Convergence of an Adaptive Scheme for the one dimensional Vlasov-Poisson system

HAL Id: inria-00070487

https://hal.inria.fr/inria-00070487

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Convergence of an Adaptive Scheme for the one dimensional Vlasov-Poisson system

Martin Campos Pinto, Michel Mehrenberger

To cite this version:

Martin Campos Pinto, Michel Mehrenberger. Convergence of an Adaptive Scheme for the one dimen-

sional Vlasov-Poisson system. [Research Report] RR-5519, INRIA. 2005, pp.49. �inria-00070487�

ISRN INRIA/RR--5519--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Convergence of an Adaptive Scheme for the one dimensional Vlasov-Poisson system

Martin Campos Pinto — Michel Mehrenberger

N° 5519

Mars 2005

Unité de recherche INRIA Lorraine

)# *+,.-/,+*0132546+*0*7*98!*:$#

;=<?>A@CBEDGFH<?IKJMLONQPBED:@CLRS;GBET%UVXWY;GVZU>)VZD9[MVZ>]\?VZ>

^ _5`AaHbc1dYegfihkjmlonp`AaHb)lqkr5aHs)tvuxwZr5b)l

yztp{}|obAnY~€}ƒ‚Zu

„$}…5…?{†tvn$‡mbtpb)ˆ‰_5b)tpˆ‰_5bŠqO‹SŒ†Œ5)Žfie†tpl$‘†’†’XŒ“f•”XŽ…–}—Xb)l

˜š™œ›)Až%Ÿ5 ††¡£¢ q!†‡5†…mnvuƒ‚†blvbAaHu¥¤§¦}—†t‰}q–—†ux}q(lpˆ‰_5bAaHb€¨©{†tlª{X¥‚kuƒq5—1np_5bY~€}r«ˆ‰_Zj(…5tv{X¬5ƒbAaXlvlv{mˆ+ux%nvb]‡

np{np_5b…?bAtpuƒ{k‡5uƒˆ{†q5bA¤§‡5u¥aHbAq«lªuƒ{†q–†M­YƒXlª{%‚Z¤§yœ{Xuƒlplv{†q.lªjmlªnvbAa®uxl3…5tv{X…«{Zlªb]‡¯}q«‡¯}q–†¥jk°)b)‡:± ¢&² bAj

¨©b]%npr5tvb£{}¨œ{†r5taHbAnv_5{m‡uƒl€np_5b†ˆ)ˆ+r5t‰%npb1bA‚X{†ƒrmnvuƒ{†q{}¨nv_5bX‡5}…mnpu¥‚XbYaHb]lª_¨©tv{Xa³{†q–b1npu¥aHblªnvbA…nv{

np_5bq5bA´Znz{Xq5b†µ†¬–†lvb)‡3{†qnv_–b$}q–†¥jmlvuƒlœ{}¨?nv_–b¥{mˆA†mtvb)—†r5x}tpu¥noj}q–‡_5{%¶·u¸nz—Xb+nplœnptp†q–lv…«{Xtªnpb)‡3¬Zj(nv_5b

qkr5aHb)tvuxˆA†%¹–{%¶±0^ _5b€†ˆ)ˆ+r5t‰†ˆAj${}¨knv_–blpˆ‰_5bAaHbSuxl9aH{†q5u¥nv{Xtvb]‡“¬kj£ …5tpb)lpˆ+tpuƒ¬«b]‡£nv{†ƒbAt‰}q«ˆ+bS…–†tp†abAnvb)t

¶_–uƒˆ‰_tvb)…5tvb]lªb)qZnplMnv_5b“ƒ{kˆ)}?u¥qZnpbAtp…«{Xƒ}nvuƒ{†qšb)tvtp{†t€%n b)Xˆ‰_QnvuƒaHblonpbA…9±^ _5b£qZr–ab)tvuxˆA†?lv{†ƒrmnpu¥{Xq–l }tpb

…–tv{%‚Xb)‡3nv{(ˆA{†qk‚†b)tv—Xb u¥qšlvr5…m¤§q5{Xtvanv{%¶ }t‰‡5lnv_–b$b+´5†ˆ+nM{†q–b)lMXlSnp_5b$np{†ƒ{Zb)tp†q–ˆ+b …–†tp†aHb+nvb)tM}q«‡Hnv_5b

npu¥aHb“lonpbA…šnvbAq«‡!nv{3°Ab)tv{3…5tv{%‚kux‡mb)‡np_5b£u¥q5u¥nvux}:‡5}np(uƒl ¦9uƒ…–lpˆ‰_5u¥nv°Y†q–‡Q_«†l€º–q5u¥nvbYnv{†np†OˆAr5tp‚C}nvr5tpb†±

hmbA‚†b)tp†Xqkr5aHb)tvuxˆA†Xnpb)lªnpluƒ¥ƒr–lªnvt‰%npbMnp_5b b+»Ob)ˆnpu¥‚XbAq5b]lvlœ{†¨«{†r–t†…5…5tp{XXˆ‰_¨©{†t—†b)q5bAt‰%npu¥q–—$nv_–b€{X…mnvuƒa!}

X‡5}…5nvuƒ‚†b“‡muƒlpˆ+tpb+npu¥°]%npu¥{Xq–lA±

¼¾½Z¿OÀpÁ( ž%ܛ†¡ÅÄ r5¥ƒj†‡–}…mnpu¥‚Xb3lpˆ‰_5bAaHb†µ:lvbAaHu¥¤§¦}—†t‰}q–—†ux}qÆaHb+nv_–{k‡9µ~€}r–ˆ‰_kjƅ5tp{†¬5ƒbAaƵO­1x†lv{%‚Z¤

y{Xuƒlplª{XqÇlªjmlªnvbAaƵ1ˆA{†qk‚†b)tv—XbAq–ˆAbÆtp}nvb)l)µ†‡5†…mnvuƒ‚†b—†t‰†‡mb]‡·aHb]lª_5b]lAµ1_5u¥b)tp†tpˆ‰_–uƒˆ)} º–q5u¥nvbÈbAƒbAaHbAqZn‰lAµ

‡5uƒlpˆ+tpb+npb“ˆAr5tp‚C}nvr5tpb)l)±

ÉmʉË}ÌÍEÊpÎÏÌÐÑÍÏÒÓ+ʉÔoÕ)ÖCÒo×ÙØÚÉkÌÖCÐÑ×ÉkÐÑÌÛC×vÜCݜÞߐà:áœßâ1ã‰äåæ]܆ÝÛCÐç+ÒoÍÙ×EÐÎÏèzé«ÐÑÒoÍÏÍÙÒzÒêΜÞʉÍÙÐÑÒMàOÖCÍÙÐÑÒ+Ü%é–ʉÍÙÐÑ×

ëÚÛC×ÙÎEÐÎÏÖ]Î9ìCÒSß9ÒoÔ§íCÒoÍÙÔ§íCÒzÞÊpÎÏíCèoî$ÊpÎEÐÑÕAÖCҜïkç‰Ê‰ÛCÔoèoÒ+ÜCݜÞßà9áߜâ$ã‰äð]ñÜ]ÝÛCÐç+ÒoÍÙ×EÐÎÏèÉkÌÖCÐÑ×Oé–ʉ×ÙÎEÒoÖCÍoÜ}â+ÎEÍÏʉ×EË}ÌÖCÍÙò

-/,+*0132546+*0*

› ¡ d$q¾lpˆ‰_5sAa!H†‡5†…mnp}nvu¥¨œ…?{†r–t$ƒHtps)lv{†ƒrmnpu¥{Xq‡mr¾…5tp{†¬–¥`)ab‡mb3~€†r–ˆ‰_kjš†lplv{kˆAu¥s†rÆlvjmlo¤

np`AaHb ‡mb­YƒXlª{%‚Z¤§y{†uxlvlv{†q“b)lªnS…5tp{†…?{Xlvszb+nz}q«}ƒjklvs†±d$q…«{Xu¥qZnSˆ+ƒs€‡mb€q5{†nvtpbMaHs+np_5{m‡mb b)lªnœs)‚†{X¥rmnpu¥{Xq

…–tvs]ˆ+uxlªb‡mrHa!}uƒ¥x}—Xb†‡5†…mnp}nvu¥¨O‡r5qH…–Xlz‡mb nvb)aH…–l“†rmnvtpb†µ}¬«†lvsAb$lvr5tS}q«}ƒjklvb‡mbx“tvs)—†r5x}tpu¸nps

ƒ{mˆA†¥bYb+n$lª{Xq!nvt‰}q«lª…?{†tvn…–†tƒb£¹–rm´Qqkr5aHsAtpuxwXr–b†±S¦0(…5tvs]ˆ+uxlªuƒ{†q‡mrlpˆ‰_5s)aH(b)lªn€—Xr5ux‡msAb£…–}t€r5q…–%¤

t‰}aH`Anvtpb ‡5bnv{X¥s)tp†q–ˆ+b$wZr5u5tpbA…–tvs]lªb)qXnpbbAtptpbAr5tz‡uƒqXnpbAtp…«{Xƒ}nvuƒ{†qHƒ{kˆ)}ƒbˆ‰_–†wZr5b…«†lz‡mb$nvbAaH…–l)±Yq

aH{XqXnptvb3wXr–b3ƒb)l£lª{X¥r5nvuƒ{†q–lYqkr5aHsAtpuƒwZr5b]l1ˆ+{Xqk‚†bAtp—†b)qZn1bAqq–{†tpab3u¥q5º–q5uƒb3‚XbAt‰l$ƒb)l£lª{X¥rmnpu¥{Xq–lYb+´5†ˆ+nvb)l

ƒ{†t‰lpwXr–bƒbQ…–†tp†anvtpbQ‡mb!nv{X¥s)tp†q–ˆ+bHbAn(¥bš…–†l3‡mb!nvb)aH…–lonpbAq–‡5bAqZn(‚†b)tpl°AbAtp{–µˆ+{Xq–‡mu¥nvuƒ{†q wZr5bQƒb)l

‡5{†q5q5s)b)luƒq5u¸npuƒ†¥b]lHlª{Xu¥b)qXn!¦9uƒ…–lpˆ‰_5u¥nv°Æb+nQ}uƒbAqZnHr5q5b¾ˆ+{Xr5tp¬5r5tpbšnp{}np†¥bº–q5uƒb†±yzƒr–lvu¥b)r5tplnpb)lªnplqkr5aHs+¤

tpuxwXr–b)luƒƒ¥r–lªnvtpbAqZnbQˆAXˆ+u¥nvs‡mbq–{}nvtpbš†…5…5tp{kˆ‰_–bQ…«{Xr5t—†sAq–sAtpbAt(ƒb)l‡muxlvˆAtvsAnvuxlv}nvuƒ{†q–l(†‡5†…mnp}nvuƒ‚†b)l

{X…mnvuƒa!}ƒb)l)±

 ]› À   ›!¡ hmˆ‰_–sAa!3X‡5}…5np%npu¸¨0ˆA{†aH…5ƒb+n]µmasAnv_5{m‡mb“lvbAaHu¸¤§x}—†t‰}q–—†uƒbAq5q5bXµ†…5tp{†¬–¥`)ab£‡mb~€}r–ˆ‰_kj†µ

lvjmlonp`AaHbH‡mb3­YƒXlª{%‚Z¤§y{†uxlvlv{†q9µ–np†rm´‡mb!ˆ+{XqZ‚XbAtp—†b)q–ˆ+bXµ«a!}uƒƒƒ†—†b3X‡5}…5np%npu¸¨—†t‰†‡mr5sXµOsAƒsAaHbAqZnplYº–q–uƒl

_–u¥s)tp†tpˆ‰_5uxwZr5b)l)µkˆ+{†r–tv¬5r–tvb]l€‡5uƒlpˆ+tp`+npb)l)±

.?30! M+

^ _–b£­YƒXlª{%‚Z¤§y{†uxlvlv{†qšlvjmlonpbAa

∂ t f (t, x, v) + v · ∂ x f (t, x, v) + E(t, x) · ∂ v f (t, x, v) = 0,

^{/†±ƒ10}

∂ x E(t, x) = Z

R

f (t, x, v) dv − 1 = ρ(t, x)

^/†±Ñ‘0

‡5b)lpˆ+tpu¥¬?b)lnp_5b¾bA‚X{†ƒrmnvuƒ{†q·{}¨ ˆA{†ƒ¥uxlvu¥{Xq5¥b]lvl…5x†lva! {}¨ˆ‰_–}tp—†b]‡ …–}tvnvuxˆ+ƒb)l

/

bAƒb)ˆ+nvtp{†q–l!†q–‡Kuƒ{†q«lAµ

¶u¥nv_q5{†tpa!}ƒu¥°)b)‡a!†lpl1}q«‡ˆ‰_–}tp—†bˆA{†q–lªnp†qXn‰l0+µ«_–bAtpbu¥q{†q5b(‡5u¥aHbAq«lªuƒ{†q9µ?uƒq¾nv_5bˆAXlªb¶_–bAtpbnv_5b

a!†—†q5bAnvuxˆYbA»?b]ˆn‰l$}tpb“q5bA—X¥b]ˆnvb]‡u¥qnv_–b¦9{†tpbAqZnp°Y¨©{XtpˆAb†±z^ _5b‚%†tvux}¬5ƒb)l

x

^†q–‡

v

‡mb)q5{}npu¥q5—!tpb)lv…«b]ˆ¤

npu¥‚XbAƒjHnv_5blv…–Xˆ+b£…«{Zlªu¥nvuƒ{†qÆ}q–‡šnv_–b£‚XbAƒ{mˆ+u¥noj†µ

f (t, ·, ·)

uƒl€np_5bbAƒb)ˆ+nvtp{†q…5tv{X¬–}¬–u¥ƒu¸nojQ‡5uƒlªnvtpu¥¬–rmnvuƒ{†qu¥q …–_–†lvblª…«†ˆ+b}nnvuƒaHb

t

±321bAtpb†µ«u¸n1uxl1Xlvlvr5aHb)‡np_–%nnp_5b(¬–Xˆ ² —†tp{†r5q–‡

/

…?{Xlvu¸npu¥‚Xb10 uƒ{†q«l$‡muxlonptvuƒ¬5rmnpu¥{Xq

f b

lv}nvuxloº–b]l

R

R f b (t, x, v)dv = 1

µlª{nv_«%n

−ρ

aHCj¾¬?bQlvbAbAq.†lYnp_5bHnv{†np}zˆ‰_«}tp—†b3‡5bAq–lvu¸noju¥q nv_–b y{Xuƒlplª{Xqb)wZr–}nvuƒ{†q

