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Periodic Solutions of the Vlasov-Poisson System with Boundary Conditions

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(1)Periodic Solutions of the Vlasov-Poisson System with Boundary Conditions Mihai Bostan, Frédéric Poupaud. To cite this version: Mihai Bostan, Frédéric Poupaud. Periodic Solutions of the Vlasov-Poisson System with Boundary Conditions. RR-3518, INRIA. 1998. �inria-00073166�. HAL Id: inria-00073166 https://hal.inria.fr/inria-00073166 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Periodic solutions of the Vlasov-Poisson system with boundary conditions Mihai Bostan et Frédéric Poupaud. N° 3518 Octobre 1998 THÈME 4. ISSN 0249-6399. apport de recherche.

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(134)  G H \ ] L  !132 "!j8 1j2 8  e1'%! 8   U ! 01 8 ! g ! 9 1  8  + /  CS U I. .

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(163) (. *.

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(171) 9Z¢¬£:flu#V3sialUWh%`bV CcjS%V kjhadcj\ ad` m V up\^›dadcj\^›lV0ž,up\^cjV k «  Z$ Z)     S$  'Y1j24/ 57/98 + 1"$PN01 /  1  57/P/62 8 / `1 657/6 / 2 868!S$    lˆ  F  ") N 1 /  1 657/6 / 2 8 / `1 657/P/62 8 8! S$  D8!Pi    M R |¨g¯flu m Vu€cef ŽEVc0adg¯Vkpcj\bU adcjVW¥²fluzceS%VWad‹?f‰›EVW\^g˜ceV Žlueal` ž1V m Vuj\ ›la•ceVZceSiV kpoqkpcjV Ufl¥ r‚SalueadrceVuj\^kpcj\brk<\^g~upV kph?Vrc#cef 5 ! « . . ).

(172). *.

(173) .

(174). *8. . .

(175). . . .  M S$   M   S$ 1j2$ 57/P/62 8  M   S$  M   S$  V1j2$ 57/P/62 8. S$  V1 657/P/62 84/ 1-9/ -0/ R R R g / -9/ R R R -8  /  1   /   / RRR A 8 !S$   1 657/P/62 84/  1j5  /65  / R R R 5 A 8 - R. + + +. . +. .  . . :  "! / 4# :  "! / 4#  lŒ. <f‰ž  _%kj\^g%ŽeanupfEg˜ž¬ad`b`7½ k V UWU a%ž¦V m V m _%rV0cjS%VnV kxce\^U¡a•ceVl«     /  : 24/ 4 1j24/657/98 # /  X F S$   1 657/6 / 2 8   $  1 657/P/62 8  )i   !      G H \   L   <alg m ceSiV ujV¥²fEujV  ž,S%V upV  S$   +   1 13 5 / /65  / / R R R R R R 5  A 8 8  A _%kj\^g%Ž  Cl ¦ž¦VnŽEVc « $ )         S$  'Y1j24/657/608 F     Z Z  N   !Pi        L  )          Xi‡  N   !Pi   F ž,S%\br‚S \^UWh%`^\bVkceSa•c S$  ' : 09 1 < => fY FZ 9 1 868 ¢2|¨g cjS%V¡kjalUWV©ž1a‰o Lž¦V*ŽEVc0ceS%adc )  ' : 09 1 < => fY FZ 9 1 868 ¢ [°o*ceS%VnflcjS%V u,Sadg m _%kp\bg%Ž  ES† ¦ž1VnSaI›lVl«  S'    N    !  '     !#"$'      !  ") '  R

(176).

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(179). . . *. 

(180)  .  . . . . 

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(183). 

(184) .  .  . . . *. hjiTWkhjl.

(185) ‡A†.  

