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On Uniqueness and stability for systems of conservation laws

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(195) >A@ )   >H0+-, DL +B(*)  : ; J E :$4QCN2 DL DL ) >H0> @3,+ @ 2%0:AC @#(B+B:8, :/ H Π  I2  0>8(B+-0-< A+-, 6  >A, G W <Q:AE . . . .  . "!. 3. . #. . #. . #. #. . &. #. . . . . .  . . . #. . . !. . #. . . . +. . . . 

(196) . . 3. . 3. . . . . 3. . . 3. . . !. . . . . . . #. . . . %.  . . . .   . . . . .   . . . . . 0Q: /2 I:8,#0F(7>A, (  >A, G0Q: /2, :A,+*,. E 2I>H05+*,96 <F@3,. (B+ :A,  L &5, G$2I2IG(*)32F0 2 I :A,GA+*(B+ :A,#0+ TJC (*)32 + G$2, (B+*( DL8L .

(197) z      . ‹‰ @ \ \^V_LOmfJUZT fJ\dU#\dcHc(VH\dcHfJOQSHckWlMV \dU S`VHM  \dLA c`MPOQfJSHTiMPUJckS`Me9 \^VH[ MPOQT %c  c`S`\dgc MW MPUJc`\^V LNSHTiMPU OQL cheJL \%L \dLNVH\dj TQU SHe#\O TiS`\^V_LNSHf#VH\X #MVkcH ^LOQLNV

(198) MPUZc`\^V NLNSHTiMPUOQL c 7SHe#\ fJUJT f#\dUJ\dcHcLUJj c`SHLN[ZT OQTiS  V_M[ZOi\dg LcK MPg Oi\^S`\dOOc`MPO \dj%[9 V_f de#M H F[JI X9;YU%SHe#\b ^Lc`\ MW>c  c`S`\dgc ZeJM \ \^V fJU SHTQO VH\d \dU SHNO LOQO L LTQO LN[ZOi\ SHe#\^MVH\dghcVH\ fJTiVH\djLjZjJTiSHTiMPUJLO V_\^nPfJOQLNV_TiSR 9e  MSHe#\dc`\dc„MPU SHe#\(cHMPOQf#SHTiMPUJcdX;YU H [H SC[ I L)fJUJT f#\dUJ\dcHc VH\dcHfZOiSmTQc VHM \dj TQU SHe#\k ^OQLcHc„MW Ti\d \ TQc`\

(199) c_gpM MSHe W fJUJ SHTiMPUZc eJTQOi\

(200) TQLU H HJH ITiS3TQc3cHe#M U SHeZLNS3TiWL TQ\d \ TQc`\

(201) VH\^nPfJOQLNVbc`MPOQf SHTiMPU \  TQcHSHc SHe#\dU TiSmTQcmfJUJT f#\ TiSHeJT U L ^OQLcHcmMW  WrfJUZ SHTiMPUJc^X(=e#\hgLT U VH\dcHfJOQStTQ3U H S I \dc`SHLN[ZO TQcHe#\dc9SHe#\pfJUJT f#\dUJ\dcHc.MW&c`MPOQf#SHTQMPUJc/fZUJj#\^V L(VH\dc`S`V_TQ SHTQMPU%MPU SHeJ\bOQM ^LNSHTiMPUZcKMWC \dU S`\^VH\dj V_LNVH\^W L SHTiMPU L \dc^X JMVE= \dg Oi\7 ^OQLcHc c  c`S`\dgc cHS`VHMPU#n\^V‡fJUJT f#\dUJ\dcHcGSHe#\^MV_\dgc> ^LUb[ \CWrMPfJUJj TQ<U H HNŒ PCŒ I X>D.\dNO  T U#nKMPUbSHe#\c`\dgTinV_MPf L VHMPL e VHM MPc`\djbTQU H ˆAˆ I PSHe#\fJUJT f#\dUJ\dcHc SHe#\^MVH\dg TQ*U H ‹NŒ I Lc.SHe#\ JV_c`S.MPU#\ eZTQ _e MPfZOQj [ \mL OQTi\dj TiSHeZTQUkSHe#\„c_Lgp\„ ^OQLc_cMW  W fJUJ SHTiMPUJc e#\^V_\LU \  T c`S`\dUJ \pSHe#\^MVH\dg TQc  U#M UGX=e#\ VH\dcH\dU StLcHc_fJg SHTiMPUZc  W f#VHSHe#\^V \dAL \dU SHe#\„VH\^nPfJOQLNV_TQS  MPUJjJTiSHTQMPUfJc`\dj%TQ*U H ‹NCŒ I LUJj)LNVH\„ ^Oi\dLNVNO cHLNSHTQc J\dj9[ SHeJ\ \dAL c`MPOQf#SHTQMPUJcM[ SHLTQU#\dj LcO TQgTiSHc„MW.SHe#\RKO TQgg,cH _eJ\dgp*\ H H KIMVpWlV_MPU SbS`V_LI  TQUJnL VHM  TQghLNSHTiMPUJ<c H ˆdJ‹ I X 69c cHe#M UkTQU H ˆ J[ I #L$U (OQTQgTiSEMW WrVHMPU S&S`V_LI  TQUJnbL VHM #TQgLNSHTQMPUJc7cHLNSHTQc J\dcCSHe#\ j#\d ^L (LcHc_fJg SHTiMPU W V_M TQj#\dj