/

†±Ñ‘01np_–%nˆ+{Xr5…5ƒb)lYnv_–bHlvbA¥¨Ú¤êˆ+{†q«lªuxlonpbAqZn“bAƒb)ˆnptvuxˆ(º–bAx‡

−E

¶u¸np_np_5bHbAƒb)ˆ+nvtp{†q–l

‡5uƒlªnvtpu¥¬–rmnvuƒ{†q

f

±54êqšnv_5b“lvb)wZr5bAEµk¶b£lª_«}ƒ–¨©{X¥ƒ{%¶Ça!}qkj!}rmnp_5{†t‰lMuƒq‡5tv{X…5…5uƒq5—np_5bYaHuƒqkr–l lªuƒ—†q–l€}q«‡

tpb+¨©b)t nv{

ρ

^†q–‡

E

†l€np_5bˆ‰_–†tv—XbY‡mb)q–lªu¥noj†q–‡šnv_5bb)¥b]ˆnvtpuxˆ1º«bAx‡:±

^ _–ba!}nv_5b)aH}nvuxˆA† lv{†ƒrmnpu¥{Xq–l3nv{ nv_5b~€†r–ˆ‰_kj yztv{X¬5ƒbAa —Xu¥‚XbAqK¬kj

/

†±ƒ60Aµ

/

†±Ñ‘0H}q–‡¯nv_5b¾uƒq5u¥nvux}

‡–%np

f (0, x, v) = f 0

^/

†±- 0

_«C‚†bš¬?bAb)q·lªnvr–‡5u¥b]‡¯¨©{Xta!†qZj.jXb)}t‰l67!nv_5b—†ƒ{†¬«} b+´muxlonpbAq–ˆAbšuƒqKnpu¥aHbÆ{}¨1¶€b)

²

lª{X¥rmnpu¥{Xq–l(nv{ nv_5b

a3r5¥nvux‡mu¥aHb)q–lªuƒ{†q«}:…5tp{†¬5ƒbAa ¶€Xl€º–t‰lon…5tp{%‚†b]‡Quƒq )Ž8

-

¬kj

¢

tplvbAq b)‚:9¥;Eµ«}q–‡¶€b£tpb+¨©b)t nv_5btpb)X‡mbAt

np{¾nv_–bQ¨©r5tvnv_5b)t3tpb)lvr5¥npl

/

{†q ¶€b)

²

†q–‡ lva{k{†nv_Klv{†ƒrmnvuƒ{†q«l0“{†¨=< %nªn>9‘;Eµ?2${Xtplªn@9ё}Žmµ

-

’;Eµ< }t‰‡m{Zl

†q–‡@AYbA—X{†q–‡)9ьB;§µ–hmˆ‰_–}bA»?b)tC9-D ;Eµ5¦0u¥{Xq–l }q–‡ybAtvnv_–†aHbE9- ‘B;Eµ5†q–‡Qnp{(np_5btpbA‚ku¥b)¶ ¬?{Z{ ² {}¨?F£ƒXlvlvbAj

9ё†ŒB;E±C<€b)ˆA†r–lvb¶b}tpb(ˆ+{Xq–lvuƒ‡mb)tvuƒq5—!np_5b{†q5b(‡5u¥aHbAq«lªuƒ{†q–†…–tv{X¬5¥b)aƵO†q–‡¾¬?b)ˆ)}r–lvb{†r–tYqkr5aHbAtpuxˆA}

lpˆ‰_5b)abYq5bAb]‡5lMnv_5b£u¥q–u¸npuƒ†Olv{†ƒrmnpu¥{Xq!nv{3¬«b£u¥q

W ^2,1

µm}q–‡!_–bAq–ˆAb£ˆA{†qZnvuƒqkr5{†r–l)µZ¶b£ˆA}qšr–lªb£}qQb]}tp¥uƒbAt tpb)lvr5¥n{}¨G4ê{†t‰‡5}q–l ² uG9- H;

/

x%npbAtuƒaH…5tv{%‚Xb)‡¬kjš}q–‡~{k{†…?bAt†q–‡JI“¥uƒa!†lC9ƒB86;K0€¶_5{Hb)lªnp†¬5¥uxlv_5b)‡uƒq

np_5bb)†tvƒjML†’Xl —†ƒ{†¬–†Ob+´muƒlªnvb)q–ˆ+b†q–‡r5q5uxwXr–bAq5b]lvl€¨©{Xt$ˆ+x†lplªuxˆA†

/

ˆ+{XqZnvuƒqZr–{†r–l0€lª{X¥rmnpu¥{Xq–l€u¥qnp_5b{†q5b

‡5u¥aHbAq«lªuƒ{†q–†9ˆAXlªbX±

hmu¥q–ˆAb1np_5b£¬«b)—†uƒq5q5uƒq5—Xl{}¨9qkr5aHbAtpuxˆA}?…5x†lva!(lvu¥a3r5ƒ}nvuƒ{†qQuƒqQnp_5bCL†’†l)µZaH†qkj3npb)ˆ‰_5q–uƒwZr5b]lM_–C‚Xb1¬?bAb)q

uƒqk‚†b]lonpu¥—Z%nvb]‡:µk}aH{†q5—3¶_5uxˆ‰_Qnv_–b“¦}—Xtp†q5—†ux}qNª…–}tvnvuxˆ+ƒb1uƒqˆAbAƒPO£abAnv_5{m‡

/

y4v~Q0¶_5uxˆ‰_ƈ+{†q«lªuxlon‰lM{†¨

†…5…5tp{C´mu¥a!%npu¥q–—Hnv_5b3…5ƒXlªa!!¬kj¾!º«q5u¸npb(qkr5a3¬«b)tY{†¨zaHXˆ+tp{}¤§…–}tvnvuxˆ+ƒb)l)±$^ _5b(nvt‰%|ob)ˆ+nv{Xtvuƒb)l{†¨Snv_5b]lªb

…«}tvnvuxˆ+ƒb)l£}tpb(ˆ+{Xa…–rmnvb]‡¾¨©tp{†a nv_–bˆ‰_–†tpXˆnpbAtpuƒlªnvuxˆˆ+r5tp‚†b]l

t → (X(t; s, x, v), V (t; s, x, v))

^{†¨znv_–b

­YƒXlª{%‚!b]wZr–%npu¥{Xq

/

†±ƒ60nv_–}n1}tpblª{X¥rmnpu¥{Xq–l€{}¨

∂ t X(t) = V (t), ∂ t V (t) = E(t, X (t))

^/X±”R0

¶u¥nv_Èu¥q–u¸npuƒ†S‚C†¥r–b)l

X (s) = x

^†q–‡

V (s) = v

µ:¶_5b)tvb]†l£lªb)¸¨Ú¤êˆ+{Xq–lvuƒlªnvb)qXnYº–bAx‡5l“}tpb3ˆA{†aH…5rmnpb)‡¬Zj

—Z%np_5bAtpu¥q–—Ynp_5b1ˆ‰_–†tv—Xb}q–‡Hˆ+r–tvtpbAqZnM‡mb)q–lªu¥nvuƒb)lz{}¨Onv_5b1…–}tvnvuxˆ+ƒb)lS{Xq!“ab]lª_!{†¨?…–_–†lvb$lª…«†ˆ+b

/

lªb)b"9ƒ)’B;

¨©{Xt“a{Xtvb‡mb+n‰}uƒƒl0± Ä {†t“{Xq5b‡5u¥aHbAq«lªuƒ{†q–†Slªuƒa(r5x%npu¥{Xq–l)µSMr5¥b)tvux}q aHb+np_5{m‡5l£¥u² b3º–q5u¥nvb!‡mu¥»?b)tvb)q–ˆ+b

aHbAnv_5{m‡5l)µº–q5u¥nvb bAƒbAaHb)qXnaHb+nv_–{k‡–l†q–‡Çˆ+{Xq–lvbAtp‚C}nvuƒ‚†b¾¹«rm´Ç¬–}x}q–ˆAbaHb+np_5{m‡5l_–C‚†b†ƒlv{¯¬?bAb)q

uƒqk‚†b]lonpu¥—Z%nvb]‡:µœ¶bQtpb+¨©b)tnp{ 9‘ - ;€¨©{†t3¾¨Ù}uƒtvƒjÈbA´m_–}r–lªnvuƒ‚†bštvb)‚ZuƒbA¶ {}¨np_5bšƒu¥nvbAt‰%npr5tpbQ{†q qkr5aHbAtpuxˆA}

lvuƒa(r5x%npu¥{Xq¯{}¨$np_5bÆ­1x†lv{%‚.b)wZr–%npu¥{Xq9± Ä {Xt3_–u¥—X_5bAtH‡muƒab)q–lvu¥{Xq–}lvu¥a3r5ƒ}nvuƒ{†q–l)µnv_–by 4v~ aHb+nv_–{k‡

¶ †l…5tpb+¨©b)tvtpb)‡np{(np_5b“—†tpuƒ‡!¬«†lvb)‡!aHb+np_5{m‡š‡mr–bYnp{np_5bAuƒtˆ+{†aH…5r5np%npu¥{Xq–}Oˆ+{Zlon]µku¥q…–}tvnvuxˆ+r5x}t€¶_5b)q

º«q5blpˆA}ƒb)lH¶€bAtpb¾tpb)wZr5uƒtvb]‡Knp{¯lªuƒa(r–ƒ}nvbXˆAˆ+r–tp}nvbAƒj np_5b‡mb)‚†bAƒ{†…?bAaHb)qXnQ{†¨“º–ƒ†aHbAqZnp}nvuƒ{†q–lHnp_–%n

{†¨Únvb)q!{kˆ)ˆ+r5t‰luƒqHnv_5b$…–_–†lvb$lª…«†ˆ+bX±œd1qm¨©{†tvnvr–q–%npbAƒj†µ}np_5uxlzaHb+np_5{m‡H…5tv{m‡mr«ˆ+b)lM}q!uƒq5_5bAtpbAqZnzqkr5aHbAtpuxˆA}

q–{†uxlªb“np_–%n£…5tvb)‚†b)qXn‰l ¨©tp{†a ‡mb)lpˆ+tpu¥¬–u¥q5—!…–tvb]ˆ+uxlªb)¥jšnv_5b(np}uƒ0{†¨Snv_5b3‡muƒlªnvtpuƒ¬5rmnvuƒ{†qƨ©r–q–ˆnpu¥{Xq

f

µ?†q–‡Æuƒq np_5bQx%npbQŽ†’†l)µnv_5bQuƒq–ˆAtvb]†lvb{†¨$ˆ+{†aH…5r5np%npu¥{Xq–}M…«{%¶€bAt†¥ƒ{%¶b]‡Ènv{¾_–†q–‡mƒb!no¶{‡5u¥aHbAq«lªuƒ{†q–†M}q«‡

b)‚†b)quƒqlv{†aHbˆAXlªb£nv_–tvb)b(‡muƒab)q–lvu¥{Xq–}9…–tv{X¬5¥b)a!l ¶u¸np_—†tpuƒ‡¬–Xlªb]‡aHbAnv_5{m‡5l)± ¢ aH{†q5—!np_5b)lvb†µ–nv_5b

lvbAaHu¥¤§¦}—Xtp†q5—†ux}q(lpˆ‰_5b)ab{Xtvuƒ—†uƒq–†¥ƒjuƒqXnptv{m‡mr«ˆ+b)‡¬kjH~_–bAq5—†q–‡,I“q5{†tptQ9¥1LB;?†q–‡tvb]ˆ+{Xq–lªux‡mb)tvb]‡(¬kj

hm{†q5q5b)q–‡mt –ˆ ² b)t)µ9„${kˆ‰_–b†µ&<€b)tªnptp†q–‡†q–‡ F£_5u¥°)°A{9”X’B;ˆA{†q–lvuxlon‰l“{}¨ˆ+{†aH…5r5nvuƒq5—¾%nb)Xˆ‰_npu¥aHbQlªnvb)…

º«q5u¸npb(b)¥b)ab)qZnnojZ…?b(lv{†ƒrmnvuƒ{†q¾{Xq¾!—†uƒ‚†b)qaHb]lª_Æ{}¨S…5_«†lvblª…–Xˆ+b¬kjÆH¬–†ˆ ² ¶ }t‰‡ˆ‰_–†tpXˆnpbAtpuƒlªnvuxˆ

aHbAnv_5{m‡:±?4§n$ˆA†qš¬?btvb)—X†tp‡mb]‡!†l†q%nvnvb)a…5nnv{Hˆ+{†a3¬5uƒq5b1np_5b†‡5‚C†qZnp}—Xb)l{}¨0¬?{}np_JSzr5ƒbAtpuƒ†q}q«‡

¦}—Xtp†q5—†ux}qX‡m‚†b]ˆnvuƒ{†qÈlvˆ‰_5b)aHb)l£u¥qÈnv_«%nu¸nC‚X{†ux‡5lnp_5bH{}¨Únvb)qnp{k{tpb)lªnvtpuƒˆ+nvuƒ‚†bQ~ Ä ¦·ˆA{†q–‡mu¥nvuƒ{†q.†l

¶€bAƒMXl£nv_5bš‡5uƒlªnv{Xtªnpu¥{XqÈ{†¨np_5bQaHb)lv_ nv_–}n(…5tpb)ˆ+uxlvbAƒj{mˆAˆ+r–tpl“¶_5b)tvbHnp_5bHº–x}aHbAqZn‰%nvuƒ{†q ‡mb)‚†b)¥{X…–lA±

e{†tpb…5tpb)ˆAuƒlvbAƒj†µ}nv_5baHb+np_5{m‡3np

²

b)l†‡5‚C†qZnp}—Xb{†¨«np_5b$ˆ+{Xq–lvbAtp‚C}nvuƒ{†q3{}¨–np_5bb+´5Xˆnzlª{X¥r5nvuƒ{†q

f

^{}ƒ{†q–—}

np_5bˆ‰_–†tpXˆnvb)tvuxlªnvuxˆ1ˆ+r–tv‚Xb)l

/

X±”R0€¬Zj‡mb+º–q–u¥q5—Qq5{Xq5ƒu¥q5b]}t bA‚X{†ƒrmnvuƒ{†q{†…?bAt‰%np{†t

S _∆t

^Xl

S _∆t f (t ⁿ )(x, v) := f (t ⁿ , X(t ˜ ⁿ ; t ⁿ⁺¹ , x, v), V ˜ (t ⁿ ; t ⁿ⁺¹ , x, v)) ' f (t ⁿ⁺¹ , x, v)

^/†±ÑŒ0

¶_–bAtpb

∆t

uxl$Hr5q5u¥¨©{†tpa nvuƒab3lonpbA…}q«‡

t ⁿ = n∆t

^±321bAtpb

X ˜

^}q«‡

V ˜

}tpb“†q¾}…5…5tp{C´mu¥a!}nvuƒ{†q{}¨

X

†q–‡

V

{X¬mnp†u¥q5b]‡¬kjtpbA…5x†ˆAu¥q–—šuƒq

/

†±”01nv_5bHb+´5Xˆn

/

r5q ² q5{%¶q0$º«bAx‡

E(X (t), t)

¬ZjÈ}qȆ…5…5tp{C´mu¸¤

a!}nvb)‡º«bAx‡ˆ+{Xa…–rmnvb]‡¨©tv{Xa

f (t ⁿ )

µ?¶_5uƒˆ‰_¾a! ² b)l

S _∆t

^P ±$^ _5uƒl£†‡5‚†b)ˆ+nvuƒ{†q¾lªnvbA…¾uxlnp_5bAq

ˆA{†r5…–¥b]‡!nv{np_5bY…–_–†lvb£lv…–Xˆ+b“‡muƒlpˆ+tpb+npu¥°]%npu¥{Xqš¬kjQº«q5u¸npb“bAƒbAaHbAqZn nojk…«b“uƒqXnpbAtp…«{Xƒ}nvuƒ{†qQ{X…«b)tp}nv{Xt

P

^±

4êqÆ{Xtp‡mb)t€nv{Htvb)—X†u¥qƈ+{Xa…–rmnp}nvuƒ{†q–†:bQˆ+uƒbAq–ˆAj†µ5lv{†aHb†‡5†…mnvuƒ‚†b“‚XbAt‰lªuƒ{†q–l€{†¨0nv_–uƒl$lvbAaHu¥¤§¦}—Xtp†q5—†ux}q

aHbAnv_5{m‡ _–C‚†b¬?bAbAqtpb)ˆAbAqZnvƒjK‡mb)‚†bAƒ{†…?b)‡·uƒq 9-

Ž;1}q–‡ 9P8Zµ‘L;Eµ€¶_5bAtpbnv_5b†rmnv_–{†t‰lHr–lªbÆaH{%‚ku¥q–—

…–_–†lvb+¤êlª…«†ˆ+b“—Xtvux‡5l${Xt$uƒqXnpbAtp…«{Xƒ}nv{XtvjQ¶ C‚†b)¥bAnpl{†¨A1b]lªx}r5tpuƒbAt1†q–‡ A1r–¬5r–ˆ}±Y^ _5b…5tpu¥q«ˆ+uƒ…5¥b3lª_–†tvb]‡

¬kj(nv_5b]lªb1X‡5}…5nvuƒ‚†b$lpˆ‰_5bAaHb)lzuƒlznv{r–lvb${†q–¥j(nv_5bQNoq5b]ˆ+b]lvlp}tpjO—†tpuƒ‡…?{†uƒqZnplz¶_–bAqQlvu¥a3r5ƒ}nvuƒq5—…5x†lva!

