Riemann problems for hyperbolic systems

Transcript

1 Riemann obems o heboic ssems A Disseaion Sbmie in aia imen FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS UNDER THE ACADEMIC AUTONOMY NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA TSWATI Une he Giance o DR RAJA SEKHAR TUNGALA DEPARTMENT OF MATHEMATICS NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA 7698 ODISHA

2 CERTIFICATE D RAJA SEKHAR TUNGALA Assisan Poesso Deamen o Mahemaics NIT Roea-ODISHA This is o cei ha he isseaion enie Riemann obems o heboic ssems bein sbmie b TSai o he Deamen o mahemaics Naiona Insie o Technoo Roea Oisha o he aa o he eee o Mase o Science in mahemaics is a eco o bonaie eseach o caie o b hem ne m sevision an iance I am saisie ha he isseaion eo has eache he sana iin he eqiemens o he eaions eain o he nae o he eee Roea Dae: D Raja SehaTnaa Seviso

3 DECLARATION I heeb cei ha he o hich is bein esene in hesis enie RiemannPobems o heboic ssems in aia imen o he eqiemen o he aa o he Deee o Mase o Science sbmie in he Deamen o Mahemaics Naiona Insie o Technoo Roea is an ahenic eco o m o caie o ne he sevision o D Raja SehaTnaa The mae emboie in his has no been sbmie b me o he aa o an ohe eee (TSai This is o cei ha he above saemen mae b he caniae is caie o he bes o he Knoee DR RAJA SEKHAR TUNGALA Assisan Poesso Deamen o mahemaics Naiona Insie o Technoo Roea Oisha

4 4 Acnoeemens I eem i a iviee an hono o have oe in associaion ne D Raja seha Tnaa Assisan Poesso Deamen o mahemaics Naiona Insie o Technoo Roea I eess m ee sense o aie an inebeness o him o sesin me he obem an iin me hoho he isseaion Wos an inaeqae o eess m eeins o hanness o a he aena cae an aecion he has shon hie m o as in oess I han a ac membes o he Deamen o mahemaics ho have aas insie me o o ha an hee me ean ne conces in o sa a NIT Roea I o ie o hans o aens o hei nconiiona ove an so The have hee me in eve siaion hoho o ie I am ae o hei so I o ie o acco o sincee aie o M ibeanana bia o his vaabe sesions iance in cain o m ojec an his sincee he I aso han o C aa an Abhiman mahaana he eamen assisan an eon esecive o hei cooeaion an he Fina I o ie o han a o iens o hei so an he ea amih o shoe his bessin on s an main eams an asiaions TSai

5 5 Absac In his eo e eine heboic ssem an iven some eames We s he behavio o heboic ssem Lae e evise he eac soion o he Riemann Pobem o he non- inea PDE hich in heboic ssem o he enea om o consevaion as hich ovens oneimensiona isenoic maneoasnamics Las e in he soion sin hase ane anasis an ineacions o eemena aves beeen he same amiies as e as ieen amiies

6 6 INTRODUCTION The Riemann obem is eine as he iniia vae obem o he ssem ih o vae ieceise consan iniia aa The Riemann obem is a namena oo o sin he ineacion beeen aves I has ae a cena oe boh in he heoeica anasis o ssems o heboic consevaion as an in he eveomen an imemenaion o acica nmeica soions o sch ssems asica he Riemann obem ives he mico-ave sca o he o One can hin o he oaaion o he o as a se o sma scae Riemann obem beeen he ave aisin om hese Riemann obems

7 7 TALE OF CONTENTS CHAPTER : Inocion o heboic ssem Deiniion an Eames Heboic ssem o consevaion as Cach obem Riemann obem Wea soion 9 5 CHAPTER : Riemann Pobem o isenoic maneoasnamics Shoc an Raeacion aves Shocs Raeacion aves Riemann obem Ineacion o eemena aves Ineacion o aves om ieen amiies 44 Ineacion o aves om same ami 48 REFERENCES

