Existence and uniqueness of the entropy solution to a ...

4 downloads 0 Views 233KB Size Report
Abstract. This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: ut + ...
              !" #%$ &% '  % $(  )*+,"!- ) .-%$ / 0 &% 1243 5768(9;:=A@)2CBD8(EFEHGJILM&K NPO 8(Q4:124R M 9TS=UHQ=V WXCYPZ\[?]7^_Z ` aJbHcd'e(fhgbHcjiTek7iTlPfhkml\nod bqphaophamlrmfheJe_steusvphamllTwxbHcypzlPk7iTl|{uk7n}mkJbH~}mlPkml\czcje_svphamllPkJphfherJd l\{_g cye_€q}mpybqekphe%phaml s‚e_€ƒ€qe_d bqkm„,kme(kJ€…bqkml\{_faxJr7lPfh†‡e_€ƒbHil\~(}7{_pybqek4ˆ ‰mŠC‹ŒJސ’‘“x”•‘“‰—–h–˜š™jbqk›yœž Ÿ¢¡£™m¤h¥ ¦¨§ d bqpha-bqkJbqpybH{?€ nm{_p©{‰=‘Tªq¤©™J–˜«‰‡¬T‘©ªƒ–)bqk­›yœ ¯® d amlPfhl‰‡¬°š±#²ž‘›yœ  –)bHc{„_bq³lPk´sµ}mk7iTpybqek4§¶b·c%{ n(bq³lPfh„lPk7iTlP¸Fs‚fhlPl †‡e}mk7nJl\n|sµ}mk7iTpybqek¹e_s iz€H{chcº)» s‚fhe¼½›“œž­Ÿ¡¾™¿¤h¥ ¦4pze*›yœv§={_k7n¹”¹bHcž{)s‚}mk7iTpybqeke_s iz€H{chc º » s‚fhe¼À›“œÁphe¹›“œ ®Â p {?€Hcye0„_bq³l\c {%fhl\cy}J€qpje_stiTekJ³lPfh„lPk7iTleust{%kx}m¼ lPfÃb·iP{&€tchizamlP¼ ls‚ef)phaml n(bHchiTfhlPpybqÄ\{_pÃbÅe(k­e_svphamlTbHcl\~}7{_pÃbÅe(k ® Æ lÇ7f©cypcyame_d½phamllTwJbHcyphlPk7iTl¹e_sD{È_rmfhemiTl\chchȁcye_€q}mpybqekɑd aJbHiza „lPkmlPf©{&€…bqÄPl\cÊphamlDiTek7iTlPrmpËe_s#lPkJphfherJjd'l\{ugcye_€q}mpybqek7cP§={_k7n¹iP{_k†‡l)e†mp©{?bqkml\n†Jr7{chc“bÅkm„pze phaml€ƒbq¼)bqp e_s'cyeu€Å}mpÃbÅe(k7ce_s#phaml)kJ}m¼ lPfybHiP{?€'chizamlP¼ l_– ® ` aml)}mkJb·~(}mlPkml\chcže_s#phaJbHcËlPkJphfherJrmfhemiTl\chccye_€q}mpybqek|b·cËpzamlPk rmfhe_³l\n=̕bqpžb·c){?€Hcye%rmfhe_³lPkpha7{upphaml*lPkJphfhe(rx%rmfhe’iTl\czcDcÃe_€q}mpybqekbHcÊbqks{iTp){lPkJphfherJ¹d'l\{ug,cye_€q}mpybqek4§ amlPk7iTlphaml lTwJbHcyphlPk7iTl{_k7n0}mkJbH~}mlPkml\czc)e_s phamljlPkxphfzerx¹d l\{_gcye_€q}mpybqek4§{_k7n0phamliTekJ³lPfh„lPk7iTlje_s phaml kJ}m¼ lPfybHiP{?€LchizamlP¼ l ® ÍËlPJd ef©nmcPˆÎekJ€ƒbqkml\{_faxJr7lPfz†7e_€ƒbHižl\~}7{_pÃbÅe(k4§7rmfhemiTl\chc cÃe_€q}mpybqek4§7lTwJbHcyphlPk7iTl){_k7nj}mkJbH~}mlPkml\chcϧ iTe(kx³lPfz„lPk7iTlve_sCÇ7kJbqphlv³(e_€q}m¼ lvchizamlP¼ l\cPÌ ÐtÑ0Ò cy}m†¿Ó“l\iTp¯iz€H{czc¨bƒÇAiP{_pybqek4ˆ ÔÕ Ð ™Õm§mÔÕ Ð¯Ö ™¿§xÔÕוØ™m§mØÕ сÙ\Ú ® Û Ü %' 0 ¯&% ­Ý Ë@ Þ M­ß 9 MàzM Q Ná GJ9©âU àã GJQ ã\M 9TQ M : á U N Þ N Þ M-M_ä U àhNTM Q ã?M Gå8Q M Q N 9TG ß 5 áËM 8â à GxEFI N UFGxQ½æ/ç è éêyëyìjíî ïµð‡ñPò*óÅô N G N Þ M åGxEFEFG á UHQ=õQ=GJQAEFUFQ M 8(9Þ75 ß4M 9©S4GJEFU ãžMö I48 N UFGxQ á U N ÞUFQAU N Uµ8(EL:=8 N 8‡÷ æ=øhê“ùúñ©ûÏôýü­þmÿ 4ê ê“æêyùñTûPôTô©ô ð‡ñ ùçëyìjí)ñûç­ó¾ð‡ñPòï æêyùñPð¿ô æ ê“ùCôPñùçëyìjí á: Þ QAM G 9 M ò Þ ð‡:‡>#U æ=ø9©õ : M Q Q=G N©MG à)N Þ9P8 MGJß 9 89 N U‚U 8(E¯Þ: 9 M 9TU x8 N U M G G(åÊÞ æ á U8 N Þ9 xM8(àh9TߕUµ8(Muã?S=N E N Gù NšUH6 êyùM Jñ 89TUµ?8(ñ;S=ù E M M < í I=MQ4: M NT9 MuN àÊÞ N M åM GxEFEFG á M UHQ=õM 8 ã?à©M à I=6 ß4M ßAN UFN GxQ à á GJQ N N Þ M Màz:=ß48 MN ã?8NòN > •>N M*#àz>4ß æ _ã?÷ M »  





















 





  

ê (ô ê ¿ô û\>þmÿ ô;>48(Q4: 





  !"$#%'&)(*&+!"-,.&0/21 +!"/*&!2(*3-45/"/7&06 &/8:9$;?&@A&0B'&$8C 9$D7EF9G1-'#H/I-+:.KJI+5#HB &"/'#%!7&G 6 1-"#%/ILI  .?8 M  -+:N& OFPIQ L QSR?Q UT$ +8$VWX-,%,6& &).UY Z[!-,%#%&$87W$\ ]$WV^?T $+8 M  +N& _ JI+5#HB &"/'#%!7&G 6 .& R FB $#%&$8C$] ]$C$W`?&Gab$47'c$&!d.4e N-8 M -+Nf&



šê < ñ ?ñ í ô¯ç < êyëyì í îó¾ð‡ñPòïyñPë“ì í ôPñ ü ñ ø à I ß 4êyùñTûPô þmÿ •ê“ùúñ©ûÏô í êyùñTûPô ð‡ñÊåGx9v8(EHE êyùñTûPô çëyìjíîó¾ð‡ñPòïyñ ç < ê“ëyì¹ñPëyì%

Suggest Documents