lax - hyperbolic systems of conservation laws and the mathematical theory of shock waves

8/12/2019 Lax - Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves

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Hyberbolic Systemsof onservation Laws andtheMathematical Theory ShockWaves


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Peter D LaxCourant Institute ofMathematical SciencesNew York University

Hyberbolic Systemsof onservation LawsandtheMathematical TheoryofShock Waves

SOCIETYFORINDUSTRI L ND PPLIED M THEM TICSPHIL DELPHI


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Copyright 1973 by Society for Industrial and Applied Mathematics.1098765All rights reserved. Printed in the Un ited States of America. No part of this book may bereproduced stored ortransmitted in any manner withoutthewritten permission of thepublisher. For information write to the Society for Industrial and Applied Mathematics3600University City Science C enter Philadelphia PA 19104-2688.ISBN0 89871 177 0

is aregistered trademark.


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ont nts

Preface xI n t r o d u c t i o n 1 Q u a s i l in e a r h y p e r b o l ic e q u a t io n s 12. Co nserva t ion laws 33. Single conservat ion laws 44. The decay of solutions as t tends to infini ty 1 75. Hyperbol ic sys tems of conservat ion laws 246. Pairs of conserva t ion laws 33No t e s 41References 47

Z


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PrefaceThe mathem atical theory of hyperbolic systems of conservation laws and thetheory ofshock wa ves presented in these lectures were started by Eberhardt Hopfin 1950 followed in a series of s tudiesb y Olga Oleinik th e au th o r and m a n yothers. In 1965 James Glimm introduced a number of s trikingly new ideas th e

possibilitiesofwh ich are explored.In add i t ion to the mathematical work reported here there is a great deal ofengineeringlore abo ut shock wave s; m uch o fthat l i teratureup to 1948is reportedin Supersonic FlowandShockWaves byC ourant and Fr iedrichs. Sub seque ntworkespecially in the sixties relieso n a great deal of computat ion .A series of lectures along the lines of these notes was delivered at a Reg ionalConference held at the Universi ty of California at Los Angeles in September1971 arrangedby theC onference BoardofM athem atical Sciences and sponsoredby the National Science Foundation Thenotes themselvesarebasedonlecturesdelivered at Oregon State U nive rsity in the summ er of 1970 and at Sta nfordUniversi ty summer of 1971.To all these insti tutions my thanks and mythanksalso to the A tom ic Energy Com mission for its generous support over a numberof years of my research on hyperb olic conservation laws. I express mygrati tudeto Julian Cole and V ictor Barcilon organizers of the regional conference forbringing together a very s t im ulat ing group of people.New York P T R D. LAX ecember1972


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Hyperbolic SystemsofConservation Lawsand the Mathemat ica lTheory ofShock W avesPeter D. Lax

Introduction. It is well k n o w n t ha t an in i t ia l va lue problem for a n o n l in e a ro rd ina ry differential e q u a t i o n m ay very well f i l to h a v e a s o l u t i o n for al l t im e ;the so lu t ion m ay b low up a f te r a f inite t i me . T he same is t r ue fo r q u a s i - l i nea rhyperbol i c pa r t i a ldif ferent ial equa t i ons : so l u t i ons m ay b r eak d own af ter a f initet imew hen the i rf irstde r ivat ives blow up .In these notes w e s tudy f i rst order quas i - l inea r hype rbol i c sys tems w hich com efrom c on serva tion laws. Since a conservat ion law is an i n tegra l r e la t ion i t maybe satisfied b y funct ions wh i ch are not d i f feren t i able not even con t i nuou s . me r e l ym easura ble and bo unded. W e shall ca l l these generalized solut ions in co ntra st toth e regular i.e. differentiable ones The breakdow n of a regula r so lu tion maymerely mean th at al thoug h a generalized solution exists for al l t im e i tceases tobe differentiable after a fini te t ime . All available evidence indicates th at th is isso. It t u r n so ut how ever tha t there are many general ized solut ions w i th the sameini t ia l data on ly one o f w hich has phy sical significance ; the task is to giv e acri ter ion for selecting the r ight one. A class of such cri teria is described in thes enotes; they are called entropy conditions for in the gas dynamical case theya m o u n t to r equi r ingthe increase o fen t r opy o f particles crossing a shock f ront .These lectures deal w i th the mathemat ica l side of the theo ry i.e. w ith res ultsthat can be proved r igorou sly.W e present wh atever iskno w n abou t ex i stence anduniqueness o f general ized solut ions of the in i t ia l value problem subject to theen t ropy condi t ions. W e also inve st igate the subt le d issipat ion in troduced by theen t ropy condition and show tha t itcauses a slow decay in signal strength.A s s ta ted in the preface no n um erical resul ts are presen ted; y et there is a verybrief i n t r od uc t i on to numer ica l methods in 7 of the Notes. The a p p r o x i m a t esolut ions that can be compu ted by these m ethods are not on ly enormously usefulquant i tat ively but t h e r e ishope that such methods can also be used to p rove the

existence of solut ions and to s tu dy them qua l i ta t ively .1 . Qu asi linear hyperbolic equations. A first order system of quas i - l inea requa t ions in two independent var iables is of the form

l


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2 PETER D. LAX

w h e r e u is a vec to r f u n c t i o n of x and t, A a m a t r i x f u n c t i o n of u as wel l as of xan d t. S u c ha sys tem is ca l l ed s t r i c t ly hyperbo l ic if for each x , t an d u the m a t r i xA = A x, f , u ) ha s rea l and d i s t i n c t e i g e n v a l u e si = T ^ X , f , w ) ,j 1, , n .Simi l a r ly , a q u a s i - l i n e a r s y s te m in k + 1va r i ab l e s x1 . x 2 , , xk , r ,

is ca l le d h y p e r b o l i cif f or each x , t, u a n d u n i t vec to r to ,th e m a t r i x

has real an d d i s t i n c t e i g enva l u e s T ^ X , r , u ,co), y = 1, , n.T he i n i t i a l v a l u e p rob l em f o r ( 1 . 1 ) o r (1 .2)is to f ind a s o l u t i o nu(x , r ) w i t hpresc r ibed va luesat t 0:

W e sha l l d educe n ow eas i lyf rom th e l i near theory t h a t t h i si n i t ia lva l u e p rob lemhas a t most on e so lu t ion in theclass ofC1 so lu ti on s . For let uan d v both solve 1 - 1 ) :

Sub t r ac t i ng the two equa t i on s we f ind t ha t th e d i f ference d = u vsat is f ies

A s s u m e t h a t A isC1;then \A u) A v}\ const \d \for thequas i - li n ea r equ a t i o n( 1 . 1 ) ,an d s im i l a r l yfor (1 .2 ) , It f o l l ow s t h a t d = 0 .Does the in i t i a l va lue prob lem a lways hav e a so l u t i o n? W e shal l sketch anargu me n t , based on l i n ear theory , t ha t the answ er i s yes i f the i n i t i a l va l u es a re

smooth enough . T he so lu t ion isobtained by i terat ing the '.ransform ation u = -T vde f ined as f o l l o w s : uis the so lu t ion of the l i near i n i t i a lv a l u e p rob l em

Let usassum e t h a tth e /4 ,a resym met r ic matrices I t i s n otha rd to show, using th eenergy es timates fo r l in ea r sym me t r ic hyperbo l ic sys tems an d the Sobo levinequa l i t i es , tha t th e t r ans fo rmat ion 3~ has the fo l l owingproperties:Suppose tha t u0 is ofclass Cw ,w h e re N > 1 + k /2. Def ine the n o r m \\u\\N< T by


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Since B* T isclosed in the | | ||0 T n o r m , it f o ll o w s t h a t 3~ has a u n i q u e fixed p o i n tin B*T c o n s t r u c t i b le b y i t e r a t ion . Thisf ixedp o i n t solves th e in i t ia l value p rob lemfo r th e q u a s i - l in e a r e q u a t i o n 1 . 2 ) .T h i s p r o o f s h o w s t h a t 1 . 2 ) , 1 . 3 )h as a s o l u t i o n in s o m e f in i te t i m e i n t e r v a l0 r T. E x a m p l e s p r e s e n t e d in 3 a nd 6 s h o w t h a t , in general , s m o o t hs o l u t i o n s d o n o t ex i s t beyond somef in i tet im e in te r v a l . S ince so lu t ion sa res u p p o s e dto desc r ibe th e s t a t e o f a p h y s i c a l s y s t e m , h o w i s o n e t o i n t e r p r e t th e n o n e x i s t e n c eof s o l u t i o n s in the large? W e s h a l l s h o w in the nex t sec t ion th a t fo r q u a s i - l i n e a re q u a t i o n s w h i c h c o m e f rom co ns er va t io n l aws th e r e i s a way o f de f in ing gener a l i zeds o l u t i o n s .