¬?b)†a!l ¶u¸np_Ælva!}ƒ9lonptvr«ˆnvr–tvb]l€nv_–}n$a{%‚XbYt‰}…–uƒ‡mƒj!np_5tv{Xr5—†_nv_–b…5_–†lvblª…«†ˆ+bX±

^ _–br–lvbÆ{}¨YX‡5}…5nvuƒ‚†bša3r5¥nvuƒtvb]lª{X¥rmnpu¥{Xq¯npb)ˆ‰_5q5uxwZr5b)l3¨©{XtH‡5b)}ƒuƒq5—ȶu¥nv_·nv_–bqkr5aHb)tvuxˆA†Mnptvb]%nvaHb)qXn

{†¨nptp†q–lv…«{Xtªn$y A Sl1_«†l$†¥tpb)X‡mjš!lvr5¬–lªnp}qZnpuƒ†9_5uxlonp{†tpj†µm¨©tp{†a †‡5†…mnvuƒ‚†baHb)lv_ÆtvbAº–q5b)ab)qZn9”;0nv{

¶ C‚†b)¥bAnª¤§¥u² b3nvb]ˆ‰_5q5uxwZr5b)l@9‘8Zµœ‘ D ;E±J4êq¯lvr–ˆ‰_ nvb]ˆ‰_5q5uxwXr–b)l)µœ ² bAj uƒlplªr–b!uƒlnv{‡mb]lªuƒ—†q.np_5bš†‡–}…mnpu¥‚Xb

‡5uƒlpˆ+tpb+npu¥°]%nvuƒ{†qu¥q¾lvr–ˆ‰_ÆH¶€CjHnp_–%n

†±^ _5b€bAtptv{Xt9¬«bAno¶b)bAq(nv_5b€qkr5aHbAtpuƒˆ)}Xlv{†ƒrmnpu¥{Xq(}q–‡nv_5b€b+´5†ˆ+nœlv{†ƒrmnvuƒ{†q(uƒl H 'R! '

@C '! H@ ±Èbtpb+¨©b)t u¥qƅ–†tªnpuƒˆAr5ƒ†t€nv{J9¥ D ;§µ5u¥q¶_5uxˆ‰_šnp_5uxl—†{Z}OuƒlXˆ‰_5uƒbA‚†b]‡

u¥qnp_5bˆ+{XqXnpb+´kn{}¨œº–q5u¥nvb‚X{†ƒr5aHb“lpˆ‰_5b)ab]l€¨©{†t_kjk…«b)tv¬?{†ƒuxˆ£ˆA{†q–lvbAtp‚%%npu¥{XqšxC¶$lA±

‘m±^ _5bšX‡5}…mnpu¥‚XbQ‡muxlvˆAtvbAnvuƒ°)%npu¥{Xq ¨©{†t(Xˆ‰_5u¥b)‚ku¥q5—Ænp_5uxl…5tpb)lpˆ+tpu¥¬?b)‡.XˆAˆ+r–tpXˆ+j_–Xl #, #E$

± ¢ …«{Zlvlvuƒ¬5¥b€¬?bAq–ˆ‰_5a!†t ² np{Y…5tp{†…?bAtpƒj“‡5b+º–q5b†q–‡b)‚C†¥r«%nvbnv_–uƒlœq5{†nvuƒ{†q({†¨–{†…5nvuƒaH†¥u¥noj

uƒlY…5tp{%‚kuƒ‡mb]‡Æ¬kj P ' #,( * µ:¶_5uxˆ‰_‡mb)lpˆ+tpu¥¬?b)l1nv_–b({†…5nvuƒaH†0nptpX‡mb+¤§{}»

¬«bAno¶b)bAqQXˆAˆ+r–tpXˆ+j}q«‡ˆA{†aH…5ƒb+´mu¥noj(uƒq3npbAtpa!lS{}¨:lª…?b)ˆAu¸º?ˆ <€b)lv{%‚X¤ohk{X¬«{X¥b)‚lªaH{k{}np_5q5b)lpl…5tp{†…m¤

bAtvnvuƒb)l {†¨np_5blª{X¥r5nvuƒ{†q9µ5lvbAb“uƒq…«}tvnvuxˆ+r5x}tC9ёm}µ«‘X‘kµ9CŒkµ5Ž;§±

4êqnv_–uƒl …–†…«b)t)µm¶€b£…5tv{X…«{Zlªb“q5bA¶=†‡5†…mnvuƒ‚†b“lvbAaHu¸¤ê¦0†—†t‰}q5—Xuƒ†qQlvˆ‰_–bAaHb†µkuƒq¶_5uxˆ‰_nv_5b“º–t‰lªn †¬«{%‚Xb

‡5b)lpˆ+tpu¥¬?b)‡Æ{X¬m|ob)ˆ+nvuƒ‚†buxlYXˆ‰_5uƒbA‚†b]‡:± b3}xlv{!—†uƒ‚†bnp_5bA{XtvbAnvuxˆA}_5bAr–tvuxlonpuƒˆ)l1†q–‡Æqkr5aHbAtpuƒˆ)}0b)‚kuƒ‡mb)q–ˆ+b

np_–%n1nv_5b(X‡5}…5nvuƒ‚†b“—†tpuƒ‡–l—†b)q5bAt‰%npb)‡¬kjQnv_–uƒlYlvˆ‰_5b)aHb_–C‚†b{X…mnvuƒa!}0ˆA{†aH…5ƒb+´mu¸nojX± 4êqnp_5uƒlYlvˆ‰_–bAaHb†µ

np_5b1X‡5}…mnpu¥‚Xb ab]lª_!uxlSb)‚†{†ƒ‚†b]‡np{†—Xb+nv_–bAtS¶u¥nv_Hnv_–b1lª{X¥r5nvuƒ{†qH¬5rmn†¥¶ Cjmltvb)a!}uƒq–lu¥qZnp{“…–}tvnvuxˆ+r5x}t

ˆAƒXlvl€{†¨_5uƒbAt‰}t‰ˆ‰_5uxˆA}«º–q5u¥nvbb)¥b)aHbAqZn nvtpuƒ†q5—†r5x%npu¥{Xq–l)±^ _–b£q–{}nvuƒ{†q{}¨œ—†{k{m‡X‡5}…5np%npu¥{Xqš{†¨œaHb)lv_

M

np{33—†uƒ‚†b)qH¨©r5q–ˆnpu¥{Xq

g

uƒlaHb)†qXn€_5bAtpb1uƒqQnp_5b“lªb)q–lªb1nv_–}nnp_5b£u¥qZnvb)tv…?{†x%npu¥{Xq!bAtptv{Xt

(I − P M )g

^uxl

lva!}ƒ¥b)tuƒqnv_5b

L ^∞

^q5{†tpa nv_–†q3…5tpb)lpˆ+tpu¥¬?b)‡Qnv{†ƒbAt‰}q«ˆ+b

ε

±?<€rmn¶_5bAtpb)Xlznp_5b‡mb)lvuƒ—†qš{†¨

M

^{uxl¶€bAƒ}

r–q–‡mbAt‰lªnv{k{m‡Æ¶_5bAq

g

^{uƒl ²} q5{%¶q9µ?…5tvb]‡muxˆnvuƒq5—!np_5b(aHb]lª_¾¨©{XtYHnp{}¤§¬«bA¤§ˆA{†aH…5rmnpb)‡¾lv{†ƒrmnvuƒ{†q¾uxl£!aH{†tpb

‡5uQˆ+r–¸n€npXl ² ±54êqQ{†r5t€}ƒ—†{Xtvu¥nv_5a nv_–uƒluƒlb+´mb]ˆ+rmnpb)‡!uƒq!no¶€{(lonpbA…–l67œnv_5b1º–t‰lon lonpbA…Qr–lvb)lMnv_–bY†‡–}…mnpu¥‚Xb

aHb]lª_!%nSnp_5b…5tpbA‚kuƒ{†r–lœnpu¥aHb1lªnvb)…3np{†—Xb+nv_–bAtS¶u¥nv_Hnv_–bqZr–ab)tvuxˆA†knvt‰}q–lv…?{†tvnœ¹«{%¶·uƒqH{†t‰‡mbAtnp{“‡5b)lvu¥—Xq

Æq–bA¶ †‡5†…mnvuƒ‚†bHaHb)lv_9µ}q–‡Ènv_–bQlªb]ˆ+{†q«‡ lªnvb)… lªƒu¥—X_ZnvƒjȈ+{Xtvtpb)ˆ+nplYnv_–uƒl(ab]lª_.¬kjƒ{mˆA†StvbAº–q5bAaHb)qXn

{Xt£ˆA{X†tplvbAq5uƒq5—Q¬«†lvb)‡{Xqnp_5b

E+! H( H

}q–†¥jmlvuƒl1{}¨znp_5bHqZr–ab)tvuxˆA†œlv{†ƒrmnpu¥{Xq{†¬5np}uƒq5b]‡¬kjÆnv_5b

lvbAaHu¥¤§¦}—Xtp†q5—†ux}qH}…5…–tv{Z†ˆ‰_H{†qHnp_5bYuƒqZnvb)tvaHb)‡5uƒ}nvb$aHb)lv_9±S^ _5uxllvb)ˆA{†q–‡!lªnvb)…QuxlˆAtvr–ˆAuƒ†–uƒqQ{†t‰‡mb)tznv{

ˆA{†qZnvtp{†–nv_5bXˆAˆ+r–tpXˆ+j{}¨9np_5b“qZr–ab)tvuxˆA†?lpˆ‰_5b)abXµk¶_5uƒˆ‰_‡mbA…?bAq«‡5l€{†qš¬?{}np_šnp_5bYnvuƒablªnvb)…

∆t

^}q«‡

np_5b“nv{X¥b)tp†q–ˆ+b

ε

^±

^ _–b¯lon‰}tvnvuƒq5—…«{Xu¥qZn{}¨!{Xr5t¾¶{Xt ² ¶€Xl¾ º–tplªn¾‚†b)tplvu¥{XqG{†¨Ha3r5¥nvuƒtvb]lª{X¥rmnpu¥{Xq=X‡5}…5nvuƒ‚†b lªb)aHu¸¤

¦}—Xtp†q5—†ux}qlvˆ‰_5b)aHb$…5tp{†…?{Xlvb)‡¬kjHhk{Xq5q5b)q–‡mt –ˆ ² b)tM}q–‡š~{X_5bAq0µX¶_5uxˆ‰_!¶ †lSuƒaH…5¥b)aHbAqZnvb]‡!u¥q‘}’†’ -

}n$nv_–b!~b)at‰†ˆ)l$lvr5aHab)tYlpˆ‰_5{k{†u¥q¦0r5aHu¥qkjXµ?lvbAb>9¥ - ;Eµ?j†bAnY¶u¥nv_5{Xrmn£†qkjtpu¥—X{†tp{†r–l†q–}ƒjmlªuxl)±$„b+¤

—Z}t‰‡muƒq5—Hnv_5b3bAtptv{Xtb)lªnvuƒa!%nvb]lAµ?¶€ba!}uƒq5¥j¾‡mtpbA¶ {Xr5t1uƒq–lv…5u¥t‰%npu¥{Xqƨ©tv{Xa&np_5b(…–tv{k{}¨S{†¨SˆA{†qk‚†b)tv—XbAq–ˆAb

—Xu¥‚XbAq3¬Zj<b]lvlvb u¥q 9L;m¨©{†tz£r5q–u¸¨©{Xtva ‚XbAt‰lªuƒ{†q3{}¨«nv_5b$lvbAaHu¥¤§¦}—†t‰}q–—†ux}qlpˆ‰_5bAaHb†µ†¬5rmnz_–X‡np{“ˆ‰_«}q5—Xb

np_5blv…5u¥tpu¥n {†¨œu¥n)± blv_–}ƒ:¬5tpu¥bA¹–jQb+´m…5x}uƒq¶_ZjX±z^ _5br5q5u¥¨©{†tpa&lpˆ‰_5bAaHb£¨©{†tpa!}ƒ¥jštvb]†‡5l †l

f _u ⁿ → f _u ⁿ⁺¹ = P h S _∆t f _u ⁿ ,

¶_–bAtpb

P h

‡mbAq5{†nvb]lnv_–bQu¥qZnpbAtp…«{Xƒ}nvuƒ{†q.{†…?bAt‰%np{†t{†q.np_5b!º–q–u¸npbQbAƒbAaHbAqZn3lv…–Xˆ+b!{}¨r5q–u¸¨©{Xtva'aHb)lv_

lvuƒ°Ab

h

^µ«}q–‡ <€b)lplªb‡mb]ˆ+{†aH…?{Xlvb)lnv_–bqZr–ab)tvuxˆA†:bAtptv{XtuƒqZnv{

f (t ⁿ⁺¹ ) − f _u ⁿ⁺¹ = [f (t ⁿ⁺¹ ) − S _∆t f (t ⁿ )] + (I − P h ) S _∆t f (t ⁿ ) + P h [ S _∆t f (t ⁿ ) − S _∆t f _u ⁿ ].

