(U) vanishes. A special case of system (1.1), (1.2) is given by the equations for compressible flow in a variable area duct, a ρv2,

Transcription

1 MEHODS AND APPLICAIONS OF ANALYSIS. c 2003 Intenational Pe Vol. 10, No. 2, pp , June HE GENERIC SOLUION OF HE RIEMANN PROBLEM IN A NEIGHBORHOOD OF A POIN OF RESONANCE FOR SYSEMS OF NONLINEAR BALANCE LAWS JOHN HONG AND BLAKE EMPLE Abtact. We decibe the geneic olution of the Riemann poblem nea a point of eonance in a geneal 2x2 ytem of balance law coupled to a tationay ouce. he ouce i teated a a coneved quantity in an augmented 3x3 ytem, and Reonance i between a nonlinea wave family and the tationay ouce. anonic compeible Eule flow in a vaiable aea duct, a well a pheically ymmetic flow, ae hown to be pecial cae of the geneal cla of equation tudied hee. 1. Intoduction. We conide a geneal 3x3 ytem of balance law of the fom (1.1) (1.2) a t = 0, w t + f(a,w) x = a g(a,w), whee (1.2) i a 2 2 ytem of conevation law, w = (u,v) (w 1,w 2 ), f = (f 1,f 2 ), g = (g 1,g 2 ) and (1.1) i incopoated to model eonance between a tationay ouce a(x) R and one of the nonlinea wave familie of (1.2). Letting U = (a,u,v), ytem (1.1), (1.2) i equivalent to the 3 3 ytem (1.3) U t + F(U) x = a G(U), whee F = (0,f 1,f 2 ) and G = (0,g 1,g 2 ). Reonance occu at tate U = (a,w ) whee an eigenvalue of Df Dw (U) vanihe. A pecial cae of ytem (1.1), (1.2) i given by the equation fo compeible flow in a vaiable aea duct, (1.4) (1.5) (1.6) a t = 0, ρ t + (ρv) x = a a ρv, (ρv) t + (ρv 2 + p) x = a a ρv2, whee ρ i the denity, p i the peue, v i the velocity, and a(x) i the diamete of the duct at poition x. Sytem (1.4)-(1.6) eult unde the aumption that p i a function of ρ alone, in which cae the enegy equation (ρe) t + (ρev + pv) x = a a (ρev + pv) uncouple fom the ma and momentum equation (1.5) and (1.6). Spheically ymmetic n-dimenional flow aie when a a = n 1 x. Sytem (1.4)-(1.6) i a 2 2 ytem with ouce of fom (1.1), (1.2), and the condition of eonance Received Febuay 18, 2003; accepted fo publication Augut 15, Depatment of Mathematic, UCLA, Lo Angele, CA , USA (jhong@math.ucla.edu). Depatment of Mathematic, Univeity of Califonia, Davi, Davi CA 95616, USA (jbtemple@ucdavi.edu). Suppoted in pat by NSF Applied Mathematic Gant Numbe DMS and by the Intitute of heoetical Dynamic, UC-Davi. 279

