On a Kinetic Equation for Coalescing Particles
38
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On a Kinetic Equation for Coalescing Particles Miguel Escobedo — Philippe Laurençot — Stéphane Mischler. N° 4748 Février 2003. N 0249-6399. ISRN INRIA/RR--4748--FR+ENG. THÈME 4. apport de recherche.
(3)
(4)
(5)
(6) ! #"$ %'&
(7) ! #" (*),+.-0/214357628.9:/2;<8 =?>@ ),1A)AB<BC/EDGFH-<IJ/2K<L28HM =:N MPOQB @ FHK0/R(*),5J6 @ 1,/2I S4TVUXWY[Z]\_^a`bW cVdfegh`jiQklY g#inmogh`jW`fpqergq`bink s YpqtapugqU7WY7p#v iQWmVdjYXwoY7p x:yqinz{Y g}|4~} GenmVmiQyug s Y[yhY7vTVY7yhvT YkHZaZQ\_<72yh`bYXyGrQn\ nZmernYJp ∗. †. ‡. <7XVnn wo`bpugqY7k v Yir:QdjiQ erdCYJer EpuiQdjcVgq`bink pGgqilepqm egh`bendjdbt`bkVTViQWinnY7kVYXiQc pG a`bkVY gh`bvWi s YXd ¡iQy vXiQendjYJpqvX`jkVm eryqgq`fv dbY7pC`fpCm yqiQY s£¢ Y7eQvTm enyugh`bvXdjYY7`jk ` s YXk2gh`¥¤ Y s at`jghp4W¦enphp ¢ WinWY7kQghcVWerk s mHiQpq`jgq`bink¨§ S4TVYdferyhnYgq`bWY v iQk2QYXyhnY7k v Y©gqi¦ª7YXyhi¦`bpGendbpqi«pqTVi#k¨§:S4TVYv inyhkVYJp{ghinkVY[ir¬iQcVyGerk endjtopq`bp `fp4ghT eg ¢ ¡inyGenk2tkVinkVk YXQergq`bnYerk s vXinkanYXw«¡cVk v gq`bink ¢ ghTVYenphpqiavX`bergqY sl yhdj`fv ª[kViQyqW®`bpGe¯$`benmVcVkVi ¡c k v°gh`jiQk erdA§ wo`bpugqY7k v Yenk s eQputaWmoghirgq`fv©Y7T ePa`jiQcVy gqTVY7kyhYXdbt«inkCYJer «enk s pugqyhink ]vXinWm env gqkVYJpqp WYXgqTVi s p#`bk L `bkghTVYpqmV`jyh`jg4inghTVY±}`bx0YXyhk e²³¯¨`binkp gqTVY7inyht¦¡iny4ghTVY|CiQd¥ghªXW¦erk k YJ´2c egh`jiQk¨§ µE¶2·.¸h¹ º »<n wo`bpugqY7k v Y ¢ pughenV`bdj`jg{t ¢ CYJer ¼enk s pugqyhink RvXinWm env gqkVYJpqp ¢ dbinkVRgh`jWY½enpqtaWmogqingq`fv HYXTeP2`biny ¢ ¯¨`fermVcVk i©¡c k v°gh`jiQkVk erd ¢ a`jk Y gq`fv v iQendjYJpqvXYXk vXY ¢r yqdb`fv ª:k inyhW ¢ WiQW]Y7k2ghp ¢ Y7k2gqyhinmV`bY7p ¢ pumVyePt ¢ s yqiQmVdjYXghp ¢Vs tak erW`fv[p{ghYXdbdbeny ¢V¾ enkVYXmingqYXk2gh`bend ¢ a`bkVY gh`bv[Y7kVYXyhnt 1. Û ãäÆGÂÊ¿<ÌÎÔoÂhÀ{å2ÁÄAòPæJÂó{çáÆGÄÅèXÂéXÂhÆ#êÅÄ,æPÌÎÀ{ô7ë{Ç æPõJÄAì ÀuÈ]Ï íhîVÉPÁrÀ[ïäÈ°éhõPÊð°Ã ñ ÂhÔfïäö Ä,éX÷äÀ{ÇPê ÆGÉPõPËhÏÓÄAÄAÌÎÍuÃÓÌÎÂÀ£ÏqÐ.À³ÄÑÔøÂÇPÌ ÕÒ À{ò ÃÓùJÏAÏ,ÌÎÉÌÎô7ÂõJÉÀ Ð7ÉPú£À{Ô$ûÕVü<Âá?ÖÎÏ Ñ<×.ÊÂü¦Ï,Í{ÈPý°ÙÐ£Ú Ø<ÞJÁÐXÂÑ$áÇPÄAÂÌ Ò ÉPÀ{ȦÃ,Ï,ÙÌ ÄAÚ°ó¨ÚPÕVШÂÛoõPÜXÔrÚ á7Ý°ÂÞ°àÝ°ÂhÞ]ÄAÌÎß.À{ÃäÌÎÜXÔÎàþoÂÈ°ÈPõPУÔÎÈ°á7õPÁÏ,ÂÀ<ÌÎÇ2ÿJâÐ Ý Ã,ÕÈ°õ7ÃÓÈ ÄAÓÀ?À{Ä ÉPÀ?ß.Ø<ûûÂÃ,CàrÈ°ÐuÇP÷¡û<ÇPÀ°ü¨Ð ÷äaØ«Ü7ü¨ÿ È7Þ°Í{Ù ôX#õPÀ{þoÇPÈ°Í{õPÈ°ÔÎõPÈ°ÃÓõPÄHÏ,ß<À¬âÍuÕoÀ{âÉPÀ Þ°©ý ÚPÐÝ ý°Ã,ÿÂÇP2Í{À£À°â£ú.Û òPÀ{ãfÆGÏAÇÂÂ{ÌÎùCÔVú£åHíÛHëh¿0ñ Û é CÐ ì{ÃAÂæ ÇJ³Íuè À°ïbÐñ°aèJêÂàn{çXÈ°í Ã,ÂhêhÄ,éoÈ°ï ÌÎÃ,À¨ÉPÀHÊÂhÄAòPó³ã ÆGÃAÂhÂÄAÇPÌÎÍ{ôXÀ©õPÀ{ÂÏÇPØɦÁP¿0ÁPÔÎÊ[ÌÎôXØ:õPó{Ð À{ú£ÏqÐXû<ÑüÇJáÌ Ò ÀuÑÃÓÊÏAÌ Ä,ü¼ó¨ÉPÝ°À£ý°ý°×HÿJÀ{ÐHÃÓÏÅÛÂÍ{ÌÎÔÎÈ°ÔÎÀ{ÔÎÀ}ÏoÜCûáJÈ°ÂÃÓÆGÌÎÇ ÄÂÔÎ<À}õPá7À{õPÇ ÁrÄAó{ÌÎÇQÃÓÌÎÐ7ÀuÚ õJý<Ã,À°ÂuÐ.Ò Ú À{Ç7ý[õJÃÓÀõPÉPÀGÀ{É2ÏHö Û Ñ$ÄAÔÎÂhÆ[ÄAÏÓÐ ã¡aÑÜ ÇPÌÎÏuý Ð °aÿ°ÞÜ ÕVÝ°Þ°Âÿ°ÃÓÌÎý<Ï ×HÍ{À{À{ÃÓÉPÏÅÀ ¦ÌÎÔÎÔÎÞ°À{ýJÏqÐÐ ÃAÂÇPÍ{À°â£ÛVãäÆGÂÌÎÔoå2æ hê uð7íé ì{æPëçð ïbñ 7ê Vï ∗. . †. . . ‡. . . .
(8). . . . . . . .
(9) . " $# . Unité de recherche INRIA Rocquencourt. . . ! . . .
(10) #] [
(11) X
(12) ® % . l
(13) CX! #" ! #"C #
(14) #"
(15)
(16) GiQc p s XWiQkQghyqiQk pd Y wo`fp{ghYXk vXY¦ndbin endjY s Y¦pqindbcogq`bink p[mHinc ycVkVY¦7´2c ergq`bink v `bkV gh`b´2cVY Wi s 7dj`fpqenk2gdfe v i2erdbY7phv Y7k v Y s YEm enyugh`bvXcVdjYJp¦` s YXk2gq`j¤ 7Y7p¦m eny¦djY7cVy«W¦enphpqY ¢ dbYXc y«`jWmVc dbpq`jiQk Y g«dbYXcVy mHiQpq`jgq`bink¨§<Gincp0W]iQk2gqyhink p<´2cVYCdbY7p¬pqindbcogh`jiQk p¨ghYXk s YXk2g¬nYXyp 0 eQputaWmoghirgq`f´2cVYXWY7kQg0YXk gqY7W]mp0nyerk s § ¯$Y«mHin`bkQg]v dbY s Y«kVingqyhY«erk endjtopqY¦Y7pug dfeEmVyhYXcVQY«´Qc Y ¢ miQcVyghincoghY¦¡ink v gq`bink mHiQpq`¥gh`jQY«v iQk2QY woY ¢ dbe k inyhW]Y s yqdb`fv ªenphpuiov `bXYY7pugc kVY]¡ink v gq`binkVkVY7djdbY s Y¦¯¨`fermVc kVi.§¯¨Y7p[yh7pqcVdjgherghp s Y wo`bpugqY7k v Y]Y g s Y vXinWmHinyqgqYXWY7kQg]enpqt2Wmoghirgh`b´2cVYpqY s s cV`fpqYXk2g]endjiQyhpmery s Y7p gqY7vT kV`b´2cVYJp s Yv iQW]menv `jgqÓer`bVdbYY g ¡iQyughY s erk p L s erkpCd Y7pqmVyq`jg s Y©dfe gqTV7inyh`jY s Y±©`bx<Y7yqke YXg4¯$`jiQk p miQcVy4d 7´2c ergq`bink s Y[| indjgqª7WenkVk¨§ º J ¸
(17) wo`bpugqY7k v Y ¢ p{gerV`bdj`jgq ¢ v iQW]menv `jgq Óer`bVdbYYXg©¡iQyughY ¢ v iQW]mHinyqgqY7WYXk2g[eQputaWmoghirgq`f´2cVY ¢ ¡iQk v°gh`jiQkVkVY7djdbY s Y¦¯¨`fermVcVk i ¢ v iQendjYJpqvXYXk vXYvX`jkVXgq`f´2cVY ¢ kVinyhWY s yhdj`fv ª ¢ WinWYXk2ghp ¢ Y7k2gqyhinmV`bY7p ¢ Qenª s YQincogqgqY7djYXgugqYJp ¢s tak erW`f´Qc Y[pugqY7djdfer`byhY ¢ mHirgqY7k2gq`bYXd s Y ¾ enkVYX ¢ 7kVYXyhn`bYv `bkV gh`b´2cVY 1.
(18) .
(19)
(20) !" #
(21) %$'& (
(22) !" #.