(202) SHeJLNSLOQO eJLNV_L S`\^V_T c`SHTQ J\dOQjJcCMWGSHe#\ c  c`S`\dg  ˆNXQˆ  LNVH\Kn\dU fJTQU#\dNO UJMPUJOQTQU#\dLNVX 2 2. &. . ,. . &. . &. . +!. . 3. . . !. . . . . . . 3. . . . . . !. . !. . . . . +!. . !. . . .  . !. . . . !. . . . . . . . .  . . . . . 3. .  5  I .   =. ;<USHeJTQcc`\d SHTQMPU „\ MPOQOi\d S.c`MPg\tS`\d _eJUZTQ ^LOGOi\dgghLc #WlMV/OQLNS`\^V/fJcH\X PS$ 0 021' G  Q2 ( / H Œ  I >   2   + 4I2><F@3,. (B+ :A, )$+ ) 0>8(B+-0 '20(*)32

(203) :8,#0Q2QE A>8(B+B:8, @>8(B+B:8,#0  M0>  +B(*) :8(7L >AEQL   ?  /  <Q:AE >AC*CK  H Π IRL )32Q,/(*)32 >FJ 2 ' ( ?  .

(204). . $. . . . . +-0  K+ J 0

(205) Q)$+*(  5 :A, (B+*,@:A@ 0 '+BLB2 L .  ˆNX ‹ . '' '  ?  (   ?  '' + - / " 2  (  2. <Q:AE"0: 2 5 :A,#0(7>A, ( " >A, G;>AC*C    *H Œ  RI L  T%  LU9  :  :. kŒ LUJj) MPUJc`S`V_fZ SKLkcHgM MSHe L V_M #TQgLNSHTiMPU S`M SHe#\ eJLNV_L S`\^V_TQcHSHTQ kWrfZUJ SHTiMPU MW/SHeJ\ TQU S`\^V LO H    I X JMVhSHeJTQc fJV MPc`\ >SHLA\L cHgpM MSHe U#MPUJjJ\d VH\dLcHTQU#npW fJUJ SHTQMPU /   2 H Œmˆ(I Zc_fJ _eSHeJLNS  TiW  / ( Cˆ     Œ ˆ TiW ! ˆ  .   !". . . $#. #. . .

(206) NH Œ /z   LUJj j#\ ZU#\      #     X%6/c  2 Œ

(207)  SHe fJcmL VHMPL _eJ\dcmSHe#\ $ \dL T cHTQj#\W fJUJ SHTQMPU X 5&MPUZcHTQj#\^VLU9cHgpM MSHeWrfZUJ SHTiMPU     iT SHe MPg L S/cHf MV_S LUZjj#\ ZU#\ &. . . #. #. . . 3. #. . . .   ()   H   (   (   (   I     #  fJcHTQUJn  TQU  ˆNX   LUJjbOi\^S`SHTQU#n  2 ŒPSHeJLU# c. #. #. 'a mSHeJ\7Lc_cHfJg SHTQMPU„MPU MW SHe#\tW fJUJ SHTQMPU 2   ?  \tM[ZSHLTQU. S`M SHe#\ P \ MPU SHT U fJTQS . . ;YSSHe fJcWrMPOQOiM c.    H  )   (% ' )   I  

(208)    . .          Œ #. . '' '  ?  (% '  ?  '' + -  . / . c_f \     H ' )  ( = MS^X F‡LNVdX   ' ( ? )  . .  )   I  . .  - *     . /   ? "   ? 2  (  2 . eJ\^VH\

(209) [ *"   \j#\dU#MS`\SHe#\?! T cH _eZTiS ( MPUZc`SHLU ShMW.SHe#\W fJUJ SHTiMPU MPU SHe#\ j#MPgLT U 2 2 / X  PS$ 0 0`1  cG cG 2Q( / I   CH2   + 4I2 >A,+-, ( 2768E >$4QCN2 <5@3,. (B+B:A, 05@ )(*) >D( D<:AE 0: /2  2I>H05@3E2 . . .

