CLASSIFICATION OF CODIMENSION-ONE RIEMANN SOLUTIONS STEPHEN SCHECTER, BRADLEY J. PLOHR, AND DAN MARCHESIN Abtract. We invetigate olution of Riemann problem for ytem of two conervation law in one patial dimenion. Our approach i to organize Riemann olution into trata of ucceively higher codimenion. The codimenion-zero tratum conit of Riemann olution that are tructurally table: the number and type of wave in a olution are preerved under mall perturbation of the ux function and initial data. Codimenion-one Riemann olution, which contitute mot of the boundary of the codimenion-zero tratum, violate tructural tability in a minimal way. At the codimenion-one tratum, either the qualitative tructure of Riemann olution change or olution fail to be parameterized moothly by the ux function and the initial data. In thi paper, we give an overview of the phenomena aociated with codimenion-one Riemann olution. We lit the dierent kind of codimenion-one olution, and we claify them according to their geometric propertie, their role in olving Riemann problem, and their relationhip to wave curve. 1. Introduction 1.1. Riemann olution. We conider ytem of two conervation law in one pace dimenion, partial dierential equation of the form U t + F (U) x = 0 (1.1) with t > 0, x 2 R, U(x; t) 2 R 2, and F : R 2! R 2 a mooth map. The mot baic initialvalue problem for Eq. (1.1) i the Riemann problem, in which the initial data are piecewie contant with a ingle jump at x = 0: U(x; 0) = ( U L for x < 0, U R for x > 0. (1.2) Thi paper i the econd in a erie in which we tudy the tructure of olution of Riemann problem. We eek piecewie continuou weak olution of Riemann problem in the cale-invariant form U(x; t) = b U(x=t) coniting of a nite number of contant part, continuouly changing part (rarefaction wave), and jump dicontinuitie (hock wave). Shock wave occur when lim bu() = U? 6= U + = lim bu(): (1.3)!?!+ Date: March, 1997. 1991 Mathematic Subject Claication. 35L65, 35L67, 34D30. Key word and phrae. conervation law, Riemann problem, vicou prole. Thi work wa upported in part by: the National Science Foundation (NSF) under Grant DMS-9205535 and DMS-9501255; the Conelho Nacional de Deenvolvimento Cientco e Tecnologico (CNPq) under Grant CNPq/NSF 910029/95-4, 53.0054/93-0, and 52.0725/95-6; the Financiadora de Etudo e Projeto under Grant 65920311-00; the NSF under Grant INT-9216753 and INT-9512873; the Applied Mathematic Subprogram of the U. S. Department of Energy under Grant DE-FG02-90ER25084; and CNPq Grant 301411/95-6. 1
2 SCHECTER, PLOHR, AND MARCHESIN They are required to atify the following vicou prole admiibility criterion: a hock wave i admiible provided that the ordinary dierential equation _U = F (U)? F (U? )? (U? U? ) (1.4) ha a heteroclinic olution, or a nite equence of uch olution, leading from the equilibrium U? to a econd equilibrium U +. By the term Riemann olution for Eq. (1.1) and (1.2) we mean a weak olution U of thi kind (or, equivalently, the cale-invariant function b U, or the equence of wave in U, or the quadruple (b U; UL ; U R ; F )). There are variou type of rarefaction and hock wave (e.g., 1-family rarefaction wave and tranitional hock wave); the type of a Riemann olution i the equence of type of it wave. Riemann olution have been tudied by many author. For ytem that are trictly hyperbolic and genuinely nonlinear, local olution were found by Lax [10], and global olution were obtained, for a certain cla of ytem, by Smoller [20]. Thi work wa extended to allow for lo of genuine nonlinearity by Wendro [23] for ga dynamic and by Liu [11] for a broad cla of ytem. Many example of ytem that fail to be trictly hyperbolic have been analyzed over the lat two decade; ee, e.g., Ref. [6, 18, 15, 5]. A common feature of thi analyi i the contruction of wave curve, one-parameter familie of Riemann olution. Special aumption about the ytem of conervation law lead to wave curve with imple geometry and permit the formulation of imple, but eective, admiibility criteria for hock wave. For general ytem, however, the wave curve geometry can be exceedingly complicated and a more fundamental admiibility criterion, uch a the vicou prole admiibility criterion, i needed. Our approach to undertanding Riemann olution i to invetigate the local tructure of the et of Riemann olution: we conider a particular olution (b U ; U L; U R; F ) and contruct nearby one. More preciely, we dene an open neighborhood X of b U in a Banach pace of cale-invariant function b U, open neighborhood UL and U R of U L and U R in R 2, repectively, and an open neighborhood B of F in a Banach pace of mooth ux function F. Then our goal i to contruct a et R of Riemann olution (b U; UL ; U R ; F ) 2 X U L U R B near (b U ; U L; U R; F ). To guide thi contruction, we view R a organized into trata of ucceively higher codimenion. The larget tratum of R, which ha codimenion zero within R, conit of tructurally table Riemann olution. For uch olution, b U change continuouly, and it type remain unchanged, when (U L ; U R ; F ) varie in certain open ubet of U L U R B. Moreover, the left and right tate and peed of each wave in b U depend moothly on (UL ; U R ; F ). In contrat, at a tratum of higher codimenion, either Riemann olution are degenerate in ome way or the parameterization of wave by U L, U R, and F loe moothne. For example, olution can change type upon croing the tratum, or the parameterization can have a fold. Structurally table Riemann olution were tudied in the rt paper [17] of the preent erie. In thi econd paper, we begin the tudy of codimenion-one Riemann olution, for which tructural tability fail in a minimal fahion. 1.2. Structurally table Riemann olution. A quadruple (b U ; U L; U R; F ) repreent a tructurally table Riemann olution if for each (U L ; U R ; F ) near (U L; U R; F ), there i a Riemann olution b U near b U uch that (1) b U ha the ame type a b U and (2) the left and right tate and peed of each wave in b U depend moothly on (UL ; U R ; F ). In particular,
CODIMENSION-ONE RIEMANN SOLUTIONS 3 we obtain a et R of Riemann olution, repreented a a graph of a function from an open ubet of U L U R B to X. In Ref. [17], we identied a et of ucient condition for tructural tability of trictly hyperbolic Riemann olution. Briey, thee condition have the following character. (H0) There i a retriction on the equence of wave type in the olution. (H1) Each wave atie certain nondegeneracy condition. (H2) The \wave group interaction condition" i atied. In the implet cae, the forward wave curve and the backward wave curve are tranvere. (H3) If a hock wave repreented by a connection to a addle i followed by another repreented by a connection from a addle, the hock peed dier. The method by which thee condition were derived trongly ugget that they are alo neceary for tructural tability. 1.3. Codimenion-one Riemann olution. In thi paper, we begin an invetigation of trictly hyperbolic Riemann olution that lie in the boundary of the et of tructurally table Riemann olution in that they violate preciely one of the tructural tability condition (H0){(H3). Under appropriate nondegeneracy condition, thee Riemann olution contitute a graph over a codimenion-one ubmanifold of U L U R B. A point (b U ; U L; U R; F ) repreent a codimenion-one Riemann olution if there exit a codimenion-one ubmanifold S of U L U R B with the following propertie. For each point (U L ; U R ; F ) 2 S near (U L; U R; F ), there i a tructurally untable Riemann olution b U near bu uch that (1) b U ha the ame type a b U and (2) the endpoint and peed of each wave in bu depend moothly on (U L ; U R ; F ). Furthermore, (3) S bound a region in U L U R B that correpond to tructurally table olution. In particular, we obtain a et R of Riemann olution a a graph of a function from a manifold-with-boundary in U L U R B to X. Finally, (4) S i ituated with a certain regularity in U L U R B: either S i in general poition relative to plane of contant (U L ; F ) and plane of contant (U R ; F ), o that (U L ; F ) and (U R ; F ) both erve a good coordinate for S; or S i a cylinder over a hyperurface in (U L ; F )-, (U R ; F )-, or F -pace. In mot cae, a codimenion-one ubmanifold S can be regarded a bounding tructurally table Riemann olution not only on one ide but alo on the other ide. Uually the number and type of wave in a tructurally table Riemann olution change at a codimenion-one boundary; a a reult, there i a change in the number and form of equation that dene olution. To accommodate uch change, we ue everal technical device; for intance, we allow hock and rarefaction wave to have zero-trength, and we ometime repreent a Riemann olution in dierent, but equivalent, way. 1.4. Claication of codimenion-one Riemann olution. Codimenion-one Riemann olution can be claied in everal way. A. They can be claied with repect to the tructure of R. The following poibilitie arie. 1. Join: R i formed from two manifold-with-boundary joined along their common boundary. A the boundary i croed, a tructurally table Riemann olution become degenerate and then turn into a tructurally table olution of a dierent type. See Fig. 1.1(a) and (b) for chematic illutration.