2 280 J. HONG AND B. EMPLE fo ytem (1.4)-(1.6) tanlate into the tatement that the flow i tanonic in a neighbohood of U, [2]. Hee we identify natual geneic condition on f and g at a point of eonance U in ytem (1.1), (1.2) that guaantee a canonical olution of the Riemann poblem in a neighbohood of U, and we then peent the olution. A a pecial cae we how that thee condition ae met by ytem (1.4)-(1.6) howeve, with a diffeent choice of ign, qualitatively diffeent olution ae poible. By identifying the geneic local tuctue of the Riemann poblem nea a point of eonance, we accomplih a peliminay tep in ou pogam to extend the eult in [7] to ytem; that i, ou pogam i to obtain a time independent etimate fo the total vaiation of the coneved quantitie nea a point of eonance by analyzing appoximate Glimm cheme olution of ytem (1.3). Note that becaue of the peence of the a tem in (1.2), the Riemann poblem, (the initial value poblem when the data i given by piecewie contant tate), appea to be ingula when a i dicontinuou. Howeve, becaue of a e-caling popety of tanding wave fo ytem of fom (1.2), the Riemann poblem eally doe give the elementay wave that povide the building block fo moe geneal olution. Fo example, in [6] it wa hown that Glimm analyi of wave inteaction and the local total vaiation bound extend to thee ingula Riemann poblem fo n n tictly hypebolic ytem of fom (1.2), and convegence of the Glimm Scheme i poven fo Lipchitz continuou a, (the cae when the weak fomulation of (1.2) applie). Inteetingly, the eidual convege weakly, by ocillation, athe than by L 1 convegence a in Glimm oiginal pape. hee eult wee extended to eonant cala balance law in [7]. 2. Geneic Condition. We dicu geneic condition on the function f(u), g(u) fo ytem (1.1), (1.2), that guaantee a canonical olution of the Riemann poblem nea a point of eonance U. hi i pepaatoy to the peentation of the olution of the Riemann poblem in the next ection. o tat, let λ i λ i (a,w) denote the eigenvalue, and R i R i (a,w), L i L i (a,w) coeponding ight and left eigenvecto, epectively, fo the 2 2 matix Df Dw ( fi w j ) at fixed a, i,j = 1,2. Let Dg Dw ( gi w j ), the diffeential holding a contant. (We will nomalize R i, L i below, but the condition to follow ae independent of nomalization). We aume the following fou geneic condition hold at the tate U : (2.1) 0 = λ 1 (U ) < λ 2 (U ), (2.2) λ i R i U=U 0, i = 1,2, (2.3) L i (g f a ) U=U 0, i = 1,2, (2.4) [ L1 (g f Det a ), L 1 Dg Dw R 1 L 2 (g f a ), L 2 Dg Dw R 1. ] 0.

3 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 281 Ou main eult can be tated a follow: heoem 1. Aume that ytem (1.1), (1.2) atifie (2.1)-(2.4) at the tate U = U. hen the olution of the Riemann poblem ha a unique canonical tuctue in a neighbohood of the tate U, thi being explicitly given in the Figue below. Befoe we contuct the olution of the Riemann poblem fo ytem (1.1), (1.2) in a neighbohood of U, unde aumption (2.1)-(2.3), we fit veify (2.1)-(2.4) fo the ga dynamic example (1.5), (1.6) when U = (a,u,v) (a,ρ,ρv), f(u) = (ρv,ρv 2 +p), and g(u) = ( ρv a, ρv2 a ). he eigenvalue and ight eigenvecto fo ytem (1.5), (1.6) at fixed a ae given by, (2.5) λ 1 = v σ, λ 2 = v + σ, (2.6) R 1 = (1,λ 1 ) t, R 2 = (1,λ 2 ) t, whee the ound peed σ i given by (2.7) σ = p (ρ). Since the left eigenvecto atify L i R j = 0, i j, we can chooe (2.8) L 1 = 1 ρ ( λ 2,1), L 2 = 1 ρ ( λ 1,1). he coodinate ytem of Riemann invaiant (,) can be defined in tem of L 1 and L 2 by (2.9) = L 1, = L 2, which yield (2.10) = v = v + ρ ρ p ρ, p ρ. Fom (2.5)-(2.7) it follow that aumption (2.1)-(2.4) epectively tanlate into the following condition at U = (a,u,v ), (we aume equation of tate p = p(ρ), p (ρ) > 0, p (ρ) > 0): (2.11) u = σ = p (ρ ), (2.12) { p λ i R i = 2σ + σ } > 0, i = 1,2, ρ