(23) $ 0 :
(24) +RY«v iQk pq` s Y7y[gqTVY-, enc vTat½mVyhinVdbYXW ¡inyel a`bkVY gh`bvYJ´Qcegq`binkRWi s YXdbdj`bkV/.,eg elWY7pqiQphv iQmV`bvdbYXnY7d10 ghTVY s tak erW`fv ine¼pqtop{ghYXW'irm eryqgq`fv dbY7pcVk s Y7yqQin`bkV vXiQendjYJpqvXYXk vXY2.¡iQy pugq`fv at30mVyhiov Y7php7§ ¾ inyhY inGm eryqgq`fv dbY7p #`¥ghT m yqYJv `fpuY7djt ¢0s Y7phv yh`jV`bkVEghTVY2enp irGm enyugh`bvXdjYJpatRgqTVY s YXk pq`jg{t W¦eQpqp m ∈ R := (0, +∞) erk s WiQW]Y7k2gqcVW p ∈ R erg}gh`jWfY (t,t ≥x,0m,erk p)s ≥mHiQpq0`jgq`bink x ∈ Ω ⊂ R ¢ Yp{ghc s tgqTVYYXwa`fpugqYXkv Yerk s dbinkV]gq`bWYY7T ePa`jiQcVy4ir¬puiQdjcVgq`bink p ghighTVYY7´2c ergq`bink `bk (0, +∞) × Ω × Y, .54n6§ 470 ∂ f + v · ∇ f = Q(f ) `bk Ω × Y. .84n§Î9 0 f (0) = f :}YXyhY?erk s HYXdbi ¢ `jkiQy s YXy.gqi©puTViQyughYXkgqT Y:kVinghegh`jiQk p ¢ Y¬`jk2ghyqi s c vXY¬gqT Y?W¦enphp{²ÅWinWYXk2gqc W eryh`benVdbY erk s ghTVYnYXdbiov `jg{tPenyq`fer djY v = p/m §0S4TVY[v indbdb`bpq`jiQk¦inmHYXyeghiny Q(f ) y := (m, p) ∈ Y := R × R `fp4n`bnY7kat Q(f ) = Q (f ) − Q (f ) ¢ #TVYXyhY Z Z .54n§ ; 0 1 Q (f )(y) = a(y , y − y ) f (y ) f (y − y ) dm dp , 2 Z Z .54n§ %Z 0 Q (f )(y) = a(y, y ) f (y) f (y ) dm dp . S4T YWY7erk `jkVir0gqTVYJpuY gqY7yqW¦pG`fp#ghTVY¡iQdjdbi#`bkV §#±©YXkVingq`bkV«at = (m, p) e¦m eryqgq`fv dbYin0W¦eQpqpu² WiQW]Y7k2gqcVW y ¢ Q (f )(y) eQvXvXincVk2ghp<¡iQy¬gqT Y#¡inyhW¦egq`binkin.meryqgq`fv {y} dbY7p {y} atvXiQerdbY7phv Y7k v Y4in£pqW¦erdbdjY7y iQkVY7p ¢ `A§ YQ§ ¢ at¦ghTVYyhY7env gq`bink ). *. 3. +. t. 3. x. in. +. 3. 1. 2. m. 1. R3. 0. 0. 0. 0. 0. 0 ∞. 0. 2. R3. 0. 0. 0. 0. 0. 1. {y 0 } + {y − y 0 }. a(y 0 ,y−y 0 ). s YJpqvXyq`bYJpgqTVY s YXmVdbY gh`jiQkRinCm enyugh`bvXdjYJp enk s `A§ Yn§ ¢ aQt¦(fgqTV)(y) YyhY7eQv°gh`jiQk −→. {y} ,. 2. y 0 = (m0 , p0 ) ∈ (0, m) × R3 , {y}. 2tv iQenncVdfegh`jiQk #`jgqT ingqTVY7ym eryqgq`fv dbY7p ¢. a(y,y 0 ). {y} + {y 0 } −→ {y + y 0 } ,. y 0 ∈ Y.. ~ gGe]W`bvXyqi2pqvXinmV`fvGdbYXQYXd ¢ gqTVYv iQdjdb`fpu`binkir$g{ i]m eryqgq`fv dbY7p enk s {y } dbY7e s pCgqi]gqTVY¡iQyqW¦egh`jiQk ¢ gqTVYW¦enphp#erk s WiQW]Y7k2gqcVW Y7`jkV Óat < !" #<
(25) < 0 ir¬e¦pu`bkVndbY[m eryqgq`fv dbY {y } #`jgqTlyegqY a(y, {y} y) vXink pqYXyhnY ss c yq`bkV]gqTVYvXindbdj`fpq`jiQk¨§>=³kingqTVY7y#CiQy s p ¢ #`jgqT y = (m , p ) = (m + m , p + p ). {y} + {y } −→ {y } =³k]v iQkQghyheQp{g$gqi©gqTVY4| indjgqªXW¦enkVk YJ´Qcegq`bink¡iny<YXdfenpugq`fv:v iQdjdb`fpu`bink p ¢ gqTV`fp0`fpenk `byqyhYXQYXypu`bVdbY?W`bvXyqi2pqvXinmV`fv m yqiov YJpqp7§ puY7yqQY ¢ `bk meryqgq`fv cVdfery ¢ gqT ergG`¥g s iaYJp4kVirgGmVyhY7pqYXyhnY©gqT Y 2`bkVYXgq`fv©Y7kVYXyhn@t ? 0. .. 00. 0. a(y,y 0 ). Φke (y, y 0 ) :=. 00. 0. 00. 00. 00. 0. 0. |p|2 |p0 |2 |p00 |2 m m0 + − = |v − v 0 |2 ≥ 0. 2 m 2 m0 2 m00 2(m + m0 ). x¬eryqgq`fv dbY7p?YXniQdja`bkVenv7v iQy s `bkVgqigqTVYJpuY#yhcVdbY7p:enyqY#WYXg ¢ ¡iny?`bk pugherkv Y ¢ `jk s YXk pqYGpqmVyhePtop0`bkanindbnY s `bk½v iQW Vcp{gh`jiQk yhY7eQv°gh`jiQk p ¢ #TVYXyhYdb`b´2cV` sEs yhinmVdbY gp}eryhYvXenyqyh`bY s 2tle¦2enpqYXincp4mVT enpqYerk s cVk s Y7yqQi üü ÇAVÚ hÚ Ý .
(26) Z. #
(27) < 9 3 7
(28) 1#
(29) 3!" . vXiQendjYJpqvXYXk vXY}mVyhiov Y7phpqY7p . s cVYghivXindbdj`fpq`jiQk p 0 §<S4T Y s t2kerW`bv7p ir$ghTVY s YXkpu`jg{t¦ir<dj`f´2cV` s s yhinmVdbY gp4eryhY ghTVYXk s YJpqvXyq`bHY s at½Y7´2c ergq`bink .54n§6470 ¢ nZ ¢ ¢ ¢ puY7Y¦erdfpui 647 ¢ 4P ¢ A§ =³kghTV`fpv iQkQghY wag ¢ gqTVY vXiQendjYJpqvXYXk vXY© QYXyhkVYXd a `fp4n`bnY7kat .54n§ 90 a(y, y ) = a (y, y ) := (r + r ) |v − v |, ¢ s Y7kVirghY]gqTVY¦ye s `b`¬irCgqTVY¦meryqgq`fv dbY7perk s ¢ ¢ #T YXyhY ghTVYX`by}QYXrdbio=v `jgqm`bY7p7§4}rirghY=gqT(megG)ghTV`bp© nY7yqk YXd$vXinyhyqYJpumHink s p gqi«ghTVY YXdbd¥²Å akVi#kEvv yh=iQphp/m pGpqY7v°gh`jviQkl=¡iny©p T /mery s pqmVTVY7yqYJp¬`bk¦gqTVY| indjgqªXW¦enkVkgqTVY7inyhtn§ =³k¦Óenv g ¢ Y7´2c ergq`bink .54Q"§ 470¬ink djtghen nY7p:`jk2ghi env7v inc kQg v iQendjYJpqvXYXk vXY enk s kVY7ndbY7v ghp©gqTVY¡yernWY7kQgegh`jiQkEir?ghTVY s yqiQmVdjYXghp s cVYgqigqTVYenv gq`bink½in?gqT Y2enp ¢ eQp} YXdbd0enp©gqTVY vXink s YXk phegh`jiQk PY7PenmiQyhergq`bink ir?ghTVY s yqiQmVdbY ghperk s gqT YY7dbeQp{gh`b-v .¡nyerª7`jk % 0#vXindbdj`fpu`binkp}in:|CiQd¥ghªXW¦erk k g{tamHYn§ :Gi YXQYXy ¢ erdbd¬ghTVY7pqY«Y .Y7v ghp WePtYW]YXggqiQnYXgqTVY7y ¢ ¡iny`bk pughenk v Y ¢ `bk v iQW Vc pugq`binkRghTVYXiQyqt 2 A§ =³k gqT `bp#v7enpqY ¢ ink Ycpuc endjdbtvXink pq` s YXyp gqTVY[¡iQdjdbi#`jk YJ´Qcegq`bink Î3 4 . 0. 1/3. 0. . . . 0. HS. . 0 2. . 0. 0 1/3. 0. 0. 0. . . . . . ¡iQy gqTVY s Y7k pu`jg{t g(t, x, r, v) ir s yqiQmVdjYXghp #TV`fvT ¢ eg4gh`jWY t ≥ 0 ¢ eryhY[eg gqTVY[mHiQpq`¥gh`jiQk x ∈ Ω #`jgqTle ye s `jc p r > 0 . s yhinm djYXghperyhY puc mVmi2puY s gqiYpqmVTVYXyhY7p<0 ¢ erk s nY7djiovX`¥g{t v ∈ R § =³kEgqT `bpY7´2c ergq`bink ¢ erk s s YXkVingqY©gqTVY s yhinmVdbY g4env7v YXdbYXyegh`jiQk«erk s YXermHinyegh`jiQk¦yhergqY ¢ yhY7pqmYJv°gh`jQYXdbt ¢ gqTVY©gqY7yqW β yhYXm yqYJpuY7ωkQgp#ghTVY Y HYJv°ghpin¬gqTVY]VyhY7er 2²ÅcVmEir:gqTVY s yqiQmVdbY ghp s cVYgqigqTVYenv gq`binkEin?gqTVYQeQp ¢ erQk s gq(g)TVY ghYXyhW Q (g) yhYXmVyhY7pqYXk2ghpGgqTVYY HYJv°ghpin?V`jkeryhtlvXindbdj`fpq`jiQk p}HY g{ YXY7k½m enyugh`bvXdjYJpX§=Åg[`jk vXdjc s Y7pHirgqT Y7dbeQp{gh`bvCv iQdjdb`bpq`bink p<erk s v iQdjdb`bpq`bink p$n`b2`bkV©yq`fpuYCgqi©gqT Y vXiQerdbY7phv Y7k v Y:irVghTVY vXindbdj` s `bkVm eryqgq`fv dbY7p7§^a`bW`jdfery YJ´2c egh`jiQk p WePterdfpuiHY[¡incVk s `bk enYXyhiQpqindHgqT YXinyht În ŧ ~}kVirghTVYXyCmVTatapq`fvXerdpu`jgqc ergq`bink¦`bk¦#TV`fvTvXiQendjYJpqvXYXk vXY#iav7v cVyp¬W¦ePt HY#¡iQcVk s `bk«p{ghYXdbdbeny s tak erW`fvXp `bkgqTVYWi s YXdbdj`bkV ir4v dbinc s pin:m eryqgq`fv dbY7p .ÓQerdfewo`bY7p 0R`bk2gqY7yheQv°gq`bkV2terkergughyheQv°gq`bnY ¾ enkVYXEm en`jy mHirghYXk2gq`ferd −α/r − ε/r .ÓpqYXY ¢ r 0erk s ghTVYyhY ¡YXyhYXkv Y7p#gqTVY7yqY7`jk 0 § =³kEmeryqgq`fv cVdfery ¢ ¡iny α = 0 erk s .Ó}YX4gqiQkV`ferklmingqY7kQgh`ben1d 0 ¢ gqT Yenphpuiov `feghY s v yhiQphp{²³puYJv°gh`jiQk«`bp ε=1 .54n§ ;0 1 m+m a(y, y ) = a (y, y ) := . mm |v − v | =³k HirghTRWi s Y7dbp s Y7phv yh`jHY s TVY7yqY ¢ gqT Y«v iQendjYJpqvXYXk vXYyhergqY v iQyqyhY7pqmHink s p©ghilgqTVY«vXindbdj`fpu`bink¡yhY ² ´2cVY7k v tQ§| cog ¢ v iQdjdb`fpu`bink p]W¦ePt kVingerdb ePtop]yhY7pqcVd¥g¦`bk e v iQenancVdfegh`jiQk¼Y7nYXk2gJ§S4TV`bp]ÓeQv°gv7erk Y eQvXvXincVk2gqY s ¡inyC2t]ghTVY}`bk2gqyhi s c v°gh`jiQk¦ir$evXiQerdbY7phv Y7k v Y}Y «v `bYXk vXt .¡yhYXmVyhY7pqYXk2gq`bkVghTVY}m yqiQ erV`bdb`¥g{t ghT eg}gqTVYg{ i«v iQdjdb` s `jk ¦m eryqgq`fv dbY7p s i¦yqYJerdbdjt p{gh`bv 0°§4S4TVY7k a(y, y E) = E(y, y ) a (y, y ) ¢ #T YXyhY a `fp gqT Y[¡yqYJ´Qc YXk vXt«ir¬v iQdjdb`bpq`bink p4erk s W¦ePt¦HYn`bnY7k ¢ ¡iny#`bk pugherkv Y ¢ at .54n§ 0 iQy .54n§ 90 ¢ puY7Y A§ ¯$Y g¬c p<kVi s YJpqvXyq`bHY:ghTVY#v iQendjYJpqvXYXk vXY:yeghY gqTeg0CY #`jdbdVv iQk pq` s Y7y`bkgqTV`fp0m ermHYXyJ§ +RY#enphpuc W]Y ghT eg a ¡cVd¥¤dbp4ghTVYpqt2WWY ghyqtenk s mi2pu`jgq`ba`¥g{tv inak s `¥gh`jiQk p e § YQ§¬ink Y , .54n§ø 0 0 < a(y, y ) = a(y , y) enk s gqTVYpugqyhc v°ghcVyhY[vXink s `¥gh`jiQk .54n§ ; 0 a(y, y ) ≤ a(y, y + y ) + a(y , y + y ) , y, y ∈ Y. ∂t g + ∇x (v g) + ∇v (β g) + ∂r (ω g) = Qcoll (g) + Qbr (g), 3. br. ! . ! . coll. " $#. % . 2. . 0. 0. 0. NP. . 0. 0. &. 0. 0. 0. 0. 0. . 0. 0. ' . 2. 0. 0. . 0. 0. 0. ÷¡û<ü¨÷äØ.