4 SCHECTER, PLOHR, AND MARCHESIN X R X R one olution S (a) U x U x B L R one olution U x U x B two olution no olution S (b) L R Figure 1.1. Riemann olution join. 2. Fold: R i a manifold homeomorphic to R 4 B, and there i no change in type of the Riemann olution upon croing the codimenion-one ubmanifold, but there i a fold in the projection to U L U R B. Thu R fail to be a graph over (U L ; U R ; F )-pace, a illutrated in Fig. 1.2(a). X R X R U x U x B L R U x U x B L R two olution no olution S (a) fold one olution S (b) frontier no olution Figure 1.2. Riemann olution fold and frontier. 3. Frontier: R i a manifold-with-boundary homeomorphic to R 3 R + B. Riemann olution exit only on one ide of the codimenion-one ubmanifold; ee Fig. 1.2(b). The reader will notice that, in each cae, the Riemann olution et R ariing naturally in our contruction i either a manifold or a manifold-with-boundary. Thi i in contrat to the ituation when one ue the Lax admiibility criterion for hock wave [10], where branching can occur [2]. We emphaize, however, that it i poible for two uch et, R 1 and R 2, to interect within X U L U R B; thi happen when the olution b U contain a hock wave that ha two ditinct vicou prole. B. The codimenion-one Riemann olution can alo be claied with repect to how In our experience, either S ha regular projection onto S i ituated in U L U R B. coordinate plane or S i a cylinder; when the projection of S onto a coordinate plane ha a fold, for example, we believe that U b hould be regarded a having higher codimenion. For
CODIMENSION-ONE RIEMANN SOLUTIONS 5 thi reaon, our denition of codimenion-one Riemann olution require S to be one of the following type of boundarie. 1. Intermediate boundary: the ubmanifold S i tranvere to both of the two-dimenional plane f(u L ; U R ; F ) : (U L ; F ) = (U L ; F )g and f(u L ; U R ; F ) : (U R ; F ) = (U R ; F )g. Thu for each (U L ; F ) near (U L; F ), S meet the correponding copy of the U R -plane in a curve; and for each (U R ; F ) near (U R; F ), S meet the correponding copy of the U L -plane in a curve. In other word, if U L and F are xed, codimenion-one Riemann olution correpond to a curve in the U R -plane; and if U R and F are xed, they correpond to a curve in the U L -plane. 2. U L -boundary: there i a codimenion-one ubmanifold ~ S in (U L ; F )-pace, tranvere to the two-dimenional plane f(u L ; F ) : F = F g, uch that (U L ; U R ; F ) 2 S if and only if (U L ; F ) 2 ~S. Thu for each F near F there i a curve C(F ) in the U L -plane uch that (U L ; U R ; F ) 2 S if and only if U L 2 C(F ). That i, for a pecic ytem of conervation law, codimenion-one Riemann olution occur when U L lie on a xed curve. Another type of boundary i obtained through duality by revering the role of U L and U R in thi denition. 3. F -boundary: there i a codimenion-one ubmanifold ~S in B uch that (U L ; U R ; F ) 2 S if and only if F 2 ~S. Remark. What we call an \intermediate boundary" i called a \U R -boundary" in Ref. [15]. We ue a dierent terminology to avoid confuion between intermediate boundarie and dual of U L -boundarie. Thee boundarie are ueful in decribing the olution of Riemann problem [19, 14, 2, 5, 4]. In olving Riemann problem for a ux function that i not on an F -boundary, the rt tep i to locate the U L -boundarie, which divide the U L -plane into everal open region. In the econd tep, for a repreentative choice of U L in each region, the intermediate boundarie are located; thee curve divide the U R -plane into open region. Finally, Riemann problem are olved for a repreentative choice of U R in each U R -region. The qualitative tructure of Riemann olution i the ame for all U L in a U L -region and U R in a U R -region. Example of U L -boundarie include the inection, double onic, and econdary bifurcation loci (ee Ref. [2] for a comprehenive lit in the context of the Lax admiibility criterion); example of intermediate boundarie include the rarefaction and hock curve drawn through U L. Codimenion-one F -boundarie eem to be new. To our knowledge, they do not occur in the ytem of conervation law that have been invetigated o far. C. The codimenion-one Riemann olution can be claied with repect to the number of olution of nearby Riemann problem. In the cae of a fold, for data (U L ; U R ; F ) on one ide of S, there are two nearby tructurally table Riemann olution; for data in S, there i a locally unique codimenion-one olution; and for data on the other ide of S, there i no nearby Riemann olution. Thi cae i depicted chematically in Fig. 1.2(a). The ame ituation can occur along ome of the Riemann olution join. In claical example, the two manifold-with-boundary, which meet along their common boundary, project to dierent ide of S, o that there i local exitence and uniquene of Riemann olution, a in Fig. 1.1(a). It i poible, however, for the two manifold-with-boundary to project to the ame ide of S, o that for nearby data there are two, one, or zero nearby Riemann olution, a in the cae of a fold; ee Fig. 1.1(b). For a frontier, there i a locally unique olution on S and on one ide of S, but no olution on the other ide, a illutrated in Fig. 1.2(b).