4 282 J. HONG AND B. EMPLE (2.13) (2.14) L 1 (g f a ) = 1 ρ ( λ 2,1) ( ρv a, ρv2 a ) = vσ a > 0, L 2 (g f a ) = 1 ρ ( λ 1,1) ( ρv a, ρv2 a ) = vσ a < 0. Finally, to veify (2.4), wite (2.15) (2.16) L 1 Dg Dw = 1 ρ ( λ 2,1) L 2 Dg Dw = 1 ρ ( λ 1,1) [ 0 1 a v 2 a 2v a [ 0 1 a v 2 a 2v a It follow that at U = U, λ 1 = 0, and o we have ] ] = 1 aρ (v2,λ 2 2v), = 1 aρ (v2,λ 1 2v). (2.17) (2.18) L 1 Dg Dw R 1 = 1 aρ ( v2 + λ 2 1) = v2 aρ < 0, L 2 Dg Dw R 1 = 1 aρ ( v2 + λ 1 λ 2 ) = v2 aρ < 0. he condition (2.4) now follow fom (2.13), (2.14) and (2.15), (2.18). 3. he Riemann Poblem. he Riemann poblem i the initial value poblem with initial data given at t = 0 by the jump dicontinuity (3.1) U 0 (x) = { UL = (a L,u L ) if x < 0, U R = (a R,u R ) if x > 0. We deive the olution of (3.1) fo ytem (1.1), (1.2) within the cla of elementay wave, hock wave, aefaction wave and tanding wave. It i eaiet to diplay the olution in a coodinate ytem of Riemann invaiant, whee, with convenient convention fo R i, L i, the olution ha a unique local tuctue. At each fixed a, ytem (1.2) i a 2 2 ytem of conevation law, and thu, ince the tate pace i the plane, thee exit a coodinate ytem of Riemann invaiant (,). Chooe (,) uch that L 1 = ( u, v ), and imilaly, L 2 =. Alo, let {U : λ 1 (U) = 0} denote the tanition uface. hi i the uface whee the peed of 1-aefaction wave change thei ign elative to the local peed of tanding wave, c.f. [8]. Since λ 1 0, it follow that i a mooth one dimenional uface paing though the bae point U = U and tanveal to R 1 at that point, and hence the one wave cuve cut the tanition uface tanveally in a neighbohood of U. Fo convenience, chooe the ign in (2.1)-(2.3) o that they agee with the ign in the ga dynamic example (2.11)-(2.13). hat i, aume (3.2) λ 1 R 1 > 0, (3.3) L 1 (g f a ) U=U > 0,

5 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 283 (3.4) L 2 (g f a ) U=U < 0. Unde thee aumption, the olution of the Riemann poblem i diagammed in the figue below. Befoe dicuing the olution of the Riemann poblem, we fit how that thee i no lo of geneality in the choice of ign (3.2)-(3.4), and dicu the ignificance of the ign in (2.4). o tat, note that we can fix the ign in (3.2) by chooing the oientation of R 1. We next how that the ign in (3.3) and (3.4) fix the convexity of tanding wave cuve elative to the and coodinate axe, and the ign in (2.4) then detemine whethe the zeo peed hock cuve, (c.f. [7]), lie inide o outide the coeponding tanding wave cuve, both of which eminate fom the ame point on the tanition uface. Moe peciely, we how that (3.3) fixe the ign of d2 a d, and then (3.4) 2 fixe the ign of d2 d at the point whee a tanding wave cuve U 2 () coe the tanition uface λ 1 = 0. hen, with the choice of ign in (3.2)-(3.4), we how that the negative, [epectively poitive], ign in (2.4) implie that zeo peed hock cuve, which emanate fom point on the tanition uface, beak to the inide, [epectively outide], of the coeponding tanding wave cuve that emanate fom the ame point on the tanition uface, in a neighbohood of U = U, c.f. [7]. o thi end, let U (x) (a (x),w (x)) denote a tanding wave cuve, obtained fom (1.3) by auming no tempoal dependence. hat i, U atifie (3.5) Df Dw dw = (g f a)da. Multiplying by L i on the left, i = 1,2, we obtain (3.6) (3.7) λ 1 L 1 dw = L 1 (g f a )da, λ 2 L 2 dw = L 2 (g f a )da. Uing L 1 dw = d, L 2 dw = d, we obtain (3.8) (3.9) λ 1 d = L 1 (g f a )da, λ 2 d = L 2 (g f a )da, which lead to (3.10) (3.11) da d = λ 1 L 1 (g f a ), d d = λ 1 {L 2 (g f a )} λ 2 {L 1 (g f a )}. Note that (3.10) implie that tanding wave cuve ae tangent to 1-wave cuve at the tanition uface λ 1 = 0. We now how that (3.4) implie that thi tangency i quadatic. (Since hock wave cuve have cubic tangency with aefaction cuve, thi implie that hock cuve emanating fom left tate on U and coing into