Yq«ˆ+b—†uƒ‚†bAq }qÈb)lªnvuƒaH}nvb¨©{†t£np_5bnvuƒab!‡muxlpˆ+tpb+nvuƒ°)}nvuƒ{†qÈbAtptv{Xt

kf (t ⁿ⁺¹ ) − S _∆t f (t ⁿ )k L ^∞

µ:nv_–b ² bAj

…?{†uƒqZn€uxlMnv{aH ² bYr–lªbY{}¨9np_5b

C ²

lvaH{k{}nv_–q5b)lplM¶_–uƒˆ‰_Quxl †lplªr–ab]‡H¨©{†tnv_–bYbA´mXˆnlª{X¥r5nvuƒ{†q9µZu¥q{†t‰‡mbAt np{Æb)lªnvuƒaH}nvbHnp_5bH…5tv{†|ob)ˆ+nvuƒ{†q bAtptv{Xt

k(I − P h ) S _∆t f (t ⁿ )k L ^∞

±,4êq.ˆ+{XqXnptpXlon]µ0‡mb)q5{}npu¥q5—Ƭkj

M ⁿ

^nv_5b

aHb]lª_Ɔlplª{mˆAuƒ}nvb)‡Qnv{nv_–b†‡5†…mnvuƒ‚†b£qkr5aHbAtpuƒˆ)}9lv{†ƒrmnvuƒ{†q

f ⁿ

µmnv_5bX‡5}…5nvuƒ‚†b“lvˆ‰_5b)aHbY¨©{Xtva!†¥ƒj!tvb]†‡5l

Xl

f ⁿ → f ⁿ⁺¹ = P _M ⁿ⁺¹ S _∆t f ⁿ ,

¶_–bAtpbnv_–baHb]lª_

M ⁿ⁺¹

_–†lY¬«b)bAq {†¬mn‰}uƒq5b)‡¾¨©tv{Xa

M ⁿ

¬kjÆnv_5bH}¬?{%‚†b3ab)qZnvuƒ{†q5q5b]‡lªnvt‰%npbA—†jX±C4êq {Xtp‡5bAtnv{u¥q–ˆA{†tp…«{Xtp}nvbnp_5uxllonptp}nvbA—XjHu¥q{†r5t}q«}ƒjklvuxlAµZ¶b£_–X‡!nv{H‡mb]ˆ+{†aH…?{Xlvb1np_5b“qkr5aHbAtpuƒˆ)}?bAtptv{Xt

XˆAˆA{†t‰‡mu¥q–—(np{

f (t ⁿ⁺¹ ) − f ⁿ⁺¹ = [f (t ⁿ⁺¹ ) − S _∆t f (t ⁿ )] + (I − P M ⁿ⁺¹ ) S _∆t f ⁿ + [ S _∆t f (t ⁿ ) − S _∆t f ⁿ ].

d1q5ƒu² bYuƒqšnp_5b£º–t‰lon‡mb)ˆA{†aH…?{Xlvu¸npu¥{Xq9µZnv_5b“uƒqXnpbAtp…«{Xƒ}nvuƒ{†qQb)tvtp{†tuƒl q–{%¶ ˆ+{†q«lªux‡mbAtpb)‡š{†q

S _∆t f ⁿ

^¶_5uxˆ‰_

uxlƒb)lpl$lva{k{†nv_nv_–†q

S _∆t f (t ⁿ )

^± b(}tpb£nv_5b)q¾¥bA¨Ún1np{Qlonpr–‡mjnv_5btpbA—Xr5ƒ†tvu¥nojQ{†¨np_5bqkr5aHbAtpuƒˆ)}0lv{}¤

ƒrmnpu¥{Xq–l }q–‡Q¨©{†t€nv_5uxl …5r5tp…?{Xlvb†µk¶b£uƒqXnptv{m‡mr«ˆ+b“3‡muxlpˆ+tpb+nvb£ˆ+r5tp‚%%npr5tvb£aHb)Xlªr5tpb

| · | ?

nv_–}nlªnp†q–‡5lM¨©{†t

Q¶b] ² b]wZr5u¥‚%†¥b)qXnY{}¨nv_5bHhk{†¬?{†ƒbA‚

W ^2,1

lªb)aHu¸¤§q5{†tpa ¨©{†t1ˆA{†qZnvuƒqkr5{†r–l1…5uƒb)ˆ+b)¶uƒlvb( Hq–b¨©r5q–ˆ+nvuƒ{†q–l {†¨

R ²

^± Yr–ttvb]lªr–¸npu¥q5—¾b)tvtp{†tb]lonpu¥a!}nvb)l(}tpb3np_5bAq b]lon‰}¬5ƒuxlª_5b]‡.r5q–‡mb)tnv_–bšlª{X¥bQXlvlvr5aH…mnpu¥{XqÈnp_–%n np_5buƒq5u¸npuƒ†0‡5}npuƒl$u¥q

W ^1,∞ ∩ W ^2,1

µ5¶_–u¥ƒb“nv_5{Zlªb“{†¨39LB;Xlvlvr5aHb

C ²

lªaH{k{}np_5q5b]lvl)±zc${†nvb£nv_–}n¨©{†t lv{†ƒrmnpu¥{Xq–l¶u¥nv_¾ƒ{kˆ)}0lvuƒq5—†r5x}tpu¥nvuƒb)llvr–ˆ‰_†l º«ƒ†ab)qZnp%npu¥{Xq–l€¤M¨©{Xt$¶_5uxˆ‰_†‡5†…mnvuƒ‚†b‡muxlpˆ+tpb+nvuƒ°)}nvuƒ{†q–l

†tvb£{†¬k‚kuƒ{†r–lv¥j!q–bAb)‡5b)‡¤Snv_5b

C ²

q5{†tpauƒl$a(r–ˆ‰__5uƒ—†_5b)t nv_–†qnp_5b

W ^2,1

^q–{†tpa±

Yr–tš…–†…«b)tlªnp†tªn‰lšuƒq ‘¯¶u¸np_GK¬5tvuƒb+¨tpbAaHuƒq–‡mb)t{†¨nv_5bÈa!}uƒq6…5tp{†…?bAtvnvuƒb)lš{}¨nv_–bÈbA´mXˆnÆlª{X¥rm¤

npu¥{Xq–l}q«‡{†¨np_5bHnvuƒaHbQlª…5ƒu¥nªnvuƒq5—lpˆ‰_5bAaHb

S _∆t

{†¨$~_–bAq5—¾}q«‡)I“q5{Xtvt]± ¢ q5bA¶ b)lªnvuƒa!%nvbHuxl—†uƒ‚†b)q

¨©{Xt£nv_5b!ˆ+{Xtvtpb)lv…?{†q–‡muƒq5—šnvuƒab!‡muxlpˆ+tpb+nvuƒ°)}nvuƒ{†qÈbAtptv{XtY¶_5uxˆ‰_{Xq5ƒju¥qk‚X{†ƒ‚†b)l1nv_–b

W ^1,∞

lvaH{k{}nv_–q5b)lplA±

^ _–blªnvt‰%npbA—†j¨©{†t£—†b)q5bAt‰%npu¥q–—Qnv_–bX‡5}…mnpu¥‚Xb(aHb)lv_5b]l1uxlYnv_5b)qȇmb]lvˆAtvuƒ¬«b]‡uƒq - nv{X—†bAnv_5b)tY¶u¥nv_Ènv_5b

qkr5aHb)tvuxˆA†œlpˆ‰_5bAaHbXµ9}q–‡nv_5bHa!}uƒqȈA{†qk‚†b)tv—XbAq–ˆAbtvb]lªr–¸n“uƒl“lªnp}nvb)‡9µ:nv{X—†b+np_5bAt£¶u¥nv_.lª{XabtpbAa!}t ² l

{Xqnv_5b{X…mnvuƒa!}ƒu¸noj{}¨œnv_5baHb+np_5{m‡:±€hmbA‚†b)tp†O…5tv{X…«b)tªnpu¥b]l {}¨œnv_5b3†‡5†…mnvuƒ‚†b“aHb)lv_5b)l†tvb“b]lon‰}¬5ƒuƒlv_5b]‡

uƒq ”5µO¶_5uƒˆ‰_†tvb(r–lªb]‡¾uƒq¾nv_5b3…5tp{Z{†¨S{}¨Snv_–b3ˆ+{Xqk‚†bAtp—†b)q–ˆ+b“tpb)lvr5¥nY—†uƒ‚†b)q¾u¥q Œk±“hkbA‚XbAt‰}9qkr5aHbAtpuxˆA}

npb)lªnpl†tvb“—†uƒ‚†b)quƒq LH¶_5uxˆ‰_uƒƒ¥r–lªnvt‰%npb£nv_5bbA»?b]ˆnvuƒ‚†b)q5b)lpl {}¨{†r–t †…5…5tp{XXˆ‰_9±

"'€3+ *³œÆ,+$# ‰œ* A# +*0œ€œ+

21bAtpbuƒlY!aH{†tpb…5tpb)ˆ+uxlvb‡mb)lpˆ+tpu¥…5nvuƒ{†qÆ{}¨S{†r–t1…5tp{†¬5ƒbAaƱ Ä {X¥ƒ{%¶u¥q–—M<b]lvlvbuƒq 9L;Eµ«¶b(ˆA{†q–lvux‡mbAtnp_–%n

np_5b!…5x†lva!Æuƒl!A¤E…?bAtpuƒ{k‡5uƒˆHuƒq nv_5b

x

‡mu¥tpb)ˆ+nvuƒ{†q9±>4êq.{}nv_–bAt¶€{†t‰‡5l)µ

/

X±¥B0§¤

/

†±

-

0“_–{†x‡¨©{Xt

x ∈ [0, 1]

¶u¥nv_Ƭ?{†r5q«‡5}tpjšˆ+{Xq–‡mu¥nvuƒ{†q–l

f (t, 0, v) = f (t, 1, v)

^/‘5±L0

†q–‡

E(t, 0) = E(t, 1).

^/‘5±80

¢ ˆ)ˆ+{Xtp‡muƒq5—3nv{nv_–byœ{Xuƒlplv{†qb)wZr–%npu¥{Xq

/

†±Ñ‘0µ5nv_5bx%nvnvbAtuxlb]wXr–u¥‚%}ƒbAqZn nv{

Z

ρ(t, x) dx = Z Z

f (t, x, v) dx dv − 1 = 0

^{/‘5±D 0}

/

r5q5ƒb)lpllv…«b]ˆ+u¥º–b)‡:µnv_–bQu¥qZnpbA—†t‰}xl£u¥q

x

^†q–‡

v

†tvb!}ƒ¶ Cjkl£n‰ ² bAq tvb]lª…?b)ˆ+nvuƒ‚†bAƒj{%‚XbAt

[0, 1]

}q–‡.}ƒ

R

0+µ–¶_5uxˆ‰_ÆaHb)}q«l nv_–}nnv_5b(…5ƒXlªa!uxl$—†ƒ{†¬«}ƒ¥jQq–bArmnptp†Ïµ–†q–‡Æ}xlª{HuƒaH…5¥uƒb)lnv_–}n$nv_5b(—†ƒ{†¬–†:aHXlvl uxl“ˆ+{Xq–lªb)tv‚Xb)‡:± Ä uƒq–}ƒƒj†µ:¶€blvbAb3nv_–}n“nv_5bHbAƒb)ˆ+nvtpuƒˆ(º–b)ƒ‡uƒl‡mbAº–q5b]‡r5…Ènv{ƈA{†q–lªnp†qZn)µ9lv{šnp_–%n“¶b

ˆ)}q5q–{}n_–C‚†b“¶b)¥¥¤§…«{Zlªb]‡š…5tp{†¬–¥b)a³r5q5ƒb)lpl ¶bX‡5‡3°)bAtp{}¤§aHb)}qbAƒb)ˆ+nvtp{Xlªnp%npuƒˆ£ˆ+{Xq–‡mu¥nvuƒ{†q

Z

E(t, x) dx = 0.

^/‘m±Ž0

¢ ˆ+x†lplvuƒˆ)}?¶€Cj3nv{tvb]†‡Hnp_5b“y{†uxlvlv{†qQb]wZr–%npu¥{Xq

/

X±‘0uxlznp{3uƒqZnvtp{m‡mr–ˆ+bYnv_–bYb)¥b]ˆnvtp{Xlªnp}nvuxˆ$…?{}npbAqZnvux}

ϕ = ϕ(t, x)

lvr–ˆ‰_np_–%n

E(t, x) = −∂ x ϕ(t, x)

±MAYbAq5{†nvuƒq5—ƬZj

G = G(x, y)

np_5b@F£tvb)bAq¨©r5q–ˆnpu¥{Xq

Xlvlv{mˆ+ux%npb)‡!np{H{†r5t…5tp{†¬–¥b)aµknp_–%n$uxl np{HlpCj†µk¨©{Xt

y ∈ [0, 1]

µknv_5blv{†ƒrmnvuƒ{†q{}¨

∂ _xx ² G(·, y) = δ(· − y) on [0, 1]

¶u¥nv_ƅ?bAtpu¥{m‡muxˆ£¬«{Xr5q–‡5†tvjˆ+{†q«‡mu¸npu¥{Xq–l

G(0, y) = G(1, y)

µm¶€b“{†¬mn‰}uƒq

E(t, x) = Z

K(x, y) Z

f (t, y, v) dv − 1

dy

^{/‘m±ƒ)’0}

¶_–bAtpb

K(x, y) = −∂ x G(x, y) =

y − 1 if 0 ≤ x < y

y if y ≤ x ≤ 1.

^/

‘m±ƒ†B0

¦0b+n

Ω = [0, 1] × R

±?4êq{†r5t$ˆA{†qZnvbA´kn)µknv_–bA{†tpbAa Œ({†¨z~{k{X…«b)t$}q–‡ I“¥uƒa!†lC9ƒ86;0tpb)X‡5l †l€¨©{†ƒƒ{%¶$lA±

½k ž ½ "!$#&%'#)(

f 0 ∈ C(Ω)

!B!H B+*-,$+ H(1()

x

^{*,#=+!1 H}

v

^*E

!!/. !

Z Z

f 0 (x, v) dx dv − 1 = 0,

K*+ K*+ 'M!H! 10 ! (

(f, E)

^{/X±¥B0 $ / †±- 0 * /‘m±L0 $/‘m±Ž0}

[0, T ]

^{2 ( (*}

K*R!=! ( ! !H K*+* ' H!H( HB!

/

†±”0 E G!!3.3H! *

/

‘m±ƒ)’0

K*+54 #=+ ' 76

∂ t E(t, x) = − Z

f (t, x, v)v dv + ¯ 

^{/‘5±¥C‘0}

6&* "K*+# H ' +!H

¯



^!@!H31.

¯

 = Z Z

f (t, x, v)v dv dx = Z Z

f 0 (x, v)v dv dx.

^{/‘5±¥ - 0}

& 89:;< =>-< ?@ A

4§nSuxlœq5{%¶¯¶€bAƒ ² q5{%¶qnv_–}nœnp_5blª{X¥r5nvuƒ{†q–l{}¨«nv_5b ­1x†lv{%‚Z¤Ey{Xuƒlplª{XqlªjmlªnvbAa lp%npuƒlª¨©jr5q5u¥¨©{†tpalvaH{Z{†nv_m¤

q–b)lplb]lonpu¥a!}nvb)lœ¨©{Xtƒ†tv—Xb

/

¬5rmnSº–q5u¥nvbB00nvuƒab]lA± ¢ lS}q(bA´5}aH…5ƒb€{†¨?lvr–ˆ‰_(tpb)lvr5¥n)µ%np_5b€¨©{†ƒ¥{%¶uƒq5—£¥b)aHaH

uxl€¨©{†r–q–‡u¥q9- LB;§±

B½ ¯ŸC!D#E!D#F(

f 0 ∈ W ^m,p (Ω)

^{!!/. ! K*+C* K*+!H! K*+ '#HG 2},I*?K*+"! (:!!/. !