(186)   !"$# %&'( ) * + ,. ™ nV SaI›EV0h%upfA›lV m ceSadc ' :VU ¢°<f‰ž :r flg%kj\ m V u  /4  c¨ž1f¡h?Vuj\^f m \br¼ž¦VIal§kpfE`b_]ce\^fEg%k ¥²fEuLceSiVU f m \ ¿V m š€`balkpfA›V ®K_adcj\bflg©alg m `bVc N: 9 1j< =>? ? < = )A 8 ‹CV,a hCV up\bf m \^r ¥²_%g%rcj\bflg~ž,\ ceS 1jNS8 .< =>L? ;? ]:T< = )A   U  ¢ ™ V m Vg%flceV‹˜o ' : U a kjfl`b_icj\bflg2¥²fEu  ˜†S3¢ M<kj\^g%ŽWceS%V¼ž¦VIal§*¥²flujU_i` adcj\bflg %ž1V m V m _%rVl« . . . /N.

(187) (. *.

(188). . *'.

(189). +. F  J FMLBFG HIJ   1j24/ 57/98 !PN 1324/657/98NMOOM5PMO2 +. F@ J F L F G HIJ   1j24/ 57/98*1 +. F  J F QS  !( K1j5Q8!  1j24/657/608 !#'Y1j24/657/608 MOMTKM2 +. F@ J F L F G H IJ   1j24/ 57/98*1 +. F@ J F L F G HIJ   1j24/ 57/98 !PN 1324/657/98NMOOM5PMO2. . alg m ceS%VujV¥²fEujV. . /N.  + . . !'] S ']  !#"$#']  !#")*'L8NMOOM5PM2. . +. . . %. !#"$". . !'] S ']  !#"$#']  !#")*'L8NMOOM5PM2  X7CS. /.  \^g ¢.     *  d0   !<"W$%& ' )(*)',+.    |¨g ceS%\^k0[]V rce\^fEgOž1V rfEg%kj\ m VuncjS%VVg m \^UWV g%kp\bfEg%al`9rIadkjV alg m \bk€cjS%VZ_%gi\^cn\^g˜ceV ux›dal` # -0/ g  ¢ .  . . 

(190)

(191)  .F "#$&. $.

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(193) ! +.  U¥²fEu,ad`b`hadupce\^r `^V k¢ ™ Vnh%upfA›lV r fEg˜ce\^gK_%\ c¨o fl¥cjS%Vnr‚SalujalrcjV uj\^kpcj\brkzfl¥š€`balkpfA› V®K_adce\^fEg)¢ °µ   ” O    1  S8

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(211).  0 1 8.

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(218)  G H \ ] Z    0 . . ! . . (. . 

(219)  G H*\] Z    0. . /. (  U. . .  U. /.  X7BS.  U. (. . “ ¹)¹ 9 ™ V ¿ujkxc©upV U alup§»ceSadc \^g ›q\^Vž fl¥  XK†S   XEXOZS%fE` m k¡al`bkpf­¥²fEu  ¢RS%V upV¥²fEupV2cjS%V r fEupujV kph?flg m \^g%Ž r‚SalujalrcjV uj\^kpcj\brk,alujV#ž1V`b` m V¿%g%V m ¢ Vc 1j24/657/608 :(< =>Q?(1-9/ g 8L?e< =>) ¢ ™ V<U_%` ce\^h%` o  ‡ CSv ‹˜)o  1  57/P/62 8  1 657/6 / 2 8 alg m 1 657/6 / 2 8 V1 657/6 / 2 8 cefWŽEVc  : 1 ! / 4 8 1'%! / 4 8 « I. . . . g M  0 1 8 U M

(220).  1 8

(221). . g M U M. +. 1 . 1 1 /. . . *. U. g.

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(224).  1 868 !1. 1 8. !

(225) . . 1 88.  +. . 1 8 . 1 868.  1 /  1 868 8!O1. 1 8.  1 8   Ug ! 0 1 8. 1 8.  1 8 .

(226) .

(227).

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(239) . 1 8. . . e1 8 . .

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(244). alg m al`^kjf%«.  *. U. (.

(245) . 1 8.  1 8 V1 8. +. *.

(246). *.

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(248)  M. +.    