6 SCHECTER, PLOHR, AND MARCHESIN 1.5. Overview of the paper. Several implifying aumption are made in the current paper. Firt, the dierential equation (1.4) ued to determine the admiibility of hock wave ha a pecial form. (More generally, the left-hand ide could be replaced by D(U) _U, where D(U) i called the vicoity matrix.) Second, we conider only Riemann olution that are trictly hyperbolic, in that all tate U() b in the weak olution lie inide the region where characteritic peed are ditinct; however, we do not require vicou prole for hock wave to lie entirely within thi region. Third, we exclude hock wave with vicou prole that form homoclinic orbit. Thee implifying aumption are adopted to reduce the number of cae to be tudied. Notice that certain wave and wave conguration are excluded by thee aumption; example include tranitional rarefaction wave and hock wave with addle-to-piral or homoclinic connection. Given thee aumption, our goal i to lit and claify codimenion-one Riemann olution. Specically, we do the following. 1. We give a precie denition of codimenion-one Riemann olution. 2. We lit the way in which hypothee (H0){(H3) can be violated on the boundary of the et of tructurally table, trictly hyperbolic Riemann olution. Violation of hypothei (H0) can be identied with violation of hypothei (H1). Alo, we can amalgamate all violation of hypothei (H2) into a ingle cae. In order to reduce the number of violation of hypothee (H1) and (H3) that mut be conidered, we amalgamate thoe cae that are analogou under a duality between low and fat wave. 3. We note that certain violation of hypothei (H1) lead to failure of trict hyperbolicity, and we argue that other have codimenion higher than one. We dicard thee from further conideration. 4. There are 63 remaining violation of hypothee (H0){(H3). It i expected that each of them give rie, under appropriate nondegeneracy condition, to codimenion-one Riemann olution. Indeed, many occur in the literature. For ome of thee degeneracie, we mention one or two of the more obviou nondegeneracy condition that are required. However, precie tatement of the required nondegeneracy condition in each cae are left to later paper. To prove rigorouly that each of the 63 violation of hypothee (H0){(H3) give rie to codimenion-one Riemann olution, one mut check, in each cae, that under the appropriate nondegeneracy condition, certain matrice of partial derivative of an appropriate mapping are invertible. In thi paper we give the mapping, but we do not make any of the check. The knowledgeable reader will realize that in may cae the neceary computation are well-known, or are at leat preent in the literature. We plan to provide the neceary computation for at leat ome of the cae in later paper. 5. We claify the 63 type of codimenion-one Riemann olution according to how they are ituated in R. (Again proof are omitted.) Four are fold; ve are frontier; and 54 form 27 pair of related degeneracie that give rie to 27 join. The ve frontier cae involve overcompreive wave [15]. Except for thee cae, whenever one arrive at a trictly hyperbolic codimenion-one boundary of a et of tructurally table Riemann olution of a type identied in Ref. [17], another et of tructurally table Riemann olution of a type identied in Ref. [17] lie on the other ide. We regard thi a further indication that the lit of type of tructurally table, trictly hyperbolic Riemann olution in Ref. [17] i complete. 6. We claify the codimenion-one manifold of untable olution with repect to how S i ituated in U L U R B. In mot cae, the type of violation of (H0){(H3) doe not
CODIMENSION-ONE RIEMANN SOLUTIONS 7 determine whether it i an intermediate boundary, U L -boundary or dual, or F -boundary; one mut alo know where in the wave equence the violation occur. Again we do not give proof. 7. We explain how the reult of thi paper are related to wave curve. Many of the Riemann olution degeneracie we decribe can be undertood in term of \junction point" in wave curve. The reult of thi paper explain how the complexity of wave curve increae and decreae a they are extended. The tructure of olution of Riemann problem i very rich, and the lit of codimenionone bifurcation of thi tructure i long and complicated. The preent paper only formulate thi lit. Supplying proof for each cae contitute a rather large program, to which we plan to contribute in later paper. We hope to have compenated for the lack of proof in the current paper by preenting a clear overview of the range of bifurcation phenomena. The remainder of the paper i organized a follow. In Sec. 2 we review terminology and reult about tructurally table Riemann olution from Ref. [17]. In Sec. 3 we introduce ome new terminology needed for our treatment of codimenion-one Riemann olution, and provide a denition of uch olution. In Sec. 4 we conider the poible violation of hypothee (H0){(H3), dicard thoe that give rie to failure of trict hyperbolicity or to phenomena of codimenion greater than one, and give appropriate mapping for the remaining codimenion-one phenomena. In Sec. 5{7 we carry out tep 5{7 decribed above. The proof of one lemma i given in Appendix A, and detail of everal cae are relegated to Appendix B. 2. Structurally Stable, Strictly Hyperbolic Riemann Solution In thi ection we review tandard terminology concerning conervation law [21] along with reult about tructurally table Riemann olution from Ref. [17]. We conider the ytem (1.1) with t > 0, x 2 R, U(x; t) 2 R 2, and F : R 2! R 2 a C 2 map. Let U F = U 2 R 2 : DF (U) ha ditinct real eigenvalue (2.1) be the trictly hyperbolic region in tate-pace. We hall call a Riemann olution b U trictly hyperbolic if b U() 2 UF for all 2 R. In thi paper, all Riemann olution are aumed to be trictly hyperbolic. (Notice, however, that we do not require that vicou prole for hock wave lie entirely within the trictly hyperbolic region.) In the following we x an open et U R 2 and an open et B in a Banach pace of mooth (i.e., C 2 ) ux function. Thee et depend on the Riemann olution whoe tability we are invetigating; they are pecied in Sec. 2.2 below. For the dicuion in the next ubection, it uce to aume two propertie of U and B. Firt, U U F for all F 2 B. Second, there exit a cloed, bounded interval I R uch that the eigenvalue of DF (U) belong to I for all U 2 U and F 2 B. In thi cae, if b U i a cale-invariant olution of the Riemann problem (1.1){(1.2), then b U() = UL for min I and b U() = UR for max I. Therefore we can regard cale-invariant olution a belonging to the open et in the Banach pace L 1 (I; R 2 ). n o X := bu : I! U j U b 2 L1 (I; R 2 ) (2.2)
