { x 2 max c ν } < 1 0 t,x 1 2, it is easy to prove that the quantities ξ ν x + x), ξ ν x + x) ξ ν x))/ x remain uniformly bounded. If we consider the polygonal line γ ν n t) obtained by connecting the points ξ ν n t, k x) with n fixed, then it follows that γ n converges in C 0 to the unique classical solution of 1.9 ). At each time step, the approximate solutions γ ν thus constructed will satisfy γ ν n t) γ ν n + 1) t). From the relations γ ν 0) γ ν 1), and the convergence γ ν 0) γ, γ ν 1) γ we conclude γ γ. We also observe that 1.9) implies that the vector ξ ν t lies in the half plane {v IR 2, v ξ ν xx 0}. Hence, by possibly modifying the function λ which has the only effect of changing the parameterization), we obtain a solution of where the vectors two ξt ν and ξxx ν are parallel and have the same orientation. In other words, ξ ν will move in the direction of curvature. γ γ s) B B γ γ n γ t) A A Figure 3. Figure 4. 4

More generally, consider a path varying with time. This is described by a map γ : [t 1, t 2 ] F. We say that γ moves in the direction of curvature if γs) γt) for all s < t, s, t [t 1, t 2 ] fig. 4). Observe that, from our definitions, it follows that the endpoints of γt) remain constant in time. The area swept by γt) during the time interval [t 1, t 2 ] is defined as Area γ; [t 1, t 2 ] ) {. n = sup Area [ γs i ), γs i 1 ) ] } ; t 1 = s 0 < < s n = t 2, n 1. 1.10) i=1 An immediate consequence of the above definitions is Theorem 1. Let t γt) F denote a curve in the plane, moving in the direction of the curvature. Then, for every t 1 < t 2 one has Area γ; [t 1, t 2 ] ) Q γt 1 ) ) Q γt 2 ) ). 1.11) Remark 2. It is possible to state Theorem 1 in more generality by enlarging the class of equivalent curves. Consider two curves γ i F, i = 1, 2, parametrized by arc length, and define γ 1 γ 2 if there exists a bijection h : [ 0, Lγ 1 ) ] [ 0, Lγ 2 ) ] such that γ 1 = γ 2 h. 1.12) Using 1.4) and 1.6), it is easy to verify that Lγ 1 ) = Lγ 2 ) and Qγ 1 ) = Qγ 2 ). In essence, we identify two curves γ 1, γ 2 of finite length if the sets { γ i x); x [a i, b i ] }, i = 1, 2, are equal. γ 0 γ 1 1 1 1 γ 1 1 0 1 γ Figure 5. Let now a continuous parameterization x γx) IR 2 be given. As before, by a cut we mean the replacement of the part of the curve γ, {γx); x [a, b] [0, 1]} with the segment connecting γa) to γb). Note however that this definition now depends on the parameterization. For example, consider the curve in fig. 5. Given two points on the curve γ, we can obtain many different curves γ depending on which part of the original curve is replaced by the cut. 5

When γ γ, the area between γ and γ can be again defined as the area of the regions in which the winding number of the curve γ γ is odd. Recall that γ γ was defined at 1.7). It is not difficult to see that the estimate 1.11) remains valid also in this more general case. Remark 3. It is possible to generalize Theorem 1 to a curve γt) in IR n : the definition of motion in the direction of curvature remains the same. However when computing the area swept by γ and the functional Qγ) in 1.6) we should use the external product on IR n. 2 - Estimates for parabolic equations In this section we consider two applications of Theorem 1 to parabolic equations. The basic idea is to associate to a solution ut) a parametric curve γt) in such a way that γt) moves in the direction of curvature. Theorem 2. Consider a scalar parabolic equation on an open interval ]a, b[, with fixed boundary conditions: { ut, a) = ū1, u t = ct, x)u xx, 2.1) ut, b) = ū 2. Assume that the function ct, x) is continuous and positive on [0, T ] [a, b]. Then every solution of 2.1) satisfies the a priori estimate T b 0 a u t t, x) dxdt 1 4 b b a a u x 0, x) u x 0, y) dxdy. 2.2) Proof. We can assume that ct, x) is smooth: otherwise we can approximate ct, x) with a sequence of positive smooth functions c ν t, x) and apply classical convergence results to get that the solution u ν t, x) tends to the solution ut, x) of 2.1) in C 0 [0, T ] [a, b]) see for example [4]). Observe that the graph of every solution of 2.1) is a curve moving in the direction of the curvature. Indeed, setting γt, x) =. x, ut, x) ), one has γ t = 0, ct, x)u xx ) = ct, x)γxx. Applying Theorem 1 to the curve γ we obtain the a priori estimate Area γ; [0, T ] ) T = 0 This yields 2.2). b a γ t s, y) γ x s, y) dyds Q γ0) ) Q γt ) ) 1 2 6 T b 0 a<x<y<b a u t t, x) dxdt 1, u x 0, x) ) 1, u x 0, y) ) dxdy.