. #

T

^J

f ∈ L ^∞ [0, T ]; W ^m,p (Ω) .

4êq {Xtp‡5bAt“nv{¾b]lon‰}¬5ƒuxlª_ }q b)tvtp{†t“b]lonpu¥a!%npb!¨©{†tnv_5b!npu¥aHbš‡5uƒlpˆ+tpb+npu¥°]%nvuƒ{†q0µ0¶€bQlª_–†¥Mq5bAb]‡ nv_–}nnv_5b

lv{†ƒrmnpu¥{XqQ}q–‡Hnp_5b£bAƒb)ˆnptvuxˆ º–b)ƒ‡š_–C‚†bYlª{Xab1tvb)—†r5x}tpu¸nojXµX†q–‡Qlvr Qˆ+uƒbAqZn tvb]wXr–u¥tpbAaHbAqZnMuxlMnv_–}n

f 0

uƒl

¦0u¥…«lvˆ‰_5u¥nv°1¤¨©{†tnv_5b“†‡5†…mnvuƒ‚†bYlvˆ‰_5b)aHb$nv{¬?bY¶€bAƒ«…?{Xlvb)‡9µZ¶€b1¶uƒƒ?x%npbAt †l

²

f 0

np{(¬?b£†ƒlv{(uƒq

W ^2,1

^±

e{†tpb1…5tpb)ˆAuƒlvbAƒj†µk¶€b£lv_–}ƒOa! ² b£r–lvbY{†¨9nv_–bY¨©{X¥ƒ{%¶uƒq5—(ƒbAaHa!

/

:K*+C!K0 L*?K*+

C

6? , +!H M6&*R(*)E#,>B J=* 1 '+*Q6&*R(* R!J+!

:!#,. #,

T

>) *" ( ! (

f 0

07

B½ ¯Ÿ !D#

# (

f 0 ∈ W ^1,∞ (Ω)

^!!3.3H!,K*+,* K*+!H! M * '# G 2

,7* * > .

#,

T < ∞

^{* * ! (J*+! !1+ H ) *}

v

^'(

Q(T ) := sup{|v| : ∃x, ∃t ∈ [0, T ], f (t, x, v) > 0} ≤ Q(0) + 2T

^{/‘m±ƒA”R0}

M !!/. !

kf k L ^∞ ([0,T];W ^{1, ∞} (Ω)) ≤ C(T ) k∂ t f k L ^∞ ([0,T];L ^∞ (Ω)) ≤ C(T ) kEk L ^∞ ([0,T];W ^2,∞ ([0,1])) ≤ C(T ) k∂ t Ek L ^∞ ([0,T];W ^1,∞ ([0,1])) ≤ C(T ) k∂ _tt ² Ek L ^∞ ([0,T];L ^∞ ([0,1])) ≤ C(T ).

/

‘m±ƒ]Œ0

Q '6

2

<€b]ˆA}r«lªb€nv_–b)lvb$lªaH{k{}np_5q5b)lplSb]lonpu¥a!%npb)lz†tvb ‚XbAtpj3lvu¥aH…5ƒb nv{{X¬mnp†u¥qHuƒqHnv_5b${Xq5b$‡muƒaHbAq–lvu¥{Xq–}

ˆ)†lvb†µm¶€b“tvb]ˆA}ƒO_5{%¶ np_5bAjš¨©{†ƒ¥{%¶6¨©tv{Xanv_–b­1x†lv{%‚X¤§y{†uxlvlv{†qlªjmlªnvbAaƱS^{!¬«b)—†uƒq¶u¥nv_9µ–¶b{X¬–lªb)tv‚Xb

np_–%n

kEk L ^∞ ([0,T ];W ^1,∞ ([0,1])) ≤ C(T )

^{/‘m±ƒ6L0}

†q–‡

k∂ t Ek L ^∞ ([0,T ];L ^∞ ([0,1])) ≤ C(T )

^{/‘m±ƒB80}

†tvb3b)lªnp†¬5¥uxlv_5b)‡È†l£lª{k{†q †l

f 0 ∈ C(Ω)

^/}q«‡¾nv_–bAjuƒq¨ÙXˆn}tpb(…5tp{%‚†b]‡Æu¥q 9ƒB81; 0+±"4êq–‡mb)b)‡:µ9¶€b3lvbAb np_–%nnp_5bˆ+{Xq–lvbAtp‚C}nvuƒ{†q{}¨

f

}ƒ{†q5—np_5bˆ‰_–}t‰†ˆ+nvb)tvuxlonpuƒˆYˆ+r5tp‚†b]l

/

†±”0€jkuƒbAx‡5l

0 ≤ f ≤ kf 0 k _L ^∞ _(Ω)

^{/‘m±ƒ D 0}

†q–‡

Q(T ) − Q(0) ≤ sup

(x,v)∈Ω

Z T 0

|∂ t V (t; 0, x, v)| dt ≤ T kEk _L ^∞ _([0,T];L ^∞ _([0,1])) .

^{/‘m±ƒ)Ž0}

dYlªuƒq5—!lvr–ˆAˆAb)lplªuƒ‚†b)¥j

/

‘m±ƒ)’0+µ

/

‘5±¥ D 0 }q«‡

/

‘m±D 0µ–¶b“_–C‚Xb1np_5bAq

kE(t)k _L ^∞ _([0,1]) ≤ kKk L ^∞

Z Z

|f (t, x, v)| dx dv + 1

≤ 2,

^{/‘m±Ñ‘}’0}

†q–‡

/

‘m±ƒA”0Y¨©{†ƒ¥{%¶$lY¨©tp{†a

/

‘m±ƒ)Ž01nv{X—†bAnv_5b)t¶u¸np_Ènp_5uxl“ƒXlon¬?{†r5q–‡9± ÈbQ‡mb)tvuƒ‚†b3nv_–bAq.tvb]lª…?b)ˆ+nvuƒ‚†b)¥j

¨©tp{†a³nv_5by{†uxlplª{Xq

/

X±‘0 }q–‡šnv_–b ¢ aH…«b)tvb

/

‘m±ƒ]‘0€b]wZr–%npu¥{Xq–l

k∂ x Ek L ^∞ ([0,T ];L ^∞ ([0,1])) ≤ Q(T )kf 0 k L ^∞ (Ω) + 1

†q–‡

k∂ t Ek L ^∞ ([0,T];L ^∞ ([0,1])) ≤ Q(T ) ² kf 0 k L ^∞ (Ω) + ¯ ,

¶_–uƒˆ‰_…5tp{%‚†b)l

/

‘m±ƒ6L0 †q–‡

/

‘m±ƒB80+±zc${%¶Gu¸¨

f 0 ∈ W ^1,∞ (Ω)

µ5ƒb+nvnvuƒq5—

(X, V )(s) = (X, V )(s; t, x, v)

(X ⁰ , V ⁰ )(s) = (X, V )(s; t, x ⁰ , v ⁰ ) and

e x (s) = |X (s) − X ⁰ (s)|

e v (s) = |V (s) − V ⁰ (s)|,

¶€b“_–C‚†b

|f (t, x, v) − f (t, x ⁰ , v ⁰ )| = |f 0 (X (0), V (0)) − f 0 (X ⁰ (0), V ⁰ (0))|

≤ |f 0 | W ^1,∞ (Ω) (e x + e v )(0).

~{XaH…5rmnvuƒq5—¨©tp{†a

/

†±”0nv_«%n

| e ˙ x (s)| ≤ e v (s)

^}q–‡

| e ˙ v (s)| ≤ |E(s, X(s)) − E(s, X ⁰ (s))| ≤ k∂ x Ek _L ^∞ _{([0,T ];L} ^∞ _([0,1])) e x (s)

¶€blªb)b£nv_–}n

/

‘m±ƒ6L0nv{†—Xb+np_5bAt ¶u¥nv_¾MF£tv{Xqk¶€†¥:}tp—†r–ab)qZn€jkuƒbAx‡

(e x + e v )(0) = (e x + e v )(t) − Z t

0

( ˙ e x + ˙ e v )(s) ds

≤ (e x + e v )(t) + (1 + C(T )) Z t

0

(e x + e v )(s) ds

≤ C(T )(e x + e v )(T )

≤ C(T ) (|x − x ⁰ | + |v − v ⁰ |) .

^ _–uƒl9lv_5{%¶$l?np_–%n

f (t)

u¥q–‡5bAb)‡“_«†l:r5q–u¸¨©{Xtva=¦9uƒ…–lvˆ‰_–u¸np°zlªaH{k{}np_5q5b)lpl:{Xq

[0, T ]

µ]lª{€np_–%n

kf k _L ^∞ _([0,T];W ^{1, ∞} _(Ω))

uxl ¬?{†r5q–‡5b)‡:µ5¶_–bAtpb)†l

k∂ t f (t)k _L ^∞ _(Ω) ≤ Q(T )k∂ x f (t)k _L ^∞ _(Ω) + kEk _L ^∞ _([0,1]) k∂ v f (t)k _L ^∞ _(Ω)

¨©{X¥ƒ{%¶$lOtpb)†‡5u¥ƒj$¨©tp{†aGnv_5bM­1x†lv{%‚$b)wZr–}nvuƒ{†q

/

X±¥B0±^r5tpq5uƒq5— nv{np_5bzb)¥b]ˆnvtpuxˆœº–bAx‡:µC¶bº«tplªnlvbAbS¬kj£‡5u¸¨Ú¤

¨©b)tvb)qZnvux%nvuƒq5—np_5b$y{†uxlplª{Xq!b)wZr–%npu¥{Xq

/

†±Ñ‘0S¶u¸np_Qtvb]lª…?b)ˆ+nznv{

x

^†q–‡

t

^nv_–}n

k∂ _xx ² Ek L ^∞ ([0,T];L ^∞ ([0,1]))

†q–‡

k∂ _tx ² E(t)k L ^∞ ([0,T ];L ^∞ ([0,1]))

}tpbtvb]lª…?b)ˆ+nvuƒ‚†b)¥jƬ?{†r5q–‡5b)‡¬kj

Q(T )k∂ x f k L ^∞ ([0,T];L ^∞ (Ω))

}q«‡

Q(T )k∂ t f k L ^∞ ([0,T];L ^∞ (Ω))

µX¶_5uƒƒb$nv_–b1¬?{†r5q–‡Q{†q

k∂ _tt ² Ek L ^∞ ([0,T ];L ^∞ ([0,1]))

uƒl{†¬mn‰}uƒq5b)‡H¬kjH‡mu¥»?b)tª¤

b)qZnvux%nvuƒq5—nv_–b ¢ aH…«b)tvbb]wZr–%npu¥{Xq

/

‘m±ƒ]‘0€¶u¥nv_Ætvb]lª…?b)ˆ+nnv{

t

^±

9:=D

Ä {†ƒ¥{%¶uƒq5— 9¥1LB;§µ 9”Z’B;0}q«‡)9LB;§µ5¶€b£q–{%¶G‡mb]lvˆAtvuƒ¬«bHlvu¥aH…5ƒb}q–‡†ˆ)ˆ+r5t‰%npb1npu¥aHblv…5¥u¥nªnpu¥q–—Q†‡m‚Xb)ˆnpu¥{Xq

lpˆ‰_5b)ab

S _∆t

^± btvb]ˆA†¥?nv_«%n

∆t

uxlr5q5u¥¨©{†tpa³npu¥aHblªnvbA…¾†q–‡š¶tpu¥nvb

t ⁿ = n∆t

^±

x

^ŸœÃ

v

^{Až%Ÿœ›  žC  ½ ž%Ÿm  ž%›#} ^0{Æ}qkj¾—†uƒ‚†b)qÈX‡m‚†b]ˆnvuƒ{†qº–bAx‡

F : Ω → R ² ,

¶€bXlvlv{mˆ+ux%npb

nptp†q–lv…«{Xtªn${X…«b)tp}nv{†t

T : g → g ◦ F ⁻¹

‡5b+º–q5b]‡¨©{†t1†qkjˆA{†qZnvuƒqkr5{†r–l€¨©r–q–ˆnpu¥{Xq

g

±34êq¾…–†tªnpuƒˆAr5ƒ†t)µ no¶€{H{†q5bA¤§‡5u¥tpb)ˆ+nvuƒ{†q–†?X‡m‚†b]ˆnvuƒ{†q«l}tpb£ˆA{†q–lvuƒ‡5bAtpb)‡u¥qnv_5b“npu¥aHblv…5ƒu¸nvnvuƒq5—Qlvˆ‰_–bAaHb7

F x : (x, v) → (x + v∆t/2, v)

^{/‘m±Ñ‘mB0}

†q–‡

F v (h) : (x, v) → (x, v + ∆t E(h)(x)), ˜

^/‘5±‘X‘0

¶_–bAtpb

h

^}q–‡

E(h) ˜

tpb)lv…?b)ˆnpu¥‚XbAƒj1‡mb)q5{}npbM}q}r5´kuƒƒuƒ†tvj1‡5bAq–lvu¸noj1¨©r5q–ˆ+nvuƒ{†q}q«‡1np_5b B!!1( H(

.3

E(h)(x) = ˜ Z

K(x, y) Z

h(y, v) dv − 1

dy.

^{/‘m±Ñ‘ - 0}

b“nv_–bAqƁ¥bAn

T x : g → g ◦ F _x ⁻¹ , T v (h) : g → g ◦ F v (h) ⁻¹

^{/‘m±Ñ‘%”R0}

¬?bnp_5b(ƒuƒq5b)†t1nptp†q–lª…?{†tvn1{X…«b)tp}nv{†t‰l†lplv{kˆAuƒ}nvb]‡nv{

F x

}q–‡

F v (h)

}q–‡Æº«q–}ƒ¥jƇmb)q5{}npb(¬kj

T v

nv_5b

P nvt‰}q«lª…?{†tvn€{X…«b)tp}nv{Xt

T v : g → T v (g)g.

^{/‘m±Ñ‘†Œ0}

^ _–bYnpu¥aHblv…5ƒu¸nvnvuƒq5—Qlvˆ‰_–bAaHb

S _∆t

†…5…5tp{C´kuƒa!%npb)lMnv_5b)q

f (t ⁿ⁺¹ )

^¬kj

S _∆t f (t ⁿ ) := T x T v T x f (t ⁿ ),

^/‘m±Ñ‘L0

¶_–uƒˆ‰_ƈ+{Xtvtpb)lv…«{Xq–‡5lnv{

/

†±ÑŒ0€¶u¥nv_

( X(t ˜ ⁿ ; t ⁿ⁺¹ , x, v) := x − v∆t + ∆t ² /2 ˜ E(T x f (t ⁿ ))(x − v∆t/2) V ˜ (t ⁿ ; t ⁿ⁺¹ , x, v) := v − ∆t E(T ˜ x f (t ⁿ ))(x − v∆t/2).