8 8 Deiniions an Eames: Chae- Inocion o Heboic Ssems The enea om o ssem o consevaion as in sevea sace vaiabes j j j Hee be an oen sbse o R : R ; : R X R j hee The se is cae he se o saes an he ncions j j j ae cae ncions he ssem ( is ien in consevaion om he consevaion o he ea qaniies We have a simes ieenia eqaion moe o a i o: This eqaion is cae invisci e s eqaion hich is aso non as one- imensiona consevaion a hich is a o imensiona eqaion Fom his eqaion e e ooin ssem o o imensiona eqaions: R an e T Le D be an abia omain o n n n bona D o D Then i oo om ( ha be he oa ni noma o he j n j s D j D This is consevaion a in inea om This eqaion has a hsica meanin ha he Vaiaion o is eqa o he osses hoh he bona D D

9 9 Heboic Ssem o Consevaion Las: Fo a j e i ij j A be an Jacobian mai o ; j eqaion (is cae a heboic ssem I o an Ω an R he mai j A j j A has ea eienvaes ih Ineenen eienvecos λ A ie ae ih eienvecos λ A ae e eienvecos I A has ea eienvaes an coesonin inea ineenen eienvecos an i ea isinc eienvaes hen he ssem is cae sic heboic Eame: Le hen A Hee he eienvae is an eienveco is Eame: ( ( = + = ; v υ v λ v v v v

10 v v v I is heboic ssem I he eienvaes heboic ae a isinc The ssem ( is cae sic Cach Pobem: Le s s s be he aia ieenia eqaion ih iniia aa o he cve We have he sace hich conains he cve is cae Cach obem : R aone an hich have iniia vae : R o > an is he ncion o Whee an ae consans hen he Cach obem is cae Riemann obem 4 Riemann Pobem: The consevaion as is iven Le an be o saes o Ω R ; e have o ieceise smooh coninos ncion :( ( soions o ( ha connecion an :ih iniia coniion is cae Riemann obem Eame: The eqaion o as namics in Eeian cooinae: In Eeian cooinaes he Ee eqaions o a comessibe invisci i in he consevaion om + j= ( i + j= ( j = ( + δ = i j i ij

11 ( ( ( = + + = j j e e hei o = ensi veoci he ene seciic inena esse heseciic oaene e λi A e e e e

12 eienvae s ae - I A o λ λ -λ -λ -λ λ _

13 assme mie( - (4 ( ( ( is ih Coesonin eienveco Theeienvae assme

14 4 ] om eqaion in eqaion ( in

15 5 heboic a sic an i is ( an is eienveco s ae ( ( Theeienveco is omeqaion so assme 5 Wea soion: Chaaceisics cve in one-imensiona case: Le R : R be a C ncion The consevaion as ih iniia aa: 8 R R Hee be a smooh soion hich oos he above eqaions

16 6 C Le be smooh soion o Eqaion ( hen he non-consevaion om We ae a( ( Fom above eqaion e have non-consevaion om a( The chaaceisics cve o above coniion; i i be eine as he soion is inea cve o he ieenia eqaion a (9 Theoem (: Assme ha is a smooh o ( he chaaceisic cve ae saih ines aon hich is consan Poo: Consie a chaaceisic cve assin hoh he oin a soion o he oina ieenia eqaion is sin he Meho o chaaceisics so ih iniia vae C Aon a cve is consan i e above eqaion is sin b chain e so Hence he chaaceisic cves ae saih ines hose consan soes eens on he iniia vae C ( in inea cve

17 7 Eame: 4 a Soion: a e i e a he chaaceisiccves ae accoin oiniia aa a a a is smooh ncion Non-smooh Soion: an esecive Eisences o non-smooh soion: We consie conve case ie Le R sch ha i is eceasin ncion hen ae o cases o conve an concave Since hen ( ( imies ha So ha chaaceisics inesec ae inie ime an om non smooh soion Eame: 5 The es eqaion (invisci eqaion is ih iniia coniion i - i i Soion: sovin chaaceisic cve e e