(30)
(31)
(32) !" #
(33) %$'& (
(34) !" # . RYerdfpqiyhY7´2cV`byqYghTVYnyhi4gqT v ink s `¥gh`jiQk p +. aδ,R (M ) := ωδ,R (R0 ). Z. sup y 0 ∈Y. :=. δ,R. n§ .84n6§ 47%0 .54 ;0. a(y, y 0 ) 1a(y,y0 )≥M dy −→ 0, M →∞. Yδ,R. sup. Z. a(y, y 0 ) dy −→ 0, R0 →∞ |y 0 |. ¡iQy enkat R > δ ¢ #`jgqT Y := (δ, R) × B § =³km eryqgq`fv cVdfery ¢ gqTVY[v i2erdbY7phv YXkv YG QYXyhkVYXd a enk s a ¡c d¥¤ d¨gqTVYeniQYeQpqpqcVWmogq`binkp ¢ eQp4`¥gG`fp#puT i#k`bkE~GmVmHYXk s `¥wl|©§ ¢ gqTVY v7enpqYEir[gqT Y² ~©p¦¡iny ¢ CY WePt v iQk pu` s Y7y¦gqTVYRvXeQpuYEinghTVY½#TViQdjYpum eQv Y s `jWYXkpu`bink enΩdogqiQyqcp Ω = T iny0gqT Y}vXeQpuY#in£e[Hinc k s Y s¦s inW¦er`bkir R Ω§>=³=kghRTVYGdfegughYXy:v7enpqY7p ¢ ink YGT enp ghi«puc mVmVdbYXWYXk2g .84n6§ 4704#`jgqTEY7`¥ghTVYXyGmHYXyh`bi s `bvHinc k s eryhtvXink s `jgq`binkp#iny ¢ ¡iny©`jkp{gerk vXY ¢ kVin²A`bk vXinW`jk cowv iQk s `jgq`bink p iQk {x ∈ ∂Ω , n(x) · v < 0} , .84n6§ 494 0 γf = 0 #T YXyhY γf p{gerk s pG¡iQy#gqTVY gqyenvXYir f inklghTVY Hinc k s eryhtenk s n(x) s Y7kVirghY7pGgqTVYincog{4ery s kVinyhW¦erd c kV`¥g#QY7v°ghinyC¤ YXd s § =³kiny s YXyCgqi¦pq`jWmVdb`¥¡tghTVY[mVyhY7pqYXk2ghergq`bink ¢ Y©iQkVdjt«vXink pq` s YXyCgqTVYv7enpqY `bk ghTVYpuYJ´2cVYXdA§ :Gi YXnY7y ¢ Y© QYXYXmgqT Yk irghergq`bink Ω gqi s ` HY7yqY7k2gq`fegqY[HY g{ YXY7kghTVYpum eQv Yir$ΩmH=iQpq`jRgq`bink p enk s gqTVYpqm envXY[irWiQW]Y7k2ghe .¡iQy#nYXdbiov `jgq`bY7<p 0°§ +RY[¤ k endjdbt¦yqYJ´2cV`jyhY©gqT erg ghTVY[`jk `¥gh`bend s eghcVWT eQp:¤kV`¥ghY[gqingherd.kacVWY7y#irmeryqgq`fv dbY7p ¢ ¤kV`¥ghYgqirgerd W¦eQpqp#enk s ¤kV`¥ghYWY7erk WinWYXk2ghcVW ¢ gqT ergG`bp ¢ .84n6§ 4J; 0 0 ≤ f ∈ L (Ω × Y, (1 + m + |p|) dydx) . cVy#W¦en`jklyhY7pqcVd¥g#`fp gqTVY[¡iQdjdbi#`bkV § ¶aº ¶ # #(
(35) ' <7!" #
(36)
(37) < ; (@ ! a <3! ! # #<#(
(38) # !#"$ p0 ∈R3 ,m0 ≥R0. δ,R. Yδ,R. R. HS. NP. 3. 3. 3. . in. . 1. . . ! ;
(39) #
(40) 1#! ( #%#! &'#)( +*;1#( # !" <#< # ! 3 <7!" #
(41) < @ < ,#!- ./#0&'21 3 #
(42) 1#3 #. !!. T >0. 41
(43) . s `b ∂ ρ+ t. x. f ∈ C([0, +∞); L1 (Ω × Y )), ΩT := (0, T ) × Ω. . j≤0. .
(44). < 5 . 1 . 0. D (ΩT ),. t 7−→. Ω. m f (t, x, y) dydx Y. 7 @!!98 #
(45) ! 3 #
(46) 1#! (#. üü ÇAVÚ hÚ Ý . f (t) → 0. . . Q(f ) ∈ L1 (ΩT × Y ). n§ 7. p f dy,. .84 64 %0. 1# '@ .
(47) < #(%$ <3
(48) 5 !
(49) ' .. .84 64 0. ρ=. 6( 3!" Z Z. f ≥ 0. L1 (Ω × Y ). Z. #. m f dy, j = Y. t → +∞. . Z. Y. n§ XZ.
(50) #
(51) < 9 3 7
(52) 1#
(53) 3!" . 7 3( /
(54) ' . #
(55)
(56) ' p ∈ [1, ∞]
(57) 1# ' . @ <#< $ <3@ 5 !
(58) .84 64;0 t 7−→ kf (t)kL (Ω×Y ) . #
(59)
(60) R > 0
(61) #( # f (t) ⊂ V 7 @!!98 #( # in R = {(x, y) ∈ Ω × Y, |v| ≤ R} R 8 t ≥ 0 f f in⊂mV|v| . . 1 # 4 # @ @ $#8 2 ∈ L1 (Ω × Y ) f in 1# '@ .
(62) < #(%$ <3
(63) 8 !