6 284 J. HONG AND B. EMPLE the convex ide of the tanding wave cuve, mut co the tanding wave cuve at a unique point on the oppoite ide of ). So conide the tanding wave cuve at a point on the tanition uface. hen (3.10), (3.11) and (3.3) imply that (3.12) (3.13) d 2 a d 2 = 1 dλ 1 L 1 (g f a ) d > 0, d 2 d 2 = L 2 (g f a ) dλ 1 λ 2 L 1 (g f a ) d < 0, give the cuvatue in a and of the tanding wave cuve at a point on the tanition uface. By (3.12), we have hown that (3.3) detemine the ign of d2 a d, and (3.4) 2 detemine the ign of d2 d along a tanding wave cuve at the point whee it coe 2 the tanition uface, a claimed. We can now how that (2.4) detemine whethe the zeo peed hock cuve beak to the left o to the ight of the tanding wave cuve at the point on the tanition uface at which they emanate. o define the zeo peed hock cuve, tat with a fixed tanding wave cuve U (). hi cuve coe the tanition uface at a unique point in a neighbohood of U, and o a not to intoduce moe notation, aume without lo of geneality that thi point i U itelf. By ou ign convention, the wave peed λ 1 inceae moving to the ight though the tanition uface, (that i, towad inceaing ), along the tanding wave cuve U () in the (,)-plane. Conide the potion of a tanding wave cuve that lie to the ight of the tanition uface; that i, conide U () = (a (),u (),v ()) fo >, whee U ( ) = U. hen λ 1 inceae fom λ 1 (U ) = 0 a inceae fom = along U (), and thu it follow that fo each >, thee i a unique tate Ũ() on the left of the tanition uface uch that the hock [U (),Ũ()] i a zeo peed hock when the left tate of the hock i U () and the ight tate i Ũ(). We call the cuve Ũ() the zeo peed hock cuve aociated with the tanding wave cuve U (). (hi i diagammed in Figue 1 and 2, in the cae when the tanding wave cuve lie to the left and ight of the zeo peed hock cuve, epectively, auming nomalization (3.2)-(3.4). he dotted line denote the zeo peed hock cuve, and it aociated tanding wave cuve i dawn a the paabolic cuve though tate U U. Ou convention i that the zeo peed hock cuve i paameteized by value of > that paameteize U on the ight of, and obeve that U () and Ũ() ae cuve that lie at the ame value of a. A a paameteized cuve, Ũ () lie to the left of, and emanate fom the tate U = U, whee it i tangent to R 1.) Note that ince hock cuve have thid ode tangency with = cont at the tate U L, and the tanding wave cuve have quadatic tangency, it follow that the hock cuve emanating fom left tate on U () fo >, alway beak tanveally into the egion below the tanding wave cuve, and hence, in a ufficiently mall neighbohood, uch hock cuve inteect the tanding wave cuve U at exactly two point: at the left tate U L = U () on the ight of, and at a unique point U R on the left of. We now how that, (with the choice of ign in (3.2)-(3.4)), the point Ũ() on the zeo peed hock cuve alway lie on the hock cuve emanating fom U L between U L and U R when the ign in (2.4) i negative, (that i, the hock wave [U L,U R ] ha negative peed), while the point U R lie on the hock cuve emanating fom U L between U L and Ũ() when the ign in (2.4) i poitive, (o that the hock wave [U L,U R ] ha poitive peed in thi cae). o thi end, conide the zeo peed hock [U ( ),Ũ( )], >. (We ue in thi agument to indicate the -paameteization of U on the ight of.)