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(251)  G H \ ] Z  .  (. . 0  U U . . U. 1 8

(252)  8 . . .  1 8  .  1 8. 1 8. 1 0  U 8 F  

(253)  01 8  1 8

(254)  

(255)  01'P8 

(256) 2   

(257) G H \ ]   Z  R

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(259). *. *. [¦o2_%kj\^g%Ž]a¼ujfEg˜ž1al`^` 1 "! / 4 8 «.

(260) . /. ) 0  U / ).  †q‡  † CS. U ¢9|¨g fEu m V u¤cefnV kxce\^U adceV,ceS%V m \ X?V ujVg%r V#fl¥?V g˜ceuxo ce\^UWV kqalkpkj_%UWV ?‹ VrIal_%kpV 2  ceSadc "! ! %S fl` m k<alg m ž,up\^cjVl«

(261).

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(263) ! U. "!.   . .  M .  . +. WTWYX ZR[ \]_^. F *.

(264).

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(266).

(267) ‡‰ˆ. =(#  #   J 

(268) ' # R +.  1 "!j8 +  1 ! 8 !.

(269). "!.

(270) *. U. *. ‹?VrIal_%kpV.  1' ! 8

(271). . ). . alg m jk fi« U . (

(272) . ! . .  '1 ! 8   . .

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(274). *.

(275). . . (. . . 7. ). . . 8 # @   ?     : +  `W        %  : + C  G#?8

(276) )

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(278)  G H \ ] Z    0  U  †  U  U. g   ! . 

(279)    U. *. . !. “ ¹)¹ 9 ™ V adkjkj_iU VZcjSadc %! ! ceS%Vg )_%kp\bgiŽ V g˜ceuxo*›EV`bfqr \ c¨o2\bk,ŽE\ ›EVg~‹˜oC« *. 1 ! 8. e1'%!j8 . *.

(280) .

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(282) 

(283) V1 ! 8.     !. .    

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(285) . . 1 "! 8 . 0  U. .   F. ). .

(286). . . 0  U ).  !. . .

(287). . !.  M. M. . . M.     . "! !  

(288). 0  U. ) R. 

(289)  † BS hjiTWkhjl.

(290) ‡‰Œ.  

(291)   !"$# %&'( ) * + ,. {zkjkp_%UW\bgiŽ©g%f‰ž cjSadc :Y FZ 9 1 < = ? fl¥cjS%Vn`bVUWU¡a _%kj\^g%Ž  KX BS¬adg m  † BS3« .

(292).  j1 24/ 57/98. . L1j24/ 57/98

(293). +. ¢vRS%V g ž¦VV alkj\^`^o r‚SiV r‚§*ceSiV0kxc‚adcjV UWV g˜c 8. 1 ! / -0/  01 ! 868.

(294) . 1'%! / -0/  1 "!j868 .

(295). ?.     G H*\] Q%S  1 ! "!  1' ! 8 ?.     G H*\] Q S       G H*\] Z    U  U 0  U. g .  . *. . .

(296). *.  . .

(297). . . ). V1 "!j8

(298) 8.

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(300) C F. =(#  #   J 

(301) ' # R. 8 : *#c 

(302) H

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(304) )  K  H

(305)   W -  & 

(306) )!# <I . . . !#"$". ) . 1 8 %. 91324/ 0- /98

(307) 2 ( : < => / - &(* 6*    91324/ g /98

(308) 2 ( : < => /   *  * ) O/. (. cjS%VnU alh)« . . . ž,S%V upVl«. Bi\:.  :  +. . . iO1j24/65Q8

(309). . .    . iO1324/08. 0 + U !  !.   G H*\] Z     /  57/V: -0/ g # 0 ! 5  /V / 5Q8 / 1j24/ 5Q8 :f< => ? -0/ g # O/.

(310) *. .

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(312) )

(313).

(314).     G H \ ] Q%S  R .  . + S$#'. 2;:@< => /.  dXO. \^kcjS%V°kjfE`^_ice\^fEg. +.FG H J L1j24/657/608 MO / 1j24/65Q8;:@< =>L?  -0/ g # /. 'Y1j24/ g 8+'  132 8 /.  l . . . ^<V upV  \^kcjS%V¦UW\^` m h?Vuj\^f m \br¦kjfE`^_ice\^fEgŽE\^›lV gZ‹˜o  ‡ alg m fl¥cjS%V0tfE\^kjkpfEg hiujfE‹%`^V U « . .   CS. +.$' / . 1 < => ? -0/ g # 8$  i. iO1324/65Q8 +8iO1327. ž,\^cjS)«.  ]‡.