8 SCHECTER, PLOHR, AND MARCHESIN 2.1. Elementary wave. Riemann olution are compoed of elementary wave. The denition of elementary wave given in thi ection are not the mot general, but they uce in the context of trictly hyperbolic tructurally table Riemann olution. In Sec. 3 we adopt more general denition. For F 2 B and U 2 U, let 1 (U) < 2 (U) denote the eigenvalue of DF (U). Alo let `i(u) and r i (U), i = 1; 2, denote correponding left and right eigenvector, normalized o that `i(u)r j (U) = ij. For uitable neighborhood N U (decribed below), we can chooe thee eigenvector to depend moothly on U 2 N. More generally, with each eigenvalue family, i = 1; 2, i aociated the mooth line eld of null direction for DF (U)? i (U)I. A rarefaction wave of type R i i a dierentiable map b U : [; ]! U, where <, uch that b U 0 () i an eigenvector of DF (b U()) with eigenvalue = i (b U()) for each 2 [; ]. The tate U = b U() with 2 [; ] contitute the rarefaction curve?. Thi denition implie that if U = b U() 2?, then ince Di (U) b U 0 () = 1, `i(u)d 2 F (U)(r i (U); r i (U)) = D i (U) r i (U) 6= 0: (2.3) Condition (2.3) i genuine nonlinearity of characteritic family i at U. The denition alo implie that b U i actually C1 and that i (U? ) < i (U + ), where U? = b U() and U+ = b U() are the left and right tate of the rarefaction wave, repectively. We will nd it convenient to aociate a pecic peed to a rarefaction wave: for a rarefaction wave of type R 1, = 1 (U + ); for a rarefaction wave of type R 2, = 2 (U? ). A hock wave conit of a left tate U? 2 U, a right tate U + 2 U (with U + 6= U? ), a peed, and a connecting orbit? R 2, i.e., an orbit of the ordinary dierential equation (1.4) leading from the equilibrium U? to the equilibrium U +. In particular, the peed and the left and right tate of a hock wave are related by the Rankine-Hugoniot condition F (U + )? F (U? )? (U +? U? ) = 0; (2.4) which tate that U + i an equilibrium for Eq. (1.4). The orbit? i the range of ome olution e U of Eq. (1.4) uch that lim!1 e U() = U. Correponding to e U are traveling wave olution U(x; t) = e U((x? x0? t)=) of the parabolic equation U t + F (U) x = U xx ; (2.5) each traveling wave tend to the hock wave a! 0. For any equilibrium U 2 U of Eq. (1.4), the eigenvalue of the linearization of Eq. (1.4) at U are i (U)?, i = 1; 2. We hall ue the terminology dened in Table 2.1 for uch an equilibrium. name ymbol eigenvalue Repeller R + + Repeller-Saddle RS 0 + Saddle S? + Saddle-Attractor SA? 0 Attractor A?? Table 2.1. Type of equilibria.
CODIMENSION-ONE RIEMANN SOLUTIONS 9 (a) (b) (c) Figure 2.1. Poible phae portrait for a repeller-addle. The dierential equation on the center manifold i _ = b k + : : :, where k 2. In cae (a), k i even and b 6= 0; in cae (b), k i odd and b < 0; in cae (c), k i odd and b > 0. (a) (b) (c) Figure 2.2. Poible phae portrait for a addle-attractor. The dierential equation on the center manifold i _ = b k + : : :, where k 2. In cae (a), k i even and b 6= 0; in cae (b), k i odd and b < 0; in cae (c), k i odd and b > 0. Our name for an equilibrium account only for the ign of the eigenvalue; it doe not necearily reect the topological type of the phae portrait if there i a zero eigenvalue. Figure 2.1{2.2 how the poible phae portrait for repeller-addle and addle-attractor. If w i a hock wave, it type i determined by the equilibrium type of it left and right tate. (For example, w i of type R S if it left tate i a repeller and it right tate i a addle.) Thee type are lited in Fig. 2.3{2.6. (In thee gure, the phae portrait are drawn for the cae that repeller-addle and addle-attractor are nondegenerate, i.e., k = 2 in Fig. 2.1 and 2.2.) Shock type are grouped into four et of four: low, fat, overcompreive, and tranitional hock wave. Slow and fat hock wave are called claical hock wave. Remark. It i helpful, when thinking about hock wave involving addle-node, to regard an RS equilibrium a a addle S on the left and a repeller R on the right, a in Fig. 2.1(a). Similarly, an SA equilibrium i an attractor A on the left and a addle S on the right, a in Fig. 2.2(a). For intance, for an RS S hock wave, the orbit leave the repeller half of the RS and connect to the addle; thu the connection i generally table, and thi i a claical wave. By contrat, for an S RS wave, the connection lead from the addle to the addle half of the RS, o that the connection i generally untable, and thi i a tranitional wave. An elementary wave w i either a rarefaction wave or a hock wave. We write w : U??! U + (2.6)
10 SCHECTER, PLOHR, AND MARCHESIN R S R RS RS S RS RS Figure 2.3. Slow hock wave. S A S SA SA A SA SA Figure 2.4. Fat hock wave. R A R SA RS A RS SA Figure 2.5. Overcompreive hock wave. S S S RS SA S SA RS Figure 2.6. Tranitional hock wave. if w ha left tate U?, right tate U +, and peed. Notice that an elementary wave alo ha a type T, a dened above. Aociated with each elementary wave i a peed interval : for a rarefaction wave of type R i, = [ i (U? ); i (U + )], wherea for a hock wave of peed, = [; ]. If 1 and 2 are peed interval, we write 1 2 if 1 2 for every 1 2 1 and 2 2 2. Alo aociated with each elementary wave i a compact et?: if w i a rarefaction wave,? denote it rarefaction curve; if w i a hock wave, then? denote the cloure of it connecting orbit. We hall ay that an open et N R 2 i a neighborhood of the elementary?! U + if? N. wave w : U? 2.2. Structurally table Riemann olution. A wave equence (w 1 ; w 2 ; : : : ; w n ) i aid to be allowed if:
CODIMENSION-ONE RIEMANN SOLUTIONS 11 (W1) for each i = 1, : : :, n? 1, the right tate of w i coincide with the left tate of w i+1 ; (W2) the peed interval i for w i atify 1 2 n ; (2.7) (W3) no two ucceive wave are rarefaction wave of the ame type. For uch a wave equence we write (w 1 ; w 2 ; : : : ; w n ) : U 1 0?! U 2 1?!?! n U n : (2.8) If U 0 = U L and U n = U R, then aociated with an allowed wave equence (w 1 ; w 2 ; : : : ; w n ) i a Riemann olution U of Eq. (1.1){(1.2). Therefore we hall refer to an allowed wave equence a a Riemann olution. The type of a Riemann olution i imply the equence of type of it wave. In the following, we begin with an unperturbed Riemann olution (w 1; w 2; : : : ; w n) : U 0 1?! U 1 2?! n?! U n (2.9) for the ytem of conervation law U t + F (U) x = 0 and initial data U L = U 0 and U R = Un; then we eek Riemann olution when the data and the ux are perturbed from UL, UR, and F. To thi end, we rt x a compact et K R 2 uch that Int K i a neighborhood of each w i for i = 1, : : :, n. Second, we chooe an open neighborhood U Int K of the tate U 0, : : :, U n and of all rarefaction wave appearing in the unperturbed olution. Third, we chooe an open neighborhood B of F in the Banach pace of C 2 function F : K! R 2, equipped with the C 2 norm. The et U and B are choen to have two propertie: (i) U U F for all F 2 B; and (ii) there exit a cloed, bounded interval I R uch that the eigenvalue of DF (U) belong to I for all U 2 U