18 8 ( In hese means he chaaceisics cve asses hoh he oin Then e have i i i i - e no ha i - i i - i - i A = he chaaceisic inesec i i No i is isconiniies ma eveo ae a inie ime i is noninea hen is smooh in i ( i (

19 9 Chae- Riemann Pobem o isenoic maneoasnamics Shoc an aeacion aves: When o o an isenoic invisci an eec concin comessibe i is sbjece o a ansvese maneic ie hen consevaion om can be ien as R ( Whee i ma eesen ensi veoci esse ansvesa maneic ie an enoe maneic emeabii esecive; an ae ncions in hich ae an hee an ae osiive consans an is he aiabaic consan hich ies in he ane o mos o he ases The ineenen vaiabes ae an hen hee

20 / ha i imies above eqaion can be ien as o smooh soions ssem ( can be ien as AU U ( hee he mai A is eine as A = an b c is he maneo-acosic see ih c is he oca son see an ( b hich is Aven see; AU U b c A hee

21 λ λi A ( b c λ λ λ λ λ λ λ ( λ λ λ -λ λ The eienvaes o A ae an Ths he ssem ( is sic heboic hen > Le an ae he ih eienvecos coesonin o he eienvaes an esecive We have j i hen he is chaaceisic ie is enine noninea Simia i can be shon ha he secon chaaceisic ie is noninea hen

22 The aves associae ih an chaaceisic ie i be eihe shoc o aeacion aves Shoc: Le an he e an ih han saes o eihe a shoc o a aeacion ave ae an enoes esecive; he ssem ( ae sin in Ranine Honis jm coniions he iven b 4 hee enoe he jm acoss a isconini cve an υ is he shoc see Lemma : Le S an S esecive enoe - shoc an -shoc associae ih an chaaceisic ies Le he saes cves sais U an U sais he Ranine-Honio jm coniions (4 an ( Then he shoc (5 Whee sch ha o e have o an on S his o an on S Poo: The -eiminaion o υ e have

23 ( ( ( ( ( ( ( ( ( ( ( ( Le ; on he ieeniain (5 ih esec o e obain - Ψ

24 4 hich is neaive o We can aea sho ha an ae osiive o an ψ ψ he ψ hisψ ; Le χ ψ ψψ so ha χ Since χ ψψ i oos o χ Hence o Ths o i e ieeniae aain e e ψ ψψ 4ψ on S Simia o an e have on aain ieeniain hen on S No hese shoc cves ae saisie he La eno coniions Lemma(: I saisies an hen he La coniion ho ie -shoc saisies Whis he -shoc saisies Poo: U U 6 U U U U 7 Le s consie -shoc cve o ove U On a -shoc e no ha an b Laane s mean vae heoem hee eis eiss a a b cb a cab a b ξ ξ ξ since sch ha

25 5 he since e have an is an inceasin ncion c ξ an hs ξ ξ c an i imies ha 8 ( c ξ ξ Aso since e have i imies ha ( This imies b ha b an heeoe

26 6 b (9 Fom eqaion (8 an (9 e have ( ( ( ( ( ( ( ( b c In (5 he above ineqai hos ha an hence U In same manne anohe coniion since an on -shoc e have η o some η an hence ( c ( Fhe since

27 7 b ( an hence om ( an ( e obain b c ( ( I imies ha - Fom eqaion ( an (5 im ha - - U υ an hence λ υ Las e sho ha U In his a he eqaion ( hich imin ha ( Fo -shoc cve sin (5 e have hich imies ha ( U Hence -shoc saisie La coniion; as e as a saisie b he a coniion o he -shoc No e i sho ha he ensi esse veoci maneic ie va acoss a shoc Ain eqaion ( o -shoc he e an ih saes have o saisies La coniions (6 Le s eine V - ; V an hence υ U i oos ha hen since Simia b sin secon coniion ie U U e e V imies ha