2 Con servation laws A c o n s e r v a t io n law a s se r t s t h a t th e ra te o f c h a n g e o f t hetotal amount o f sub s tanc e con ta ined in a f ix ed dom ain G i s equ a l to th e f lux o ft h a t s u b s t a n c e ac ross th e b o u n d a r y o f G .D e n o t in g th e density o f t h a t s u b s t a n c eb y M and the f luxby/ th e c o n s e r v a t i o n l aw i s

w h e r eeach/'iss o m en o n l i n e a r func t ion oful , u .C a r r y in go u t t h e d i f feren t ia-t i ons in 2 .3 )we ge t the f i rs t o r der quas i - l inear sys tem

W e s h a l l d e a l w i t h sys tem s o f conser va t ion l aws

D i v i d i n g by vol G) and s h r i n k i n g G to a p o i n t w h e r e a ll p a r t i a l d e r i v a t i v e s o fua n d/are c o n t i n u o u s w eo b t a i n th e d i f f e ren t i a l c o n s e r v a t i o n la w

h e r e n d e n o t e s th e o u t w a r d n o r m a l to G a nd dS th e s u r f a c e e l e m e n t on c G, theb o u n d a r y o f G, so t h a t th e i n t e g r a lo n t he r i g h t in 2 . 1 ) m e a s u r es o u t f l o w h e n c eth em i n u s s ig n . A p p l y i n g th ed iver gence th eo r em a n d t a k i n gd/dt u n d e r th ei n t e g r a ls ign w e o b t a i n

H Y P E R B O L I C S Y S T E M S O F C O N S E R V A T I O N L A W S 3 i ) F o r R l a r g e e n o u g h a n d T s m a l l e n o u g h T m a p s th e ba l l o fr a d i u s R:

i n t o i tself . i i ) J7 is a c o n t r a c t i o n of B* T w i t h r e spec t to th e || 0T n o r m :


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P E T E R D . L A XI n t r o d u c i n g the vec to r and m a t r i x n o t a t i o n

Sincethe j a r e non l i ne a r f u n c t i o n s o f u , the m a t r i ce s Ai as def in ed by 2.5) a ref u n c t i o n s o fu.W e a s s u m e t h a t the q u a s i - l i n e a r s y s t em 2 .6 )is s t r i c t lyh y p e r b o l i c .u is called a generalized solution of the system of conservation laws 2.3) itsatisfies the integral form of these laws,i.e., if

holds for every smoothly bounded domain and for every time interval t l , t 2 ) . Thisis equivalent to requiring 2.3)to hold in the sense of distribution theory.Let S t) be a s m o o t h s u r f a c e m o v i n g w i t h r u a c o n t i n u o u s l y d i f fe ren t i ab lesolut ion o f 2.3)on ei ther sideo f Swh i ch isd i s c o n t in u o u sacrossS;the con d i t ionw h i c h m u s tbe sa t i s f i eda t each po in t o fS ifu is a ge ne r a l ize d so lu t i o n a c ro s s S is

we can w r i t e 3 .1) in the f o rm

w e can w r i t e 2.4)in the f o rm

Here [u ] and [ / ] d e n o t e the d i f fe rence b e t w e e n v a lu e s o f u and / r e s p e c t i v e l y o nthe two s ides of5;n i s the normal to S and s the speed w i t h w h ic h S p r opa ga t e sin the d i r e c t i onn .Re la t ion 2 .8)isca l led the R a n k i n e - H u g o n i o t j u m p c o n d i ti o n ;w e shal l prove it for the o n e - d i m e n s i o n a l case in 3.W e leaven o w these f o r m a l c o n s i d e r a ti o n s an d t u r n to s o l v i n gth e i n i t i a l va luepr ob le m w i t h i n the class o ft h e se ge ne r a li ze d so lu t i on s .

3 Single conservation laws A s i ng l ec onservation law is an e q u a t i o n of the fo rm

where is s o m e n o n l i n e a r f u n c t i o n o fu .D e n o t i n g


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HYPERBOLIC SYSTEMS OF CONSERVATION L A W S 5which asserts that u i s con stant a long trajector ies x = x t) which propagate wi thspeed a:

For this reason a is called the signal speed, the trajector ies , satisfying 34) , arecalled characteristics. No t e t h a tif is a nonl inear func t ion of u , both s ignal speedand c harac te r i s t ic depend on the so lu t ion u .The constancy of u along character ist ics combined with 3.4) shows t ha t th echaracter is t ics propagate wi th cons tan t speed ; so t hey are straight l ines. Thisleadsto th efo l lowinggeo metr ic solu t ionof thein i t ia lv a lue problem u x,0) = u0 x ) :Draw s t ra igh t l ines i ssu ingfrom po in ts yof the x-axis, w ith speed uQ y) see Fig. 1).

F I G l

As wesha ll show ,ifu0 is a C 1 func t ion , these straight l ines simply cover a neigh-borhoodof the x-ax is. Since th eva lueofualong th e line issuing from thepoint yis u0 y), u x, t)isu n i q u e l y determined near th e x-axis.

An analyt ical formo fthis constru ct ion isshown inFig. 2. Let x,t)be any point,ythe intersect ion of the cha racter is t ic thro u gh x,tw ith the x-axis.Then u = u x, t)satisfies


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PETER D . L A XAssu m e u 0 d i f f e r e n t i a b l e ; t h e n , a c c o r d i n g to the i m p l i c i t f u n c t i o n theor em , (3 .5)can be so lved for u as a d i f f e r e n t i a b l e fun c t ion o f x and f fo r fsm a l l e no u g h , an d

Then i f u Q is^ for a l l x , u, a n d u x a s g iven b y f o r m u l a s ( 3.6 ) r e m a i n b o u n d e d fora l l t > ; on the o the r h a n d , if u 0 is 0. I n the second case t h e r e a re t w o - p o i n t s

} 'j a n d y2 such that y\ < v2, an d Uj = u0(yl) > l;2) 2^ tnen by (3-7) alsot f t = a W j ) > a (u 2)= 2 so t h a t t h e c h a r ac t e r i s t i c s i s su i ng f rom t h e se po i n t si n t e r sec t a t t i m e t = (y 2 y l)/(a 1 a 2 ) (see Fig. 3). At the p o i n t of in t e r sec t ion ,u h a s t o t a k e o n b o t h v a l u e su a n d u2 ,a ni m p o s s ib i l i t y .

Both th e geomet r i c a n d t h e a n a l y t i c a r g u m e n t p r o v e b e y o n d the s h a d o w o f ad o u b t t h a t if a(u 0(x}) is no t an i nc r e a s i ng fu nc t i o n of x , t h e n n o function u(x, t)exists for all t > 0 with initial value u0 which solves (3.3) in the ordinary sense\W e s a w h o w e v e r in 2 t h a t b o u n d e d , m e a s u r a b l e f u n c t i o n su w h i c h satisfy(3.1)in the sense ofd i s t r ib u t i o n s c a n b e r e g a rde d a s sa t i s fy i ng the i n t e g r a l f o rmof thec o nse rv a t i o n l aw o fw h i c h (3.1) is the d i f f e r e n t i a l f o r m .W e t u r n now to thes t u d yof su c h d i s t r i bu t i o n so l u t i o n s , s t a r t i ng w i t h the s i m p l e s t k i ndt h o se sa t i s fy i ng

6

S u b s t i t u t i n g (3 .6) in to (3 .3)we see i m m e d i a t e l y t h a t u de f i n e d by (3.5)sat isf ies(3.3).Let us assum e t ha t (3 .3 )i sg e n u i n e l y n o n l i n e a r , i .e ., t h a ta u 0 fo r a l l u , say

F I G . 3


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3 . 1 ) in the o rd ina ry sense on each sideof a s m o o t h c u r v ex y t) acrossw h i ch uis d i s c o n t i n u o u s .W e sha l l denote b y u ,and ur the v a l u e so f u on the l e f t and r i gh tsides, respec t ive ly ,o f x y . Choose a and b so t h a t the cu rve y in t ersec t s the

F I G . 4i n t e rva l a yobta in ing after car ry ing out the in tegra t ion tha t

w h e r e w ehav e used the abb rev i a t i on

w eh a v e

H Y P E R B O L I C S Y S T E M S O F C O N S E R V A T I O N L A W S 7


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PETER D. LAXCombining th i swi th the above relat ionw ededuce thejump condition

w h e r e u ] = u r u, and[ ] = fr ft denote th e j u m p in u and in across yW e show now in an example tha t pre vious ly unsolvableini t ial va lue p rob lemscan be solved for all t wi th the aid ofd i scon t inuous so lu t i ons . Take

The geome tric solution see Fig. 5) is single-valued for t 1 but double-valued

thereaf ter . N o w w edefine for t 1,

The discont inui ty s tar ts at 1.1); it separates the state u t = 1 on the left from thestate u r = 0 on the r i g h t ; th e speed ofpropagat ion w as chosen according to thejump condi t ion 3.10), w i t h / u )=\u :

In t roduc ing generalized solutions makes it possible to solve in it ial va lueproblems which could not be solved within the class of genuine solutions. At thesame t ime i t threatens wi th the danger tha t the enlarged class of solutio ns is solarge that there are several generalized solutions with the same init ial data. Thefol lowing example shows that this anxiety is well f o u n d e d :

8

FIG. 5


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HYP ERBOLIC S YS TEM S O F C O N S E R V A T I O N L A W S 9The geom etric solutio n see Fig.6) issingle-valued fo r t > 0 butdoes notdeter

mine the valueofu in the wedge 0 < x


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PETER D. LAXWe shall f irst treat the case when condition 3.7)is satisfied, i.e.,a u) is aninc reas ing f unc t i on of u . Clea r ly th i s is so whe ne ve rf u) is a convex func t ion ofu

F I G . 7

seeFig.7 ). Such a func t ion l i e s above all its t a n g e n t s :

Let u be a ge nu ine i.e., c o n t i n u o u s and d i f f e ren t iab l e ) so lu t ion of 3 . 1 ) , andsuppose t ha t w0 x ) = w x ,0 ) i s 0 fo r x large e nou gh n e g a t i v e ; then the same ist r u e o f u x ,t) for any t > 0 for which u i s de f ined . We in t roduce the in tegra tedf unc t ion U x, t)defined as fo l lows :

Denote by y the po in t where the l ine dx/dt a v) t h r o u g h x , t intersects thex - a x i s ; clearly,

In tegra t ing 3.19)alongthis line from0 to we obtain, for 0,

t h e n

Apply ing inequ a l i ty 3 .14 ) w i th U = uand any n u m b e r v w e ob ta in t h a t

w h e r e w ehaveadjusted/so t h a t

In tegra t ing 3 .1) f rom oo to x and us ing 3 .16)w e ob t a in


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Suppose u x,t) is a generalized solution. Relation 3.17) is the integral form ofth e conservation law 3.1),so it follows that when/isconvex, inequality 3.24) isvalid fo r generalized solutions as wel l .

alldiscontinuities of the generalized solution u are shocks, then every point x,t)can be connected to a pointyon the in i t ia l l ineby a backward characteristic.

ana

Here b is the inverse function of a , and g isdefined by

where y = y x, t) isthat value which minimizes

Thisinequality holds for allchoices of y; for that va l u e of y forwhich v,given by 3.22),equalsu x, r) ,thesignofequal i tyholdsin 3.19)alongth ewhole characteristicdx/dt = uissuing from x,f ) ;therefore equalityalsoholds in 3.24).Wesummarizethisresult asfollows.