Remark 4. Recalling Remark 2, one easily checks that the same estimate 2.2) holds in the periodic case, i.e. for solutions of u t = φt, x)u xx, { ut, a) = ut, b), u x t, a) = u x t, b), assuming that the continuous function φ satisfies φt, a) = φt, b) for every t. 2.3) Our second application is concerned with viscous scalar conservation laws. Let f, c : IR IR be smooth functions, with c 0, and consider the conservation law u t + fu) x cu)u x = 0. 2.4) )x Let u = ut, x) be a smooth solution of 2.4), with initial data having bounded variation. This of course implies ut, ) BV for all t 0, because the total variation cannot increase in time. For each t 0, consider the curve γt) F defined as γt, x) =. ) ut, x) f ut, x) ) c ut, x) ) u x t, x) x IR. 2.5) u ut ) 1 fu) cu)u x γ t ) 1 ut 2 ) γ t ) 2 fu) x Figure 6. u By a direct computation, one checks that γ moves in the direction of curvature fig. 6). Indeed, it satisfies the equation γ t + f u)γ x cu)γ x = 0. 2.6) )x If we define amount of interaction Iu; [t 1, t 2 ]) during the interval [t 1, t 2 ] as the area swept by γ = γu), then Theorem 1 implies I u; [t 1, t 2 ] ). = Area γ; [t1, t 2 ] ) Q ut 1 ) ) Q ut 2 ) ), 2.7) where the interaction potential Qu) is now defined as Qu) =. Q γu) ) = 1 γ x x) γ x y) dxdy 2 x<y = 1 u x x) u x y) ηx) ηy) dxdy, 2 x<y 2.8) 7

η. = f u) cu)ux ) x u x. 2.9) Observe that, for a solution of 2.4), the above quantity η = u t /u x can be regarded as the speed of a viscous wave. Indeed, for a given x IR, ηt, x) is the speed of the intersection of the graph of x ut, x) with the horizontal line u u x). Clearly Qu) = 0 if and only if this speed η is constant. The functional Q thus seems to be suited for studying the stability of traveling waves. By 2.7), Q provides a Lyapunov functional for all BV solutions of 2.4). Indeed d dt Q ut, ) ) 0. 2.10) Remark 5. An entirely similar result holds for periodic solutions of 2.6), say with ut, x + p) = ut, x). In this case γ : [0, p] IR 2, defined as in 2.5), is a closed curve. The inequalities 2.7), 2.10) remain valid in connection with the functional Qu). = 1 2 0 x<y p u x x) u x y) ηx) ηy) dxdy, 2.11) 3 - Application to scalar conservation laws Consider a scalar conservation law u t + fu) x = 0, 3.1) with f sufficiently smooth. Given an initial data u 0 BV, let u = ut, x) be the corresponding unique entropic solution. In this section we will show that to ut, ) one can associate a parametric curve γt) such that γt) moves in the direction of curvature, i.e. γs) γt) whenever 0 s < t. Given a map u : IR IR with bounded variation, define the function U BV as Ux). = x { } Du = Tot.Var. u;, x]. 3.2) Here Du is the measure corresponding to the distributional derivative of u. For θ ] 0, Tot.Var.u) [, we define xθ) to be the point x such that Ux ) θ Ux+). 3.3) Let now two points u, fu ) ), u +, fu + ) ) IR 2, be given, with u > u + or u < u +, respectively). We then define the curve R θ; [u, u + ] ), where θ [ 0, u + u ], as the graph of the 8