(64) ' .84 64$%0 2 f in ∈ Lp (Ω × Y ). n§. p. n§ SiEincVy[ akVi#dbY s QY ¢ ghTVY¦Y wo`fp{ghYXk vXYir#puiQdjcogh`jiQk pgqi ghTVY«v i2erdbY7phv YXkv Y]Y7´2c egh`jiQk .84n§6470T eQp[kVirg HYXY7kpugqc s `jY s tQY gC#T YXkgqTVY s `fp{ghyq`bVcogh`jiQk«¡cVk v gq`bink f s YXmHYXk s p4inkgqTVY©¡incVy4eryh`ferVdbY7p (t, x, m, p) § =³k ghTVY pqm egh`bendjdbtTViQWinnY7kVYXiQc p#vXeQpuY f = f (t, m, p) ¢ YXwa`fpugqYXkv Yerk s cVkV`f´Qc YXkVYJpqp#in0pqindbcogh`jiQk pGeryhY YJp{gerVdb`fpuTVY s `bk ¢ #TV`bdjYekacVWY7yq`fvXend¨pqvTVY7WY`bp s YXnY7djiQmY s `bk ŧC¯¨YXgGc p#mHin`bk2gGincog#ghT eg ¢ `bk ¢ ¢ gqT YG¡inyhW c dbergq`bink¦ir .84n§64 0¬`fp s ` .YXyhYXk2g enk s `bk2QindbnYJp0gqTVY©eryh`benVdjYJp (r, v) .¡#`jgqT r = m ¢ 0?`jk pugqYJe s in §¬GYXQYXyqgqTVY7djYJpqp ¢ gqTVY©g{Ci¡inyhW cVdfegh`jiQk pCenyqY}Y7´2cV`bPendjY7k2g enpC`jgC`fp pqTVi#k v`bk =~Gp/m mVmHYXk s `¥w ~§ + TV(m, YXk ghp)TVY s `fp{ghyq`bVcogh`jiQk ¡c k v°gh`jiQk f = f (t, m) s iaY7p¦kVing s YXmHYXk s iQk (x, p) ¢ ghTVY Y7´2c egh`jiQk .84n§64 0yqY s c v YJpghi ghTVYEv dfenphpu`fvXend4^oW]iQdjcvTViGpu a` v iQenncVdfegh`jiQk Y7´2c ergq`bink #T `bvT T enp HYXY7kYXwagqYXkpu`bnY7djtEpugqc s `bY s pq`jk vXY ghTVYmV`binkVY7YXyh`jkV inyh in ¾ Y7djªJer n ŧ +RYyhY ¡Y7ygqigqTVY¦pqcVyqQYXt at ~Gd s incp 64 enk s gqT YHi2iQ at ±}c iopq 2` n ¡iny eEWiQyqY s Y ghen`jdbY s s Y7phv yh`jmogh`jiQk¨§ + TVY7k gqTVY s `bpugqyh`j cogq`bink¡cVk v°gh`jiQk s iaY7p kVing s Y7mY7k s iQk«ghTVY©WiQW]Y7k2gqcVW ¢ eyhYXdfeghY s Y7´2c egh`jiQk ¢ ghTVY s ` .c pu`bnYCv iQenncVdfegh`jfiQk©=YJ´2fc (t,eghx,`jiQk m)¢ T enp£yhY7vXYX`bnY s Wc vTegqgqY7kQgh`jiQkyqYJv Y7kQghdjtQ§ =³k[gqT `bp$Y7´2c egh`jiQk ¢ gqTVY Y7niQdjcogh`jiQkir f #`jgqTyqYJpumHY7v g¬gqighTVYGmHiQpq`¥gh`jiQk x ∈ Ω `fp?Wi s YXdbdjY s at]e s ` .c pu`binkghYXyhW −d(m)∆ f `bk pugqYJe s inVgqTVY4e s nY7v gq`bink[ghYXyhW f §<^a`bk vXY:ghTVYC inyh op¨at| XkV`bdbenk +Ryqª7iQpqYX aenk s ,CiQdjdbY g x0iQcVm erc s 6494 vXink vXYXyhkV`bkVGghTVY s v·∇ `fpqvXyqYXgqY s ` .c pq`jQY#v iQenncVdfegh`jiQkY7´2c ergq`bink ¢ YXwo`bpugqY7k v Y4ir.puiQdjcVgq`bink pgqi ghTVY©vXink2gq`bkacVincp s ` Hcpu`bnYv iQenncVdfegh`jiQkY7´2c egh`jiQk¦T enpCHYXYXk`jkanYJp{gh`j2eghY s `bk ¢ 4J ¢ Q ¢ 4 ŧ¬¯¨YXgCc p mHin`bk2gCiQcogCTVY7yqYGgqT erg:gqT Y© QYXt]Y7pugq`bWergqYJp?c pqY s `bk«in$S4TVYXiQyqY7W 4Q"§ 4}enyqY}`jkgqTVYpumV`byh`¥g ir Î ¢ Q ¢ 4 A§ +RY¬¤ kerdbdjtWYXk2gq`binkghT eg ¢ nYXyht}yhY7vXYXk2gqdbt ¢ e# a`bkVY gh`bv:Y7´2c egh`jiQ' k .¡#`jgqTQYXdbiavX`¥g{teryh`benVdb7Y 0¡inym eryqgq`fv dbY7p c k s YXyhniQ`jk db`jk Y7ery4¡yerQWYXk2ghergq`binkTenp4HYXYXkEpugqc s `jY s `bk 4 ¢ Z A§ ¶ o 1# ! 9
(65) !98 *91#( @< #<3! #< ( < -
(66) #! t 7−→ kf (t) m |v| kL1 (Ω×Y ). .. . . . . . 1/3. . . . . . x. . x. . . . ". . . . . . . . . . . . . *;
(67) # < #
(68) 1
(69) ' !"
(70) # 9<<
(71) !" $5 ' '; / ;< 3 1# (βf ) + ∂ (ωf ) 1#'; - !" < . #( 9 8 #!$ 3 9
(72) #(3 !6 #<#(
(73) . # ∇ v 9<< !"
(74) m ' -5 #$ 7 !!6 1 %$ & ! < < $
(75) '
(76) /
(77) < 3! ! #
(78) 9
(79) #!- 3 9
(80) #( @ #<#(
(81) @ 5 1 < @ (#<# 5 <
(82) #$
(83)
(84)
(85) < 1/#
(86) 1 / # < < !" #<
(87) <
(88) < 1#/
(89) 2 2
(90) @ #
(91) 9! 8 #(1# ; # @ <(/ 8' 5 #
(92) 1
(93)
(94) !" #( <!! 1#( # ! #
(95)
(96) ! 9 < !! 1#( /
(97) 5
(98) ! $
(99) ' 8 1# ; ( $ . #( 9 5 #!- - < #
(100) p . @/
(101) # @ !6%$%
(102) 3@ 5 <3@ @! # 1 ; @ 1 12 -
(103) ' 8 31# ; L 4 64 -= 4 64 . ;0 (fn ) '.54 6470 1 L .84 6470 &. ¯$Y g c p kVi n`bnY¦puiQWY«v inWWYXk2gpink S4TVY7inyhYXW n§ r§ Åg gqcVyhk piQcoggqTegS4TVY7inyhYXW n§ ¦`bpe Xv ink pqY7´2cVY7k v Y¬inVe#p{gerV`bdj`jg{tyqYJpucVdjg ÓpqYXY ^aY7v gq`bink £#TV`bvTeQpqpqYXyqghpHghT eg<eGpuYJ´2cVYXk vXY irVpqindbcogq`bink p ghi n§ 0phegh`bpu¡ta`jk kegqc yhendoiQcVk s p?vXinkanY7yqQY7pCYJer adbt`jk ¢ c mghie[pucVpuYJ´Qc YXk vXY ¢ ghiepuiQdjcogh`jiQk ghi n§ §?#incVQTVdbt«pumHY7en 2`bkV ¢ ghTVY[W¦er`bkW¦eghTVYXW¦egh`bv7erd s ` «v cVdjg{t«`fp gqi]mVyqY7nY7kQg gqT Y©¡iQyqW¦ergq`binkin ÷¡û<ü¨÷äØ.
(104) .
(105)
(106) !" #
(107) %$'& (
(108) !" #. e±©`byheQvCW¦eQpqp?erg:puiQWY#miQ`jk2g?irHgqT YGmVT eQpuYGpqm envXY §0S4T YGp{ghyqc v gqcVyhYGenphpqcVWmogq`bink .54Q§ %00erdbdjiGp cp#gqi¦m yqiQY[gqT erg ¢ ¡iQy©enk2tkVink kVYX2egq`bnYenk s v iQk2ΩQY ×wY¡cVk v°gh`jiQk ¢ ghTVYeQpqpqiov `feghY sE yqdb`bvXªkVinyhW `bpGe ¯$`benmVcVkVi¡c k v°gh`jiQk erd ¢ #TV`fvT m yqY7nYXk2gp vXink vXYXk2gqyegh`jiQkat¦ghTVY±}cVkV¡iny s ²Ax0YXgugq`fp4ghTVYXiQyqY7W §>=³k Óenv g ¢ YvXerkRmVyhinY gqTVY Y7er ½vXinWm eQv°gqk Y7php©`bk L ir:HirghT (f ) erk s (Q(f )) §¦^2gqyhink vXinWm env gqkVYJpqp `bk ir ²³ePnYXyerQY7p#in ghTVYXk¡indbdbiGp}atEghTVYQYXdbiov `jg{t½ePnYXyerQ`jk ¦djY7W]W¦eir4puiQdjcogh`jiQk p}gqi gqTVY ghyhenLk pqmiQyug y Y7´2c egh`jiQk¨§ S4fTVYRyqY7Wen`jk s YXy irghTVYRmVyqiain `bpgqT YXk mY7yu¡iQyqWY s `jk gqTVY pqmV`jyh`jg ingqTVY ±©`bx<Y7yqke²³¯¨`bink pCgqT YXinyht¦¡iQy4gqTVY| indjgqª7W¦erkVk Y7´2c ergq`bink "4 A§ ¯$Y gGc p4¤k erdbdjtyqY7Wenyq ghT eg ¢ ¡iny4ghTVY nY7yqk YXd a ¢ pughergq`bink enyqt«pqindbcogq`binkp gqi .54Q"§ 4 04enyqY .84n6§ 4P9 0 S(m, p) = µ(m) δ #`jgqT µ ∈ M (0, +∞) `fpGe]iQcVk s Y s W]YJenpqcVyhYenk s u ∈ R §:S4T YXinyhYXW 4Q"§ 4[`bWmVdj`bY7p#gqT erg#gqTVYª7YXyhi pqindbcogh`jiQk]`fp¬ghTVY}iQkVdbtpughegh`jiQk eryhtpughergqYG#T `bvT¦`fp?yhY7eQvTVY s `jk«gqTVYGdbink gq`bWYG#TVYXk«pughenyugh`jk ¡yqiQW enk `jk `¥gh`bend s egeV § +RY[gqTac pG` s Y7k2gq`j¡tWinyhYenv7v cVyeghYXdbt¦gqTVYeQputaWmoghirgq`fvpugheghY[gqT enkl`jk ŧC¯$Y gGc p L endbpqi]W]Y7k2gq`binkgqT erg ¢ `j S `fpGe]puiQdjcogh`jiQkghi .54n6§ 470C#TV`fvT `bpGe]pughegh`jiQk eryht«pqindbcogq`bink¡iny4Y7eQvT (t, x) ¢ ghT egG`fp ¢ S `bp#Q`jQYXk2t .54Q"§ 4 0 #`¥ghT µ = µ erk s u = u(t, x) ¢ `¥g}phegh`bpu¤ Y7p 1. 1. n. n. n. . HS. v=u. 1. 3. . 1. t,x. t,x. t,x. ∂t St,x + v · ∇x St,x = 0.. ³k2ghyqi s c vX`jk . ¢ Y¬yqYJerdb`jª7Y0gqT erg (%, u) pqergq`fp{¤Y7p.ghTVY?mVyhY7phpuc yqY7djYJpqp.2enpqY7p£pqtapugqY7W .84n6§ 47% 0 ∂ % + ∂ (%u) = 0, ∂ (%u) + ∂ (%u ) = 0, pqYXY ¢ ¢ n enk s gqTVYyhY ¡Y7yqY7k v YJpCgqTVY7yqY7`jk¨§ +RY:kVi iQcogqdb`bkVY?ghTVYCvXink2gqY7kQgp¨inogqTV`fp$mermHYXyJ§ =³kghTVY:kVYXwag<pqY7v°gh`jiQk ¢ CYCv indbdbY7v°g0puiQWY:´2c erdb`¥gegh`jQY enk s ¡iQyqW¦erd `bko¡inyhW¦egq`bink¼ink ghTVY pqindbcogh`jiQk pX§ S4TVY7tRdbY7e s ghi½gqT Yk ergqcVyerd HincVk s pghT eginkVY vXenk YXwomYJv°g0inkghTVY pqindbcogh`jiQk pX§ =³k]^aYJv°gh`jiQkG Y4pumHY7vX`¥¡t[ghTVYCk irgq`bink irpqindbcogq`bink CY s Y7end2#`jgqT ¢ erk s p{gegqY ghTVY4 nY7tp{gerV`bdj`jg{tghTVYXiQyqY7W .ÓS4TVY7inyhYXW § 90#TV`fvT]`bp0mVyhinY s `bk^aYJv°gh`jiQkZ§0^aYJv°gh`jiQk ©`bp s YXningqY s gqi ghTVY}m yqiair£in£gqTVYv inkaQYXyhnYXkv Y4ir£ghTVY©pqindbcogq`binkp¬gqi ªXYXyhi¡iQy:dferyhnY#gh`jWY7p7>§ +Y}gqTVY7k«Vyh`jY t]Y womVdfer`bk `bk ^aYJv°gq`bink ¦T i ghTVY pugher `jdb`¥g{tyqYJpuc d¥g}e s ermogp4gqimVyqiQYS4TVYXiQyqY7W 4n6§ 4r§ +Y¤ k erdbdbtY7pughenVdb`bpqTlgqTVY YJ´2cV`jendjY7k v Y©YXg{CY7YXkgqTVY©g{Ci¡inyhW c dbergq`bink p:yhe s `jcp{²ÅnY7djiov `jg{t (r, v) erk s W¦enphp{²ÅWinWYXk2gqc W (m, p) in .84n6§ 4 04`bkE~GmVmHYXk s `¥wE~erk s ghTVYXkEvTVYJv ¦gqTegGghTVYv i2erdbY7phv YXkv Y[ nY7yqkVY7dbp a enk s a n`bnY7kat .84n§ 04erk s .84n§ ;0 s i¦phegq`fpu¡t .54Q§Î 0³<² .84n6§ 479 0 `bkE~GmVmHYXk s `¥wl|§ ºa¹ ¶ » ¶ ¶ 7 +RYnyeghY ¡cVdbdjt«eQv akVi#djY s nY©gqT Ym eryqgq`ferd¨pqcVmVmHinyqg#ir<gqTVY c yqiQmYJerkl#Y ² pqY7enyhvT S$yer`bkV`bkV GYXg{CiQyq :}x:G#8² , S ²{rn2P²³n2rQ s cVyh`jkV ghTV`fp¦CiQyq .§ S4T Yl¤ yhpugerk s ghTV`by s encogqTViQyhp}CY7yqYmeryqgq`ferdbdjt pucVm miQyughY s at , ©}^Eenk s
(109) x :
(110) gqTVyhincVQTEe¦ x = ,4^lHY g{ YXY7k ghTVY
(111) k `jQYXypu` s e sRs Y7d:x¬e fp ?enphv iEenk s gqTVY vXindbY«GiQyqW¦erdbY^acVmHXyh`bYXcVyhYn § +RY«erdfpqighT erk x?`bYXyhyhY ±©YXQink s ¡iQy0¡yqcV`jgu¡c d s `fpqvXc phpu`bink p?#TV`bvTWirgh`jergqY#c p0ghiY wagqY7k s S4TVY7inyhYXW 4Q"§ 4 gqi`bkV`jgq`ferd s ege©#`jgqT mHiQphpq`jVdbt«`bko¤ kV`jgqY a`bkVY gh`bv[Y7kVYXyhntQ§ =. %(t, x) :=< m, µt,x > t. '. x. t. 2. x. . . . . . HS. . . üü ÇAVÚ hÚ Ý . NP.