/

‘m±Ñ‘80

^ _–b“¨©{†ƒ¥{%¶uƒq5—¥b)aa!Hb]lon‰}¬5ƒuƒlv_5b]l€nv_–}n$nv_5bXlvlv{mˆ+ux%npb)‡Q—X¥{X¬–}Onpu¥aHb‡muxlpˆ+tpb+nvuƒ°)}nvuƒ{†qbAtptv{Xt‡mb)ˆ)Cjkl

ƒu² b

∆t ²

^±

B½ ¯ŸC!D# #F(

f 0

!

W ^1,∞ (Ω)

^{* *}

kf (t ⁿ⁺¹ ) − S _∆t f (t ⁿ )k L ^∞ ≤ C(T )∆t ³ .

^ _–b£…–tv{k{}¨œuƒl$—†uƒ‚†bAquƒqšnp_5b}…5…?bAq«‡mu¸´:±

½ ¯Ÿmž !$#$#¢ lªuƒaHu¥x}t“tpb)lvr5¥nYuxllª_5{%¶qu¥q 9L;Eµ:¶u¥nv_Ènv_5bHlªnvtp{†q5—XbAt“†lplªr5aH…mnpu¥{Xq¾nv_«%n

f 0

uƒl£uƒq

C ² (Ω)

^±

½ ¯Ÿmž !$#D# ^ _5bƈ‰_5{†uxˆ+b{}¨1r–lvuƒq5—np_5uxl(npu¥aHb¾lv…5¥u¥nªnpu¥q–— lpˆ‰_5bAaHbuƒlaH{Xlªnvƒj aH{}nvuƒ‚%%npb)‡¯¬kj u¥npl

lvuƒa…–¥uxˆ+u¥noj

/

}q«‡¾u¸n‰lYXˆAˆAr5tpXˆ+j†l$¶€bAƒ 0± 4§nlª_–{†r5x‡Æ¬«b3bAaH…5_–Xlªuƒ°Ab]‡:µ?_5{%‚†b)¶b)t)µ–nv_–}n1nv_–b3†‡–}…mnpu¥‚Xb

lv…–Xˆ+bƇmuƒlpˆ+tpb+npu¥°]%npu¥{Xq·lpˆ‰_5bAaHbnv_–}nH¶€bq5b+´knQ‡mb)lpˆ+tpuƒ¬«b¾ˆA†qK¬?b¾}…5…5ƒuƒb)‡Knv{.{}np_5bAtnojk…«b]lH{}¨$npu¥aHb

‡5uƒlpˆ+tpb+npu¥°]%nvuƒ{†q«lA± Ä {Xtb+´5}aH…5ƒb†µm{Xq5bˆ+{Xr5ƒ‡ˆ+{Xq–lvuƒ‡mb)tHr5q5uxwXr–bYnptp†q–lv…«{Xtªn{†…?bAt‰%np{†t {}¨nv_5b“¨©{Xtva

T : g → g ◦ F(g) ⁻¹

¶_–uƒˆ‰_ƈ+{Xa(¬5uƒq5b)l€np_5b†‡5‚†b)ˆ+nvuƒ{†qu¥qnv_–b

x

^}q–‡

v

‚%}tpux}¬5ƒb)l)±

"' Æ# œA,K*0"$#

Yr–t†‡–}…mnpu¥‚Xblpˆ‰_5bAaHbz¬–XlªuxˆA†¥ƒj“ˆ+{†q«lªuxlon‰l0{†¨mnv_5tpbAb€¦0†—†t‰}q5—Xb+¤§yztv{†|ob)ˆ+nvuƒ{†q“lonpbA…–l¶_5uxˆ‰_tvb]lª…?b)ˆ+nvuƒ‚†b)¥j

¨©{X¥ƒ{%¶Šnp_5b(np_5tpbAb!†‡m‚Xb)ˆ+nvuƒ{†qlªnvb)…–l“{}¨znp_5bnvuƒaHbHlv…5ƒu¸nvnvuƒq5—–± Ä {Xt£b]†ˆ‰_È{}¨znp_5bAaƵ:np_5bHab]lª_

M _i ⁿ

^{Xq

¶_–uƒˆ‰_Ænp_5buƒqXnpbAtpaHb)‡mux%npb(lª{X¥r5nvuƒ{†q

f _i ⁿ

^{uƒl ²} q–{%¶quxlº«tplªn3Nonvt‰}q–lv…«{Xtªnpb)‡O“u¥qZnp{šHq5bA¶ {Xq5b

M _i+1 ⁿ

^µ?uƒq

lvr–ˆ‰_š(¶€Cj3nv_«%n‡mb+npbAtpauƒq5uƒq5—

f _i+1 ⁿ := P M _i+1 ⁿ T i f _i ⁿ

}aH{†r–qXn‰lznp{np_5b£ˆA{†aH…5rmn‰%npu¥{XqQ{}¨

T i f _i ⁿ

^%n€nv_5b

q–{k‡5b)l {}¨

M _i+1 ⁿ

^± blv_–}ƒ:q5{%¶Glv…?b)ˆ+u¥¨©jQnp_5btpr5¥b]l€¨©{†t$‡mb]lªuƒ—†q–u¥q5—3nv_–b)lvb†‡5†…mnvuƒ‚†b“aHb)lv_5b)l)±

=$

Yr–tMlª…«†ˆ+b$‡5uƒlpˆ+tpb+npu¥°]%nvuƒ{†qHuxlz¬–Xlªb]‡{†q!_5uƒbAt‰}t‰ˆ‰_5uxˆA}kº«q5u¸npb$bAƒbAaHb)qXn‰lA± 6? K*+ ' * CB!!H $

( # 61, (C0P (! B "!H #= !H '+* ¶€b$¶uƒ¥

ˆA{†q–lvux‡mbAtƅ–†tªnpuƒˆAr5x}tˆ+x†lpl“{}¨†‡–}…mnpu¥‚XbHnvtpuƒ†q5—†r5x%npu¥{Xq–l£nv_«%n(}tpb!†lplª{mˆAuƒ}nvb)‡nv{ ' (

0 'R (+!

}q«‡È¶_–uƒˆ‰_ ˆA†q }xlª{Ƭ?bQlªb)bAq †lƅ«}tvnvuxˆ+r5x}tˆAXlªb!{†¨np_5bQlv{ˆA}ƒƒb)‡ Noq–bA¶€b)lªn

‚XbAtvnvbA´“¬–uƒlvb)ˆ+nvuƒ{†qROMaHb+np_5{m‡3‡5b)lpˆ+tpu¥¬?b)‡(u¥q>9Ž;

/

¶b€tpb+¨©bAtnv{E9ƒ]Œ1;5{Xt 9”–H;k¨©{XtœaH{Xtvbu¥qm¨©{Xtva!}nvuƒ{†q–lœ}¬?{†rmn

_–u¥b)tp†tpˆ‰_5uxˆA†«¬«†lvb)l0±

˜Hܟ œ ½ à ¿ Ÿ–ÃÏ Ÿ–Ã0ž%Ÿ ٟ5  œ›I# ¦0b+nMr–lMº–t‰lon€‡mbAq5{†nvb$¬kj

Q `

nv_5bYr5q5u¥¨©{†tpa wZr–X‡mtp†q5—†r5¤

x%npu¥{Xqa!X‡mb“{}¨S}ƒ9‡mjXX‡muƒˆ†µ5lvwZr–†tvb“ˆ+b)¥xl{†¨tpb)lv{†ƒrmnvuƒ{†q¥b)‚†bA

` ∈ N Q ` :=

[j 2 ^{−`</sup> , (j + 1)2 <sup>−`} ] × [k 2 ^{−`</sup> , (k + 1)2 <sup>−`} ] : j, k ∈ Z

†q–‡Æ¬kj

Q := ∪ ` Q `

nv_5b3lªbAn1{}¨z}ƒ‡5jX†‡5uƒˆwZr–X‡mtp†q5—†ƒb)l)±Q2$bAtpbq5{QƒbA‚XbAuƒl$lvr5…5…?{Xlvb)‡np{Q¬?b(ƒ{%¶b)t

np_–}q.…5tvb]lvˆAtvuƒ¬?b)‡

` 0 > 0

µ0†q–‡¶€bHlª_–†¥zˆA{†q–lvux‡mbAt£nv_«%n

Q ` := ∅

^{¨©{†t†qkj}

` < ` 0

± Èb!{X¬–lªb)tv‚Xb

np_5bAqQnv_–}nznv_5b£ˆ+b)¥xlM†tvb$b)a(¬?b)‡5‡5b)‡:µklª{nv_–}nb)Xˆ‰_

Q `

ˆA}q!¬?bYlvbAb)qQ†ltvbAº–q5b)ab)qZnM{}¨Onv_–bYlvaH†¥ƒbAt

lvb+n

Q `−1

µ?}q–‡¶€b{†¬mn‰}uƒq¾!wXr«†‡knptvb)b“lªnvtpr–ˆ+nvr5tpb¬kj‡mbAº–q5uƒq5—H¨©{†t1†qkjš—†uƒ‚†b)q

α ∈ Q

{†¨œƒbA‚XbA

`(α)

u¥npl * ' ˆ+b)¥xl$Xl

C(α) :=

β ∈ Q `(α)+1 : β ⊂ α ,

†q–‡!u¥npl€…–†tvb)qXn ˆ+b)¥O†l

β ∈ Q `(α)−1

lvr–ˆ‰_!np_–%n

α ⊂ β

±Èb“}xlv{(‡mb+º«q5b1np_5b !H ˆ+b)¥xlM{†¨

α

^†l

A(α) :=

β ∈ [

`<`(α)

Q ` : β ⊃ α .

b¶uƒ¥:np_5bAq¾ˆ)}ƒ

Λ ⊂ Q

^{ !!H '} u¸¨u¥n1lv}nvuxloº–b]l

Q ` 0 ⊂ Λ and [

β∈A(α)

C(β) ⊂ Λ for any α ∈ Λ.

^ _–uƒlQlvb)ˆA{†q–‡·…5tp{†…?bAtvnvuƒb)lHuƒaH…5¥uƒb)lnp_–%nq5{ ˆ+bAƒ${†¨

Λ

uxl!…–†tªnpuƒ†¥ƒj¯tpb+º–q5b]‡:µ lª{ nv_«%nš}qkj

α ∈ Λ

lp%npuƒlªº–b]l

C(α) ∩ Λ = ∅

^{†t

C(α) ⊂ Λ

^{± ¢} lHˆ+{Xq–lvb)wZr5bAq«ˆ+b†µk¶€b“{†¬–lvbAtp‚†b1nv_–}nnv_5b

!

{}¨

Λ

^µknp_–%n

uxl€nv_–blªbAn

L(Λ) := { α ∈ Λ : C(α) ∩ Λ = ∅ },

¨©{Xtva!l$Q…–}tvnvu¥nvuƒ{†qÆ{}¨Snp_5b…5_«†lvblª…–Xˆ+b

/

b+´5ˆ+b)…mn$¨©{Xt$nv_–bb)‡m—Xb)l0± b¶u¥ƒlvCjšnv_–}n

M ⊂ Q

^uxl1}q

B0 ' ( u¥¨znv_–bAtpb3uxl“ˆA{†q–lvuƒlªnvb)qZn1nvtpbAb

Λ

lªr–ˆ‰_nv_–}n

M = L(Λ)

^±"4êq †‡5‡5u¸npu¥{Xq9µ

M

¶uƒ¥$¬?bˆ)}ƒ¥b]‡ ' u¥¨£u¥nplš¥{mˆA†tvb]lª{X¥r5nvuƒ{†q _–†l!q–{ NÚ|or5aH…–l Oµ{XtQu¥q{†nv_5b)tQ¶{Xtp‡–lAµu¸¨“no¶€{

q–bAuƒ—†_k¬«{Xtvuƒq5—Hˆ+b)¥xl

α

^}q«‡

β

^/lª_–†tvuƒq5—H%n$ƒb)Xlon{†q–b£b]‡m—†bB0€lv}nvuxlo¨©j

|`(α) − `(β)| ≤ 1

^±

½ ¯Ÿmž

#&%'# 4êa…?{Xlvuƒq5—np_5uƒl3—†t‰†‡5b)‡ ˆ+{Xq–‡mu¥nvuƒ{†q¯uxlH¾tpb)†lv{†q«}¬5ƒbQtpb)wZr5uƒtvb)ab)qZn)µSlªuƒq–ˆAbQ¨©{†tH}qkj

X‡5}…5nvuƒ‚†b¯wZr–†‡5tp†q5—†r5x%npu¥{Xq

M

µnv_5b)tvb·b+´muxlon‰lǗXtpX‡mb)‡=tpb+º–q5b)aHbAqZn

M ⁰

^{}¨

M

^nv_–}n lp%npuƒlªº–b)l

#(M ⁰ ) ≤ C#(M)

^¶u¸np_

C

}q¾†¬–lª{X¥r5nvbˆ+{Xq–lon‰}qZn

/

lvbAbƒbAaHa!H‘5±”Huƒq9ё}’B;9¨©{Xt$…5tv{k{†¨0±

  ž .Až ٟ ϟmÂ œ›# ^{†qkj—XtpX‡mb)‡¾X‡5}…mnpu¥‚Xb3wZr–X‡mtp†q5—†r–ƒ}nvuƒ{†q

M

{}¨Snp_5b3†¬«{%‚Xb nojk…?b†µ}¶€b lª_–†¥kq5{%¶·‡mbAtpu¥‚Xb£ˆ+{Xqm¨©{†tpaHu¥q5—Ynvtpux}q5—Xr5ƒ}nvuƒ{†q¨©{Xtœnv_5b

P 1

uƒqZnvbAtp…?{†x%nvuƒ{†q(¬«b ¶€bAƒ5‡mb+º–q–b)‡:±

btpb)ˆ)}ƒ:nv_–}n£nptvux}q5—Xr5x%nvuƒ{†qÆuxl1lp}ux‡nv{Q¬?b(ˆA{†qm¨©{XtvaHuƒq5—!u¸¨M†qkjšb)‡5—†b{†¨S†qkjQnvtpux}q5—X¥buxl$b)u¸np_5bAt

lvr5¬–lvb+n{}¨nv_5bH¬?{†r5q–‡–}tpj

/

¶_5b)qÈnp_5bAtpbuƒl“{Xq5b10+µ9{†t}q b)‡5—†b{}¨€†q5{}np_5bAt£nptvux}q5—X¥bX±3^{‡5{np_5uxlAµ

¶€bYˆA{†q–lªnvtpr–ˆ+n º–tplªn€nvtpuƒ†q5—†r5x%npu¥{Xq

M ˜ t

¬kjQlª…–¥u¥nªnpu¥q5—b)Xˆ‰_šˆ+b)¥

α ∈ M

uƒqQno¶€{nptvux}q5—X¥b]lAµZ¶u¥nv_nv_5b

¨©{X¥ƒ{%¶uƒq5—“tvr5ƒb7u¸¨

α

uƒlz†qHr5…5…?bAtzƒb+¨ÚnM{XtM£¥{%¶€bAtStpuƒ—†_ZnMˆ‰_5uƒƒ‡

/

{†¨?u¥nplM…–}tpbAqZnzˆ+b)¥ 0µZu¥nzuƒlzlv…5ƒu¸nvnvb]‡Hu¥qZnv{

u¥npl¥{%¶€bAt“ƒb+¨Ún3†q–‡ r5…5…?bAttpu¥—X_Zn“_«}ƒ‚†b)l)µ†q–‡Ènv_5bQlv…5ƒu¸nvnvuƒq5—¾uxllªjkaHb+nptvuxˆHuƒqÈnp_5b!{}np_5bAtno¶{¾ˆAXlªb]lA±