and F 2 B. (Notice that we do not require U to be a neighborhood of the hock wave appearing in the unperturbed olution; thu we allow hock orbit to leave the region of trict hyperbolicity.) Let H(Int K) denote the et of nonempty, cloed ubet of Int K, which we equip with the Haudor metric. Denition 2.1. We hall ay that the Riemann olution (2.9) i tructurally table if there are neighborhood U i U of U i, I i I of i, and F B of F and a C 1 map G : U 0 I 1 U 1 I 2 : : : I n U n F! R 3n?2 (2.10) with G(U 0 ; 1; U 1 ; 2; : : : ; n; U n; F ) = 0 uch that: (P1) G(U 0 ; 1 ; U 1 ; 2 ; : : : ; n ; U n ; F ) = 0 implie that there exit a Riemann olution (w 1 ; w 2 ; : : : ; w n ) : U 1 0?! U 2 1?!?! n U n (2.11) for U t + F (U) x = 0 with ucceive wave of the ame type a thoe of the wave equence (2.9), with each w i contained in Int K, and with each rarefaction wave contained in U; (P2) DG(U 0 ; 1; U 1 ; 2; : : : ; n; Un; F ), retricted to the (3n?2)-dimenional pace of vector n ( _U 0 ; _ 1 ; _U 1 ; _ 2 ; : : : ; _ n ; _U n ; _ F ) : _U 0 = 0 = _U n ; _ F = 0 o, i an iomorphim onto R 3n?2. Condition (P2) implie, by the implicit function theorem, that G?1 (0) i a graph of a function dened on a neighborhood of (U 0 ; Un; F ), which we may a well take to be U 0 U n F. Therefore for each wave w i we can dene a map? i : U 0 U n F! H(Int K); namely,
12 SCHECTER, PLOHR, AND MARCHESIN? i (U 0 ; U n ; F ) i the rarefaction curve or the cloure of the connecting orbit of the wave w i. We further require that (P3) (w 1 ; w 2 ; : : : ; w n ) can be choen o that each map? i i continuou. The map G will be aid to exhibit the tructural tability of the Riemann olution (2.9). Aociated with (w 1; w 2; : : : ; w n) i a olution U (x; t) = b U (x=t) of the Riemann problem (1.1){(1.2) with U L = U L := U 0, U R = U R := U n, and F = F. Similarly, for each (U L ; U R ; F ) near (U L; U R; F ), there i a Riemann olution b U near b U aociated with the point in G?1 (0) that ha left tate U L, right tate U R, and ux F. 2.3. Local dening map. To contruct map G that exhibit tructural tability, we ue local dening map for each type of elementary wave. Let w : U??! U + be an elementary wave of type T for U t + F (U) x = 0. The local dening map G T ha a it domain a et of the form U? I U + F (with U being neighborhood of U, I being a neighborhood of, and F being a neighborhood of F ). The range i ome R e ; the number e depend only on the wave type T. The local dening map i uch that G T (U?; ; U+; F ) = 0. Moreover, if certain wave nondegeneracy condition are atied at (U?; ; U+; F ), then there i a neighborhood N of w uch that:?! U + of (D1) G T (U? ; ; U + ; F ) = 0 if and only if there exit an elementary wave w : U? type T for U t + F (U) x = 0 contained in N ; n (D2) DG T (U?; ; U+; F ), retricted to the pace ( _U? ; _; _U + ; F _ ) : F _ = 0 o, i urjective. Condition (D2) implie, by the implicit function theorem, that G?1 T (0) i a manifold of codimenion e. Therefore we can dene a map? from thi manifold to H(Int K) (jut a above). We have that (D3) w can be choen o that? i continuou and reduce to? at the point in G?1 T (0) correponding to (U?; ; U +; F ). The ytem of equation G T (U? ; ; U + ; F ) = 0 i called a ytem of local dening equation. We now dicu local dening equation and nondegeneracy condition for each type of elementary wave.?! U + i a rarefaction of family 1 for the 2.4. Rarefaction wave. Suppoe that w : U? equation U t + F (U) x = 0. Then there exit neighborhood F of F and N of w uch that for all F 2 F: (a) the eigenvector r 1 (U) of DF (U) correponding to the eigenvalue 1 (U) can be choen to depend moothly on U throughout N ; and (b) D 1 (U)r 1 (U) 6= 0 for all U 2 N. We can therefore normalize r 1 (U) o that D 1 (U)r 1 (U) 1. For each U? 2 N, dene 1 to be the maximal olution of the initial-value problem @ 1 @ (U?; ) = r 1 ( 1 (U? ; )); (2.12) 1(U? ; 1 (U? )) = U? : (2.13) Then, for F 2 F and U? ; U + 2 N, there exit a rarefaction wave of type R 1 for the equation U t + F (U) x = 0 that lead from U? to U +, ha peed, and lie within N if and only if U +? 1 (U? ; ) = 0 (2.14) = 1 (U + ) > 1 (U? ): (2.15)
Similarly, we can dene 2 to be the olution of CODIMENSION-ONE RIEMANN SOLUTIONS 13 @ 2 @ (; U +) = r 2 ( 2 (; U + )); (2.16) 2( 2 (U + ); U + ) = U + : (2.17) Then there i a rarefaction wave of type R 2 for U t + F (U) x = 0 from U? to U + with peed if and only if U?? 2 (; U + ) = 0 (2.18) = 2 (U? ) < 2 (U + ): (2.19) Equation (2.14) and (2.18) are dening equation for rarefaction wave of type R 1 and R 2, repectively. The nondegeneracy condition for rarefaction wave of type R i, which are implicit in our denition of rarefaction wave, are the peed inequality (2.15) or (2.19), and the genuine nonlinearity condition (2.3). To dene codimenion-one Riemann olution, we mut allow thee nondegeneracy condition to be violated. Therefore the denition of rarefaction wave will be generalized in Sec. 3. 2.5. Shock wave. If there i to be a hock wave olution of U t + F (U) x = 0 from U? to U + with peed, we mut have that: F (U + )? F (U? )? (U +? U? ) = 0; (E0) _U = F (U)? F (U? )? (U? U? ) ha an orbit from U? to U + : (C0) The two-component equation (E0) i a dening equation. In the context of tructurally table Riemann olution, condition (C0) can be regarded a a nondegeneracy condition except for tranitional hock wave, for which it i a dening equation. Codimenion-one Riemann olution can violate condition (C0), neceitating the generalized denition of hock wave given in Sec. 3; in thi context, the role of condition (C0) i more ubtle, a we dicu in Sec. 4. In Table 2.2{2.4 we lit additional dening equation and nondegeneracy condition for hock wave of variou type. The additional dening equation are either equality of the hock peed with a characteritic peed or, for tranitional hock wave, the vanihing of a eparation function. The wave nondegeneracy condition are open condition. The table omit everal type of nondegeneracy condition, which we aume implicitly: (a) U? 6= U + ; (b) inequality condition on the eigenvalue that are implied by the hock type (e.g., for an R S hock wave, 1 (U? ) < 2 (U? ) < and 1 (U + ) < < 2 (U + )); and (c) condition (C0) when it i an open condition (given the dening equation and the lited nondegeneracy condition). The additional dening equation and nondegeneracy condition for claical hock wave are given in Table 2.2{2.3; the reader hould alo refer to Fig. 2.3{2.5. In thee table, condition (C1){(C4) are that the connection? i not ditinguihed; thi mean the following. For RS hock wave, the connection? hould not lie in the untable manifold of U? (i.e., the unique invariant curve tangent to an eigenvector with poitive eigenvalue). For SA hock wave, the connection? hould not lie in the table manifold of U +.