(210). . ':8,2) >H0. . .          /  1 H \   I 8   . )32Q, . . H \   I. . )32,M2 A2QE . .         /  1I   C H 8 #  I9[Zc`\^V \3SHeZLNS TQU  ˆNX ‹‰   MPU#\p ^LU. TQSHeLUM \dUMPU#\XE89Lgp\dON. . . . .  ! !. . .    . -.   . > . \> . . > K #. . .            OQTQ g             /  1 I \   H 8 - . .      ( &       / M # .  ˆNXIHNŒ . VH\ OQL \bSHe#\p ^OiMPc`\dj TQU S`\^V LO. 8/\ S #LU$ :Œ XE=e#\dUkSHe#\^V_\t\ T c`SHcL TQ\d \ TQc`\ MPUJc`SHLU S.W fJUJ SHTQMPU & cHfZ _eSHeJLNS  .  ˆNX ‹‰ . .  ˆNXIH ˆ .

(211) z 57LOQO TQU#n 2 2. . &. .   ,. . &. . .SHeJ\ MPT U SHcMW jJTQc_ MPU SHTQU fJTiSRhMW & \t MPg f#S`\.    >5?A?A? >   . .       / . . .  . Hˆ.   . &. . .

(212). .

(213). / .

(214). /. .

(215). . .    ( &      

(216)    &           &               &                   

(217)     1 &    (    8             1 I   \   MH 8

(218)   &   (       .  . . /. .    . ‹

(219)  1 I   CH 8 # @ TQUZ \ Lc.LNVH[ZTQS`V_LNV #SHeJTQc VHM \dc.SHe#\mOi\dgghL X PS$ 0 021' G  2Q( / H Œ   I 2  452 + J 0)$+B( )I:A, (B+-,@:A@ 0QL( > 6D+ 82,?J#:A+-, (    TCO2Q( (*)32 5:A, GA+B(B+B:8,#0 DL  DL

(220) ) :ACNG <Q:AE0: /2  :  + ;: L )32, <Q:AE 2> ) ; :Œ :A,2)3>H0 OQ TQg  cH f  ˆ     

(221)  ,  ( 

(222)     2    Œ #  ˆNXIH‹   69cHcHfJg\ >MPU SHe#\( MPU S`V_LNV  CSHeJLNSbSHe#\^VH\k\ #TQc`SHc : Œ LUJj c`\ f#\dUJ \dc  2 Œ 2  2 >  Zc_fJ _eSHeZLNS      

(223)     ( 

(224)     2   :   ˆNXIHH   WrMV\ \^V ! ˆNXE= M 

(225) SHe#\ TQj#\dLc Lc_cHfJgp\  ! ŒbWrMVLOQO  SHe#\ MSHe#\^V ^LcH\dc[ \dTQU#n3\dU SHTiV_\dNO  cHTQghTQOQLNVdX 'a (SHeJ%\ ! T c_ _eJTiS MPU SHTQU fJTiS  V_M \dj TQU  ˆNX ‹  \meJL \   ' ()  (  ()  (  )  (  )    / " 2 ( 2  ˆNXIH  WrMV/c`MPgp\3 MPUJc`SHLU S " JWlMV/\ \^V <() !ŒhLUJj \ \^V  TQU S`\^V LO H   I X7;YS.TQc.UJMSVH\dc`S`V_T SHT \tS`M LcHcHfZgp\ " : ˆ XC=.e#\tLcHcHfJg SHTiMPU  ˆNXQˆ^Œ  T g OQTi\dc OQ TQg  ˆ      ' ()  (  ()      Œ #  ˆNXI8H S  /. . . . . . 3.

(226). .      . !. . . .   . .  . .  ! . . . . . . . . #. . . . . . . . . 3. . . . . . . (. (. . . . . . . . . . . . . . . . .  . . .  .   . . . . . . . . . (. . . . . (. . . . .