Ä {†t np_5uƒl1tvr–¥b¬?b}…5…–¥uƒb)‡np{Hnv_5bˆAbAƒƒl{†¨np_5bƒ{%¶b]lonƒbA‚XbA

` 0

µ5¶bˆ)}q¾}ƒ¶ Cjkl ˆA{†q–lvuƒ‡5bAt$º«ˆ+nvuƒ{†r–l

ƒ{%¶€bAtƒbA‚†b)

` 0 − 1

µ?lª{Hnp_–%n1b]†ˆ‰_¾ˆ+b)¥{}¨SƒbA‚†b)

` 0

_«†l$H…–†tvb)qZn1ˆ+bAƒ

/

q5bA‚XbAtvnv_5b)¥b]lvl)µmnp_5b¶ CjQnv_–bAj

†tvb“lv…5¥u¥nªnpb)‡¾‡m{kb]l q5{}na!%nvnvbAt1a(r–ˆ‰_¨©{†t np_5blªb]wZr5bA 0±

bYˆ)}qQ{X¬–lvbAtp‚†b{†qQº–—†r5tpbnp_–%nnv_5b£tpb)lvr5¸npu¥q–—

M ˜ t

uƒlq5{Xq–ˆ+{Xqm¨©{†tpaHu¥q5— 7lvu¥q«ˆ+b

M

uxlMq–{}nr–q5u¸¨©{XtvaƵ u¥nMˆ+{XqZnp}uƒq–lS}nM¥b]†lªnS{†q5b$ˆAbAƒ

α

lv_–}tpuƒq5—}qb)‡m—Xb¶u¸np_Hno¶{ˆ+b)¥xl

β

^†q–‡

λ

lªr«ˆ‰_3np_–%n

`(β) = `(λ) =

`(α) + 1

^{± ¢} q–‡Æu¸¨z¶€b‡mb)q5{}npb¬kj

β t

}q–‡

λ t

np_5bno¶€{Hnvtpuƒ†q5—†ƒb)l$tvb]lªr5¥nvuƒq5—H¨©tp{†a&nv_5blª…5ƒu¥nªnvuƒq5—Q{†¨

β

†q–‡

λ

np_–%nlv_–}tpb3†qb)‡5—†b(¶u¥nv_

α

µ:u¸nuxlYtpb)X‡muƒ¥j¾lvbAbAqnv_–}nYnp_5bAjÈ}tpb(q5{†n“ˆA{†qm¨©{XtvaHuƒq5—š¶u¥nv_Ènv_5b

X‡C|ª†ˆAbAqZn$nptvux}q5—X¥b(tvb]lªr5¥nvuƒq5—š¨©tv{Xa np_5b3lv…5ƒu¸nvnvuƒq5—{}¨

α

^± <€rmn“lªuƒq–ˆ+b

M

uƒlY—†t‰†‡mb]‡:µ5np_5uxlYuxl1nv_5b3{†q5ƒj …?{Xlplvu¥¬5ƒbYˆA{†qmº«—†r5t‰%npu¥{Xq¶_–bAtpb$nv_–b1nptvux}q5—X¥b]lM†tvbYq5{†q–ˆA{†qm¨©{XtvaHuƒq5—–µZ}q–‡Q¶b£lªb)b$nv_«%n(ˆ+{Xqm¨©{†tpaHu¥q–—

nptvux}q–—†r5x%npu¥{Xq

M t

uxl$lvu¥aH…5ƒjš{X¬mnp†u¥q–b)‡¬Zjšab)tv—Xu¥q–—†qZjlªr–ˆ‰_ƈ+{Xr5…5ƒb{}¨nvtpuƒ†q5—†ƒb)l

(β t , λ t )

^±

β t

λ t

M M ˜ t M t

β λ

α

Ä u¥—Xr5tpb7Slª…5ƒu¥nªnvuƒq5—!{†¨œq5{Xqkr5q5u¥¨©{†tpa³ˆ+bAƒxl$}q–‡aHbAtp—†uƒq5—H{}¨œq5{†q–ˆA{†qm¨©{XtvaHuƒq5—3nvtpuƒ†q5—†ƒb)l)±

^{Kb)aH…5_–†lvuƒ°AbÈnv_5b.lªuƒaH…5ƒb lªnvtpr–ˆnpr5tpb{}¨(nv_5b {†tpu¥—Xu¥q«}“wXr«†‡mt‰}q5—Xr5x%nvuƒ{†q

M

ˆA{†aH…–†tvb]‡nv{·nv_5b

Xlvlv{mˆ+ux%npb)‡šˆA{†qm¨©{XtvaHuƒq5—3nvtpuƒ†q5—†r5x%npu¥{Xq

M t

µ–lvr–ˆ‰_ÆHˆ+{†r–…5¥buxl tpbA…5tpb)lvbAqZnvb]‡Q{Xqº«—†r5tpb‘k±

½   ½ZÁ ٛ ½ Ÿ ½ O ½ ž  ٟ5Â œ›I# <€b)ˆA†r–lvbQ¶b}tpbQuƒqXnpbAtpb)lªnvb]‡.u¥q

L ^∞

bAtptp{†tb]lonpu¥a!%npb)l)µSu¸n

¶uƒƒœ¬?b_5bAƒ…m¨©r5np_–%n“nv_–b3…5tp{}|ob]ˆnpu¥{Xq{†…?bAt‰%np{†t‰l1ˆA†q5q5{†n£uƒq–ˆ+tpb)Xlªbnv_5b

L ^∞

^q5{†tpaƱ b!lª_«}ƒ_5b)tvb

ˆA{†q–lvux‡mbAt

P 1

¦0†—†t‰}q5—Xb(uƒqXnpbAtp…«{Xƒ}nvuƒ{†q ¶_5uxˆ‰_ {X¬k‚Zuƒ{†r«lªƒj¾aHbAb+n‰l“nv_–uƒltpb)wZr5uƒtpbAaHbAqZn)±MAYbAq–{}nvuƒq5—¾¬kj

N (M t )

nv_–b£‚XbAtvnvuxˆ+b]l {}¨

M t

µ«}q–‡¬kj

V M :=

g ∈ C ⁰ : g |K ∈ Π 1 , ∀K ∈ M t

np_5bXlvlv{mˆ+ux%npb)‡Yº–q5u¥nvbbAƒbAaHbAqZnœlv…–Xˆ+b†µ]¶€bz¥bAn0np_5bAq

P M

¬«bMnv_5bMq–%npr5tp†

P 1

uƒqZnvbAtp…?{†x%nvuƒ{†qXlvlv{mˆ+ux%nvb]‡

np{nv_–b1ˆ+{Xqm¨©{†tpaHu¥q5—“nptvux}q–—†r5x%npu¥{Xq

M t

± b$a!Cj(tpb)ˆA†¥5nv_–}nz¨©{†t€}qkj3ˆA{†qZnvuƒqkr5{†r–lœ¨©r–q–ˆnpu¥{Xq

g

^µ

P M g

uxl€nv_–br5q5uxwXr–bbAƒbAaHbAqZn${}¨

V M

nv_–}n1lv}nvuxloº«b)l

P M g = g on N(M t ).

M M t

Ä uƒ—†r–tvb‘ 7S{Xq5b“—†t‰†‡mb]‡šwZr–†‡5tp†q5—†r5x%npu¥{Xq†q–‡šu¥npl1†lplª{mˆ+ux%npb)‡ˆ+{†q5¨©{†tpauƒq5—3nvtpuƒ†q5—†r–ƒ}nvuƒ{†q9±

½ ¯Ÿmž

#E!D# 4§n(uxl…?{Xlplªuƒ¬5ƒbHnv{¾r–lvbQ_5uƒ—†_–bAt{Xtp‡mb)t

P k

¦}—†t‰}q–—†bHu¥qZnpbAtp…«{Xƒ}nvuƒ{†q {†…?bAt‰%nv{Xtpl

P _M ^k

Xlvlv{mˆ+ux%npb)‡Ènv{¾np_5blp}aHb!nptvux}q–—†r5x%npu¥{Xq9±¾^ _5bAj.ƒb)†‡ np{lvu¥aHuƒƒ†t3†‡–}…mnpu¥‚Xbšlvˆ‰_–bAaHbQ¶_5uxˆ‰_ aHu¥—X_Xn

bA´m_5u¥¬–u¸n€¬«bAnªnvb)t tvb]lªr5¥npluƒqQ…5t‰†ˆnpuƒˆAb†µZ¬5rmn€¨©{†t€¶_5uxˆ‰_Q¶b£‡m{(q–{}n_«C‚†bY(lv}nvuxlo¨ÙXˆnv{XtvjbAtptp{†tM†q–}ƒjmlªuxlA±

& =< A

b$q5{%¶…5tpb)lvbAqZnSnp_5bno¶€{aHb)lv_Q}ƒ—†{†tpu¥nv_5a!lœnv_«%n€†…5…?b)}tMu¥qHnp_5b1†‡–}…mnpu¥‚Xb1lvˆ‰_–bAaHb†± ( *=!K0 *

K*+ 6 'K#,H!'*F6? 6 B! ( J 0 'R (* ' * !E!

#,(

2 bQ¬?bA—Xu¥q ¬kj u¥qZnvtp{m‡mr–ˆAu¥q5—no¶{ƨ©r5q«ˆnvuƒ{†q«}xlnv_«%n(…5xCj.ƅ–}t‰}aH{†r–qXntp{†ƒb!uƒq nv_5b

‡5b)lvu¥—Xq{†¨0nv_–bab]lª_–b)l)±

ٛ] }ž ½  ½   ž 5Ÿmž ½ ›# 4êq.{†t‰‡mb)t“nv{¾—†r«}t‰}qZnvb)blva!}ƒzu¥qZnpbAtp…«{Xƒ}nvuƒ{†q.bAtptv{Xtpl)µ:¶€b!¶u¥ƒMq5b)b)‡Ènv{

ˆA{†qZnvtp{†9nv_5b #, E HB {}¨Snp_5b(qkr5aHb)tvuxˆA†lv{†ƒrmnvuƒ{†q–l)± Ä {Xt$np_5uƒlY…5r5tp…?{Xlvb†µ?¶b(Xlvlv{mˆ+ux%npb

np{!}qkjQaHb)lv_

M

}q«‡†qZjQ¨©r5q–ˆnpu¥{Xq

g

np_5bwZr–}qZnpu¸noj

µ(g, M ) := sup

α∈M

curv(g, α)

^{/- ±‘ D 0}

¶_–bAtpb

curv(g, α)

uƒlYQwZr–}qZnvu¥nojš¶_–uƒˆ‰_¾uxl$b]wXr«}:np{

|g| W ^2,1 (α)

¨©{†t

g ∈ W ^2,1

±^ _5uxl1wZr–†qXnpu¸noj¶uƒ¥

¬?b!…5tpb)ˆAuƒlvbAƒjȇmbAº–q5b]‡Èuƒq

/

”«±ŒXŒ0“}q«‡Èu¥npl3‡mb+º«q5u¸npu¥{Xq ¶u¥ƒz¬?bQb+´knpbAq–‡mb]‡Ènv{ƨ©r5q–ˆ+nvuƒ{†q–l

g

^{¶_5uxˆ‰_ }tpb}

q–{}n“u¥q

W ^2,1

¬5rmn†tvbˆ+{†qZnpu¥qkr5{Xr–lY†q–‡…–u¥b]ˆ+bA¶uxlvb Hq–b3{XqÈlv{†aHb†tv¬–u¸nptp†tvjnvtpuƒ†q5—†r–ƒ}nvuƒ{†q9±^ _5b

‡5uƒlpˆ+tpb+npbˆ+r5tp‚%%nvr–tvbˆ+{XqZnvtp{†xl€nv_5bb)tvtp{†t {†¨

P 1

uƒqXnpbAtp…«{Xƒ}nvuƒ{†qu¥qÆnp_5b¨©{X¥ƒ{%¶u¥q–—HlvbAq«lªb7Snv_5b)tvbbA´muƒlªnpl

šr5q5u¥¨©{†tpa•ˆ+{Xq–lon‰}qZn

C

lªr–ˆ‰_nv_–}nY¨©{Xt£†¥

α ∈ M

†q–‡¾¨©{Xt

K

Qnptvux}q5—X¥b3{}¨

M t

ˆ+{†qZn‰}uƒq5b)‡¾uƒq

α

^µ

¶€b“_–C‚†b

kg − P M gk L ^∞ (K) ≤ Ccurv(g, α).

^{/- ±‘†Ž0}

b…?{Xlªnv…?{†q5b£nv{ ”–±Ñ‘np_5b…5tp{k{}¨{†¨np_5uxlb)lªnvuƒa!%nvbX±

½  ½ à B œ›] 

ٝ› ½ À  ž # 4êq…«}t‰}ƒ¥b)0¶u¥nv_¾np_5b(ƒ{mˆA}ˆ+r–tv‚%%npr5tpb)l)µ–¶€b(¶uƒ¥œq5bAb]‡Ænv{

ˆA{†qZnvtp{†:Hlvb)ˆ+{Xq–‡wXr«}qZnvu¥noj†µmq–†aHbAƒj

π(g, M ) := sup

α∈M 2 ^−2`(α) |g| _W ^{1, ∞} _(α) .

^{/- ±- ’0}

bq5{%¶=‡mb)lpˆ+tpu¥¬?b£nv_5ba!†u¥qaHb)lv_Ɔ¥—X{†tpu¸np_5a!lA±

½

›

Ÿ5ܟ ]Ÿm

Â

#

F£uƒ‚†bAq.¨©r5q–ˆ+nvuƒ{†q

g

¨©{†t¶_5uxˆ‰_

µ

^}q–‡

π

†tvbº–q5u¥nvb

/

lªr–ˆ‰_.†lš¨©r5q–ˆ+nvuƒ{†q

{†¨

W ^2,1 ∩ W ^1,∞

µ9{Xtš…5uƒb)ˆAbA¶uxlªbH !q5b3¨©r5q–ˆ+nvuƒ{†q–l0“}q«‡…5tvb]lvˆAtvuƒ¬?b)‡¾np{†ƒbAt‰}q–ˆAb

ε > 0

^{µ9¶€b†tvb}

uƒqZnvb)tvb]lonpb)‡u¥q¾ˆ+{Xq–lªnvtpr–ˆnpu¥q5—HHaHb)lv_

A _ε (g)

µknv_5blva!}ƒ¥b]lon1Xl€…«{Zlvlvu¥¬–¥bXµ–lªr«ˆ‰_šnp_–%n

µ(g, A _ε (g)) + ∆t π(g, A _ε (g)) ≤ ε.