(315) )

(316). *. . . “ ¹)¹ 9 ™ V m Vg%flceVn‹˜o.  

(317)   F.  '    G H*\   W ! !      G H*\] QS     R  K W   + , C  G# #  (J#   ,S $ \ 

(318)    ;. I. . . . 5. # .  E†S. 2;:@< => R. hjiTWkhjl.

(319) C]‡.  

(320)   !"$# %&'( ) * + ,. ª’ µ 'Vc .  , #   ( ?  T    J C  1  8  G ; :  ¢ < M kp\bg%Ž    $# +  F iž¦V0ŽEVc «  

(321)  G H \ ]  Z   U !  !     G H \ ] Q S  R RS%VnV `^V rcjuj\^r0¿%V ` m  ž,up\^cjV k « 4. 5. %. . . *. . .  1j24/65Q8 +' 1j2 8 . alg m ceS%VujV¥²fEujVd«. F. . .   

(322)  G H \ ] Z   . . 1g . *. .  '   '     R. *. .  1j24/ 5Q8.  1j24/ 8. . . !   U  W !  !. . +. . . . *. 0 ! 5. +. . .

(323).

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(325).

(326). .   BS. . .  $     G H \ ] Z   ! 5     G H \ ] Z   ! 5 U !  !     G H\F] QS  !. *

(327). 1j24/08NM / . .

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(341)    9  +  \bg 1j< => ?  -0/ g # 8 ¢ zVg%flce\^g%ŽZ‹˜o  cjS%VzUW\^` m kjfE`^_ice\^fEg~ŽE\^›lV g ‹˜o  ‡ ,rfEujupV kph?fEg m \bgiŽ©cef  '¢ M<kp\bg%Ž  †Ed†    †] ‡ ,Vc  † CS ž¦Vhadkjk cef ceS%Vn`^\bUW\ c,¥²fEu g  \bg  ‡ ¦ž,S%\^r‚S g%f‰ž upVIa m k « . . . . . . . . . 5.  0 / '. . %. . . . . . + FK J F )  F  FK J F )  F . . M<kj\^g%Ž 7 ,S, #. . .  . . . . .  B F. .  !  j1 24/ g /98 !'Y1 / . 1  g /P/62 8 /  91  g /P/62 868 M MOOM2. .  !  1j24/-0/98 !'Y1 / . 1 -0/P/62 8 /  91 -0/P/62 868 M MOOM2 R. €ž1Vnr alg~h%alkjk¬cjfWcjS%V¼`^\bUW\ c<\^gceS%\^kVs]h%upV kpkj\^fEg)¢[qf  \^k,aWrfEg˜›EVuxœ ŽEV g˜cvkjV®K_%V g%rVz\^g¡kpV g%kpV<fd¥ m \^kpcjuj\^‹%_icj\bfEgikvž,SifEkjV#`b\^UW\^c  \^k¤ceS%V,UW\b` m kjfE`^_icj\bfEg r flujujV3œ kjhCfEg m \^g%ŽncefnceS%V¿V` m  ¢v}~flujV f‰›lV uK¥²ujflU ceS%V,_%gi\^¥²fEupU ‹CfE_%g m \bg 09 1 < => ?]1 -0/ g 8 8 ž1V m V m _%r Vd«   /  Bq‡ ž¦VIal§ \^g 0 9 1j< =>? 1-9/ g 8 ?@< =>) 8 R |¨g ceS%VkealUWV0ž¬aIo CceS%V m V gikj\ ce\^V k r fEg˜›lV ujŽlVž1V al§K`^o )‹CV rIad_%kjVcjS%Vkp_%h%hCfEupczfl¥  alujV¼‹CfE_%g m V m ž,\ ceS2ujVkjhCV rc<fd¥L›EV`bfqr \ ce\^V k« + F G H J  1j24/657/608 MO + F G H J L1j24/ 57/98NMOP/ ž¦VIal§ \^g 0 9 1j< =>? 1-9/ g 868 R  B CS Š¼g~ceS%VZfdceS%Vu€Salg m cjS%Vž¦VIal§~r fEg˜›lV ujŽlV g%rV fd¥ \^U hi`b\^V k€cjS%Vž¦VIal§~r fEg˜›EVujŽEVg%r V fl¥  1  S8 cef‰ž¬adu m k  1  8  ¥²fEu¤\^g%kpcealg%rV<\^g 0 9 1 < =>0?  -9/ g # 8 ¢¤[]\bgir V ƒ\bkvrfEUWhalrc  \^c¦\bUWh%`^\^V k1cjSadc  1  d8   1  8 \^g 1 < =>Q?  -0/ g 8 ¢9RSi\bk¬hiujf‰›EVzcjS%VzrfEg˜ce\^gK_%\^c¨o¡fl¥ ceS%VZU alh  ¢€{¬c<ceS%\^k€hCfE\^gKc )_%kp\bg%Ž*cjS%V [qr‚Sal_ m Vu€# ¿is]V m hCfE\bg˜c<ceS%VfEujVU ?ž1V h%upf‰›EV Vs]\bkxceVg%r V fl¥°h?Vuj\^f m \br kpfE`b_]ce\^fEgOfl¥#‡  š€`balkjf‰›˜œtfE\bkpkjfEg¯h%upfE‹%`^V U ž,S%\^r‚S¯r fEg%r`b_ m V k ceS%Vnh%upf]fl¥fl¥ceSiV ) ,  ¢ 7. . . . . . . 5. hjiTWkhjl.