(112) . #
(113) < 9 3 7
(114) 1#
(115) 3!" . "$ 4 ©[
(116) X ! " l _ :
(117) !X" =³k]ghTV`bp:puYJv°gq`bink Y s YXyh`bnY#puiQWY.¡¡inyhW¦erd00v inkpuY7yqQY s ´2c enk2gq`jgq`bY7p?enk s kVinko²Å`bk v yhY7eQpu`bkV}iQkVY7p:enp0 YXdbd,§ ¯$Y gGc p#pughenyug##`jgqT gqTVY[¡iQdjdbi#`bkV]¡cVk s enW]Y7k2gher d .,erk s ¡iQyqW¦erd0C` s YXk2gq`jg{t@?0¡inyGerkat φ : Y → R ghTVYXyhY[TVind s p Z Z Z .Ao6§ 4'%0 1 Q(f ) φ dy = a(y, y ) f f (φ − φ − φ ) dy dy. 2 S4T `bp[` s YXk2gh`¥g{t`bp[iQoghen`jkVY s erägqY7yvT erkVQ`jk eryh`ferVdbY7p[erk s enmVmVdbta`jkV .Ó#`¥ghTVincog#z{cp{gh`¥¤v7egh`jiQk 0©gqTVY cVV`bkV`gqTVY7inyhYXW gqi § :}YXyhY erk s Y7dji ¢ CY mVcog g = g(y) ¢ g = g(y ) enk s g = g(y + y ) ghi¦puTViQyughYXk kVirgegh`jiQk QpX§ (f ) ^ac `¥gerVdbYvTViQ`bvXY7p[ir ¡cVk v gq`bink p `bk .,o6§ 4';0djYJe s gqi½pqYXQYXyerd¬´2c erdb`¥gegh`jQY]`bko¡iQyqW¦egh`jiQk inkRgqTVY pqindbcogh`jiQk f gqigqT YvXiQerdbY7phv Y7k v Y©Y7´2c erφgq`bink .84n6§ 4 0 ¢ enk s iQk«gqT Y©yhY7eQv°gq`bink«gqYXyhW Q(f ) enpC YXdbd,>§ +RYdb`fp{g pqinWY[irgqTVY7W®kVi§ ¾ enphpvXink pqYXyhegh`jiQk¨§ + `jgqT ghTVYvT in`fv Y ¢ ghTVY«ghYXyhW φ − φ − φ enkV`bpqTVYJp erk s CY • s Y s c vXYgqTVY[ghirghend¨W¦enphp4v inkpuY7yqegh`jiQk ? φ(y) = m Z Z Z Z .Ao§Îr% 0 m f (x, y) dydx , t ≥ 0. m f (t, x, y) dydx = ¾ inWY7kQghcVW v iQk puY7yqergq`bink¨§©^o`jW`bdbenyqdbt ¢ #`¥ghT½ghTVY]vT in`fv Y = p ¢ ghTVYgqYXyhW φ − φ − φ erdfpqi • enkV`bpqTVYJp#erk s CY s Y s c vXY[gqTVYWY7enkWiQW]Y7k2gqcVW v inkpuY7yqeφ(y) gh`jiQk ? Z Z Z Z .Ao§Î3 4 0 p f (t, x, y) dydx = p f (x, y) dydx , t ≥ 0. =ÅgG`fp#kVYXw2gGvXdjYJery4¡yhinW .Ao6§ 4'90CghT eg ¢ `¥ φ : Y → R `fp#puc e sVs `¥gh`jQY ¢ ghT eg#`fp ¢ ¡iQyGerdbd (y, y ) ∈ Y , .Ao§În; 0 φ(y + y ) ≤ φ(y) + φ(y ) ghTVY[¡indbdbi#`jkV]W¦erm Z Z +. 0. Y. Y. 0. 00. 0. 0. Y. 0. 1. 0. 00. 00. 0. 0. in. Ω. Ω. Y. Y. 00. 0. in. Ω. Y. Ω. 0. Y. 0. 0. t 7→. 2. f (t, x, y) φ(y) dydx. `fpGekViQko²Å`jk vXyqYJenpq`jk ¡c k v°gh`jiQkEir<gq`bW]YQ§+Yk i ` s YXk2gq`j¡tlpuY7nYXyerd¨vXdbeQpqpqY7p4in0¡cVk v°gh`jiQk p}phegh`bpu¡ta`jk .Ao§În 0?} Y ¤yhpug[vXink pq` s YXy©¡cVk v gq`bink p s Y7mY7k s `bkV pqindbYXdbtEink ¢ ghTVYXk¡cVk v gq`bink p s Y7mY7k s `bkVEpuiQdjY7djt iQk v ¢ enk s ¤k erdbdjt¡cVk v gq`bink p#HYX`bkV]gqT YmVyqi s cv°gGir0e¡cVk v°gh`jmiQklir m enk s e¡cVk v°gh`jiQklir v § SCt2m `bv7erd0Y wVerWmVdbY7pinCpuc e sVs `¥gh`jQY ¡cVkv°gq`binkp[`bp φ(y) = m ¡iny α ∈ (−∞, 1] § =³kRm eryqgq`fv cVdfery ¢ •ghTVYvTVin`fv Y pqTViGp#ghT eggqTVY #
(118) .8<!!6 ghirghendkacVWY7y©in?m eryqgq`fv dbY7p s YJv yhY7enpqY7p}#`¥ghTEgq`bWY φ(y) = 1 .,enp4YXwamHY7v gqY s 0 Z Z Z Z Z Z Z Z .Ao§Îr% 0 1 a f f dy dydxdt = f (x, y) dydx. f (t, x, y) dydx + Ω. Y. α. . t. Ω. Y. 2. 0. 0. Ω. Y. Y. 0. in. Ω. Y. ÷¡û<ü¨÷äØ.
(119)
(120)
(121) !" #
(122) %$'& (
(123) !" #. ~}kVirghTVYXy]v iQk puYJ´2cVYXk vXY«`fpgqTeg ¢ `j}pqcVmVm f (x, y) ⊂ M := {(x, y) ∈ Ω × Y , m > δ} ¢ gqTVY7k pqcVmVm f (t) ⊂ M ¡iQy:erkat t ≥ 0 >§ =³k s YXY s£¢ `jgC`fp?pqc «v `bYXk2g?ghikVirgh`bvXY4gqT erg φ(y) = 1 (m) phegq`fpu¤ Y7p .Ao§În 0 §#S4T eg}WYJerk p4ghT eg©kVi«m enyugh`bvXdjYin¬pq`bªXYpqWendjdbYXyGgqT enk v7erkEHY v yhY7ergqY s `j¬gqT YXyhYeryhYkVinkVY `bkV`jgq`ferdbdjtQ§ ~ Y7en nYXyQYXypu`bink irgqTV`fpÓeQv°g«`fpgqTVY½¡indbdji#`bkV ?lδ ¡iQy«erkat kViQko²Å`jk vXyqYJenpq`jk ¡cVkv°gq`bink .,pucvTleQp φ(m) = m ¢ α < 00 ¢ YTePnY φ:R →R Z Z .Ao§ÎZ 0 1 Y (f (t, .)) + Y (f (τ )) dxdτ ≤ Y (f ) , 2 #`jgqT in. δ. &. δ. +. [0,δ]. α. +. t. φ. φ. 0. Ω. Z Z. Yφ (f ) :=. Y. •. f (x, y) φ(m) dydx,. ZΩ ZY. Yφ (f ) :=. in. φ. a φ(m) f f 0 dydx .. Y. inyGerkat«k inko² s Y7vXyqYJenpq`jk erk s kViQkVkVY7Qegh`jQY}¡c k v°gh`jiQk. φ : R + → R+. ¢ Y[kVirgh`bvXY[gqT erg. |p + p0 | ≤ |p| + |p0 | ≤ (m + m0 ) max(|v|, |v 0 |),. enk s gqTac p |v | ≤ max(|v|, |v |) §S4TVYEWinkVingqink `bvX`¥g{t in φ gqTVY7k `bWmVdj`bY7p¦gqT erg φ ¡cVdj¤ dfp ,. V§ Q 0°§ ,CiQk pqY7´2cVYXk2ghdjt ¢ Z Z 00. 0. f (t, x, y) φ(|v|) dydx. t 7−→. f` pekViQko²Å`jk vXyqYJenpq`jk ¡cVk v°gh`jiQk in?gq`bWYn§~}pe¤ yhpugvXink pqY7´2cVY7k v Yir¬ghTV`bp©ÓeQv°g ¢ Y yhY7endj`bªXY gqT erg ¢ `j ¢ ghTVYXkRpuc mVm f (t) ⊂ V ¡iny[erkat t ≥ 0 .ÓenmVmVdbtlgqTVY pqcVmVm m yqY72`binfcpC⊂yhY7pqVcVdjg#:=#`¥gh{(x, TghTVy)YvTV∈iQΩ`bvXY ×φY,=|v|1 ≤ R}0 § ,Cink pq` s YXyCkViekViQkVkVY7Qegh`jQYGerk s kVinkV²A`bk v yhY7eQpu`bkV[¡cVkv°gq`bink φ : R → R erk s ekVink kVYX2egq`bnY • s enk v inkaQY w¦¡cVk v gq`bink φ : R → R §?S4TVYXk Ω. in. Y. R. R. [R,+∞). 2. 1. 3. t 7−→. Z Z. +. f (t, x, y) m φ1 (m) φ2 (v) dydx. `fpGekViQko²Å`jk vXyqYJenpq`jk ¡c k v°gh`jiQkEir<gq`bW]YQ§ =³k s YXY s£¢ mVcVgugq`bkV φ(y) = m φ (m) φ (p/m) ¢ y ∈ Y ¢ ghTVY vXinkanYXwo`¥g{t¦in φ erk s ghTVYWinkVingqink `bvX`¥g{t«in φ YXkpucVyhYgqT erg Ω. Y. 1. 2. 2. 1. m0 m v + v0 φ(y + y ) = (m + m ) φ1 (m + m ) φ2 m + m0 m + m0 ≤ φ1 (m + m0 ) (m φ2 (v) + m0 φ2 (v 0 )) 0. 0. 0. . ≤ m φ1 (m) φ2 (v) + m0 φ1 (m0 ) φ2 (v 0 ) ,. enk s gqTVY[¡cVkv°gq`bink φ pqergq`fp{¤ YJp ,. V§ Q 0°§ üü ÇAVÚ hÚ Ý . .