14 SCHECTER, PLOHR, AND MARCHESIN type of hock additional dening equation nondegeneracy condition R S none none R RS 1 (U + )? = 0 (E1) D 1 (U + )r 1 (U + ) 6= 0 (G1) RS S 1 (U? )? = 0 `1(U + )(U +? U? ) 6= 0 (B1) (E2) D 1 (U? )r 1 (U? ) 6= 0 (G2) not ditinguihed connection (C1) RS RS 1 (U? )? = 0 (E3) D 1 (U? )r 1 (U? ) 6= 0 (G3) 1 (U + )? = 0 (E4) D 1 (U + )r 1 (U + ) 6= 0 (G4) `1(U + )(U +? U? ) 6= 0 (B2) not ditinguihed connection (C2) Table 2.2. Additional dening equation and nondegeneracy condition for low hock wave. type of hock additional dening equation nondegeneracy condition S A none none SA A 2 (U? )? = 0 (E5) D 2 (U? )r 2 (U? ) 6= 0 (G5) `2(U? )(U +? U? ) 6= 0 (B3) S SA 2 (U + )? = 0 (E6) D 2 (U + )r 2 (U + ) 6= 0 (G6) not ditinguihed connection (C3) SA SA 2 (U? )? = 0 (E7) D 2 (U? )r 2 (U? ) 6= 0 (G7) 2 (U + )? = 0 (E8) D 2 (U + )r 2 (U + ) 6= 0 (G8) `2(U? )(U +? U? ) 6= 0 (B4) not ditinguihed connection (C4) Table 2.3. Additional dening equation and nondegeneracy condition for fat hock wave. 2.6. Tranitional hock wave. Referring to Fig. 2.6, uppoe that w : U??! U + i a hock wave for U t + F (U) x = 0 of type S S, S RS, or SA RS. Thu we uppoe that, for the dierential equation _U = F (U)? F (U?)? (U? U?); (2.20) U? i an equilibrium of addle or addle-attractor type, U + i an equilibrium of addle or repeller-addle type, and there i a olution e U : R! R 2 uch that lim!1 e U() = U and eu() 2? for all 2 R. If U? i a addle of Eq. (2.20), let W? (U?; ) denote it untable manifold; if U + i a addle of Eq. (2.20), let W + (U?; ) denote it table manifold. Similarly, if U i a repelleraddle or addle-attractor, let W (U?; ) denote one of it center manifold. The manifold W (U?; ) both perturb moothly to invariant manifold of Eq. (1.4), denoted W (U? ; ). When U? i a addle, W? (U? ; ) i jut the untable manifold of the addle U? of Eq. (1.4); when U + i a addle, W + (U? ; ) i the table manifold of the addle of Eq. (1.4) near U+. Let be a line egment through e U(0) tranvere to _e U(0) in the direction V. W (U? ; ) meet in point U (U? ; ), and Then U? (U? ; )? U + (U? ; ) = S(U? ; )V: (2.21)
CODIMENSION-ONE RIEMANN SOLUTIONS 15 The function S i called the eparation function; it i dened on a neighborhood of (U?; ), and, of coure, S(U?; ) = 0. The partial derivative of S are given a follow [16]. The linear dierential equation ha, up to contant multiple, a unique bounded olution. contant, @S @ (U?; ) =? D U? S(U?; ) =? h i _ + DF (e U())? I = 0 (2.22) Z 1?1 Z 1?1 For the correct choice of thi ()(e U()? U? ) d; (2.23) () d DF (U?)? I : (2.24) One can treat SA S hock wave analogouly to S RS wave; one obtain a eparation function ~S(; U + ) [17]. The additional local dening equation and nondegeneracy condition (T1){(T4) for tranitional hock wave are given in Table 2.4. Condition (T2) i that there i a vector W uch that R `1(U + ) 1?1 () d W and R `1(U + )(U +? U? ) 1?1 ()(U()? U?) d Condition (T3) i that there i a vector W uch that R `2(U? ) 1?1 () d W and Condition (T4) i that R `1(U + ) 1?1 () d r 1 (U? ) and R `2(U? )(U?? U + ) 1?1 ()(U()? U +) d R `1(U + )(U +? U? ) 1?1 ()(U()? U?) d are linearly independent. are linearly independent. (T2) (T3) are linearly independent. (T4) type of hock additional dening equation nondegeneracy condition S S S(U? ; ) = 0 (S1) DS(U? ; ) 6= 0 (T1) S RS 1 (U + )? = 0 (E13) D 1 (U + )r 1 (U + ) 6= 0 (G13) S(U? ; ) = 0 (S2) tranverality (T2) SA S 2 (U? )? = 0 (E14) D 2 (U? )r 2 (U? ) 6= 0 (G14) es(; U + ) = 0 (S3) tranverality (T3) SA RS 2 (U? )? = 0 (E15) D 2 (U? )r 2 (U? ) 6= 0 (G15) 1 (U + )? = 0 (E16) D 1 (U + )r 1 (U + ) 6= 0 (G16) S(U? ; ) = 0 (S4) tranverality (T4) Table 2.4. Additional dening equation and nondegeneracy condition for tranitional hock wave.