TH EO R EM 3.1. Let u be agenuine solution of 3.1); then

Clearly, sinceaand bare inverse functions,

Denote a 0)by c;then b c)= 0 and in viewof the normalization 3.18), wehavefrom 3.23) that g c)= 0.Introducingth e func t ion g on theright side of 3 .21) w eobtain

Denote by g the func t ion

H Y P E R B O L I C SYSTEMS O F C O N S E R V A T I O N L A W S Denote by b th e inverseof the funct ion a; from 3.20)w eobtain


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12 PETER D. LAX

For t h a t v a l u e of y the sign of equ a l i ty ho lds in (3.24); thus Theorem 3 .1 appliesalso to generalized solu t ions o f (3.1) whose d i s con t i nu i t i e s a re shocks . W e showt h a t a lso t he converse ho lds , thereby proving the ex i s tence of so lu t ions , wi thshocks, wi t h a rb i t ra r i ly prescribed in tegra b le ini t ial d a t a .

T H E O R E M 3.2. Formula (3.25H3.26) defines a poss ibly d iscon tinuous functionw(x, t)for arbitrary integrable init ial values u 0 ( x ) ; th efunction u so de f ined sat is f ies(3.1) in the s en se ofd is tributions, and the d iscontinuties ofu are shocks.Proof, (i) If UQ is i n tegrab le , L J is bounded . As may be seen from (3.27), g is aconvex func t i on wh ich ach i eves its m i n im u m a t z c; therefore, the f u n c t i o n Gdefined by (3.26) achieves its m in im u m in y a t some poin t o r po in t s .

L E M M A 3.3. For f ixed t, denote by y x ) any value of y where G (x , y ) achieves it smin imum. Then y(x) is a nondecreasing function o f x .Proof. W e have to show that fo r x2 >x G (x 2 ,y ) doesno t take on i ts m i ni m u mfo r y


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and similarlywe see that

which shows that at points o f d i scont inui ty ur oo weobtain in the limit relation 3.1)for u. iii) Since Lemma 3.3, y x , t is an increasing function of x, and since b is anincreasing function,weh a v e fo rxt


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14 PETER D. LAX hen

is adecreasing function o f t where th e norm is the L norm with respect to the xvariable.C OR OL L AR Y.Ifu = v at t = 0,u = vfor all t > 0. Thisis theuniqueness theoremw ewere looking for.)

Proof. We can write th eL, norm of u v as

wherethepointsare so chosen that

fo r yn < x


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H Y P E R B O L I C SYSTEMS O F C O N SE R V T I O N L W S 15

Su bsti tuting this into (3.35),weobtain at the endpointyn 1 wi th u= u;,

Since/is a convex function and since by (3.36) v lies in the in terval ( u r , u , ) , itfollows from Jensen s in equ al i ty that the right side of (3.37) is negative. Sim ilarly,also the cont r ibut ion at the lower endpoint to (3.35) i s negative. This shows that d/dt)\\u v\\is always f S O sothatas tincreases \\u v \ \decreases.Thiscompletesthe proof of the theorem.

The case when y is a disco ntinu ity for both u and v can be treated similarly.In th e der ivat ion o f Theorem 3 .4 we can omi t th e r equi rement that/be convexif w e replace th e entropy condition (3.13) by the fo l lowing:

(i) If u r


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is not adecreasing function oft.The uniqueness theorem statedabove is not very interesting unlesswe canshowthat the general ized entropy condit ion is not too restr ict ive, i.e., that everyini t ia lvalue problem u x ,0 ) = u0 x ) has a d is t r ibut ion so lu t ion u of which the dis-continuit ies satisfy th e generalized entropy condit ion. Such a solut ion u can beconstructed as the l imi to fsolut ions U A of the parabol ic equat ion

It follows easily from them ax im um principle that 3.39)has at most oneso lu t ion ;it is also true that a solut ion U A exists for all r and that as A - 0 these solutionsconverge in theL jsense to al imitu .W e sha ll no t presen t these proo fs, b ut w e shallshow thatthis l imit is a distr ibu tion solution of 3.1)which satisfies the generalizedentropy condit ion. To show the tru th of the first statement we m ult ip ly 3.39) bya Q test function < > and integrateby p a r t s ;we obta in

16 PETER D. LAXA d i scon t inu i tyinu iscalledashocki furand u satisfyo ne o f theent rop y condi t ions3.38).The proof ofTheorem 3 .4yields the fo l lowing more general result .

T H E O R E M 3.5. Let f be any lfunction u and v two distribution solutions o/ 3.1)of whicha lld iscontinuities are shocks. Then

is a decreasing function oft.It followsin par t icular that tw o such solutions whicha reequalat t = 0areequalfor all t.The proof ofTheorem 3.5a lso d em onst ra testhe fo l lowing converse.C O R O L L A R Y . Ifu is adistribution solutiono/ 3.1)o f whichone of thed iscontinuities

fails to satisfy the entropy condition 3.38),then thereis a genu ine solution v such that

Let A ->0;th e l e f t side converges to

th e right side converges to 0. This proves thatu is a dis t r ibut ion solut ion of 3.1) .To prove that u satisfies the entropy condition, we show first that for any twosolut ions U A and v^ of 3.39),


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H Y P E R B O L I C S Y S T EM SO F C O N S E R V A T I O N L A W S 17isadecreas ing f u n c t i o n of / . To seet h i sw ew r i t e

O n a ccoun t o f 3 .40) th e second t e rm on t he r i gh t isz e r o ;s u b s t i t u t i ng f rom 3.39)in t o t h e f i r s t term and ca r ry ingo u t t h e i n t e g r a t i o n w eo b t a i n

Again th e second s um on t her igh tiszero,o n a ccou n t o f 3.40).W ec la im th a t eachte rm in the f i rs t sum i s nonpos i t i ve ; f o r I T ^ w ^ uj = p is nonnega t ive inth e in te rva l and by 3.40) zero a t the e n d p o i n t s ; so px is nonnega t ive a t the leftendpoin t , nonpo s i t ive a t the righ t endp o in t . Th is com ple tes th e proof th a t

isa decreasing f unc t i on o f /. Ifu is the L^ l i m i to f u - and vth e Lv l im i t o fi^,also||w 0 0 0 1 1 is decreasing.I t is nothard to show tha t every genuine so lu t io n vo f 3.1) is the l imit as X-* 0o f the s o l u t i ons v of the pa rabol ic equa t ion . There fore it f o l l ows t h a t if u is adis t r ibu t ion so lu t ion o f 3 .1) w hich is an Ll l imi t of s o l u t i o n s U A o f 3.39) and ifv is any genu ine s o l u t i on o f 3.1), t hen \\u t) 0 0 1 1 is a decreasing f un c t i on o f t .According to the coroUary to Theorem 3.5 it f o l low s then thatalldiscontinuitiesof usatisfy the general ized entropy condi t ion,as asserted.Being r id o f the convex i ty cond i t i ons m akes it possible to ex tend these no t ionsand the existence and un iquenes s t heo rems to single conservation laws in anyn u m b e r o fspace variables.4 The decay of solutions as tends to infinity Suppose/ w)is a convex f unc t i on ;thenTheorem 3 .2givesan expl ic i t express ion for the s o l u t i ons u o f 3.1)in te rmso f their ini t ia ldata:

wherey min imizes

where u ^ v ^ changessign at th e p o i n t syn Sinceso lu t io ns of 3 .39) are c o n t in u o u s ,

Dif feren t ia te \\u^ uj|:


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com bining this w ith 4.8) we obtain

Suppose that the initial va lue u0 x) is zero outside the interval A,A); thenU Q(y ) is zero for x 0. We fur ther assume that b liesbetween tw o positive constants for allu:

It follows from th isand 4.3) tha tthen

Suppose the initial value u0 ofu is inL t ; then for every y, U 0(y ) = Jy _ ^u0 dxis bounded in absolute valueby | | M O | | = M. U sing 4.6)we seethat for ally,

G x,x + ct,t) U0x + cr ) is 5 S M ; th is shows that G x,y , t ) M at the mini-mizingpoint.Combiningthis w ith 4.7)we see that

It fol lowsf rom 4.5)and b(c) = 0 tha t

Thus by 4.1) ,


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H Y P E R B O L I C S Y S T E M S O F C O N S E R V A T IO N L A W S 19If x < ct const , ^/t A y lies in the in te rva l y oo , where \\ \\ is the L\ norm an d

W e sha l l not p resen t a proof o f th i s theorem but we shall present a ver i f icat ionof one of its consequences.W ein t ro duc e th e f o l lowing abb rev ia t io n s :