28 8 - υ v so - V hence V Fom ( e have V V Since an ae osiive boh V an V ms have same sin; since V e have V Fo -shoc he as see on he boh sies o shoc is eae han shoc see an heeoe he aices coss he om e o ih In case o -shoc ain La coniions (7 his imin V since ( U o eqivaen υ hich oos ha V an hencev In case o -shoc aices coss om ih o e Le he saes ahea o an behin he shoc be esinae he - sae an -sae esecive Then o -shoc an hencev W an V W ; o -shoc so V W an V W Ths o boh shocs e have V W an V W To his coniions saisie he eqaion (4 hos ha V V hich imies ha ; V V ; so c ( c ( he above ineqaiies oos ha an heeoe an an om ( e have ; V V

29 9 Since i oos ha V V Since o -shoc V an an i oos ha V V imies ha an so Simia o -shoc V an is imin hav V Raeacion aves: The U hich ae o he ieceise smooh coninos soions o ( sch ha U λ U( U( λn U λn U n U U λn U ( I e ae η hen he eqaion ( is a ssem o oina ieenia eqaions an i can be ien as A- I hee I is ieni mai an he ieeniain ih esec o he vaiabe is enoe b o I ( hen an become consans I hen hee eis a eienveco o he mai A coesonin o he eienvae Since i has o ea an isinc eienvaes so i has o amiies o he aeacion aves R an R hich ae -Raeacion aves an -Raeacion aves esecive; Le s consie -aeacion aves since A-I an ih e have

30 ( Whee -Riemann invaian is (4 eesens R cve Simia -Riemann invaian o he -Raeacion ave cve is eesen R cve (5 Theoem On R esecive R he Riemann invaian esecive ( is consan Lemma:Acoss-aeacion aves (esecive -aeacion aves an (esecive an i an on i chaaceisic see inceases om e han sae o ih han sae

31 Poo: Since e no ha c b is an inceasin ncion o ; o i can be ien as - These ineqaiies ae an - sho ha λ U λ U Simia e can ove λ U λ U o -aeacion aves Then he convese o -aeacion aves since U e have U so i om o -aeacion aves - (6 In he since -aeacion ave eion is consan hen e e - he eqaion (6 shos ha - hich imies ha - an Hence an Simia i can be shon ha he -aeacion aves an Inocin a ne aamee hee obain om (5 he ooin omas o shoc cves o -shoc cve an -shoc cve (Resecive aeacion cves in he em o aameeizaions Fom eqaion (5

32 θ θ i iminheeb hee ha imies so ha (7 θ θ A θ θ ( ( ( ( Fo -aeacion aves ( since -Riemann invaian is consan e have θ θ θ so ha i e se θ θ θ A A (8 Simia o -aeacion ave e have hen as e as e se

33 A A θ Ths o -ami eihe shoc o aeacion ave e have A θ A θ θ i θ i θ (9 In he simia a θ A θ A θ i θ i θ ( hee A an ; is as o eession in above eqaions (9 an ( Theoem : The R cve is conve an monoonic eceasin hie R cve is concave an monoonic inceasin Poo: We no ha -aeacion ave is i ( On ieeniain ih esec o e have -

34 4 ( We no ha c b since in he ooin eqaions Aain on ieeniain ih esec o e have cc bb cc bb cc bb cc bb cc bb We no ha b an c ae ieeniain e have so ha an hee i imies cc cc c bb b cc bb b c cc bb b c

35 5 o ho an heeoe is conve ih esec o-aeacion aves Simia e can sho o -aeacion aves No e ove ha he shoc cves ae saie ih esec o o an hese has a oo eome in Riemann invaian cooinaes heneve an Theoem : The -shoc an -shoc cve ae saie ih esec o hen an o vaes o in in he ane Poo: We have o ove ha an a hoh he oin be inesece -shoc cve in a amos one oin o his is sicien o ove he as hoh he o ieen oins on he -shoc cve an hose soe ae ieen The soe o he ine joinin ih is - an Fo he -shoc eqaion (5 ae imies ha