(124) J. #
(125) < 9 3 7
(126) 1#
(127) 3!" . 4. ~ ¤ ypug `bk2gqYXyhY7pugq`bkVv iQk puYJ´2cVYXk vXY©in$ghTV`bp4yhY7pqcVdjg#`bp ghTVY s Y7v7ePt¦ir$ghTVY[ a`jkVYXgq`fvYXkVY7yqQt ¡. #`jgqT gqTVY vT in`fv Y φ(y) = |p| /m ¢ ghT egG`fp ¢ φ (m) = 1 enk s φ (v) = |v| 0? Z Z Z Z Z Z .Ao§Î ;0 |p| 1 |p| f (x, y) E (f (t, x, .)) dxdt = dydx + dydx, f (t, x, y) m 2 m #T YXyhY Z Z .Ao§Î %0 mm a E (f ) := |v − v | f f dy dy . ~ WinyhYQYXkVY7yhendm yqiQmY7yug{tE`fp[env°ghc erdbdbtEmPendj+` s m § ,Cink pq` s YXy[e kVinkVk YXQergq`bnY]erk s kVinko² s YJv yhY7enpq`bkV vXinkanYXw¡c k v°gh`jiQk λ ∈ C ([0, +∞)) pucvT½gqTeg λ(0) = 0 § + `jgqTgqTVY¦vTViQ`bvXY φ(y) = m λ(|p|/m) .¡gqT ergG`bp ¢ φ (m) = 1 erk s φ (v) = λ(|v|)0 ¢ gqTVY[¡c k v°gh`jiQk φ `fp#pucVe sVs `jgq`bnYenk s CY[T ePQY 2. 1. 2. 2. t. 2. Ω. 0. Y. Ω. Ω. 0. ke. Y. 2. in. ke. 0 2. . Y. 0. . 0. 0. Y. 1. 1. Z Z. #T YXyhY Ω. Y. 2. 1 f (t, x, y) m λ(|v|) dydx + 2. Eλ (f ) :=. Z Z. ³k s YXY s£¢ mVcogqgq`bkV =. Y. am Λ. Y. . Z tZ 0. |p| |p| + |p0 | , m m + m0. Eλ (f (t, x)) dxdt ≤ Ω. . Z Z Ω. 0. 0. f f dy dy ,. Ao§ÎQ90. f in (x, y) m λ(|v|) dydx,. .. Y. Λ(A, B) :=. Z. B. Ao§Îr%0. (λ0 (B) − λ0 (z)) dz .. .. ¢ gqT YWiQkVirghinkV`fv `jg{t«ir λ Y7kQger`bdbp ghT eg w = (|p| + |p |)/(m + m ) 0. A. 0. φ(y) + φ(y 0 ) − φ(y + y 0 ) ≥ m0 (λ(|v 0 |) − λ(w)) + m (λ(|v|) − λ(w)).. S4T Y[vXinkanYXwo`¥g{t¦in λ enk s gqTVY` s YXk2gh`¥g{t. ghTVYXkl`bWmVdjt. m0 (|v 0 | − w) = m (w − |v|). φ(y) + φ(y 0 ) − φ(y + y 0 ) ≥ m0 (|v 0 | − w) λ0 (w) − m ≥ m. Z. w. Z Z. w. λ0 (z) dz. |v|. (λ0 (w) − λ0 (z)) dz = Λ(|v|, w) . |v|. ~}kVirghTVYXyGvXink pqY7´2cVY7k v Y[`fp gqTVY[¡iQdjdbi#`bkV §¬ViQyGerkat«v iQkanY w¡cVk v°gh`jiQk t 7−→. Z. ω : R3 → R+. ¢ gqTVY[¡c k v°gh`jiQk. f (t, x, y) ω(x − v t) dydx. `fp?ekViQko²A`bk vXyqYJenpq`jkV©¡cVk v gq`bink¦ingh`jWYn§ =³k s Y7Y s£¢ ¡iny¬YJenvT (t, x) ∈ R × R ¢ φ : y 7→ ω(x − v t) `fp?e vXinkanYXw¡cVk v gq`bink¦in v erk s `fp¬gqTac p:pucVe sVs `jgq`bnYQ§¬S4TVYXyhY ¡iQyqY ¢ eäghYXy?WcVd¥gh`jm djta`bkV .54Q§"4 00at ω(x − v t) enk s `bkQghYXQyhergq`bink ¢ gqTVY v ink2ghyq`bVcogh`jiQk ir4gqTVY v i2erdbY7phv YXkv Y]gqY7yqW`bpkViQkVkVYX2egh`jQY ¢ #T `jdbY«ghTVY¦gqY7yqW¦p yhY7pqcVdjgq`bkV¡yhinW gqT YG¡yhYXYGghyhenk pqmiQyugCv7erk vXYXdA§ =³km eryqgq`fv cVdfery ¢ ger a`bkV ω(z) = |z| ¢ θ ≥ 1 ¢ z ∈ R ¢ CY iQoghen`jk Z Z Z Z .Ao§Î %0 f (t, x, y) |x − v t| dydx ≤ f (x, y) |x| dydx , t ≥ 0 . Ω. Y. +. 3. θ. θ. Ω. Y. in. Ω. 3. θ. Y. ÷¡û<ü¨÷äØ.
(128)
(129)
(130) !" #
(131) %$'& (
(132) !" #. 4;4. <`jk endjdbt ¢ dbY g Φ : [0, +∞) → [0, +∞) HY e¼kVink kVYX2egq`bnYRerk s v iQk2QY w ¡cVk v gq`bink pqergq`fp{¡ta`bkV §¬S4TVYXk Z Z. • Φ(0) = 0. t 7−→. Φ(f (t, x, y)) dydx. `fp#e]kVinkV²A`bk v yhY7eQpu`bkV¡cVk v gq`binklirgq`bW]YQ§ ¾ iQyqYmVyhY7v `fpqYXdbt ¢ YTePnYgqTVY[¡iQdjdbi#`bkVyhY7pqcVdjg C¶ 8 #
(133) ! 3 f !/ ./#0&' #
(134) 1#! (# Ω. Y. . Z Z Ω. 1 . Φ(f (t, x, y)) dydx +. Z tZ 0. Y. Ω. Ω. . Ao§ n%0 .. Φ(f in (x, y)) dydx , Y. 1 2 DΦ (f ) := DΦ (f ) + DΦ (f ) Z 1 1 DΦ (f ) := a(y, y 0 ) (f ∨ f 0 ) Φ(f ∧ f 0 ) dydy 0 ≥ 0 , 2 Y2 Z 2 a(y, y 0 ) Ψ(f ) f 0 1(m,+∞)×R3 (y 0 ) dy 0 dy ≥ 0 , DΦ (f ) := Y2. . DΦ (f (τ, x)) dxdτ ≤. Z Z. . . Ψ(u) :=u Φ0 (u) − Φ(u) ≥ 0 u≥0 max {f, f 0 } f ∧ f 0 = min {f, f 0 }.
(135) !6 1 1 #
(136) # . Ao§ 470 .. f ∨ f0 =. ¢ ¢ gqTV`fpyqYJpucVdjgTenpY7YXkmVyhinY s `jk 4 ¡inygqTVY ^aWindbc vTViGpq 2` + TVY7k YJ´2c egh`jiQk¨§ Φ(u) = u p > 1 º¨º +RY[¤ yhpugGyhY7vXendjd¨gqT ergGgqTVYvXinkanYXwv iQkz{cV2egqY Φ ir Φ `fp}ek inkVkVY7Qergq`bnYv iQk2QY w¦¡cVkv°gq`bink Q`jQYXk2t . p. . . ¡iQy. ∗. Φ∗ (u) := sup {u w − Φ(w)}. § =³kEe sVs `jgq`bink ¢ Y[T ePnY©gqT Y ?incVkV`bkVY7´2c endj`jg{t u≥0 w≥0. . enk s gqTVY`bkVY7´2c endj`jg{t | t .Ao§ Q 04enk s .,o§ n;0 CY[T ePQY Z. a f f 0 Φ0 (f 00 ) dy 0 dy Y. . =. 2. ≤. u w ≤ Φ(u) + Φ∗ (w) ,. Φ∗ (Φ0 (u)) ≤ Ψ(u) , u ≥ 0 .. Z. a (f ∧ f 0 ) (f ∨ f 0 ) Φ0 (f 00 ) dy 0 dy. ZY. 2. Y2. ≤. üü ÇAVÚ hÚ Ý . Ao§ Q;0 .Ao§ n%0 .. u, w ≥ 0 ,. Z. Y2. a (f ∧ f 0 ) {Φ(f ∨ f 0 ) + Φ∗ (Φ0 (f 00 ))} dy 0 dy Z 0 0 0 a (f ∧ f ) Φ(f ∨ f ) dy dy + a (f ∧ f 0 ) Ψ(f 00 ) dy 0 dy. Y2.