In terms of these (4.12) can be w ri t ten as


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In w o r d s: / _ and I + are time invarian t functionals of solutions.W e shal l presen t n ow a direct proof for the in var iance o f / _ .Let u be some solution of (4.11), possibly with shocks ; denote by M ( r) :

here we have used the fact t ha t u ( oo, t) = 0 and that/(0) = 0. Taking y to beX i) y y h) ^ 2 respectively, a n d u s i n g the definit ion o f 3 as m in i m u m , w eobtainthe inequalit ies

and by y ( f ) any of the va lues y whe re th e i nd icated minimum is t aken on . Ouraim is to show thatM i s independent of t.According to the in tegral form of the conservat ion law, for any f t an d t2 an danyy,

which im ply that

where

here/(y,t)abbreviates f u y, t)).L E M M A . y t), t is apoint of continuity o/u,and

Proof. By the min imizing proper ty of y we m u s t have

It follows then from Theorem 4.1 t ha t \ \ u t +T\p,q v t,p,q)\\ -> 0 ; app ly ing(4.12) to u (x , t + T) in place of u (x , t)we conclude that for any T

20 PETER D. LAXIt follows easily from th e def ini t ion o f v in (4.10) that, for any T,


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H Y P E R B O L I C S Y S T E M S O F C O N S ER V A T I O N L A W S 21Since/was assumed convex, the ent ropy condit ion is t ha t u, > ur so i t f o l l o wsfrom (4.17) t ha t y t) c anno t be a po in t of d i scon t inu i t y , and tha t u(y, / ) = 0 .Since the set of m inim izing po ints y t), ti s c losed, it fo l low s that in any compactport ion of the (x, / ) -plane, y t), t has a posit ive distance f rom any shock of s t rength\u r u , \ = . F rom this it fo l lows t h a t as t2-* i < th e oscil lat ion o f u y^ t) andof u y2, t) over t 1 , t 2 )tends to 0. According to (4.16), u y{, t { )=u y2, t2 ) = 0, sou y l , t ) a n d u y2,t ) tend to 0 u n i fo rml y o n ( f i , / 2 ) s ~ + i - S i n c e / ( O )= 0 , i tfo l lows that l ikewiseF,the m a x i m u m of/over this interva l , tends to 0 as t2-* ? i -But then w e deduce from (4.12) t ha t

It is a consequence of the in tegral form of the co nserva t ion law t ha t for so lu t ions uwhich are zero at x = 0 0 , I0u) i s independent of / ; t hus the invar iance of 1 +fo l l ows from t h a t o f / _ a n d/0.T he q u a n t i t y 7 0 i s a na tura l inva r ian t bu i l t in to the conserva tion l a w ; it isremarkable that there exist other, unnatural invariants .T H E O R E M 4.2. Equation ( 3 .1 ) has exactly 2 independent invariant functionalcontinuous in theLt topology.Proof. L et/ be any i n v a r i a n t func t iona l con t inuous in the Ll t o p o l o g y ; byTheorem 4 .1 ,

A s r emarked in 3, any d i f f e r en t i a te so lu t ion u o f (4 .11) is cons tant a longcharacterist ics

which proves the con stancy o f M and shows the invariance of the func t iona l / _ .T he i nva r i ance o f / + fo l lows s i m i l a r l y ; a l t e rna t ive ly w e observe that the m i n i m u ma n d m a x i m u m in (4.13) occur for the same value of y; this imp l ies that

The va l ue of the right side is de te rmined by the va lues of p and q which in t u r nare de te rmined by / _ ( u ) and I + u). Therefore, th e v a l ue of I u) is a f unc t ion of thevalues of 1+ u) and I- u), as asserted.Using th e explicit formula (4.1) w ededuced, in (4.9) and Theorem 4.1, thatsolut ions whose in i t ia l va lues l ie in Li decay to 0 as A -> oo . W e shal l presentnow ano the r me thod for s t udy ing th e behavior o f s o l u t ions as t -> oo, one t ha tdoes not rely on the exp l i c t f o rmu l a ( 4 .1 ) . W i t h the aid of this method we canshow that , as t - oo, solu t ions w hose ini t ia l values are per iodic tend u n i fo rml yto the i r m ean v a lue U Q:


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P E T E R D L A X

Suppose there is a shock y p resent in u b e tween the character is t icsxl and xs e e F ig. 10) . Since acco rding to ( 3 .13) cha rac te r is t ics on e i ther s ide o f a shock

F I G . 1 0

run into the shock, there exist a t any given t ime T, two character is t ics yl and y2which intersect the shock y a t exact ly t im e T . As sum ing tha t there a re no o thershocks present w e conc lude tha t the inc reas ing va r ia t ion o f u on ( x j r ) , y , f ) ) , aswel l as on x2 t),y2 t)), i s independe nt of t. According to condit ion ( 3 .13 ) , a u)decreases across shocks, so the increasing variation of a u ] a long [x 1 ( T ) , x 2 ( T ) ]equals the sum of the increasing varia t ions o f a u ] a long [ X j 0 ) , > i(0)] and a longL y 2 (0 ) , x 2(0 ) ] .This sum is in general less than th e increasing varia t ion of u a long[x j (0 ) ,x 2(0 ) ] ; t h e re f o re w e conc lude t ha t if shocks a represent,th e tota l increa singvar iat ion of a u) between two characterist ics decreases with t ime.

L et x t ( r ) and x2 t) be a pa ir o f character is t ics , 0 5 S t T. Then there is a w h o l eone -pa ra me te r f ami ly o f cha rac te r is t ics connec t ing the po in t s of the i n t e rva l[X j (0 ) , x 2(0 ) ] , t = 0 wi th poin ts of the in te rva l [x^T , x^T ], / = T; since u isconstant a long these character is t ics , u x,0 ) on the f irst interval a nd u (x , T) on thesecond interval a re equivar iant . More gene ra l ly , i f a and T are noncharac te r is t iccurves each connec t ing x, to x 2 ,u a long a and T are eq uiv ar ian t . S ince equ ivar ia n tfunc t ions have the same to ta l increas ing and decreasing varia t ions, w e conc lud et h a t the tota l increasing a nddecrea sing va ria t ions of a dif ferentia te solution betweena ny pair of characterist ics a reconserved.Denote byD t) t he w id th of the s t r ip bounded by xt and x 2 :

Differen t ia t ing (4 .20) with respect to t and u s ing (4 .19) w e ob ta in

In t eg ra t ing w i threspect to / w e obtain


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HYPERBOLIC SYSTEMS OF C ONS ER VATION LAWS 23W e give now a qu an titat ive estimate of this decrease. Let 7 0 be any interv al ofth e x -axis ; we subdiv ide it in to subintervals [ ) > _ / _ i , . y , - ] , j 1 , , n, so t ha tu(x, 0) is alternately increasing and decreasing on these intervals (w e assume fo rsimplicity t ha t u0 is p iecewise monotonic) . W e denote by yj t) the characterist icissuing f rom the;th po in t y , , with the unde r s t and ing tha t i f y / f ) r uns in to a shock,

y j t ) is cont inued as that shock.It is easy to s how tha t for any t > 0, u(x, t )i s a l te rnate ly increasing and decreasingon the intervals j; ^ r ) , yj t )) . Since across shocks u decreases, the to tal increasingvariat ionA+T) of a u) across th e i n te rva l 1 T) = [y 0 T),y n T) ] is

S u m m i n g over / odd and u s ing (4.23) w e obtain

Since th e i n te rva l s [ x _ , . _ t ( T ) , X j T ) ] = [ y j _ ^T),y }T) ] are d is joint and lie in / ( T ) ,the sum of their lengths cannot exceed the l ength L T) of 7(0- So we deduce tha t

Suppose th e ini t ial va lue of u is periodic, with period p . Then by un iqueness i tfol lows t ha t u is periodic for all p .Take L0 to be of length p\ then L T) also haslength p and we deduce f rom (4.24) t ha t A+ T ) p/T. Since th e increasingvaria t ion of a periodic function p er period is twice its total variation, we haveproved the fo l lowing t heorem.T H E O R E M 4.3. F or every space periodic solution u of 4 .11 ),

Relat ion (4.25) shows that the to ta l var ia t ion of a u) p er period tends to zeroas T tends to oo . Since th e mean va lue u 0o f any periodic solution of a con servat ionla w is i ndependent of t ,it fol lows t ha t u(x, T) t ends t o u 0un i formly as T tends to oo.Suppose | f l ( w 0 ) l = h T ^ 0; it follows f rom (4.25) t ha t the total variation o f u(T)per p eriod is ^2p/hT; so we conclude that for T l a rge enough,

where u }_ t (T ) deno te s th e value of u on the r ight edge of y ^_ t ( T ) , U j T ) denotesth e va lue of u on the left edge of y }{T) ; in case that y ^_^T)a nd yj T) are the same,the y th t e rm in (4.23) is zero. Suppose, y j_ 1 T ) > ; _ { T ) , and y^T) respect ively. The va lue o f u a long x ;(r) is U j T).Denote xj t) X j _ j ( ? ) by D- t ) ; according to (4.22)


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where the i th row of the m a t r i xA -is thegrad ien t of w i th respect to u .W e as sum et h a t the sys tem 5 .2)ish yp erb ol ic , i .e ., th at fo r each va lue of u th e m a t r i xA hasn real , d is t in ct e igen valu es/ I , , , /., labeled in increasing order. Since A dependson u , so do the e ige nva lue s/ .k and the co r r e s pond ing r igh t and l e f t e igenvec torsrk and lkG e nu ine no n l in e a r i t y p l ayed an impor t an t r o l e fo r s ingle conserva t ion laws ;th is was the r e qu i r e me n t t h a t A be a n o n c o n s t a n t func t ion of u, i.e., t h a t A u 0 .T he ana logous cond i t i on for sys tem is not m erely that grad uAk be nonzero , bu tthat it be not or thogonal to r f c , the corresponding eigenvector . I f this is so, wecall the /cthfield genuinely nonlinear and normal ize rk so that