(227) . . . . . . . . . . .

(228) H‹. /z  . . M \ \^V TiW  ˆNXIHH  e#MPOQjJc [   ˆNXIH  TiSWrMPOQOiM c. $. . . . . &. . . . . .      .   .  )   (. .  )      .    !       . .           '

(229)    (   

(230)       ( " ( 2 

(231)  ( ,2     !        ( " ( 

(232)  (        ( "      JTQU MPU S`VLjJTQ SHTiMPU TiSHe  ˆNXIH8S  X7=eJTQc VHM \dcSHe#\„Oi\dggL X .  .   .  . .  ˆNXIH[ . . . WrMV.LOQO D"%"N16(  cG W , G$2QE (*)32 0> /2 >H0505@3 J(B+ :A,#0 :=< 2

(233)   > DL L (*)32 0F(7>D( 2F0 . . DL  DL 0>8(B+-0-< . . .

(234).   . .     

(235) . +-,  ˆNXIH .   '

(236)   (     

(237) 

(238)  # ;<UJj#\^\dj#SHLA TQU#n  ŒLUZj Oi\^S`SHTQU#n 2 Œ

(239) TQU  ˆNXIH‹  MPU#\tM[ZSHLTQUJc   

(240) 

(241) D (%    ˆ  / OQT  g  cHf     '

(242)  

(243) D (%  )    

(244)    ' )   (%       Œ#. .= e#\ MSHe#\^V.TQjJ\dU SHTiSR(T c VHM \dj%cHTQgT OQLNV_ONX PS$ 0 0`1  cG cX 2Q( 452> 2>H.%0Q:AC @3(B+ :A, : < DL 8L 2Q( (*)32 5:A, GA+B(B+ :A,#0 DL  DL

(245) ) :ACNG >8( 3. 0Q: /2 J3:A+*, ( 2 @>8(B+ :A,#0. . 

(246)    L. . )32, (*)320F(7>D( 2F0.  .      . . >A, G (*)320 J#2I2IG; 0>8(B+-0-<  (*)#2 A> ,#.H+-,2   @D6$:A, +B:8(.   .  ˆNXIH . ;  (%   5   (*    #. . = Mn\^SHeJ\^V TiSHe SHe#\„\dU S`V_M  MPUZjJTiSHTiMPU T U G !G\dghgLˆNX ‹ XKS(cHe#M c/SHeJLNS.SHe#\3c`SHLNS`\dc  LNVH\t MPUJU#\d S`\dj\dTQSHe#\^V[9(LULjJgTQc_cHTi[ZOi\/cHe#M (MV[9(Lp MPU SHL S9jJTQcH MPU SHTQU fJTiS  VHM LNnPLNSHTQUJn TiSHec \^\dj:;GX = M VHM \tSHeJ\mOi\dggL JjJ\ ZU#\KSHe#\tW fJUJ SHTiMPU  TiW  > ;.M   7   # TiW 6:L;.M#. .  . . . . . . .  !

(247) !. 3. . . . .

(248) z    a'  ˆNXQˆ^Œ  TiSWrMPOQOiM c 2 2. &. . ,. . &. O TQg  WrMV\ \^Vk[ MPfJUJj#\djkc`\^S . . WrMV.LOQO . &. HH.   . .    

(249)  7 

(250)    (   7        4    X MV_\^M3\^V  ˆNX   TQg OQTi\dc. Œ Œ.   

(251)  M  

(252)      7 

(253)  1 

(254)  7 

(255)    8    7          \ LUJjLOQO  : ΠXE=eJ\ S MhLN[ M3\mV_\dOQLNSHTiMPUJcTQg ON     

(256)            Œ  TiSHc`\dOiWZT c LKc`MPO f#SHTiMPU3MW  ˆNXQˆ  XE=.e#\^VH\^WrMVH\CSHeJ\D.LU#. cHe#M QT U#n9SHeJLNS>SHe#\7W fJUJ SHTiMPU TQU#\ $ f#nMPUJTiMS \ fJLNSHTiMPUJc  ˆNXIH  g3fJc`Se#MPOQjGX 6 \ MPUJ ^O fJj#\SHeJT cCc`\d SHTQMPUh[ pVH\d ^LOQOQT U#ntS MbOiM ^LO]TQU S`\^nV_LO \dc`SHTQgLNS`\dc&SHeJLNS& _eJLNV_L S`\^VT ^\ SHe#\ S`V_>L <`\d S`MV_Ti\dc/MW L