16 SCHECTER, PLOHR, AND MARCHESIN 2.7. Riemann number. For the Riemann olution (2.9), let w i have type T i and local dening map G Ti, with range R e i. For appropriate neighborhood U i of U i, I i of i, F of F, and N i of w i, we can dene a map G : U 0 I 1 I n U n F! R e 1 ++e n by G = (G 1 ; : : : ; G n ), where G i (U 0 ; 1 ; : : : ; n ; U n ; F ) = G Ti (U i?1 ; i ; U i ; F ): (2.25) The map G i called the local dening map of the wave equence (2.9). Auming the wave nondegeneracy condition, if G(U 0 ; 1 ; : : : ; n ; U n ; F ) = 0, then for each i = 1, : : :, n, there i an elementary wave w i : U i i?1?! U i of type T i for U t + F (U) x = 0 contained in N i, for which? i i continuou. In view of the requirement in Denition 2.1 that the local dening map have range R 3n?2, a neceary condition for G = (G 1 ; : : : ; G n ) to exhibit the tructural tability of the wave equence (2.9) i that nx i=1 e i = 3n? 2; i.e., nx i=1 (3? e i ) = 2: (2.26) We are therefore led to dene the Riemann number of an elementary wave type T to be (T ) = 3? e(t ); (2.27) where e(t ) i the number of dening equation for a wave of type T. For convenience, if w i an elementary wave of type T, we hall ometime write (w) intead of (T ). Becaue of Eq. (2.26), a neceary condition P for an allowed equence of elementary wave (w 1 ; : : : ; w n ) n to be tructurally table i that (w i=1 i) = 2. The Riemann number of a rarefaction wave i 1. The Riemann number of hock wave are given in Table 2.5. U? nu + RS S SA A R 0 1 0 1 RS?1 0?1 0 S?1 0 0 1 SA?2?1?1 0 Table 2.5. Riemann number of hock wave. 2.8. Wave group. A 1-wave group i either a ingle R S wave or an allowed equence of elementary wave of the form (R RS)(R 1 RS RS) (R 1 RS RS) R 1 (RS S); (2.28) where the term in parenthee are optional. If any of the term in parenthee are preent, the group i termed compoite. A tranitional wave group i either a ingle SS wave or an allowed equence of elementary wave of the form or S RS(R 1 RS RS) (R 1 RS RS) R 1 (RS S) (2.29) (S SA) R 2 (SA SA R 2 ) (SA SA R 2 ) SA S; (2.30)
CODIMENSION-ONE RIEMANN SOLUTIONS 17 the term in parenthee being optional. In cae (2.29) and (2.30), the group i termed compoite. A 2-wave group i either a ingle S A wave or an allowed equence of elementary wave of the form (S SA) R 2 (SA SA R 2 ) (SA SA R 2 ) (SA A); (2.31) where again the term in parenthee are optional. If any of the term in parenthee are preent, the group i termed compoite. An SA RS wave i called a doubly onic tranitional wave. The reader hould note a ymmetry between the wave group (2.28) and (2.31), a well a between the group (2.29) and (2.30). The wave group R S, (2.28), and (2.29) are termed low; the wave group S A, (2.31), and (2.30) are termed fat. A olution U for the equation U t + F (U) x = 0 that conit of a fat wave group correpond to a olution U for the equation U t? F ( U) x = 0 that conit of a low wave group; the correpondence i U(x; t) = U(?x; t): (2.32) Under thi duality, rarefaction wave of type R 1 correpond to thoe of type R 2. Shock wave of type R RS, for example, correpond to thoe of type SA A; in general, to nd the dual of a hock type, one revere it name and interchange the letter R and A. Thi ymmetry will be exploited throughout thi paper to horten the treatment. 2.9. Wave group interaction condition. The wave group interaction condition i a tranverality condition appearing a a hypothei in the tructural tability theorem. In order to tate thi condition preciely, we recall ome reult from Ref. [17] concerning wave curve and tranitional wave group. Firt, conider a 1-wave group U 0 1?! k?! U k (2.33) for U t +F (U) x = 0 with local dening map G (1). Aume that each wave in the 1-wave group atie it nondegeneracy condition. Then the olution of G (1) (U 0 ; 1 ; : : : ; k ; U k ; F ) = 0 can be parameterized locally by U 0, F, and one additional parameter. (Thi i to be expected becaue the um of the Riemann number for the wave equence (2.33) i 1.) More preciely, there exit neighborhood U 0 U of U 0, F B of F, and J R of a parameter value, and mooth mapping f i : U 0 F J! R and U f i : U 0 F J! U for i = 1, : : :, k, with f i (U 0 ; F ; ) = i and U f i (U 0 ; F ; ) = U i, uch that G (1) (U 0 ; f 1(U 0 ; F; ); : : : ; f k (U 0; F; ); U f k (U 0; F; ); F ) = 0 for each (U 0 ; F; ) 2 U 0 F J. In particular, there exit a family f 1 U (U 0;F;) 0?! : : : f (U k 0;F;)?! U f (U k 0; F; ) (2.34) of wave equence for U t + F (U) x = 0 with ucceive wave of the ame type a thoe of the 1-wave group (2.33). The curve U f (U k 0 ; F ; ) parameterized by i called the forward wave curve through U k aociated with the wave equence (2.33). Similarly, if a 2-wave group U k k+1?! n?! U n (2.35)
18 SCHECTER, PLOHR, AND MARCHESIN for U t + F (U) x = 0 ha local dening map G (2), and each wave atie it nondegeneracy condition, then olution of G (2) (U k ; k+1 ; : : : ; n ; U n ; F ) = 0 can be parameterized locally by U n, F, and a parameter, giving a family Uk(U b n ; F; ) b k+1?! (Un;F; ) : : : b (Un;F; n?! ) U n (2.36) of wave equence. The curve Uk b(u n; F ; ) parameterized by i called the backward wave curve through U k aociated with the wave equence (2.35) For the implet cae of a tructurally table Riemann olution U 0 1?! k?! U k k+1?! n?! U n (2.37) compriing only a 1-wave group and a 2-wave group (joined at the tate U k ), the wave group interaction condition i that the forward wave curve U f (U k 0 ; F ; ) and the backward wave curve U b(u k n; F ; ) hould be tranvere, i.e., the tangent vector @U f =@ at (U k 0 ; F ; ) and @Uk b=@ at (U n; F ; ) hould be linearly independent. Next conider a tranitional wave group U k k+1?! `?! U ` (2.38) for U t + F (U) x = 0 with local dening map G (t). Aume that each wave atie it nondegeneracy condition. Then there exit a ubpace of U k -pace, of dimenion 0 or 1, uch that the following tatement are equivalent for any V 6= 0 in U k -pace: (a) V =2 ; (b) the linear map DG (t) (U k ; k+1; : : : ; `; U ` ; F ), retricted to the ubpace n o ( U _ k ; _ k+1 ; : : : ; _`; U`; _ F _ ) : Uk _ i a multiple of V, F _ = 0 ; (2.39) U k i urjective onto R 3(`?k) and the projection of it one-dimenional kernel onto U`-pace i one-dimenional. The ignicance of thi reult i that, if V i tangent to the forward wave curve through for the 1-wave group (2.33), and if V =2, then olution of the pair of equation G (1) (U 0 ; 1 ; : : : ; k ; U k ; F ) = 0; (2.40) G (t) (U k ; k+1 ; : : : ; `; U`; F ) = 0 (2.41) can be parameterized by U 0, F, and an additional parameter, giving a family f 1 U (U 0;F;) 0?! : : : f (U k 0;F;)?! U f (U k 0; F; ) f k+1 (U 0;F;)?! : : : f` (U 0;F;)?! U f` (U 0; F; ) (2.42) of wave equence. In thi manner, a forward wave curve can be extended by attaching a tranitional wave group, provided either that = f0g or that i tranvere to the forward wave curve. The curve U f` (U 0 ; F ; ) parameterized by i likewie called the forward wave curve through U `. Similarly, uing a dual verion of the reult jut quoted, a backward wave curve can be extended by attaching a tranitional wave group. Suppoe that a tructurally table Riemann olution conit of a 1-wave group g 0, r 1 tranitional wave group g 1, : : :, g r, and a 2-wave group g r+1. Then the wave group interaction condition i that (a) inductively for j = 1, : : :, r, the tranitional wave group g j hould atify the tranverality condition allowing it to be attached to the forward wave curve aociated with group g 0, : : :, g j?1 ; and (b) the forward wave curve aociated with