(137) P. #
(138) < 9 3 7
(139) 1#
(140) 3!" . 4. RY k i c puY .54Q§ 90#gqiiQcVk s ghTVYpqY7vXink s gqYXyhW ir¬ghTVY yh`jQT2gu²ÅT erk s pq` s Y in¬gqTVYerHinY`bkVY7´2c endj`jg{t enk s s Y s cv Y[gqTeg +. Z. 0. 0. 00. a f f Φ (f ) dy dy. ≤. Y2. CiQk pqY7´2cVYXk2ghdjt ¢. Y2. Y2. ,. Z. Z. a (f ∧ f 0 ) Φ(f ∨ f 0 ) dy 0 dy Z (a(y, y 00 ) + a(y 0 , y 00 )) (f ∧ f 0 ) Ψ(f 00 ) dy 0 dy + Y2 Z Z a(y 0 , y 00 ) f 0 Ψ(f 00 ) dy 0 dy a (f ∧ f 0 ) Φ(f ∨ f 0 ) dy 0 dy + 2 ≤ Y2 Y2 Z a (f ∧ f 0 ) Φ(f ∨ f 0 ) dy 0 dy ≤ Y2 Z +2 a f 0 Ψ(f ) 1(0,m)×R3 (y 0 ) dy 0 dy .. 0. Q(f ) Φ0 (f ) dy Y. Z 1 a f f 0 (Φ0 (f 00 ) − Φ0 (f ) − Φ0 (f 0 )) dy 0 dy 2 Y2 Z Z 1 0 0 0 ≤ a (f ∧ f ) Φ(f ∨ f ) dy dy + a f 0 Ψ(f ) 1(0,m)×R3 (y 0 ) dy 0 dy 2 Y2 Y2 Z Z 1 − a Ψ(f ) f 0 dy 0 dy − a (Φ(f ) f 0 + Φ(f 0 ) f ) dy 0 dy , 2 2 2 Y Y. =. # T YXk vXY .Ao§ n90 § t u +RYCpqcVWW¦eryh`jª7Y ¢ `jkghTVY?kVYXwagyhY7pqcVd¥g ¢ ghTVY @(( Y7pugq`bW¦eghY7p¨iQk[gqTVY puiQdjcVgq`bink p.ghi .54Q§"4 0.inVgher`bkVY s `bk gqTV`fp#pqY7v°gh`jiQk¨§ ¶aº ¶ # #(
(141) ' f # 1#3 (# . . in. K(f ) :=. Z Z . in. in. in. Φ(f (x, y)) + f (x, y). . r(m) + 1 + m + |p| + m λ. . |p| m. . Ao§ rZ. dxdy < ∞,. . 0 #
(142)
(143) ' $% !5 . ;
(144) < #( $ < 5 +*><3@ # Φ ∈ C 1 ([0, +∞)) λ ∈ C 1([0, +∞)) #( Φ(0) = 0 λ(0) = 0 #
(145) / $% !5 2 .
(146) < #(%$ <3
(147) ( #
(148) ! 3 f !/ ./#0&' /
(149) !!98 #
(150) 1#3 # r ∈ C((0, +∞)) Ω. sup t≥0. Z. Z Z Ω. ∞. Z. Y. Y. Φ(f (t, x, y)) + f (t, x, y). . r(m) + 1 + m + |p| + m λ. |p| m. . . φ=1. . φ=r. . Ao§ ;0 .Ao§ %0. dxdy ≤ K(f in ),. (DΦ (f (τ, x)) + Y1 (f (τ, x)) + Yr (f (τ, x)) + Eλ (f (τ, x))) dxdτ ≤ 2 K(f in ),. 1 0 Y Ω Y E D ; ' & 0& #1 & 1"$ r # λ < !5 !98# Φ . . .. . (# < !5
(151) !98 & 0& ' . ÷¡û<ü¨÷äØ.
(152) J.
(153)
(154) !" #
(155) %$'& (
(156) !" #.
(157) < X!
(158) . 4. 0 #"$ !
(159). ¶ º
(160) f in @ $% !5 <3
(161) #
(162) 1# 8%$ ( ! & ) 1 <' #
(163) ! 3 !/ ./#0&''1# $% !5 <3
(164) . f ∈ C([0, +∞); L1 (Ω × Y )) ,. f (0) = f in ,. #
(165) 1# 8%$ & $ & - #
(166) @ $% !5 @ . 9 < #(%$ < 5 * <3@ # Φ ∈ λ ∈ C 1 ([0, +∞)) #
(167)
(168) ' $% !5 < 5 +* - .
(169) < #( $ < 3@-. C 1([0, +∞)) #< 1 . . r ∈ C ((0, +∞)). Φ(0) = 0 λ(0) = 0. lim λ0 (u) = lim. Φ(u) = +∞ u. . ,V§ 290 .. lim r(m) = +∞ ,. $;
(170) 1 < !/ #
(171)
(172)
(173) # 5 1#( ( 3 #$
(174) #( ! #
(175) -# 1# 8- ' !9. ! 5 @ ( # ! ' #!/# < ' 3 # & $ & - !
(176) ! 8
(177) !/6
(178) '' ; (##
(179) #
(180) #( < 8 T ∈ R + . %0 Qi (f ) ∈ L1 (ΩT × Y ), i = 1, 2. + .54 "470 (.54 0 1 < #
(181) ! 3 !$ ./0&$21 ! ;
(182) f (0) !"
(183) 7 6 n. #<#<
(184) ' <3( n & ≥ 1 $ fn & ! 3 /
(185) 9! 81 ' (#
(186) > n ≥ 1 1 K(f (0)) n #
(187)
(188) ' K > 0 <3@ # (Φ, λ, r) #
(189) 1# 8%$ ( - *91#( >#( #
(190) <
(191) @≤ K0 5 0 . <3
(192) ( # 1 # 1.
(193) # ! 3 . . ! / / . # 0 ' & (fn ) (fn ) f f C([0, T ); w − L1 (BR × Y )), fn −→ f . #%0 Qi (fn ) * Qi (f ) L1 ((0, T ) × BR × YR ) 8 1 u→+∞. u→+∞. m→0. . . . RYkVi*pughergqYe] Y7er «pughenV`bdj`jg{t«mVyh`jkv `bmVdjY[¡iQy#CYJer «pqindbcogq`binkp gqi Q§ Ų n§Î § ¶aº ¶ . ,V§ n. . . . . k. ,V§ . k. k. T, R ∈ R+ Z. ψ ∈ D(Y ) # Y L1(Ω × Y ) . i ∈ {1, 2}. ψ(y) fnk. YR := (0, R) × BR Z 1 dy −→ ψ(y) f dy L ((0, T ) × BR ). ,V§ ZQ%0 .. 9
(194) # # 9< 8 C ∞ . #
(195) '7 <
(196) 9
(197) ! 8 #( #( <3 $. D(Y ) #9< 5 1 <' !98 < # < 3@ # <
(198) [0, T ) C([0, T ); w − L1 (Ω × Y )) .
(199) . Y. . Y. & <¦ # 0 =³kgqTV`fp#pqY7v°gh`jiQk ¢ CY[v iQk pq` s Y7y#epqY7´2cVY7k v Y (f ) ir0puiQdjcVgq`bink p:gqi .54Q§"470Cphegh`bpu¡ta`jk gqT YyhY7´2cV`byhYXWYXk2ghp in[S4T YXinyhYXW § V§ +Yl¤yhpug«mVyhinYgqT erg .,o§ 90erk s .,V§ 90]nc enyhenk2gqYXYgqT YECYJer ¼vXinWm env gqkVYJpqp ¢ `bk ir erk s §lS4TVY«kVYXwagp{ghYXm `fp[gqi c puY«gqTVY«m yqiQmY7yugh`jYJpir4gqT Y«dj`bkVYJery ghyhenLk pqmiQyug[(fY7´2)c ergq`bink(QgqiE(finVgh))er`bki gq=TVY«1,pugq2yhinkV L ²ÅvXinWm eQv°gqk Y7php©in y²³ePnYXyerQY7p}`jk e ePt½pq`bW]`bdfery[gqi ghTVYl|CiQd¥ghªXW¦erk k Y7´2c ergq`bink 64 ŧ S4T erk ap gqigqT Y pugqyhinkVv inWm eQv°ghkVY7phpghT2cpiQoghen`jk Y s£¢ YW¦ePt ` s Y7kQgh`¥¡tgqTVYdb`bW]`jgGir<gqT YkVinkVdb`bkVY7eny4vXiQenncVdfegh`jiQk«gqY7yqW (Q(f )) § . n. . 1. n. . i. n. . . 1. n. üü ÇAVÚ hÚ Ý .
(200) 7Z. #
(201) < 9 3 7
(202) 1#
(203) 3!" . 4. .
(204)
(205) !"
(206) #$%
(207) $
(208) &' (). RY¤ yp{g4mVyhinYGghT eg .Ao§ 0 erk s .Ao§ 90C`bW]m djt¦ghTVY[CYJer L ²ÅvXinWm eQv°gqk Y7phpCin (f ) enk s (Q (f )) § C¶ +* 7 <9 T > 0 R > 0 #
(209) <
(210) < (f ) 1# 1 < !98'<
(211) ; > L ((0, T ) × +. . . . 1. . #
(212) <
(213) < B 1# R;× Y )' -,$#. . (Q1 (fn )). . n. 1. . 1#1 <' !98
(214) 9
(215) n. 1. L ((0, T ) × BR × YR ). . n. 1
(216) . 1. YR. º¨º +Y¤Vw erk s § pqYXyhnY¤ yhpuggqT ergghTVY¤ yhpugenphpuY7yugh`jiQkEirC¯$YXWW¦eZ §ø`fpe pugqyer`bnT2gq¡inyh eny s TvXin>k pqY70´2cVY7k v Y[Rir>.,0V§ 0 ¢ .,V§ 204enk s gqTVY±©cVko¡iQy s ²Åx0Y gqgq`fp4gqTVY7inyhYXWl§ +RYkVYXwagp{ghc s tRgqT Y«CYJer Rv inWm eQv°ghkVY7phpmVyhinmHYXyqgq`bY7p ir §½¯$Y g HYelWY7eQpuc yhenVdjY pqcV pqY g]ir (0, T ) × B × Y erk s djYXg M ¢ N ¢ δ enk s R (QY mi2(fpu`jgq))`bnYyqYJerd EkacVW HYXyppqc vT ghT eg §¬x?cVgugq`bkV ϕ = 1 erk s R ≥ R + 2R /δ . . . 1. R. 2. 1. R. n. 1. E. A = An (t, x, y, y 0 ) := a(y, y 0 ) fn (t, x, y) fn (t, x, y 0 ) ,. enk s mY7yu¡iQyqW`bkV]gqTVYvT enkVnYir0eryh`benVdbY7p (y, y ) → (y , y − y ) ¢ Y[T ePnY 0. Z. #`jgqT 2I1. =. Z. δ. enk s. Z. δ. {|p0 |>R1 }. Z. R3 0 RZ. Z. Z. 0. + +. R. Z. R. δ. Z. Z. R. Z. kgqT YinkVY[T enk s£¢ iQ puY7yqa`bkVgqT erg δ. Q1 (fn ) ϕ dy = I1 + J1. B(−p0 ,R) Z RZ. 1}. 0. YR. A ϕ00 dydy 0 +. {|p0 |≤R. 1 J1 = 2. 0. δ. Z. R. δ. Z. Z. R δ. Z. R3. Z. δ 0. Z. A ϕ00 dydy 0 B(−p0 ,R). A ϕ00 dydy 0. ÓZ § Z. B(−p0 ,R). . 4 0. A ϕ00 1a≥N dydy 0 B(−p0 ,R). {|p0 |≤R1 }. Z. R δ. Z. A ϕ00 1a≤N dydy 0 . B(−p0 ,R). |p| R R1 |p0 | + |p| ≤ ≤ ≤ , m δ 2R m + m0. pqi]gqT erg mΛ. . |p| |p0 | + |p| , m m + m0. . ≥ Λδ,R (R1 ) := δ Λ. . R R1 , δ 2R. . ÷¡û<ü¨÷äØ.