36 6 - - hee e ove ha an When in in an ieeniain ih esec o an e have an is aain ieeniain ih esec o e have I aso ove ha hen in an ieeniain ih esec o an e obain ( on ieeniain e have

37 7 β ( ( Le hen Then Since i above coniion ae oos ha he vaes o in e have The above eqaion o be he in -shoc an hen

38 an imin ha is a eceasin ncion o ; an his oos ha i heeoe Ths - is a eceasin ncion o ; e hence -shoc cve - is saie ih esec o as sa in same a -shoc cve an is aso a saie ih esec o 8 Lemma Π Π an he ineqaiies an ho aon -shoc an - Π Π Wih shoc esecive ih Π Π ( (4 Poo: Fom ( an (4 e have Π an Π We no ha he above heoem as on a -shoc cves Fhe as aon -shoc cves Π Π in sicien a Π i has ha Π Π Π e have In oe o ove ha Π Π < Fo a coniion -shoc cve eqaion (5 i im ha

39 9 hence he above coniion hos ha an hese imies ha Π Simia e sho ha Π Π aon -shoc cves 4 Riemann Pobem: The ssem ( be he iniia coniion as i U i U U (4 is cae as Riemann obem Whee U be he sae o he e o an U be he sae o he ih o he consan saes ae seaae b in boh aves eihe a shoc aves o aeacion ave The Riemann invaian cooinaes ae Π Π an Lemma (4: The main Π Π is one o one an he Jacobian o his main is nonzeo hen

40 4 Poo: Since Π an Π On ieeniain ih esec o e e Π Π Π Π Ths he Jacobian o he main Π Π Π Π Π Π This is one-one an ono We consie Riemann invaians as cooinae ssem Le s i ae a ane ΠΠ in ha ane e a he cves S S R an R hich ivie he ane can ino o isinc eions I II III an IV Le U ae e sae Fiin U an vain U Le s consie U beon o an o he o eion as i4 (a Fo U R S R n n U Π Π : Π Π Snn U Π Π : Π Π R n an T U S U R U n n n In an above ave cves he ane ivies ino a o eion To sove he Riemann obem consie he ave cve U m T hee U T U U m Um m cves o enie ha sace T o m n n U T An e have o vei ha o cves U T U m an so hese a o cves ae non-inesecin an he se o a sch in he ane Π Π in one-one ashion I U I a a veica ine Π Π in i4 (a Which i be inesecs S niqe a a oin U m The soion o Riemann obem is no obvios; e ain on consan sae ou m b a -shoc an hen om U m o he consan sae U b a -aeacion ave

41 4 Le U II eion a a veica ine Π Π in i4 (a hich is inesec R niqe a a oin U m The soion is oin om U an U m b R an o om U m o U b R I U III eion e eine he conce o invese shoc cve The invese cve enoe b S consiss o hose saes Π Π hich can be connece o he sae Π Π on he ih b S shoc in i4 (a These eesene om (5 b Fi 4(a Raeacion cves (R an R an shoc cves (S an S in he ane Π Π The above cve inesec he R niqe a oin U m Theeoe U can be connece ih U b R ae ooe bs Π I U IV eion (see i4(c (as om emma Π ons I aso eines in on S Π Π This mean ha he S an S i inesecin niqe a he oinu heeoe he soion consiss o -shoc an -shoc Ths e have shon ha set U m :Um T U sace in he ane Π in a one-one a Π m 4 coves he enie ha When he vacm sae i s no saisie he same coniion Lemma (4: I Π Π he vacm occs