For systems w e requ i re tha t fo r some index k I k n ,

m u s t ho ldacross e ve ry d i s con t inu i ty , w h e re s is the speed ofp ropagat ion of thed iscon t inu i ty .Ne xt w e fo rmu la t e an entropy condition for systems. For single con vex orconcave) equa t ions th i s condi t ion requ i res tha t the characteristics o n either s ideof a discont inui ty ru n into th e line of discont inui ty, which is the case if the

characterist ic speed on the l e f t isgreater, on the r ight less, thans:

I f on the o ther hand rk grad A k= 0, we call th e / c t h charac te r i s t ic field linearlydegenerate.W et u r nnow to thes t u d yofp iecewise con t inu ou s so lu t ionso f 5 . 1 ) in thein tegra lsense ; each of the n conserva t ion laws mus t satisfy the R a n k i n e -H u g o n io t j u m pcond i t ion ,i.e.,

24 PETER D. LAXCompar ison of 4.26) w ith 4.9)shows t h a tperiodic solut ions decayfaster t hanso lu t ions whose in i t ia l va lu es a re in tegrable .The es t im ate 4 .26) is , in co ntra s t to 4 .9) , abso lute , inasm u ch as the r ig ht s ide

is inde pe nde n t of the a m p l i t u d eof the s o l u t ion .5 Hyperbol ic systemso f conservation l aws I nt his section w eshal l s tudy sys tems

ofcon serva t ion laws ,

wh e reeach ; i s a f unc t ion o f u t , , u n ; w e shal l denote th e co lumn vec torformed byul , u n by u . C a r r y i n g ou t the d i f fe ren t i a t ion in 5 .1 )w e ob t a inthe qu as i - linear sys tem


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These ineq ual i t ie s asser t that k character is t ics im pinge on the l ine of d isc on t inu i tyfrom th e left and n k + 1 f rom th e r ight , a to ta l of n + 1. Th i s in fo rmat ioncarried by these characterist ics plus the n 1 re la tions ob ta ined f rom (5.4) aftere l imina t ing s a re sufficient to determine the In va lues which u t akes on on bothsides of the l ine of d i scon t inu i ty .

A d i s c o n t i n u i t yacross w hic h (5.4) and (5.5) are satisfied is called a k-shock.W e give now a description of all weak /c-shocks, i.e.,thosew h e r e ur and ut differl i t t l e ; it is unders tood tha t ul is to the left o f ur.T H E O R E M 5.1. The set of states ur near u ,which are connected to some given stateu , through a k-shock form a smooth on e-parameter family ur = u(e), e < 0,u(0) = u ; the shock speed s also is a smooth function ofs.W e shal l omit th e p roof o f th is resul t b u t shal l ca lcula te the first two de r iva t ivesof U E ) a t e = 0. Dif fe rent ia t ing the j ump re la t ion

whileH Y P E R B O L I C S Y S T E M S O F C O N S E R V A T I O N L A W S 5

we obta in , us ing the sym bol = d/ds,

A t e = 0 we have [u] = 0, so there

wh ic h can be satisfied with u 0 o n ly if s(0) is an e igenvalue o f A:

and ti(0) a n e igenvector :

B y r eparame tr izing w e can m a k e a = 1 . D if ferent ia t ing (5 .6) once m ore and set t inge = 0 we obta in (om it t ing the subscript k

Subst i tu t ing (5 .7) and (5 .8) , with a= 1 w e obtain

T o determine s and uw e t u r n to the e igenvalue relation

restricted to u = u(e) and d i f ferent ia te w i threspect to e :

Subtractthis f rom (5.9):


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where A = A k is one of the eigenvalues of ; h is called a k rarefaction wave.Set t inga = 1 in 5.14)w eobtain

Us ing relat ion 5.3)w e obta in , after dif ferent ia t ing the first relat ion in 5.14) andusing thesecond, t h a t

This issatisfied by

which is the s ame as

Let usdenote x /t by and differentiat ion w i th respect to by ;s u b s t i tu t i n g 5.13)in to 5.2)w e obtain

is satisfied for e < 0 and not forE> 0; t h a t is why inTheorem 5.1 th e pa rame t e r eis restricted to e 0.Next we tur n to an im po rtan t class of con t inu ou s solut ions , centered rarefa ct ionw a v e s ;theseare solut io ns which depend only on the rat io x x0)/ r f 0 ,x 0,r0being the center of the wave.L et u be a rarefa ct ion w ave centered at the o r igi n:

Thus the en t rop y condi t ion 5 .5)

I t follows f rom 5.11) tha t , no dulo t e rms 0 e2) ,

Equat ion 5 .10)has the solut ion u r = f ir; by a change of pa rame t e r e we canaccompl ish tha t / ?=0;so

Sinceby 5.3),1= ugrad = rgrad / I = 1 , weobta in

26 PETER D LAX

Taking the scalar product wi th the l e f t e igenvecto r /be lo n g i n gto / I w e obta in


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H Y P E R B O L I C SYSTEMS OF CONSERV TION L W S 27Abbreviate A u , ) by A; the d i f feren t ia l equa t ion (5 .14 ) has a un ique so lu t ionsat isfying the in i t ia l condi t ion

h is de f ined for all ^ c lose enough to LLet e be a n u m b e r ^0 so smal l tha t h is de f ined at A + e ; denote by u r thevalue

W e construct now the f o l low ing p iecewise smooth fun c t ion u(x , f ) , de f ined fort 0(see Fig. 1 1 ) :

F I G . 1 1

This f unc t ion u satisf ies the dif fere n tial equ ation (5.2) in each of the three regions,and i s c o n t i n u o u s across the l i nes separating th e regions. W e shall say t ha t in uthe s tates u{ and u r are connected by a centred k-rarefaction wave.T H E O R E M 5.2. Given a s tate u there is a one-param eter family of s tates u r = u(e),0 E 0 which can be connected to u , through a centered k-rarefaction wave.Theorems 5 .1 and 5.2 can be c ombin ed .

T H E O R E M 5.3. Given a s tate u , , i t can be connected to a one-parameter family ofstates u r = u(e), e0 < e < e0 on the right of M through a centered k-wave i.e.ei ther a k-shock or a k-rarefaction wave; w e ) is twice continuously differentiatewith respect to e.The on ly part that needs proof is the con t inu i ty o f du/de an d d 2u/de2 a t e = 0.From(5.8) and(5.14) we see, sin ce a = 1 in bo th cases t h a tdu/ds = r (u , ) fo re = 0;to show t h a t d 2u/de 2 is con t inuous a t = 0 w e d i f feren t ia te (5.14 ) w ith respect to e

to obtain


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PETER D LAX

whereu 0 and un are two vectors.T H E O R E M 5.4. If th estatesu 0andunare sufficiently close,th e initialvalueproblem(5.19)has asolution. This solution consistsofn \ c onstant statesu0ul ,un,separated by centeredrarefaction or shock waves,one of each family (see Fig. 12).

since th e r - are l ine arly independe nt, i t follows f rom th e implicit function theoremthat a sm allg-ball i s m apped one-to-one onto a neighborhood of u0;this com pletesthe proof of Theorem 5.4.Wedescribenow a method developedbyJamesGl im m for solving any initialvalue problem u(x, 0) = u 0(x ) where the oscillation of u0 is small. The solution uis obtained as the l imit as h-> 0 of approx im ate solutions uh constructed asfollows:(I ) uh(x ,0) is a piecewisec onstant approximation to u 0(x ) :

wherem} i s som e kind of m ean value of u 0(x ) over th e in terval (jh,0 + 1)h).

Since w (0) = ti(0), we see, comparing (5.18) and (5.12), t h a t u = uat e = 0. Thiscompletes the proof of Theorem 5.3.If the fcth characteristic field of (5.1) is degenerate, then (5.1) has d iscont inuoussolutions whose speed of propagation is

These are called contact discontinuities.W e turn now to the so-called Riemann ini t ial value problem, where the init ialfunction u 0 is

Proof. The state u0 can be connected throug h a 1-wave to a one-parameterfamily of states u^ej to the r ight of M O ; u{ in t u rn can be connected through a2-wave to a one-parameter family u2i a)fstates to the right of U j etc. Thusu 0 can be connected through a succession of nwaves to an n-parameter family ofstates u n (e , , } . B y (5.8H5.14),

FIG 12


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HYPERBOLIC SYSTEMSOF CONSERVATION LAWS 9

(II) For 0 t 0)givenby (5.20);here A is an upper bound for 2|/llt()|.This exac t solu tion isc onstruc ted by solving the Ri emann ini t ia l valu e problems

F I G . 1 3

( I I I ) W e repeat the process, with t = h / A . as new in i t i a l t imein place of t 0.Itis not at all obviou s that thisprocessyields an approx im ate solu tion uhwh i c his defined for all to prove this one must show that the oscil lation of u ( x ,nh)remains small, uni formly for n = 1,2 , so that the Riemann problems (5.2lj)c anbesolved,and sothat doesnot tend to oo.Thisest imate tu rnsou t to dependvery sensitively on the kind of averaging used to compute th e mean values nij.In th e scheme introduced by Glimm the quant i t i es m - are computed as fol lows:A sequence of r andom numbers a t a2 , un i fo rmly distr ibuted i n [0 1] i schosen;m ,the mean valueofu x,n h / fy over the interval jh, y + l)h) istaken tobe

Glimm shows:(A ) Givenany e, we canchoose r \sosmallthatiftheosc illationandtotalvariationofuQ are


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30 PETER D. LAX C ) For almostallchoicesof the random sequence{aB},thislimit uis asolutionin the integral senseof the conservation law 5.1).For proof werefer to Glimm s paper; here we merely point out howGlimm s

scheme tre ats ap articularly simpleinitial value problem ,a Rie m ann initial valueproblem:

where u t and u r are so chosen that th e exact solution u consists of the two states u rseparated by a single shock

where 5is^the speed with which th e shock separating the two states propagates.By assumption, A > |s|. Let us assume that 0 < s; then Glimm s recipe 5.23)gives

where

Repeatingthis analysisntimesw eobtain

whereJn = numberofa,, = 1, , n,w hich satisfy

Since {o^} is a uniformly distributed random sequence,

with probability 1 ; this shows that the approximate solution given by 5.25)tends almost certain ly to the exact solution given by 5.24).Note that if , instead of using random sequences we use a single sequenceofequidistributed numbers {a^}, i.e., numbers fo r which 5.27) holds, we concludethat u htends to as h-0.W e conclude by stating precisely the existence theorem whose proof w asoutlinedabove, and by stating some open problems.