7 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 285 Since the hock ha zeo peed, the Rankine-Hugoniot jump condition imply that f(u ( )) = f(ũ( )), and a ( ) = ã ( ), o we can wite (3.14) f(a ( ),ũ ( ),ṽ ( )) = f(a ( ),u ( ),v ( )). Uing the condition fo a tanding wave df = gda, we can expe the incement df on the left and ight ide of (3.14), coeponding to an incement d, by (3.15) df = f a (Ũ)da + D Dw f(ũ)d w = g(u )da, which lead to (3.16) But D Dw f(ũ)d w = [g(ũ) f a (Ũ)]da + [g(u ) g(ũ)]da. (3.17) [g(u ) g(ũ)] = D Dw g(ũ) (W W ) + O( W W 2 ), and (3.18) W W = ǫr 1 (Ũ) + O( W W 2 ), whee (3.19) ǫ = O( W W ). Multiplying (3.16) by L i, i = 1,2, uing L 1 Df Dw = λ 1 L 1 = λ 1, L 2 Df Dw = λ 2 L 2 = λ 2, and uing (3.17) and (3.18), we obtain the deivative of a and with epect to the value of on the zeo peed hock cuve to the left of, (3.20) (3.21) da d = λ 1 ( ) + O(ǫ 2 ), L 1 (g f a )da + ǫ L 1 Dg Dw R 1 ( ) d d = λ L 2 (g f a ) + ǫ L 2 Dg 1 Dw R 1 ( ) + O(ǫ 2 ). λ 2 L 1 (g f a ) + ǫ L 1 Dg Dw R 1 Note that etting ǫ = 0 in (3.20), (3.21) give the fomula (3.10), (3.11) fo the coeponding deivative of the tanding wave cuve at U = U. Futhemoe, taking the deivative of (3.21) with epect to ǫ at ǫ = 0, we ee that the zeo peed hock cuve will beak tanveally to one ide o the othe of the tanding wave cuve at U = U if the deteminant in (2.4) i non-zeo. Uing the choice of ign in (3.2)- (3.4), it follow that Ũ() lie to the ight, [epectively left], of U () fo < if the deteminant in (2.4) i poitive, [epectively negative], (diagammed in Figue 1 and 2, epectively), a claimed.

8 286 J. HONG AND B. EMPLE he olution of the Riemann poblem i diagammed in Figue 3-10, auming the nomalization (3.2)-(3.4). A an entopy condition we take the condition that the change in a i monotone along tanding wave in a olution. hi allow fo what we call tiple compoite tanding wave, and alo entail a non-uniquene of olution, and a coeponding lack of continuou dependence of the olution on U L and U R. hi wa obeved in the cala cae in [14, 7], and epeent an inteeting complication in the tuctue of the poible time aymptotic wave patten. Even o, the analyi hee how that thi inteeting tuctue i canonical in a neighbohood of a tate of eonance. he analyi of the Glimm cheme fo the cala cae teated in [14, 7] lead to additional entopy condition that futhe etict the admiible olution of the Riemann poblem. In paticula, the L w minimization pinciple intoduced in [7], detemined a unique olution of the Riemann poblem, except fo an inheent duplicity of olution at boundaie whee the qualitative wave tuctue of olution change. Fo ou pupoe hee, we make no futhe entopy etiction, c.f. [7]. Figue 3-6 give the olution of the Riemann poblem in the cae when the zeo peed hock cuve, (dawn a a dahed, downwad paabolic cuve to the left of in each diagam), lie to the ight of the tanding wave cuve, (the cae of a poitive ign in (2.4)), and Figue 7-10 give the cae when the zeo peed hock cuve lie to the ight of the tanding wave cuve, (the cae of a negative ign in (2.4)), accoding to the fou cae a L le than o geate than a R, and U L left o ight of. (he ga dynamic ytem (1.4)-(1.6) i of the type diagammed in Figue 3-6.) o keep the diagam a imple a poible, we make the following idealization. Fit, the 1- wave cuve and 2-wave cuve ae dawn a hoizontal and vetical line, epectively. Moe pecifically, the thid ode tangency of hock cuve and aefaction cuve i neglected a a highe ode effect elative to the quadatic tangency of tanding wave cuve and 1-wave cuve in each diagam. Alo, the tate at level a = a R ae dawn along a ingle hoizontal line. In fact, the 1- and 2-wave cuve lie at contant a, o the change in a in a Riemann poblem ente a a jump aco the tanding wave. hu, the tate at level a = a R would actually be a cuve of tate obtained by maintaining the condition a=a R a L along the tanding wave in the olution. he value of a change quadatically along tanding wave cuve, and by ou convention, a take a minimum at the point of inteection of the tanding wave with the tanition uface. he dak cuve in Figue 3 10 epeent tate that can be eached by a 1-wave, o by 1-wave and the citical tanding wave that mak the place whee the 1-wave and tanding wave change thei elative peed. In each diagam, the olution i a combination of 1-wave and tanding wave, followed by a 2-wave, whee the ode of 1-wave and tanding wave i detemined by taking the lowe wave fit. In paticula, by (3.2)-(3.4), 1-wave with left tate U L to the ight of have poitive peed, (and hence come afte the zeo peed tanding wave moving fom left to ight in the xt-plane), unle the 1-wave i a hock wave with a left tate to the left of the zeo peed hock cuve. In the cae of Figue 3-10 excluding the cae of Figue 5 and 9, the olution of the Riemann poblem can contain what we call tiple compoite tanding wave. hi i a tanding wave that conit of a tanding wave to the ight of, followed by a zeo peed hock wave, followed by a diffeent tanding wave to the left of, whee the um of the change in a along the tanding wave i a = a R a L. Since all wave move with zeo peed, a tiple compoite tanding wave can be teated a a ingle wave. In each diagam, the egion between the two vetical dahed line,