42 4 Poo: Fom i4 (a Π m Π an Πm Π; i Π Π hen Π Π Π Π m m i i be m Π m Πm Which imin ha ha m Hence vacm occs Fi4(b ave cves in ane( Theoem (4: Assme ha an ha e ae iven iniia saes U an U hee o he Riemann obem o ssem ( Assme ha Π Π Then hee eiss a soion o he Riemann obem o ssem ( Moeove he soion is iven b - ave ooin b a -ave saisin an he soion is niqe in he cass o consan saes seaae b shoc aves an aeacion aves

43 4 i4(c -shoc ave an -shoc ave i4( vacm cve 5 Ineacion o Eemena aves: The ineacion o eemena aves obainin om he Riemann obem (4 ives ise o ne emein eemena aves An hen o jm isconiniies a an i as oos: U U i 5 U i U i The choice o U an U in he ems o U an an abia an R Wih he iniia aa e have o Riemann obem oca The is Riemann obem o he eemena ave ma ineac he secon Riemann obem o he eemena ave an he ime o ineacion a ome a ne Riemann obem a one imensiona Ee eqaion I ma be on o he ineacion o he eemena aves Hee e ie RS R i means ha a -aeacion aves o R o he is Riemann obem ineacs ih -shoc S o he secon Riemann obem o Then i ineacs o ne Riemann obem o via m S R In ieen amiies ae ossibe o ineacion o eemena aves an as e as he same ami ae esecive ( SS SR RR RS an (SS SS RS SR SR RS

44 44 5 Ineacion o Eemena aves om ieen Famiies: (a Coision o o shocs (S S : Le U is connecion o U b he - shoc S is a is Riemann obem an U is connece o b a -shoc S o he secon Riemann obem Fo a iven U e consie U an U in sch a a ha ien ( om (5 e have U in ohe a ha hen e have i 5(a S S coision Since see o -shoc o he secon Riemann obem is neaive S an see -shoc o he is Riemann obem is osiive S oveaess Then i shos ha o an abia saeu he sae U ies in he eion IV (in i4 (b I in sicien o ove ha o an Le ae in cona ha I in his hen Imin heeb ha

45 45 P P P P The above eqaion (5 is sic osiive hich is a conaicion Hence (5 ie he cve S U ae ies beo he cvess U heeoe U ies in he eion IV Ths an i oos ineacion ess isss SS ineacion ess in case o isae in i5 (a (b Coision o a shoc an aeacion (S R : Hee U SU an U R U eqaion (5 e have ie o a iven U Le U an U sch ha an om eqaion (5 e have om Since -shoc o he Riemann obem is osiive an - aeacion ave o he secon Riemann obem is neaive veoci i oos ha RoveaesS Since o an ivenu o an can be oos ha he cve U in he eion III sbseqen SR RS The come ess his case in i5 (b R ies beo he cve R U hence U ies i 5(b S R coision

46 46 (c Coision o o aeacion aves (R R : We consie R an R U U In an ohe a o a ivenu Le U an U U U sch ha an hen Since he aiin en o - aeacion ave has a osiive veoci (bone above in - ane an ha -aeacion ave has a neaive veoci (bone above ineacion i ae ace Since an I oos ha he cve R U ies above he cve R U ; hence U ies in he eion II an he ineacion ess II an he ineacion es is R R R R Then come ess in i5 (c i 5(c R R coision ( Coision o a aeacion ave an a shoc (R S : Hee R an S U U an ie o ivenu e choose U an U sch ha U U an Since he secon Riemann obem o -shoc see is ess han -aeacion ave o is Riemann obem o he see o aiin en in ( - ane an heeoe S eneaes R Fo an iven ha U I sho ha U I hen i o sho