(257) @ SHLUJjJLNVjD.Ti\dgLUZU@ \dgTinVHMPf X = MRS  \dc MW OQM ^LOPL VHM #TQgLNS`\Cc`MPOQf#SHTiMPUZc WlMV  ˆNXQˆ  TQO Oy[ \E MPUZcHTQj#\^VH\djGX IKU#\‡TQc j#\^V_T \dj WrVHMPg SHeJ\/c`\dOiW wcHTQghTQOQLNVEc`MPO f#SHTiMPUMWGL3D.TQ\dgLUJU VHM[ZOi\dg SHe#\.MSHeJ\^V7T c‡M[JSHLTQUJ\djh[  <WlV_\^\ dTQU#n  SHe#\K M A\ B

(258) ^Ti\dU SHc&MWGSHeJ\9 MVHV_\dc MPUJjJTQU#n fJLcHT OQTQU#\dLNV&$e  \^VH[ MPOQTQ c  c`S`\dg TQU

(259) L3U#\dTinPe [ MV_e#M M jMW LpnPT \dU MPT U S^X ! \^S /  ;2   + []\(L$U   WrfJUZ SHTiMPU LUJj # L9U  MPT U S L  e#\^VH\ eZLc„L <HfJg X‡57LOQO     ()  SHe#\ fJUJT f#\.cH\dOiW wc_TQgTQOQLNVE\dU S`VHM 

(260) cHMPOQf#SHTiMPU

(261) MWGSHe#\ D.TQ\dgLUJU VHM[ZOi\dg   ( TiW  > GΠ 

(262)     Œ    Œ )    ˆNXIH‰  TiW  : ŒG# 

(263) D !G\^ S ; [ \tLU f \^V7[ MPfJUJjkWrMVSHe#\„LN[Zc`MPOQf#S`\ NLOQf#\dc7MW>LOQO L \3c \^\djJc^X #MV !Œ Zj#\ ZU#\ Ti0W 2  ( 2 / ;G  #      ( 

(264)  ()    ˆNX PŒ  TiW02  ( 2 : ;G X   I9[ZcH\^V \

(265) SHeZLNS„SHe#\kW fJUJ SHTiMP*U 2   ?  TQc ! T cH _eJTQS

(266) MPU SHTQU fJMPfJc X VdX S^X SHe#\ P]\ jZTQc`SHLUJ \ LUJj L VHMPL _eJ\dc 1L# c 2 Œ

(267) X 8/\  S ^LOQO  9 1   8 SHe#\ 8PL M[ TQLU gLNS`V_T )MW MPg f#S`\djLNStSHe#\ MPTQU S   X #M,V ! Œ j#\ U#\   )  Lc/SHe#\3c`MPOQf#SHTiMPU MW>SHe#\„OQT U#\dLNV/9e  \^VH[ MPOQT 5LfJ _9e  VHM[ZOQ\dg TiSHe MPUJc`SHLU S/ M A\ Bh ^Ti\dU SHc   Œ   #  ˆNX Jˆ . 

(268)    Π !. . #.

(269). . . . . . !. . . +!. . . . . . . . . . . . . . . . . . .

(270). . . . . . . . . .

(271) . . .

(272) H /z   SP $ 0 0`1  cG 2Q( 4I2>  7( >A,GD>AE G +72

(273)  >A, ,  2 +N68E :A@ J 692,M2E >8( 2G;4(*)32"0 H0F( 2 $L +B(*) GD:>8+-,  >H0;+*, DL AL )32Q, (*)32E2 2 8+-0F(*0?>?I:8,#0F(7>A, (  \ 05@ ) (*) >8(%(*)32%<Q:AC-CN: +*,96 ) :8C GH0QL :AE 2 A2QE <F@3,. (B+ :A,   2 A2E     >A,G   : Œ +B(*) (*)32/>$4: 82 G$2 ,+B(B+B:8,#0 :A,M2) >H0 ˆ   1  8    (   )       ˆNX ‹  /  \ ?  :D(7L >8EL 1  I (  GH I 

(274) H 8  ˆ   1  8    (    )       ˆNX FH  /  \ ?  :8(7L >AEQL 1 = I ( 

(275) H 8 # #MVKL VHM MW Jc`\^\ H ˆˆ XE‹ ˆ I X . &.  

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