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(345)  G H\F] Z   . . U !   ! *. . !. iž¦V m V m _%rVl« 1 8 

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(347)  G H\F] Z       G H\F] QS  !   . . . . g . . !. . . 0  U.  B•XO. ). . . ( U   ( U. 

(348)  G H*\] Z  . 3. 

(349)  G H*\] Z   . 0  U ). . R  Bl†S.

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(361) %  *  . . & . 'Vc . . ‹?V a¡h?fEkp\^cj\^›lVWr fEg%kxc‚alg˜cnadg m `bVc ‹?VcjS%V kjVc¼fl¥vceSiV¥²_%g%rce\^fEg%k. M  g. ž,S%\br‚S2›EVuj\ ¥ o?« ". ' : 0 9 1 < =  fY  Z 1  8 8 S' : 0 9 1 < =  fY FZ 1  8 8  "$ '   G H \ ] L ]dG HOIJ. . ' 1j2 .  '     G H*\   L    S'     G H*\  L g    ! 9 1 8 /. /. . /. . . . . / . * .

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(363). *.  BdŒ  ˆ F. /.  ˆ]‡  ˆ CS. . /!b8 +. .  *. . '. ' 1j24/!b8 /. ž,S%V upV adg m  adujVn¿is]V m r flg%kpcealg˜cek<ž,Si\br‚S ž,\^`b`'‹CVr‚S%flkjV g ` adcjV u<fEg)¢ ™ \ ceS~cjS%V m V¿gi\^cj\bfEgik  BlŒ   ˆ F qž1V0SaI›EVn¥²fEu,alg˜o*¥²_%girce\^fEg ' :  « . . .  "$'  . . . ]L .   ZJ.  "%$#'     Z  ] L  J ]L A  S$ "$'    .  Z. +. . . . !

(364). . *. . . 1 /. M<kj\^g%ŽWceS%VnrfEUWhalrcjg%V kpk<upV kp_%`^cjk<VU‹CV m%m \bg%Ži« Y FZ 1 1-0/ 8Y? 8  1 -9/ # ? 8 1 . . . ž1V m V m _%rVzceS%adc "%$#' ' : ceSadc \bka,r fEUWhadrckjVcfd¥ U¡adh%h%\^g%Ž%«. =. .

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