(217)
(218)
(219) !" #
(220) %$'& (
(221) !" #. ¡iQy. erk s. y ∈ (δ, R) × B(−p0 , R). c y 0 ∈ (δ, R) × BR 1. 4. ¢ erk s. A = a (fn ∨ fn0 ) (fn ∧ fn0 ) ≤ a (fn ∨ fn0 ) M 1fn ∧fn0 ≤M + ≤ a (fn + fn0 ) M +. Y[nYXg 2I1. ÓZ § Z2;0 .. M a (fn ∨ fn0 ) Φ(fn ∧ fn0 ), Φ(M ). Z RZ Z δZ 1 r(m0 ) A dydy 0 r(δ) δ 0 0 B(−p0 ,R) 0 B(−p0 ,R) Z RZ Z RZ |p| |p| + |p0 | 1 , A dydy 0 mΛ + Λδ,R (R1 ) δ {|p0 |>R1 } δ B(−p0 ,R) m m + m0 ZZ a 1a≥N (fn + fn0 ) dydy 0 +M 1 r(δ). ≤. δ. M a (fn ∨ fn0 ) Φ(fn ∧ fn0 ) 1fn ∧fn0 ≥M Φ(M ). Z. 2 Yδ,R+R. M + Φ(M ). R. Z Z. ZZ. Z. r(m) A dydy 0 +. 1. 2 Yδ,R+R. a (fn ∨ fn0 ) Φ(fn ∧ fn0 ) dydy 0 1. 2 1 ≤ Yr (fn (t, x)) + Eλ (fn (t, x)) r(δ) Λδ,R (R1 ) Z M D1 (fn (t, x)) . +2 M aδ,R+R1 (N ) fn (t, x) dy + 2 Φ(M ) Φ Y. kgqT YirgqT YXy#T enk s£¢ at .84n§ 904erk s Ó. Z § Z20 ¢ CYT ePQY J1. ≤. Z. Yδ,R+R1. ≤ M N ≤ 2M N. Z. Z. A 1a≤N ϕ00 dydy 0 Yδ,R+R1. Yδ,R+R1. Z. Z. Yδ,R+R1. (fn + fn0 ) ϕ00 dydy 0 +. fn ϕ0 dy 0 dy +. 2M D1 (fn (t, x)) Φ(M ) Φ. 2M D1 (fn (t, x)) . Φ(M ) Φ. RY[kVi2egqT YXy gqTVY[mVyhYXa`binc pCY7pugq`bW¦eghY7p#erk s `bkQghYXQyhergqYinYXy (0, T ) × B § #`jkVgqi A. o§ 0 erk s .Ao§ 90 ¢ YY7k s cVm #`¥ghT +. Y2. . R. . 2. Z. T. K0 4M 2 K0 + + K0 r(δ) Λ (R ) Φ(M ) δ,R 1 0 B R YR Z TZ Z fn ϕ0 dy 0 dydxdt . +2 M aδ,R+R1 (N )K0 + M N Z. Z. Q1 (fn ) ϕ dydxdt ≤. 0. üü ÇAVÚ hÚ Ý . BR. Y2.
(222) #
(223) < 9 3 7
(224) 1#
(225) 3!" . 4.
(226) pu`bkVgqTVYGendjyhY7e s tY7pughenVdb`bpqTVY s Y7er vXinWm eQv°gqk Y7phpin `jk erk s .54Q§ %0 ¢ T )×B ×Y ) ¢ s Y¦W¦ePtmenphp[gqiEghTVY«db`jW`jg ¤ ypugenp |E| → 0 ¢ gqTVY7k¼enp N(f→) +∞L ¢ ((0, r e k ¤ kerdbdjt enp R → +∞ s s r e k q g ¦ i. v Q i k. v b d c Y q g T e g M → +∞ δ→0 1. n. R. 1. lim sup. Z. T. Z. Z. Q1 (fn ) 1E dydxdt = 0 ,. erägqY7yGkVirgh`bvX`jk ¦gqT erg .Ó § a04`bWmVdj`bY7pGgqT erg enp §#S4TVYXyhY ¡iQyqY ¢ `fp4 Y7er adbtv iQWm env g4`bk L ((0, T ) × B ×ΛY )(Rat«)ghTV→Y±}+∞c ko¡iny s R²Åx<YX→gugh`bp4+∞ ghTVYXiQyqY7Wl§ (Q (f ))tu #`bkVRgqiRgqT Ylpugqyhc v gqcVyhYlir}ghTVY½v i2erdbY7phv Y7k v YY7´2c egh`jiQk .84n§64 0 ¢ `jg¦gqcVyhk pincVg]ghT egghTVYECYJer ²L ³v iQW]menv°ghkVY7phpir (Q (f )) mVyhia` s Y7ppqinWY¦e sVs `jgq`binkerd?`bko¡inyhW¦egh`jiQkRiQk (Q (f )) §lS4TV`bpÓenv°g 4enp#¤ yhpug}iQ puY7yqQY s `jk $¡iQyGgqT Y s `fpqvXyqYXgqY s ` .c pu`bnYv iQenncVdfegh`jiQklY7´2c egh`jiQkEerk s e¦yhY7pqcVdjg}`bkEgqTVY pherWYpqmV`jyh`jg4`fpGendbpqi]ePen`jdferVdbY©¡iny .54Q"§ 4 0°§ C¶ 7 8 R, T > 0
(227) +*;1#( # @ $% !5 2 . 9 <
(228) < #(%$ <3@ |E|→0 n≥1. 0. YR. δ,R. 1. 1. BR. R. 1 n . . θ = θR,T ∈ C 1 ([0, +∞)). 1. #( . 8. δ ∈ (0, 1). T 0. Z. 1. 2. . . . . 1 . θ0 (u) → +∞ Z. 1. BR. Z. n. R. n. . u → +∞. ÓZ § ZQ%0 .. θ0 (fn ) Q2 (fn ) dydxdt ≤ C(δ, R, T ) Yδ,R. º¨º VyhinW ¯¨YXWW¦e#Z§Î ¢ Y¬ 2k i ghT eg ¢ (f ) enk s (Q (f )) HYXdbinkV#ghi#CYJer adbt©v iQWm env g ¢ yhY7pqmYJv°gq`bnY7djtQ§<S4T Y}±}c ko¡iny s ²Åx<YXgugh`bp:gqTVY7inyhYXW qp cV pqY gp?in L (Ω × Y ) erk s L ((0, T ) × B (f ×(0)) enk s eyqYX¤ kVY s QYXypu`bink ir<gqT Y s Ydfe ?erdbdj7Y ²Åx0inc Yphpu`bk) gqTVY7inyhYXW Îr ¢ x?yhinmHiQpq`¥gh`jiQk-=°§64n§64 `jWmVdbtghT eg ghTVYXyhY«`fpeEkViQkVkVY7Qegh`jQY ¢ v inkaQY wRenk s kViQko² s Y7v yhY7eQpu`bkV¡cVk v gq`bink θ = θ ∈ C ([0, +∞)) pqc vT ghT eg enp u → +∞ enk s u θ (u) ≤ C θ(u) , u ≥ 0 , θ (u) → +∞ ¡iQyGpuiQWY[vXink pughenk2g C > 1 enk s . . 1. 1. n. n. 2R. 1. 2R. n. . . R,T. 0. 0. 1. θ. θ. K1 := sup. RY½¤Vw. n≥1. +. (Z. Z. θ(fn (0)) dydx +. Z. T. Z. Z. ¢ ¢ pucvT gqT erg § +Y`bko¡Y7y4¡yqiQW .54Q§"4 0CgqT erg B2R. Y2R. 0. B2R. χ ∈ D(R3 × Y ) 0 ≤ χ ≤ 1. (θ(fn ) + θ(Q1 (fn ))) dydxdt. Y2R. ink. Y. ΩT. Y. BR × Yδ,R. < +∞.. erk s pqcVmVm. χ ⊂ B2R × Yδ/2,2R Z Z Z Z θ0 (fn ) Q2 (fn ) χ dydxdt θ(fn (T )) χ dydx + Ω Y ΩT Y Z Z Z Z Z Z 0 θ(fn (0)) χ dydx. θ(fn ) v · ∇x χ dydxdt + θ (fn ) Q1 (fn ) χ dydxdt + = ΩT. χ ≡ 1. ). Ω. Y. ÷¡û<ü¨÷äØ.
(229) .
(230)
(231) !" #
(232) %$'& (
(233) !" # 4. ^o`jk vXY θ enk s θ enyqY[kViQkVkVYX2egh`jQYenk s 0. θ∗ (θ0 (u)) ≤ u θ0 (u) − θ(u) ≤ Cθ θ(u) ,. u ≥ 0,. ghTVY ?inc kV`jkVYJ´2c erdb`¥g{tYXk2ghen`jdfp gqT erg . Z. ΩT. Z. Z. 0. θ (fn ) Q2 (fn ) dydxdt ≤ Yδ,R. Z. ≤. ΩT T 0. 2 + δ. BR × Y R. B2R T. Z. 0. Z. {θ∗ (θ0 (fn )) + θ(Q1 (fn ))} dydxdt Y2R. Z. B2R. Z. θ(fn ) p · ∇x χ dydxdt + K1 Yδ/2,2R. 4R k∇x χkL∞ ≤ 2 K 1 + Cθ + δ. Z. T 0. Z. B2R. Z. θ(fn ) dydxdt , Y2R. t u . . Z. θ0 (fn ) Q2 (fn ) χ dydxdt Y. . #T YXk vXY .ÓZ § ZQ90 § C¶ 7 - <9 ). Z. R, T > 0. # <
(234) @< . (Q2 (fn )). 1# 1 <' !98
(235) 9
Documents relatifs
[r]
Les résultats de l’étude de niveau pour un processus respectant un modèle AR(2) avec l’innovation d’un modèle GARCH(1,1) sont présentés dans les tableaux 4.3 et 4.4..
En réalité, son auto- rité repose sur un contrôle communautaire et une hiérarchie politique : dans les prisons et les camps, les comités de détention sont mis en place par
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
Même si l’armée de Terre française intègre déjà de nombreux dispositifs de simulation numérique ou d’aide à la décision dans ses systèmes d’information, il s’est
In this way, we derive two simple models for large strain viscoelasticity: the first one referred to as a Convolution Integral Model (CIM) is a solid extension of the K-BKZ model,
Dans le cadre du LabEx RESSOURCES21, les premiers groupes de métaux sur lesquels un accent a été mis sont les suivants : les métaux de la filière photovoltaïque
Dans le cas a), l’´el´ement passif stocke l’´energie cr´e´ee par le moteur durant la phase d’actionnement et la restitue pendant la phase de rappel. C’est un m´ecanisme qui