9 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 287 (which actually epeent 2-wave cuve), define the value of U R fo which thee i a olution [U L,U R ] that contain a tiple compoite tanding wave. We now dicu the diagam individually. Dicuion of Figue 3: he dahed line emanating fom tate U i the zeo peed hock wave. he coeponding tanding wave cuve i dawn to the ight of a the dak, downwad paabolic cuve emanating fom the tate U, continuing to the left of a the olid paabolic cuve emanating fom U. o obtain the olution fo given ight tate U R that lie to the ight of tate D in the figue 3, tat at U L, take the 1-wave cuve, (the hoizontal line though U L ), to an intemediate tate left of U, follow thi by the tanding wave cuve to the level a = a R, (dawn a a hoizontal line), and then take a two wave cuve at level a R to U R. When U R lie between the 2-wave cuve though D and E, take the 1-wave to U, (a aefaction wave), take the tiple compoite tanding wave to C, and then the two wave fom C to U R ; that i, the olution i U L U A B C U R. he tiple compoite tanding wave i the tanding wave fom U to A followed by the zeo peed hock fom A to B followed by the tanding wave fom B to C. Finally, if U R lie to the ight of tate E, take the aefaction wave fom U L to U, the citical tanding wave fom U to level a R, and then a poitive peed 1-wave at level a R to a tate ight of E that connect by a 2-wave to the tate U R. In thi cae the olution of the Riemann poblem i unique. Dicuion of Figue 4: he olution i a negative peed 1-hock followed by a tanding wave followed by a two wave when the tate U R lie to the left of tate D. When U R lie between D and E, the olution conit of a tiple compoite wave to a tate between D and E, followed by a 2-wave to U L, fo example, U L A B C U R, diagammed in the figue. Fo U R to the ight of tate E, the olution conit of a tanding wave followed by a poitive peed 1-wave followed by a 2-wave. In thi cae the olution of the Riemann poblem i unique. Dicuion of Figue 5: he olution conit of a 1-wave followed by a tanding wave followed by a 2-wave. In thi cae the olution of the Riemann poblem i unique. Dicuion of Figue 6 : he olution conit of a negative peed hock wave, (to a tate left of D), followed by a tanding wave fo tate U R which lie to the left of tate F. Howeve, tate between F and G can alo be olved by a tiple compoite tanding wave followed by a 2-wave, a in the olution U L A B C U R, o by a negative peed hock to a tate left of E, followed by a tanding wave, and then a 2-wave. When U R lie to the ight of tate G, the olution conit of a tanding wave followed by a poitive peed 1-wave followed by a 2-wave. In thi cae thee i a tiple non-uniquene of olution when U R lie between tate F and G. Dicuion of Figue 7: Fo tate U R to the left of tate D, the olution conit of a negavive peed 1-wave followed by a tanding wave to level a = a R, followed by a 2-wave. Fo tate U R between D and F, the olution ha a tiple multiplicity: a 1-wave followed by a tanding wave followed by a 2-wave; o a 1-aefaction wave to U followed by the citical tanding wave to B followed by a poitive peed 1-wave to tate between D and F, followed by a 2-wave to U R ; o ele olution with a tiple compoite tanding wave like the olution U L U A C E U R. When U R lie to the ight of tate F, the olution educe to the ingle olution U L U B 3 We ay the tate lie to the ight of a tate if it lie to the ight of the 2-wave cuve though that tate, dawn a a vetical line in the diagam.