47 47 (5 Since is a eceasin ncion ih esec o he is vaiabes o he cve S U an hen e i have Hence he eqaiies (5 hee im ha cves S (U ies above U ies in he eion I Ths he ineacion es isrs SR ; an is come ess in i 5( i 5( R S coision 5 Ineacion o Eemena aves om same ami: (a - shoc ave oveaes anohe -shoc ave (S S : We consie he siaion in hich U is connecion o U b a shoc o he is Riemann obem an U is connece o U b a - shoc o he secon Riemann obem In ohe siaion a iven e saeu he inemeiae sae U an he ih sae ih La coniions sais U U U U U U U an U ae chose sch ha (54 an ih La sabii coniions U U U U U U U (55 an

48 48 hee U U is he see o shoc connecion U o U an simia U U is he see o shoc connecin U o U Fom (54 an (55 e obaine U U U U ie he secon Riemann obem o -shoc oveaes he is Riemann obem o -shoc a a inie ime hen is ive ise o ne Riemann obem ih aa U an U To ove his obem We ms have o eemine he eion in hich U ies esec o U Le be caim ha U va ies in eion III so his have soion o he ne Riemann obem consiss o R an S In an moe a o sho ha o o caim: e have o ove ha S U ies in enie in he eion III; o ove his eqie o sho ha o - We consie on he conaicion ha - o Then he oo ha i e ae hen (56 Imin hee b ha Povin ha (57 hich is conaicion on as e han o ineqaiies (57 is osiive Hence R S S S ; an come ess in above siaion i5(a i 5(a S oveaes S

49 49 (b -shoc ave oveaes anohe - shoc ave (S S : Le U ies in a eion I so ha SS SR is simia o he evios case an is above siaion isae come ess i 5(b S oveaes R (C -shoc ave oveaes -Raeacion ave (R S : U is connece o U b - aeacion ave an he secon In case he Riemann obem o he Riemann o he U is connece o U b -shoc ie a ivenu Le U an he an R U o in ohe a o So e sho ha (58 U in sch a a ha S U ies beo o Le s eine F so ha F On ieeniain F ih esec o e obain F imin ha F F ie F an hence beo he cve RU o In anohe o sho ha i is sicien o U he cve S U o ; he sicien a he caim has S U ies S ies above

50 5 ; (59 Le s eine F So ha F F Le s consie ha o imin ha imin heeb ha ; P P P P o eqivaen P P P P (5 he e han sie o ineqai (5 is osiive hich eaves s ih a conaicion Hence o - imin ha F We eine a ne ncion o F A some oin ~ ~ inesece in U S an S U o ~ Since F an F i is inemeiae vae oe hee eiss a ~ beeen an sch ha ~ F b vie o monoonici Ths U S an S U is niqe eemine o he

51 5 inesecion an he come ess in i 5(b We isinishe hee cases o eenin on he vae o (a I ~ inee -shoc is ea as comae o - aeacion ave hen U III an he ineacion ess is RS RS (b I ~ inee o aves o is ami ineac he annihiae each ohe an ive ise o ave o secon ami henu ies on S U an he ineacion es is RS S (c I ~ an he ineacion es is RS SS on U IV; inee he -aeacion o he is Riemann obem is ea as comae o he -shoc o secon Riemann obem hich is sone oveaes an he aiin en o -aeacion ave a eece shoc S U U m an a connecion ne connecion consans sae U m on he e o U on he ih is oce The ansmie ave ae ineacion is he -shoc ha joins sae U on he e an U m on he ih i5(c R oveaes S ( - Raeacion ave oveaes -shoc ave (S R : Hee o a iven U e consie U an U sch ha U SU an U R U om eqaion (5 e have - an om eqaion (5 e have In he ane ( he see o aiin en o U U U ie is ess han -shoc see an heeoe he -aeacion ave om ih oveaes -shoc om e a inie ime We sho ha he cve R U ies beo he cve S U o ; o his e have o ha