^{/- ±- 10}

hmu¥q–ˆAb£nv_5uxluxl$†ˆ‰_5uƒbA‚Xb)‡š¬Zj†l ² uƒq5—nv_«%n

ν(g, α) := curv(g, α) + ∆t 2 ^−2`(α) |g| W ^{1, ∞} (α) ≤ ε/2

^{/- ±- ‘0}

_–{†x‡5l9¨©{Xtœ†qkj

α ∈ A _ε (g)

µ}$q–}nvr5t‰}klª{X¥r5nvuƒ{†q(ˆA{†q–lvuƒlªnpl{}¨5…?bAtv¨©{†tpaHu¥q–—Y†‡–}…mnpu¥‚XbMlª…–¥u¥nªnpu¥q5— 7lªnp†tªnpu¥q5—

¨©tp{†a np_5b“tv{k{†n€wZr–†‡5tp†q5—†r5x%npu¥{Xq

Q ` 0

µm†qZjQˆAbAƒ

α

¨©{†t ¶_5uxˆ‰_

/- ±- ‘0€‡m{kb)l€q5{†n _–{†x‡Quxltpb+º«q5b)‡šu¥qZnv{

u¥nplQ¨©{†r5tˆ‰_5u¥x‡mtpbAq ˆAbAƒƒl)µ$}q–‡·np_5uƒluxl!…«b)tª¨©{XtvaHb]‡tvb]ˆ+r5t‰lªuƒ‚†b)¥jXµ€lv{.nv_«%nQnp_5btpb)lvr5¥nvuƒq5—

A ˜ _ε

^uƒlQnv_5b

lva!}ƒ¥b]lon

/

q5{†q!—XtpX‡mb)‡ 0aHb)lv_Qlv}nvuxlo¨©jkuƒq5—

/- ±-

60

/

¨©{†tlp ² b{}¨9lvu¥aH…5ƒuƒˆAu¸nojXµX¶€b$ˆ‰_5{k{Xlvbq5{†nznv{uƒaH…«{Zlªb

aH}´mu¥a!}?ƒbA‚†b)

L

¨©{†tnv_5b£ˆ+bAƒxl0+± b£¥bAnMnv_–bAq

A _ε (g)

¬«b1nv_5b“lvaH†¥ƒb)lªn—XtpX‡mb)‡tpb+º–q–bAaHbAqZn{†¨

A ˜ _ε

^±

hmu¥q–ˆAb

A _ε (g)

^†q–‡

M

‡m{Hq5{}n$‡mu¥»?b)t‚†b)tvjHa(r«ˆ‰_uƒq…–tpXˆnvuxˆ+bXµm3aH{Xtvb£bQˆ+uƒbAqZn}ƒ—†{Xtvu¥nv_5a

/

jku¥b)ƒ‡muƒq5—

np_5blv†aHbYaHb)lv_ 0ˆ+{Xq–lªuxlªnpl€{}¨}…5…5ƒjku¥q–—nv_–b“†¬«{%‚XbYtpb+º«q5bAaHbAqZn …5tp{kˆAb)lplMq–{}n{†q

Q ` 0

¬–rmntp}nv_5b)t{Xq

†quƒqXnpbAtpaHb)‡mux%npb3aHb)lv_

M ˜

ˆ+{†q«lonptvr–ˆ+nvb]‡¾¬kj‡mbAtpb+º–q–u¥q5—

M

u¥qnv_5b3¨©{†ƒƒ{%¶u¥q5—¶€Cj 7$lªnp†tªnpu¥q5—š¨©tv{Xa np_5ba!%´muƒa(r5a³ƒbA‚XbA

`(M )

^{†¨

M

^{µm¶€blªbAn}

Λ ^a _`(M) := M

}q«‡š¨©{Xt$}qkj

` ≤ `(M )

^µm¥bAn

Λ ^a _{`−1</sub> := Λ <sup>a</sup> <sub>`} \ { α ∈ C(β) : `(α) = `, C(β) ⊂ L(Λ ^a _` ) and ν (g, β) ≤ ε/2 }

r–…np{

M ˜ := Λ ^a _{` 0}

±3^ _–uƒl†¥—X{†tpu¸np_5a/ˆ+ƒb)}tpƒj—Xr–}t‰}qZnpbAb)l /- ±-

60£}q–‡

!'*+ 76G! R!: '

K*+E ' P K*+" 'H! P @#,H!'*+!

/

lªb)b“…–}t‰}—Xtp†…5_

-

±ÑŒ}q–‡ L0±

½

›

)žCŸ › Â

ž]#

F£u¥‚XbAq†q†‡m‚Xb)ˆnpu¥{Xqƺ–bAx‡

F

µ«¶€b(q5{%¶=—†uƒ‚†b3Qlonptp}nvb)—†jQ¨©{Xt3Nonvt‰}q–lv…?{†tvnvuƒq5—O

†qkj!ab]lª_

M

u¥qZnv{!3q5bA¶={†q–b†µ5‡mb)q5{}npb)‡

T (M, F)

µ5{Xq¶_5uxˆ‰_ 6? EB!,@ ' * H '

H! P '# *E (

g ◦ F ⁻¹

^{* B!!1 B!}

g

^{! 76? /}¨©{†t£Q…5tpb)ˆAuƒlvb(lªnp}nvbAaHb)qXn {†¨ nv_5uxl(uƒaH…«{Xtªn‰}qZn(…5tp{†…?bAtvnoj†µ¶€bQtpb+¨©bAt(nv{ˆA{†tp{†ƒƒ†tvj”–±”}q–‡.ƒbAaHa!¾”«±¥ - 0±Èhknvtpuƒˆ+nvƒj.lv…?b) ² uƒq5—–µ

T (·, F )

^{!: } nptp†q–lv…«{XtªnQ{X…«b)tp}nv{†t]µlvuƒq–ˆ+b¾np_5bq5b)¶ ab]lª_–b)lQ†¥¶ CjmlH¬«b)¥{Xq5—.nv{.np_5bˆ+x†lpl!{†¨

—XtpX‡mb)‡‡mjXX‡muxˆYwZr–X‡mtp†q5—†r–ƒ}nvuƒ{†q–l)µX¬–rmnu¸n 1! P 3nvt‰}q–lv…«{Xtªn {†…?bAt‰%np{†t]±œ^ _–b£aHbAnv_5{m‡u¥q«‡mbAb]‡

ˆA{†q–lvuxlon‰l“{}¨ ¥{k{ ² uƒq5—¬–†ˆ² ¶ }t‰‡5l“%nnv_–b!¥{mˆA†Stvb]lª{X¥r5nvuƒ{†q {}¨

M

7“¨©{†t}qkjˆAbAƒ

α ∈ Q

µ0¶€b!ƒb+nu¥npl

6 ' B

M

¬?b“nv_5buƒqZnvb)—†bAt

` <sup>∗</sup> (M, F, α) := max{`(β) : β ∈ M, F ⁻¹ (c α ) ∈ β} ≤ ¯ `(M),

^{/- ±-- 0}

¶_–bAtpb

c α = (x α , v α )

uxl“np_5bšˆ+b)qZnvbAt{†¨

α

±>21bAtpb

β ¯

‡mbAq–{}nvb]l£nv_5bˆ+ƒ{Xlvr5tvbH{†¨np_5bQˆ+b)¥

β

±hkbAnªnpu¥q5—

np_5bAq

Λ ^t <sub>` 0</sub> := Q ` 0

µ–¶€b…5tp{kˆAbAb]‡¾†l ¨©{Xt

A _ε

¬Zju¥nvbAt‰%npu¥‚Xblª…–¥u¥nªnpu¥q5—}q–‡{†¬5np}uƒq

Λ ^t _`+1

¬kjtvbAº–q5uƒq5—!uƒq

Λ ^t _`

b)Xˆ‰_ˆAbAƒ:¶_5{Xlvb“¬–†ˆ ² ¶ }t‰‡QƒbA‚†b):uƒl x}tp—†b)t€nv_–†qu¥npl{%¶qƒbA‚†b)Ï±G4êqÆ{}np_5bAt npbAtpa!lAµm¶€b¥bAn

Λ ^t _{`+1</sub> := Λ <sup>t</sup> <sub>`} ∪ { β ∈ C(α) : α ∈ Λ ^t <sub>`</sub> , ` ^∗ (M, F , α) > `(α) }.

^ _–uƒluxl‡m{Xq5br5…nv{Hnp_5b_5uƒ—†_5b]lon$ƒbA‚XbA

`(M)

^{†¨

M

µ5†q–‡¶b£º–q–†¥ƒjšƒb+n

T (M, F)

¬«b“np_5blªa!†¥ƒb)lªn

—XtpX‡mb)‡Htpb+º«q5bAaHbAqZn€{}¨9np_5bYq–{†qQ—XtpX‡mb)‡

T ˜ (M, F) := L(Λ ^t _`(M) )

^{± ¢} qšuƒqZnvbAtpb)lªnvuƒq5—…–tv{X…«b)tªnoj

/

lªnp}nvb]‡

uƒq ƒbAaHa!¾”–±ƒ]‘0uƒlnv_–}n(¨©{†tnv_5b†‡m‚Xb)ˆnpu¥{Xq º–bAx‡5lr«lªb]‡.u¥q¯{Xr5t(lpˆ‰_5bAaHbXµ0np_5bˆ)}t‰‡mu¥q«}ƒu¸noj{†¨nv_5b

tpb)lvr5¥nvuƒq5—HaHb)lv_

T (M, F)

uƒl {†¨np_5blv†aHb£{Xtp‡5bAt†l€np_–%n{}¨

M

^±

= < ='< 9:

b(†tvb“q5{%¶ }¬5ƒb“nv{!¶tpu¥nvb“nv_–b…5tpb)ˆ+uxlvb“¨©{†tpa&{}¨œnv_–bqkr5aHbAtpuƒˆ)}0lpˆ‰_5bAaHb

/

r5…Ænv{Qlv{†aHb‡mbAº–q5u¥nvuƒ{†q–l

np_–%nQ†tvblon‰%nvb]‡¯¬?bAƒ{%¶=0±hku¥q«ˆ+bbA‚†b)tvj uƒqZnvb)tvaHb)‡5uƒ}nvbqZr–ab)tvuxˆA†€lv{†ƒrmnpu¥{Xq

f _i ⁿ

¬?bAƒ{†q5—Zl3nv{ lª{XaHb º«q5u¸npb“bAƒbAaHbAqZn1lv…–†ˆAb

V M _i ⁿ

µknv_5btpb)X‡mbAtlv_5{†r–ƒ‡š¬«bC¶ }tpb$np_–%n$Hˆ+{Xa…–¥bAnvb“qkr5aHbAtpuƒˆ)}:lª{X¥r5nvuƒ{†quƒl

£…–†u¥t{†¨«np_5b ¨©{†tpa

(M ⁿ , f ⁿ )

µX†q–‡(nv_5b$X‡5}…mnpu¥‚Xb qZr–ab)tvuxˆA†mlvˆ‰_5b)aHb€uxltpbA…5tpb)lvbAqZnvb]‡¬kj(£a!}…5…–u¥q5—

S _∆t,ε : (M ⁿ , f ⁿ ) → (M ⁿ⁺¹ , f ⁿ⁺¹ ).

Ä

{†tnv_5b“lp

²

b1{}¨0lvu¥aH…5ƒuƒˆAu¸noj!¶€bYlv_–†¥?¶tvu¥nvbYu¥qšnv_5b“lvb)wZr5bA

f ⁿ⁺¹ = S _∆t,ε f ⁿ

^{µ ²} b)bA…5uƒq5—(uƒqšaHuƒq–‡!np_–%n np_5blpˆ‰_5bAaHb†ˆ+nvr–†¥ƒjQ{†…?bAt‰%npb)l€{†qnv_–bˆ+{†r–…5¥b

(M ⁿ , f ⁿ )

^±

½ ٝϟ †Ÿm  ›) ½ ;# d1lvu¥q5—Ènv_–bÆX‡5}…5nvuƒ{†qK†¥—X{†tpu¸np_5a

A _ε

µM¶_5uxˆ‰_ˆA}q·¬«bƂkuƒbA¶€b)‡KXl3

# 'H!!H( H K*R# µ?¶b‡mbAº–q5b“nv_–b“º–tplªn$lª{X¥r5nvuƒ{†q…–}uƒt

(M ⁰ , f ⁰ )

^Xl

M ⁰ := A _ε (f 0 ) and f ⁰ := P M ⁰ f 0 .

^{/- ±- ”R0}

blv_–}ƒ9‡mbAq–{}nvb£¨©{Xt nv_5blvb)wZr5b)

S ⁰ _ε : g → P _{A ε} (g) g

^±

½  0ž›A ½ ›HŸ5ܟ  ½ ›] 

½ ½ #'Ä {†t$†qZjHnpu¥aHblªnvb)…

n ≥ 1

µm¶b“ƒb+nnp_5bAq

M ₁ ⁿ := T (M ⁿ , F x ) and f ₁ ⁿ := P M ₁ ⁿ T x f ⁿ

^{/- ±- Œ}0}

M ₂ ⁿ := T (M ₁ ⁿ , F _v ⁿ ) and f ₂ ⁿ := P M ₂ ⁿ T ⁿ⁺¹ T _v ⁿ f ₁ ⁿ

^{/- ±- Œ%¬0}

M ₃ ⁿ := A _ε (f ₂ ⁿ ) and f ₃ ⁿ := P M ₃ ⁿ f ₂ ⁿ

^{/- ±- Œ}ˆ10}

M ⁿ⁺¹ := T (M ₃ ⁿ , F x ) and f ⁿ⁺¹ := P M ⁿ⁺¹ T x f ₃ ⁿ

^{/- ±- Œ}‡ 0}

¶_–bAtpb

F x

†q–‡

T x

}tpb3‡mbAº–q5b]‡uƒqÈlvb)ˆnpu¥{Xqȑm±- µ

F _v ⁿ

uxlY†qÈ}…–…5tv{C´muƒa!%nvuƒ{†qÆ{†¨znv_5bH†‡5‚†b)ˆ+nvuƒ{†q¾º«bAx‡

F v (f ₁ ⁿ )

^{{†¨ /}‘m±Ñ‘†‘0¶_5uxˆ‰_šuxl ‡mb+º–q–b)‡¬«b)¥{%¶µ

T _v ⁿ

uƒlnv_–b£Xlvlv{mˆ+ux%nvb]‡Hnvt‰}q«lª…?{†tvn{X…«b)tp}nv{†t]µk}q–‡

T n+1

uxlHlª{†¨Únnvtpr5q–ˆ)%nvuƒ{†quƒqnp_5b

v

¤§‡muƒtpb)ˆnpu¥{Xq¶_–uƒˆ‰_uƒl1}xlª{H‡mb+º«q5b)‡¬?bAƒ{%¶±

hkr–aaHuƒq5—!r5…9µ–¶b“_–C‚Xb

f ⁿ⁺¹ = S _∆t,ε f ⁿ = P M ⁿ⁺¹ T x P M ₃ ⁿ P M ₂ ⁿ T n+1 T _v ⁿ P M ₁ ⁿ T x f ⁿ = ( S _∆t,ε ) ⁿ⁺¹ S ⁰

ε f 0 .