10 288 J. HONG AND B. EMPLE followed by a poitive peed 1-wave followed by a 2-wave. In thi cae thee i a tiple multiplicity of olution when U R lie between tate D and F. Dicuion of Figue 8: Fo tate U R to the left of tate D, the olution conit of a negative peed 1-wave followed by a tanding wave to level a = a R, followed by a 2-wave. Fo tate U R between D and E, the olution ha a tiple multiplicity: a negative peed 1-hock followed by a tanding wave followed by a 2-wave; o the citical tanding wave fom U L to E followed by a poitive peed 1-wave to a tate between D and E, followed by a 2-wave to U R ; o ele a olution with a tiple compoite tanding wave like the olution U L A B C U R. When U R lie to the ight of tate E, the olution educe to the ingle olution U L H followed by a poitive peed 1-wave followed by a 2-wave. In thi cae thee i a tiple multiplicity of olution when U R lie between tate D and E. Dicuion of Figue 9: hi agee with Figue 5. Dicuion of Figue 10: Fo tate U R to the left of tate F, the olution conit of a negavive peed 1-wave followed by a tanding wave to level a = a R, followed by a 2-wave. Fo tate U R between F and G, the olution conit of a tiple compoite tanding wave to a tate between F and G, followed by a 2-wave. And fo ight tate U R to the ight of G, the olution conit of a citical tanding wave to level a = a R, followed by a poitive peed 1-wave, followed by a 2-wave to U R. In thi cae the olution of the Riemann poblem i unique. REFERENCES [1] G. Chen and J. Glimm, Global olution to the compeible Eule equation with geometical tuctue, Comm. Math. Phy., 179 (1996), pp [2] R. Couant and K.O. Fiedich, Supeonic flow and hock wave, John Wiley & Son, New Yok, [3] G. Dal Mao, P.G. LeFloch, and F. Muat, Definition and weak tability of nonconevative poduct, J. Math. Pue Appl., 74 (1995), pp [4] J. Glimm, Solution in the lage fo nonlinea hypebolic ytem of equation, Comm. Pue Appl. Math., 18 (1965), pp [5] S.K. Godunov, A diffeence method fo numeical calculation of dicontinuou olution of the equation of hydodynamic, Mat. Sb., 47 (1959), in Ruion, pp [6] J. Hong, he Glimm cheme extended to inhomogeneou ytem, Doctoal hei, UC-Davi. [7] J. Hong and B. emple, A bound on the total vaiation of the coneved quantitie fo olution of a geneal eonant nonlinea balance law, SIAM J. Appl. Math., to appea. [8] E. Iaacon, Global olution of a Riemann poblem fo a non-tictly hypebolic ytem of conevation law aiing in enhanced oil ecovey, Rockefelle Univeity pepint. [9] E. Iaacon, D. Machein, B. Ploh, and B. emple, he Riemann poblem nea a hypebolic ingulaity: the claification of olution of quadatic Riemann poblem I, SIAM J. Appl. Math., 48 (1988), pp [10] E. Iaacon, B. emple, he tuctue of aymptotic tate in a ingula ytem of conevation law, Adv. Appl. Math., 11 (1990), pp [11] E. Iaacon, B. emple, Analyi of a ingula hypebolic ytem of conevation law, Jou. Diff. Equn., 65 (1986), pp [12] E. Iaacon, B. emple, Example and claification of non-tictly hypebolic ytem of conevation law, Abtact of AMS, Januay [13] E. Iaacon, B. emple, Nonlinea eonance in ytem of conevation law, with E. Iaacon, SIAM Jou. Appl. Anal., 52 (1992), pp [14] E. Iaacon, B. emple, Convegence of the 2 2 Godunov method fo a geneal eonant nonlinea balance law, SIAM Jou. Appl. Math., 55:3 (1995), pp [15] B. Keyfitz and H. Kanze, A ytem of non-tictly hypebolic conevation law aiing in elaticity theoy, Ach. Rat. Mech. Anal., 72 (1980), pp