convex concave) envelope of the function fu) on the interval [u, u + ]. To a function u BV we associate the parametric curve γ : [ 0, T.V.u) ] IR 2 defined as γu; θ) =. R u ), f u ) )) θ = 0, uθ), f uθ) )) u is continuous at xθ), ) θ Uxθ) ); [uxθ)), uxθ)) + ] u ), f u ) )) u has a jump in xθ), θ [Uxθ) ), Uxθ)+)], θ = Tot.Var.u). 3.4) u fu) 3 γ u) 2 1 4 5 x Figure 7. 5 4 1 2 3 u We claim that γu) F. Indeed, if κ provides a Lipschitz constant for the function f restricted to the range of u, then We can thus associate to γu) the functional L γu) ) 1 + κ 2 Tot.Var.u). Qγu)) = 1 2 θ<θ γu; θ) γu; θ ) dθdθ. 3.5) Given u BV, let γu) F be the corresponding parametric curve. We say that γ is obtained from γu) by a Riemann cut if, { for some points x 1 < x 2, the curve γ can be constructed by replacing the part of the curve γu; θ); θ [ Ux 1 ), Ux 2 ) ]} { with Rθ); θ [ Ux 1 ), Ux 2 ) ]}. Note that this definition depends on the choice of the flux function fu). It essentially corresponds to the substitution of u with ux) x < x ûx) =. 1, ux 1 ) x 1 x x, ux 2 +) x < x x 2, ux) x > x 2. 9 3.6)

All waves of u located inside the interval [x 1, x 2 ] are thus collapsed to a single point x, with x 1 < x < x 2. Lemma 3. Let u be a scalar BV function. Fix any three points x 1 < x < x 2 and construct the function û as in 3.6). Then the corresponding curves, defined by 3.4), satisfy γu) γû). Proof. Consider first the case where u is piecewise constant. By an inductive argument, it suffices to prove the result in the case where u contains two adjacent jumps, say at y 1 < y 2, which are replaced by a single jump of û. Consider the left, middle and right states u l = uy 1 ), u m = uy 1 +) = uy 2 ), u r = uy 2 +). In the various possible configurations, it it easy to check that the portion of the curve γû) corresponding to the jump u r, u l ) is contained in the convex hull of the portion of the curve γu) corresponding to the two jumps u r, u m ) and u m, u l ). The two main cases u l < u m < u r and u l < u r < u m are illustrated in fig. 8. fu) fu) γ u) γ u) ^ ^ γ u) γ u) u u l u m r u u u u l r m u Figure 8. This proves the lemma in the case of piecewise functions u. The general case is handled by a standard approximation argument. We now establish a relation between the L 1 convergence of a sequence of functions u n and the convergence of the corresponding parametric curves γu n ). The main ideas in the proof are taken from [1], where a similar construction was used to prove the lower semicontinuity of the Glimm functional. Lemma 4. Let u n ) n 1 be a sequence of scalar functions with uniformly bounded total variation. Assume that u n u in L 1 and γu n ) γ 0 in F. Then γ 0 γu). Proof. By possibly taking a subsequence, we can assume the pointwise convergence u n x) ux) 10

and the weak convergence of measures Du n µ, for some positive measure µ. The lemma will be proved by showing that, for every ε > 0, one can construct a sequence of curves ˆγ n such that lim sup d ˆγ ) n, γ 0 ε, γun ) ˆγ n for each n. 3.7) n Let ε > 0 be given. Consider first a point x such that µ {x} ) = 0. We can then choose δ ε > 0 such that µ [ x δ ε, x + δ ε ] ) < ε, Du n [ x δ ε, x + δ ε ] ) < ε for all n sufficiently large. Choose a point y ] x δ ε, x + δ ε [ where u and all u n are continuous, so that the point u n y), fu n y)) ) lies in the range of the map γ u n ), for each n 1. Let Lipf) be a Lipschitz constant for f on an interval [u min, u max ] containing the range of all functions u n. From the relations lim sup u n y) u x) ε, n + lim sup n + f u n y) ) f u x) ) ε Lipf), it follows that the point u x), fu x) ) lies in the range of γ 0. In other words, points x such that µ{ x}) = 0 correspond to the same point on the image of γ 0 and on the image of γu). Next, let x 1 < < x N be all the points such that µx α ) ε α = 1,..., N. Choose δ ε > 0 such that µ [x α δ ε, x α [ ]x α, x α + δ ε ] ) < ε, 3.8) Du n [x α δ ε, x α [ ]x α, x α + δ ε ] ) < ε, 3.9) for every α and all n sufficiently large. Choose two points y α, y + α with x α δ ε < y α < x α < y + α < x α + δ ε, where all u n are continuous and such that µ {y α } ) = µ {y + α } ) = 0. These conditions imply that the points u n y ± α ), fu n y ± α )) ), uy ± α ), fuy ± α )) ), belong to the range of γu n ), γu), respectively. Moreover, lim sup u n yα ) ux α ) ε, n lim sup u n y α + ) ux α +) ε. n We now define ˆγ n as the curve obtained from γu n ) by performing N Riemann cuts, in connection with the intervals [y α, y + α ], α = 1,..., N. By construction, γu n ) ˆγ n. Moreover, the above analysis yields d ˆγ n, γu n ) ) Cε, for some constant C independent of ε and all n sufficiently large. This establishes 3.7), proving the lemma. 11