THEOREM 5.5. The initial value problem for the system of conservation laws 5.1)has a solut ionfor all t provided that th e initial fun ction u0 has sufficiently smalloscillation and total variation


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HYPERBOLIC SYSTEMS O F CONSERVATION LAW S 31W hat is lacking at present is a proofthat the solution t constructed by Glimm sscheme satisfiesthe entropy condition, and that it is uniqu ely determined by u0Some rem arksabout both points wil l be made in 6.Another outstanding problem is to remove the r equ i rement that u 0 have smalloscillation.Ineq ualities (3.14), (3.38) and (5.5) are c riteria w hich reject certain disco ntinuitiesas physically unrea lizable even thoug h the cons ervation law s are satisfiedacrosst h e m ;w e designated these criteria as entropy conditions. We shall introduce nowa not ion of entropy w h ic h can be related to these criteria.W e s tar t with a system of conservat ion laws (5.1). Let U be some function of

ul ,unW h e ndoes U satisfy a co nserv ation law , i.e., a law of the form

where the gradient is w i th respect to u.To deduce this from (5.2),

w em ult iply (5 .2) on the left wi th grad U;(5.28) results if and only if the relation

holds. This is a system of n partial differential equat ions for U and F; for n 2it is overdetermined and has no solution in general ; there are, however, specialcases of some im portance w ith a no ntriv ial solution, for exam ple, in gas dynam ics.A general class of equations where a solution exists are the symmetric ones, i.e.,when A i s a sym m etric m atr ix . In thiscase

Relation (5.30) is the compatability relation for the existence of a function g u)satisfying

It is then easy to verify that

satisfy (5.29).The role of entropy condit ions is to distinguish those discontinuous solutionswhich are physically realizable from those wh ich are not. Another way tocha racterize the physica lly realizable so lution s is to identify them as limits ofsolutions of equations in w h ic h a small dissipative mechanism has beenadded to

where F is some function of ul ,unl Car ry ing out the differentiation in (5.28)w e obtain


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Suppose U isconvex, i.e., the matrix of its second derivatives is posit ive defini te ;then w ededuce tha t

32 PETER D. LAXthe laws already em bodied in 5.1) . A part icular example of such a dissipat ivemechanism isartificial viscosity; here th e augmented equation is

M ul t i p l y this by grad ;if 5.29) is satisfied we ob tain

W ehave the identity

where

sub st i tut ing this into 5 .33)w e obtain, since > 0 , tha t

Let ->0;th e right side tends to zero in the sense ofdistributions and wededucethe fol lowing theorem.T H E O R E M 5.6.L et 5.1)b e asystem of conservation la ws which impliesan additionalconservationlaw 5 .28) ;suppose that U is strictly convex. Let u x, t) be a distributionsolution of 5.1) which is the limit, boundedly, a.e., of solutions of 5.32) containingth e artificial viscous term . Then u satisfies the inequality

Thefollowing a re immediate consequenceso/ 5.34):

finite, is a decreasing function oft. b) Suppose u ispiecewise continuous; then acrossa discontinuity

W e shall call conditions 5.34) and 5.36)entropy conditions; to jus t i fy the namewe have to show compatabi l i ty with the previou s designations.T H E O R E M 5.7. Suppose that th e system of conservation laws 5 .1) is hyperbolicand genuinely nonlinear in the sense of 5.3). Suppose there is a strictly convexfunction U of u which satisfies th e additional conservation la w 5 .28) . Let u be asolution of 5 .1 ) in the integral sense which has a discontinuity propagating with


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HereV is the transposeof the right eigenvector rand V is the matr ix ofsecondderivativesofU .SinceUisconvex,Eispositive; this shows tha tE(s) is anincreasingfunction of near = 0. But then e)isnegativefor Enegat ive;sinceb y Theorem5.1 the shock cond ition 5.5) restricts un(s) to negative valuesof ,Theorem 5.7follows.Next we show that th e entropy condition 5.34)is equivalent to the en tropycondition in the large 3.38) impo sed on solutions of single equations. W e notethat in thiscase Ucan b e taken to be any funct ion ofu;F can b e determined from 5.29)by integration.T H O R M 5.8. The entropy condition 3.38) is satisfied if an d only if 5.36) issatisfied for all I/ F whichsatisfy 5.29)an d whereU isconvex.Proof. Suppose u ,


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W e assume that 6.2) is hyp erbolic, wh ich m ean s that the m atrix A has real anddis t inct eigenvalues; we denote them by A and p, so arranged that A < p; ofcourse A and p both are funct ions of u and v . W e denote th e corresponding leftand right eigenvector by / andr;t ha t is

34 PETER D. LAXW ewri te the system as

where/and g are func t ions of u and v . Car ry ing out the differentiation in 6.1)we obtain

where

and

The eigenvectors, too, depend on u and v . W e shall consider functions w of uand i whichsatisfy the first order equat ion

Thisequation assertsthat w i sconstant alongthetrajectories of thevectorfieldrA .W e can construct solut ionsof 6.3)by tak ing a curveC whichis not tangent to rAat any point, and assigning arbi t rary values for walong it . We shallchoose w tobe strictly increasing along C . The value of u is then determined along everytrajectory in tersec ting C, and w has distinct values along d istinct trajectories.The function z ofw,visdefinedanalogouslyassolutionof

Since w has dist inct values along dist inct rA -trajectories, and z has dist inct valuesalong rp-trajectories, and since in a simply connected domain of u v space anrp-trajectory intersects an rA -trajectory in at most one point, it follows that themapping

isone to oneoverany simplyconnected domain.


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C u r v e s s a t i s fy in g (6.6) a n d (6.8) a re called A a n d p characteristics r espec t ive ly .Rela t ion s (6.5) and (6.7) can be s ta te d in w ord s as fo l lows.A s f u n c t i o n s o f x and / , w is constant along p-characteristics, z is constant along^.-characteristics. A f t e r t he i r d i scover er , w a n d z a re called Riemann invariants.In 3 about s ing le conser va t ion l aws we gave a g eom et r ic a r g um en t fo r then o n e x i s t e n c e o f c o n t i n u o u s s o l u t i o n s b e y o n d a c e r t a i n t i m e . O n e i n g r e d i e n t o ftha t p r oof was the cons tancy o f u a l o n g c h a r ac t e r i s t i c s ; t he o ther w as the fact t h a tchar ac ter is t ics a r e s t r a igh t l ines . The f i r s t i ng r ed ien t i s p r esen t h e r e : w a n d z a r econs tan t a long char ac ter i s t ics , bu t i t i s no longer t r u e tha t char ac ter i s ti cs a r est ra ight lines; so the s imple g e o me t ri c r e a s o n i n g g iven i n 3 c a n n o t be ex tendedto sys tems. We pr esen t now a d i f f e r en t n o n e x i s t e n c e p r o o f f o r a s i n g l e e q u a t i o nwhich is c apab le of g e n e r a l i za t i o n .The equ at ion (3.3) sa t isf ied by u is

d i f f e r en t i a te th is wi th respect tox

A b b r e v i a t eux b yq;t h e a b o v e c a n b e w r i t t e n t h e n a s

w here q a b b r e v i a t e s the d i r e c t i o n a l d e r i v a t i v e

w h e r e i s d i f f e r en t i a t i o n in the d i r e c t i o n

Simi la r ly ,

w here d e n o t e s d i f f e r e n t i a t i o n i n the d i r e c t i o n

M u l t i p l y (6.2) b y g r a d w ; u s i n g t he a b o v e r e l a t i o n , a n d t h e c h a i n r u l e , w e o b t a i n

Simi la r ly ,

H Y P E R B O L I C S Y S T E M S O F C O N S E R V A T IO N L A W S 35I t i s well k n o w n t ha t the l f t a n d r i g h t e i g e n v e c t o r s o f a m a t r ix w i t h d i s t i n c te igenva lues a r e b io r thogona l . I t i s her e tha t we e xp l o i t t h a t n 2: s ince by (6.3)grad w i s o r t h o g o n a l to r^ , i t fo l lows t h a t g ra d w is a l f t e i g e n v e c t o r w i t h e i g e n v a l u e


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36 PETER D. LAXEquat ion 6.9) is an o r d i n a r y di f fe rent ia l e q u a t i o n f o r q a long th e charac t e r i s t i cand can be in t eg ra t ed e x p l i c i t l y :

This is an o rd ina ry d i f f e r en t i a l e q u a t i o n f o r q a lon g each p-charac t e r i s t ic , s imi la rto 6 .9) ex cep t tha t the coef f ic ien t k o fq2 is no t cons tan t . Ne ve r the l es s an exp l i ci tf o r m u l a f or q can be w r i tt e n :

th e r e su l t i ng e qua t i on can be r ewr i t t en a s

M u l t i p l y th is byeh;u s ing t h e abbr e v i a t i on s

Subs t i tu t ing th i s in to 6.13)w eobta in

Since according to 6.5) , w 0 ,

Le t h d eno t ea func t i onof w and zw hich sa t is f ies

substi tuting this into 6.11)w eobta in

From 6 .7 ) we deduce tha t

Abbrev i a t i ng w x by p we can put th i s as

w h e r e q Q = q Q ) and k = a (u) cons tan t a long the charac t e r i s t i c . Th i s fo rm ulashows tha t if q Qk > 0 , g(t) is b o u n d e d for al l t > 0 , wh i l e if q Qk < 0 , q(t) b l o w sup at t = \/q0kW e im i t a t e th e above proo f for the syste m 6.5) , 6 .7)as fo l lows . Di f f e r en t i a t e6.5)w i th respe c t tox


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Subs t i tu t ing th is into 6.16) w e conclude tha t ifq0 > 0 , q(t) s tays bounded, ifqQ < 0 , q(t) becomes unbounded after a finite t ime. We see from 6.14)that thesign ofq0 is the same as that ofp0 the initial value ofwx.So we can summarizewhat wehave proved as follows.