11 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 289 [16] S.N. Kuzkov, Fit ode quailinea equation with eveal pace vaiable, Mat. USSR Sb., 10 (1970), pp [17] P.D. Lax, Hypebolic ytem of conevation law, II, Comm. Pue Appl. Math., 10 (1957), pp [18] P.D. Lax and B. Wendoff, Sytem of Conevation law, Comm. Pue Appl. Math., 13 (1960), pp [19] L. Lin, J. Wang and B. emple, A compaion of convegence ate fo Godunov method and Glimm method in eonant nonlinea ytem of conevation law, with L.Lin and J. Wang., SIAM J. Nume. Anal., 32:3, pp [20] L. Lin, J. Wang and B. emple, Suppeion of ocillation in Godunov method fo a eonant non-tictly hypebolic ytem, SIAM J. Nume. Anal., 32:3, June [21].P. Liu, Quailinea hypebolic ytem, Comm. Math. Phy., 68 (1979), pp [22].P. Liu, Reonance fo a quailinea hypebolic equation, J. Math. Phy., 28:11 (1987), pp [23] D. Machein and P.J. Pae-Leme, A Riemann poblem in ga dynamic with bifucation, Compute and Mathematic with Application, 12A (1986), pp [24] O.A. Oleinik, Dicontinuou olution of non-linea diffeential equation, Upekhi Mat. Nauk (N.S.), 12 (1957), no.3 (75), pp (Am. Math. Soc. an., Se. 2, 26, pp ). [25] J. Smolle, Shock wave and eaction diffuion equation, Spinge-Velag, Belin, New Yok, [26] B. emple, Global olution of the Cauchy poblem fo a cla of 2 2 nontictly hypebolic conevation law, Adv. in Appl. Math., 3 (1982), pp [27] A. veito and R. Winthe, Exitence, uniquene and continuou dependence fo a ytem of hypebolic conevation law modelling polyme flooding, Pepint, Depatment of Infomatic, Univeity of Olo, Noway, Januay, 1990.

12 290 J. HONG AND B. EMPLE U * U * R 1 (a R S, u R S,vR S) R 1 (a R S,uR S,vR S ) ( a ~ S, u~ S,v ~ S) (a ~ S,u ~ S,v ~ S ) Figue 1 Figue 2 UL U * a=a L B A D C E a=a R Figue 3 ( a > a, U < ) R L L

13 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 291 F U L a =a L B A D C E a =a R Figue 4 (a > a,u >) R L L a=a R a=a L UL Figue 5 (a < a, U < ) R L L

14 292 J. HONG AND B. EMPLE U R F C G a =a R B A D E U L a =a L Figue 6 (a < a, U > ) R L L U L U * a=a L U R C A D E F B a=a R Figue 7 (a > a, U < ) R L L

15 RESONAN NONLINEAR SYSEMS OF CONSERVAION LAWS 293 F G UL a=a L B A D C E H a=a R U R Figue 8 (a > a, U > ) R L L U * a=a R U L a=a L Figue 9 (a < a, U < ) R L L

16 294 J. HONG AND B. EMPLE F G a=a R D E U L a=a L Figue 10 ( a < a, U > ) R L L