As a consequence of the above lemma we have: Theorem 3. Assume that f : IR IR is locally Lipschitz and let ut) be the unique entropy solution to 3.1) with initial data u 0 BV. Call γt). = γut)) the parametric curve associated to ut), as in 3.4). Then γt) moves in the direction of curvature, i.e. γs) γt) whenever s < t. Proof. By the semigroup property [7], it is not restrictive to assume s = 0. Following [3], the solution u can be obtained as limit of a sequence of front tracking approximations u n. More precisely, for each n 1 we let f n be a piecewise affine function which coincides with f at each. node p i,n = 2 n i. Moreover, we choose a piecewise constant approximate initial data u 0,n taking values inside the grid 2 n Z so that u 0,n u 0 in L 1 loc, d γu 0 ), γu 0,n ) ) 0. 3.10) Calling u n = u n t, x) the corresponding entropy solution of u t + f n u) x = 0, u n 0, x) = u 0,n, it is known that each u n is piecewise constant, with jumps on a finite number of straight lines in the t-x plane, and takes values within the grid 2 n Z. The corresponding curves γ n t) = γ u n t) ) all move in the direction of the curvature. Indeed, each γ n remains constant except at finitely many times 0 < t 1 < t 2 < < t mn where two fronts interact. At each of these times t j, the curve γ n t j +) is obtained from γ n t j ) by a Riemann cut, hence the relation γ n 0) γ n t) clearly holds for every t > 0. Using Lemma 4 we can now pass to the limit as n. By 3.10) this yields γ0) γt), proving the theorem. Remark 6. As in Section 2, define the amount of interaction Iu; [t 1, t 2 ]) during [t 1, t 2 ] as the area swept by γut)) over the time interval [t 1, t 2 ]. Theorems 1 and 3 then yield Iu; [t 1, t 2 ]) Qut 1 )) Qut 2 )), 3.11) where Qu) is the interaction potential defined at 3.5). One can interpret the quantity Iu; [t 1, t 2 ]) as measuring the change in the shape of the solution u = ut, x) within the time interval [t 1, t 2 ]. By 3.11), this is bounded in terms of a Glimm type potential, similar to the one introduced by T.P. Liu in his paper on hyperbolic systems [9]. Remark 7. For an arbitrary flux function f, there may not exist a Borel function λ the wave speed), such that Qu) = 1 2 x<y λx) λy) Du x) Du y). 3.12) 12

However, if f is strictly convex and u has no upward jumps, i.e. ux ) ux+) for every x IR, then 3.12) holds with f λx) =. ux) ) u is continuous in x, f ux+) ) f ux ) ) u has a jump at x. ux+) ux ) 3.13) In particular the entropic solution of a scalar conservation law with strictly convex flux satisfies 3.12)-3.13) for every t > 0. Acknowledgment. This research was partially supported by the European TMR Network on Hyperbolic Conservation Laws ERBFMRXCT960033. References [1] P. Baiti and A. Bressan, Lower semicontinuity of weighted path length in BV, in Geometrical Optics and Related Topics, F. Colombini and N. Lerner Eds, Birkhäuser 1997), 31-58. [2] L. Cesari, Optimization - Theory and Applications, Springer Verlag, New York, 1983. [3] C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 1972), 202-212. [4] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. [5] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 1965), 697-715. [6] B. Gustavsson, H-O. Kreiss, J. Oliger, Time dependent problems and difference methods, Wiley, New York, 1995. [7] S. Kruzhkov, First-order quasilinear equations with several space variables, Mat. Sb. 123 1970), 228 255. English transl. in Math. USSR Sb. 10 1970), 217 273. [8] S. Lang, Complex Analysis, Springer-Verlag, New York, 1977. [9] T-P. Liu, Admissible solutions of hyperbolic conservation laws, Amer. Math. Soc. Memoir 240 1981). 13