T H E O R E M 6.1. Suppose condition 6.17) is satisfied for a system o f equations 6 .2) .Letu v be a solution whose initial value s a re bounded , then ifw x(x ,0) is 0 and A 2 > 0. Suppose u, v is a solution o f

6.2)who se initial va lues a re bounded, a nd suppose that both w x,0) and z(x, 0) areincreasing functions o f x. Then the first derivatives of the solution remain uniformlybounded, and the solution exists and is differentiable for all t > 0.Remark. It iseasy to verify tha t th e condition p w 0 is the sameas thegenu inenonl inear i ty condit ion 5.3).W e turn now to so lu t ions wi thshocks. Since there are two families of character-istics, there are two families ofd iscont inu i t ies ;w eshal l refer to them asp-shocksand A-shocks.H ow does the R iemann i nva r i an t w change across a A-shock? According toTheorem 5.1, th e states u rwhichcan b econnected to u tacrossa A-shock form onone-parameter family U B ) , g


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Consider a solution containinga finitenumberofweak shocks. Let x,t)be an ypoint, t >0;draw a backward p-characteristic C through this point. Accordingto the shock condition 5.5),Ccannot run intoap-shock;soCcan be continuedall the waydown to the initial line. C will intersect a finite number ofA-shocks;between two points of intersection w is constant. Since /I < p, it follows from6.19)that w increases along Cas tdecreases. So weconclude:

6.1S) holds, then

wherey is the point whe re the p-characteristic through x, t) intersects the line t = 0.We turnnow to theasym ptotic behaviorofsolutions forlarget.This problemwas studied for single conservation laws in 4; the main tool there was theconservation of increasing and decreasing varia tion of con tinuo us solutionsbetween two characteristics. Thanks to the existenceof Riemann invariants ,wehave aconservation ofvariation of w between p-characteristics and of z between^-characteristics, valid for continuous solutions. We saw that for solutions of asingle conservation law with shocks the variation between two characteristicsdecreases as t increases;the same argument applied to the Riemann invariantsshows that the presence of p-shocks causes the v ariation of w between character-istics to decrease w ith increasing time, and sim ilarly the presence of A-shockscauses the variation of z to z todecrease. We have however theadditional taskofassessing the effect ofA-shocks on the variation ofw and ofp-shocks on z.Thishas been carried out in Glimm-Lax for solutions whose oscillation is small.The precise result proved there is the follow ing.

THEOREM 6.2. Suppose condition 6.18), and an analogous condition for z, issatisfied for a system of conservation laws 6.1). Then the initial value problem for6.1) has a solutionfor all bounded, measurable initial data whose oscillation issmall enough.The total variationof this solution on an interval of length t at timetis bounded by a constant. For periodic solutions, the total variation of u and v perperiod decays as

weconclude that, at least for weak shocks,

H owever is ingeneral not zero;if weimpose the requirement that 0, say

whichaccording to 6.4J iszero.A similar calc ulatio n, based on 5.12), shows that also

38 P E T E R D. LAXLet us calculate v v


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W e show now that the com patibility equation 6.21) has solutions w here U isconvex,provided that a certain condition, se e 6.31) below ,issatisfied;we do notclaim that this condition is necessary.W e shall construct families of solutions depending on a parameter k in thisfashion:

HYPERBOLIC SYSTEMSOF CONSERVATIONLAWS 39Not a great deal is kn o w n abou t u niqu en ess ; O leinik has studied solutions ofsystemsof the special form

where/isan increasingfunction ofu.She has shown th at s olutions which containa finite num be r of shocks and centered ve rifaction waves are u niqu ely determinedby their initial data.W etu rnnow to theentropy condition 5.36)decribed in the lastsection:

for all U u, r)and F u,r )whichsatisfy 5.29):

We can eliminate F from this system ofequations by differentiation; we get ahomogeneous second order equation fo r U :

It is easy to verify that if the original nonlinear system 6.1) is hyperbolic, so isthe linear system 6.21),and the derived equation 6.22).The question is:doesasecond orde r hype rbolic equation 6.22) have convex solutions? It is fairly easyto show that in the small the answer is yes, on the basis of this observation:If the real sym m etric ma trix atj isindefinite, there existsapositive definite matrixU i j such that

where

p V jand H j are funct ionsof u and v. UN and FN are functionsof u, v and k aswell,of order k~Nek ,i.e.,

Expansions ofthissort,with i p inplaceof < p are customary in geometrical optics.


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j = 1,2, ,N. W e solve these recursively. Equation 6.27) asserts that g r a d c pisa lefteigenvector ofA, wi theigenvalueH0/V0.Such a func t ion is called aphasefunction ingeometrical optics;according to 6.4) this condition means that q > is aRiemanninvariant.

Having found < p ,wesubstituteitinto 6.28)whichwesolve recursively, assigningarbitrary initial valuesfor U j on a noncharacteristiccurve.If the initial valueforU Q is chosen to be positive, t/0 wil lbe positive everywhere.We havetosolvea firstorder systemtodetermine U NandFN;the inhomogeneousterm in this system is0 ekv/kN) and, with the proper choice for an initial curve,U N and V N will satisfy 6.26).When is the f unc t i on U defined by 6.24), 6.25) convex for k large enough,i.e., when isQ =2Uun + ^rjU uv + r j 2U vv positivefor all < ;, 7 7 ? The answer can beread of f f romthe first two leading terms ofQin its asymptotic expansion in powersof k. The coeff icient ofk2 ek f> is

40 PETER D LAX

The fo rma l construction in the real case is the same as in the imaginarycase:W e substitute 6.24), 6.25) into 6.21), divideby e k< p and equate like powers o fktoobtain

whichis equal to

Aswe have remarked before,U0can be chosen to be positivethroughout;therefore,the above formis positive except along

The coefficient of k ek t> consists ofthree terms;two ofthemare zero along 6.30);the remainingone isW e make the assumption that there exists a Riemann invariant p fo r which6.31) ispositive;ifthis is so, U givenby 6.24)isconvexfor klarge enough.

If 6.31) ispositive, the func t ion

is convex; since is a func t ion of the Riemann invariant < > it is itself anotherRiemann invariant, and so our result can be formulated thus: // there exists aconvexRiemann invariant i f in adomain of the w ,v) -p lane , there exist functions o fth e form

which satisfy (6.21) ,furthermore U is convexfor k large enough.


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H Y P E R B O L I C S Y S T E M S O F C O N S E R V A T I O N L A W S 41W hat can wededuce f r o m theen t r o py cond i ti on

the kih r o o t of the left side o f 6.34) t endsas k- oo to the m a x i m u m of ] / on S2whilethekthr o o t of the righ t side tends to the m a xim u m o f \ l o n S ^ Theresu l t inginequal i ty proves Theorem 6.3, and even a l i t t le m o r e .Remark. The con clusion o f Theo rem 6.3 agrees w ith the sta tem ent 6 .20),deduced undertheassum pt ion 6.18).I tt u r n so u ttha t inequa l i ty 6.18)isequ iva len twith th e posi t iv i tyo f 6.31). otes1. Quasilinear equations. The energy inequali ty fo r symmetr ic hyperbo l icsystems is due to Fr iedr ichs and Lew y, for no nsym m etr ic hyperbo l ic system s,see LerayandGard ing ,andCa lde ron .Theexistence theorem usingtheco ntractivecharacter o f 9~ is due to Schauder . For a m o re detai led discussion o f these

approaches,seeChapterVI ofCou ran t and Hilbert.For the case o ffunc t i onso f one space variable one can em plo y es tim ates in them a x i m u m no rm ins tead o f the ene rgy no rm .This isdoneas f o l l ows : differentiate

If S - is so chosen thatFIG. 14

fo r funct ions of the f o r m 6.32)?T O R M 6.3.Let u v be a solution in the integral sense of the conservation laws 6.1),which satisfies the en tropy condition 6.33)/0r al l 17,F of th e form 6.32),k largeenough. Then

is a decreasing function oft.Sketch of proof. In teg ra te 6.33) ove r a lens-shaped region contained betweenStandS2 see Fig. 14).We obtain


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I n t eg r a t i ng 3 )a long t h e j t h charac te r i s t ic connect ingthe p o in t x ,/ )t o some po in ton the in i t i a l l ine / = 0 weobtain

Using this est imate for the f i rs t der ivat ives of solu t ions one can show that theini t ialv alue problem 1 .1) , 1 .3)has a solut ion in the t ime in te rva l 7 ) , whereM 0)is defined asThe n one xisten ce Theorem 6.1 shows that the restr ict io n 7) is , rough ly , necessaryunless on e pu ts cond i t ions on the in i t i a l values as inTheo rem 6.2.