52 5 G o Le s a ne ncion G o sho ha hen in his a i eine he o o an G o ho ha on - G ieeniain e have ha Imin heeb ha G G since G e have RU ies beo he cve R U o hen o o be G hich is G RU an S U inesec ino niqe a some oins ~ ~ G hen e ove ha Since he e han sie o his ineqaiies o is osiive so he concsion above So e sho ha o ; o sho ha o his a niqe oo has ~ sch ha ~ To sho a ne ncion e eine G G hee imin ha G i aes neaive as cose o zeo hen R U an U hee cases he vae o G ; an e no The cves ae inees niqe a he S i oos ha he inemeiae vae oe an in vie o monoonica; hee on eenin e isinish ie (i hen ~ an U IV ; he ineacion es is SR SS ; inee in sicien case he boh cves ae ineacion an hen he -aeacion ave is ea comae o he -shoc is sone hich is oce a ne eemena ave (ii hen ~ an U SU he ineacion es issr S ie ineacion o he is ami o eemena aves Gives ise o a secon ami o a ne eemena ave (iii hen ~ an U III he ineacion es is SR RS

53 5 Fi5( R oveaes S (e -Raeacion ave oveaes - shoc ave (S R : When U an U R U iven S U o he U e have consie U an an U S R ineacion aes ace in an anohe o in a U ae in a sch a ha om (5 e have S ies above he cve R U ie om (5 e have We sho ha o (5 We eine o sho ha M M Since hee imin b ha S U ies above R U On ieeniaion M since M e have M M since M i oos ha M e ove ha he R U ies above he cve R U o ; o sho ha o his i is enoh M o an he cve R U ies above he cve R U o M ; han sie o his ineqaiies is M hich o osiive e sho ha U niqe S U a oin ~ ~ o ~ M - o so ha M he e R o he inesec We eine an e consie a

54 54 consan K sch ha M o a K Then hee eiss a ~ sch ha M ~ Ths R U an S U ae inesec niqe a ~ ~ as R U an S U in a ems o monoone an he come ess shon in i5(c Hee hee cases ae ooin (i I ~ U IV he ineacion es issr SS inee he senh o R is sma comae o he eemena ave S an S annihiaes R in a inie ime The senh o he eece S is sma comae o he incien aves S an R (ii When ~ an U S( U he ineacion es issr S inee S is eae han R comae o he incien aves R ans (iii I ~ he ineacion ess issr SR ; inee R is sone hans ( -Shoc aves oveaes -Raeacion(R S : Fo a iven U e have U an S U o U hee U an R U U S U sch ha an We ove ha U R ies above he cve To sho ha e have a ne ncion (5 N o ; so ha N This in vie o he eession o an ies N Thee imin b ha Hence his es N N e sho ha U cve ; o sho i is sicien o his S U o S ies above he

55 55 o I N hen We consie hich is conaicion ha Ths i e have ha Thee imies b ha - ems o an ies o eqivaen ; his eession in (5 Which is conacion he above eqaion (5 is osiive o hence o e ove ha a oin ( ~ ~ 4 4 a SU an S U inesec niqe o ~ 4 Hee aain e isinish hee cases eenin on he vae o (i I ~ 4 U I he ineacion ess is R S SR inee he eemena ave R is sone comae o S he senh o eece S is sma comae o he incien aves S an R (ii I ~ 4 an U S U he ineacion es is R S S (iii I ~ 4 an U IV he ineacion es is R S SS ; inee S is sone han comae o he eemena ave R is eae ;

56 56 REFERENCES [] E Goesi PA Ravia Nmeica Aoimaion o Heboic Ssems o Consevaion Las Sine-Vea Ne Yo 996 [] T Raja Seha VD Shama Riemann Pobem an Eemena ave Ineacion in Isenoic Maneoasnamics Noninea Anasis: Rea Wo Aicaion ( [] YehaPinchoveJacob Rbinsein An Inocion o Paia Dieenia Eqaions Cambie Univesi Pess UK 5 [4] JKevoian Paia Dieenia Eqaions(Anaica Soion Techniqes Secon Eiion Sine-Vea Ne Yo