CODIMENSION-ONE RIEMANN SOLUTIONS 19 group g 0, : : :, g r hould be tranvere to the backward wave curve aociated with group g r+1. Remark. The wave group interaction condition could be formulated in alternative way. Under a certain tranverality condition, imilar to the one for attaching a tranitional wave group to a forward wave curve, a tranitional wave group can be attached to a backward wave curve. For each r, an equivalent formulation of the wave group interaction condition conit of: (a) the tranverality condition allowing wave group g 1, : : :, g to be attached to g 0 to form a forward wave curve; (b) the tranverality condition allowing wave group g r, : : :, g +1 to be attached to g r+1 to form a backward wave curve; and (c) the tranverality of thee forward and backward wave curve at the tate between g and g +1. The mot general tructurally table Riemann olution conit of wave of type SA RS eparating wave equence of the form already dicued (ee Theorem (2.2) below). In thi cae, the wave group interaction condition i that each of thee wave equence hould atify it wave group interaction condition. 2.10. Wave tructure and tructural tability. In Ref. [17] the following reult are proved. Theorem P 2.2 (Wave Structure). Conider the allowed wave equence (2.9). Then n (A) P i=1 (w i ) 2; n (B) i=1 (w i ) = 2 if and only if the following condition are atied. (1) Suppoe that the wave equence (2.9) include no SA RS wave. Then it conit of one 1-wave group, followed by an arbitrary number of tranitional wave group (in any order), followed by one 2-wave group. (2) Suppoe that the wave equence (2.9) include m 1 wave of type SA RS. Then thee wave eparate m + 1 wave equence g 0, : : :, g m. Each g i i exactly a in (1) with the retriction that: (a) if i < m, the lat wave in the group ha type R 2 ; (b) if i > 0, the rt wave in the group ha type R 1. Theorem 2.3 (Structural Stability). Suppoe that the allowed wave equence (2.9) ha total Riemann number P n i=1 (w i ) = 2. Aume that: (H1) each wave atie the appropriate wave nondegeneracy condition; (H2) the wave group interaction condition i atied; (H3) if w i ha type S and w i+1 ha type S, then i < i+1. Then the wave equence (2.9) i tructurally table. In fact, more can be concluded: not only can the connecting orbit? i of the perturbed hock wave w i be choen to vary continuouly, but alo there i a neighborhood N i uch that if? i N i, then? i i unique. 3. Codimenion-One Riemann Solution In thi ection we dene codimenion-one Riemann olution. To decribe thee olution conveniently and to relate them to tructurally table Riemann olution we mut generalize the denition of rarefaction and hock wave in Sec. 2, for the following reaon. When a tructurally table Riemann olution become a codimenion-one Riemann olution, the number and type of wave can change. However, the number and type of wave determine
20 SCHECTER, PLOHR, AND MARCHESIN the dimenion of the domain and range for the dening map. To keep thee dimenion xed, we broaden the denition of rarefaction and hock wave o that a codimenion-one olution can be regarded a having the ame number and type of wave a the tructurally table Riemann olution bordering it. 3.1. Generalized elementary wave. The denition in Sec. 2 of a rarefaction wave require the olution b U to be dierentiable a a function of = x=t, which entail that genuine nonlinearity hold all along the rarefaction curve?. For codimenion-one Riemann olution, however, we mut allow genuine nonlinearity to fail within rarefaction wave. In thi ituation,? i not parameterized moothly by the eigenvalue i. To contruct a rarefaction wave of family i, we conider a cloed egment? of an integral curve of the line eld of null direction for DF (U)? i (U)I, and we aume that i (U) i trictly monotone along?. Then? can be parameterized by a continuou function b U of = i (U), but thi parameterization fail to be mooth at the critical value of i, viz., peed = i (U) correponding to tate U 2? at which D i (U) r i (U) = 0. Neverthele, b U i a cale-invariant weak olution of Eq. (1.1), i.e., it atie the equation d d F ( b U())? d d b U() = 0 (3.1) in the ene of ditribution. (To ee thi, let U = U() parameterize? moothly, o that bu() = U() when = i ( U()). Then, in the weak formulation of Eq. (3.1), change the variable of integration from to.) Therefore we dene a generalized rarefaction wave of type R i to be a continuou map bu : [; ]! U, where, uch that (i) the rarefaction curve? = fb U() : 2 [; ] g i an integral curve for the line eld aociated to family i and (ii) = i (b U()) for all 2 [; ]. Thu we generalize the denition of Sec. 2 in two way: (1) may equal, giving a rarefaction wave of zero trength; and (2) b U may fail to be dierentiable. In other word, we allow the trength of a rarefaction wave to vanih, and we permit genuine nonlinearity to fail, a a codimenion-one boundary i approached. A generalized hock wave conit of a left tate U?, a right tate U + (poibly equal to U? ), a peed, and a equence of connecting orbit ~? 1, ~? 2, : : :, ~? k of Eq. (1.4), k 1, from U? = ~U 0 to ~U 1, ~U 1 to ~U 2, : : :, ~U k?1 to ~U k = U +. Notice that ~U 0, ~U 1, : : :, ~U k mut be equilibria of Eq. (1.4). We allow for the poibility that ~U j?1 = ~U j, in which cae we aume that ~? j i the trivial orbit f ~U j g. (A explained below, we exclude homoclinic orbit.) In particular, we generalize the denition of Sec. 2 to allow: (1) zero-trength hock wave, which we dene to have U? = U + and a ingle trivial orbit ~? 1 = fu? g; and (2) multiple orbit (k > 1). A a codimenion-one boundary i approached, the trength of a hock wave can vanih, jut a for rarefaction wave. Moreover, the orbit? connecting U? to U + can, in thi limit, break into two orbit: ~? 1 from U? to an equilibrium ~U 1 and ~? 2 from ~U 1 to U +. (More preciely, the cloure? tend, in the Haudor metric, to ~? 1 [ ~? 2.) Remark. The denition of generalized hock wave allow nontrivial cycle of orbit, when ~U m coincide with ~U n for ome n m + 2 and the union of the orbit ~? m+1, : : :, ~? n i not imply f ~ U m g. For example, a 2-cycle of hock wave can occur in a codimenion-one Riemann olution [1]. However, for implicity, and in keeping with Ref. [17], we do not conider hock wave with homoclinic orbit (e.g., joining a addle point to itelf or a repeller-addle to