2. Conservation laws.A n important class ofhype rbo l ic systems ofcon serva t ionlaws are the ones governing the flow of compressible , nonviscous non-heatconductivef luids . The re are fiveconserved q uan t i t i e s : mass, momenta and energyper u n i t v o lu me:p = mass pe r u n i t v o lu me = dens i ty ,M = m o m e n t a pe r un i t vo lume = pV, where u ,v,w) = V is flowveloci ty ,E= energy per u n i t v o l u m e= in t e rna l kine t ic energy = pe jpV2,wheree = interval energyper uni t massandV2 =uz v2 w2.

42 PETER D. LAX 1 . 1 ) wi th respect to x; d e n o t i n g ux by p w e ob ta in a re la t ion of the fo l lowingsort:

wher e Q is a q u a d r a t i c funct ion of p. D e n o t eby l} th e l e f t e igenvector of A

Mul t i p l y i n g 1) by / - w e obtain the fo l lowin g re la t ion forp , = I j p :where q } is some quadra t ic funct ion of the vector p. The first two te rms in 3) area d i rec t iona l der iva t iveofp , in a so-cal led character is t ic d irect ion, givenby

valid as long as

From this we deduce easi ly tha t

Deno te by M T) the m a x i m u m of all p/x, r ) , j = 1, , n, oo < x


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HYPERBOLIC SYSTEMS OF C O N S E R V A T I O N L A W S 43The fluxes are part ly due to material being transported across the boundarywi th th e velocityof the flow; for the momenta the re is an addi t ional flux due toth e momentum impar ted by the pressure force at the boundary, and there is anenergy flux due to the w ork done by the pressure force at the bounda ry . Ifthereis no h eat co nd uction this accounts for all energy changes. For a nonv iscous fluidthe pressure is a scalar p exerted equally in all directions. The formulas for thef luxes a re :

Using this the fluxes can be expressed in terms of p,M and E .The jump re la t ions for shocks in gas dynamics were first stated by R i e m a n n ,incorrectly, for he conserved entropy instead of energy. The correct relationswere found by R a n k i n eand byHug onio t.Other important nonlinear hyperbolic systems of conservation laws are theequat ions governingthe mot ionof a shallow layerofw ater, and theequa t ionsofhydrodynamicf low s see Co urant and H ilbert , Ch apter V I ) .

3. Single conservation laws.The starting point of the rigorous theory ofsingleconservation law s has been a paper by H opf in 1950, w here the exp licit for m ulastated in Theorem 3.1w asgiven in the special case of the quadratic conservationl a w / u ) = \u2 T he formula fo ra rbi t ra ry c o n v e x / i s stated in Lax 1954),andanalyzed in Lax 1957) .The revealing Theorems 3.4 and 3.5about the decrease of theLl norm of thedifference of two solu t ions a re due to B arbara Q ui n n ;Ll contraction also playsa role in the w o r k of Oleinik 1957 ). C ond ition 3.38) is due to Oleinik 1959);she showed that solutions satisfying 3.38) are uniq uely determ ined by theirinitial data , and Kalashnikov proved that solutions of 3.39) converge as A- 0to a solution which satisfies 3.38).There is a parallel theory of single conservation laws in n space variables.Existence theorems are contained in Conway and Smoller , Volpert , K r u s h k o v 1969), and Ko t low. A uniqueness theorem for piecewise continuous solutionshas been given by Doughs and by Quinn; a more general uniqueness theoremhas been given byK r u sh k o v .

4. The decay of solutions.In his 1950paper Hopf studied the large time behaviorofsolutions ofquadrat ic conservat ion law s;the extension to any convexpisgivenin Lax 1957) .T he more refined Theorem 4.1,and Theorem 4.2about the two and

In te rna l energye pressurep and densi typ are related by an equa t ion ofs ta te:

m ass f lux

momentum f l u x

energy flux


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Conley and Smoller have studied viscous profiles for strong shocks.The main tool in G limm s existence theorem beside th e difference scheme, is afunctional which measures the potential interaction contained in the Cauchydata along any space-like curve. Glimm shows that this functional decreaseswith t ime.The notion of entropy discussed here has been proposed by Lax (1971), andKrushkov (1970).The theory of the symmetric case is due to Godunov and wasapplied by him to the compressible f low equations.6. Hyperbolic systems of tw o conservation laws The nonexistence Theorem 6.1andTheorem 6.2 are from Lax (1964); anotherversion has been given by Zabusky(1962).Johnson and Smoller have shown that under assumption (6.18) the R iemanninitial value problem can be solved uniquely for two arbitrary initial states notnecessarily close. They have shown how to solve th e initial value problem for suchsystems under a monotonici ty assumption for the initial values. Nishida hasshown that for the system

th e initial value problem can be solved for arbitrary initial values u (x) ,y 0 (x ) ,u 0 0. Nishida s wo rk has been extended by Bakhvalov, DiPerna Greenbergand Nishida and Smoller.

where v i s independent of A and satisfies the o rdinary differential equat ion

with artificial viscous term. B y plane wave we mean a solution of the form

44 PETER D . LAXonly conserved qu antitie s is given in Lax (19 70). The law of decrease of increasingvariation, and the method for proving it is taken from Gl imm and Lax (see alsoLax (1972)). For the nonconvex case, see Dafermos (1972).

5. Hyperbolic systems of conservation laws The shock condition (5.5) an dTheorems 5.1, 5.2, 5.3, 5.4 are given in Lax (1957). In case the /cth field is l inearlydegenerate, i.e., grad l f c rk = 0, there exist discontinuous solutions where th ediscontinuity is a ^-characteristic wi th respect to either side. Such discontinuitiesare called contact discontinuities It can be shown that solutions with contactdiscontinuities only are the l imits o f continu ous solutions.Foy has shown that if u, and ur can be connected by a weak shock, then theycan be connected by a viscous profile, i.e., a plane wave solution of the equation


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HYPERBOLIC SYSTEMS O F C O N S E R V A T I O N L A W S 45The Riem ann ini tial value problem in gas dynam ics for abroadclass of equatio nsofstate has been studied by Wendroff .Theorem 6.2 is due to G limm and Lax. Further uniq uen ess theorem s, in the

absence of rarefact ion waves have been given by Rozhdestvensk i iand by H u r d .The const ruct ion of the en t ropy funct ion 6.24) is carried out in greater detai linLax 1 9 7 1 ) .7 . Dif ference sch emes . No set of lectures on hyperbol ic conservat ion lawsshouldend w i thout ment ion of the var iouseffective difference schemesfo rcompu t -in g solut ionso fconservat ion laws.Theseare used to com pute so lu t ionsof specificinitial value problems which come up in scientific and technological p rob lems ;problems involving tw o space dimensions can be handled as well. In addition toproviding numer ical answers to specific questions, one hopes t ha t numer i ca lcalculat ions will reveal patterns which playa role in the theory to be developedabout solut ions ofthese eq uat ions.If itwere possibletop rove r igorously that so lut ionsoffinite differenceequat ionsconverge, this would provide a proof of the existenceofsolut ions w ith arbi t ra r i lyprescribed data. So far this has been accomp lished on ly for single cons ervationlaws, and for a very crude difference scheme proposed by Lax 1954) :

Here u k ab breviatesan approximat ion to ua t t = nAr,x = /cAx,and/2abbreviates/ u ) .T he convergenceofthis scheme for aspecialcasewasverified by Lax 1957) ;convergencefor anyconvex/ wasproved byVvedenskaya. Convergencefor anyn u m b e r of space variables was shown by Conway and Smoller , and lso byKot low.The approximat ion 8) is in conservat ion f o r m : that is, if we th ink of u n as anapproximat ion to the average valueof uoverthe cell[ k | Ax, k +| Ax]att imet nAf, 8) is of the generic form

i.e.,whereth e average valueof u in the / c t h cell at t ime t n + l A f differs fromth e average at t ime Ar by the average of the amoun t tha t has entered and l e f tat the endpoin ts dur ing the time elapsed. The conservat ion character of theapp roxim ate equ ation 9 ) is expressed by the fact that the amount that enters thekth cell during th e t ime interval {Af , n + l )Af} th rough th e left endpoint isexactly equal to the amount which leaves th e k l)st cell th rough it s rightendpoint during thesame t ime interval.In 8 ) , / k + 1 / 2 w astaken to be


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46 PETER D. LAXth is is a rather poor ap proxim ation to the average flux at k + du r ing {n, n + 1 } ,since the flux instead of being an average is evaluated at the earliest time n. Inaddition, fo r A f small , the presence of a r a ther large amoun t of artificial viscosityproport ional to A x )

2M J C X / A f causes additional errors. This second term has to beincluded to s tabil ize (8) ; t o ensure stabil i ty one has to impose in addit ion th eC FL con dit ion (see C ourant , Fr iedr ichs and Lewy) :

A further interest ing modification which has been introduced by R. W.MacCormack has been especially efficient in the case of several space variables.Recently, still more accurate schemes have been devised by Rusanov and byBurstein and M irin.Another type of difference scheme has been introduced by Godunov; h isstarting point is the same as in Glim m s scheme, but the approximate averagevalue at time (n + l)Af over the fcth cell is defined to be the average of the exactsolution computed there. This average value is computed from the flux relation,i.e., (9) is used, w i th / k + 1 / 2 taken as the exact va lueof/at the interface betw een thekth and k + l)st cell.Calculations performed w ith th e meth ods described above produce approxim atesolutions in w h ich a shock is spread over a finite number usually tw o to fourof

where

Since this fo rm ula centers the flux properly at t ime t = n + | A,it is more accuratet han (8) ; it can be shown that th is formula i s stable if the CFL condit ion (11) issatisfied.The following modification of (12), proposed by Richtmyer (see Richtmyerand Mor ton) , turns out to be more p ract ical:

set in (9)

and the formulas

A more accurate choice o f / k + 1 / 2 has been proposed by Lax and Wendroff(1960),(1964). Starting w i th the Taylor series


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lax - hyperbolic systems of conservation laws and the mathematical theory of shock waves

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