两类非线性发展方程初值问题

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1、万方数据InitialValueProblemsforTwoTypesofEvolutionEquationsAthesispresentedbyFengbaiLitoTheDepartmentofMathematicsinpartialfulfillmentoftherequirementsforthedegreeofDoctorofPhilosophyinthesubjectofMathematicsShanghaiJiaoTongUniversityApril2012万方数据万方数据ឋḄ⚪᤺⌕ᦻὃ

2、⇋ឋḄ⚪-ᑖᦪ▤ᱥឤḄCauchy⚪!ḄRiemann⚪，#⌕%ᵨ'⚣ᑖ)Ḅ*+ᩭ-ᑮᑖᦪ▤ᱥឤḄ)Ḅ⊦012、ᵨGreen4ᦪḄ+ᩭ-ᑮᱥឤḄ)Ḅ⌲612，7ᱯ9ᑖ)Ḅ+ᩭ-ᑮ!Ḅ)Ḅ᦮;<ᙠឋ.>?@ABC.ᙠDE，ᡃGHIᑖᦪ▤ᱥឤ7ẆKᔊM、!ḄᩭN7OᐵQR，S▰UV⌕ẆKḄḄ#⌕WX.>Y@Z，ᡃGẆKᑖᦪ▤Ḅᱥឤ.✌ᐜᙠ]^_`Z，abLaplacecdḄe1/2

3、)ᑖ)Aj⚣kᑖl⚣kᑖ.ᡃGᑖma)Ḅj⚣kᑖl⚣kᑖn⊦012.ᡃG✌ᐜ-ᑮ)ᙠL2_`ZḄᨬq⊦012.ra)ᙠstSobolev_`ZR⊦012.ᡃG#⌕%ᵨ'⚣cdḄᱯឋ7᩽vᳮ.ᙠ᦮xyz{Z

4、}?~ᢈ.%ᵨ'⚣ᑖ)Ḅ*+ᙢẆKᜧ⚪Ḅ⊦012.ab，G#⌕ẆKḄ)Ḅᜧ'`A，abᜧḄ)Ḅ⊦012ᩭᑮḄW.>@Z，ᡃGὃ⇋Ḅᱥឤ，ᓽ>Y@ZḄᙠa=1Ḅ.ᡃGẆKḄᱥឤᐹᨵ᡾ḄḄCauchy⚪Ḅ)Ḅ⌲612，#⌕%ᵨGreen4ᦪḄ+.ᡃGV，-ᑮ

5、ḄGreen4ᦪ，SᵫDuhamelvᳮ，ᡃG-ᑮ)Ḅ⊤.%ᵨGreen4ᦪḄ+-ᑮ)ᙠL2_`ZḄᨬq⊦012， -ᑮ)ᙠ_`ZḄ⊦0ឋ.ᙠ☢QRḄ¢Ạ¤，ᑭᵨGreen4ᦪ+，ᡃG-ᑮ)Ḅ⌲612.ab)Ḅ⌲612，G#⌕ẆKḄ.aᜧ᡾ḄᑮOᐵW.>¥@Z，ᡃGὃ⇋ḄY^!ḄRiemann⚪.ᡃGẆKḄ?¦ᱯ§ḄRiemann⚪，ᓽ¨a©Ḅ¥ª«¬ḄORᵨ⚪.ᡃG᪀⌼?x᦮;°±)，Dx)

6、²ᨵ³´ᡈ6，S¶v6·?Ḅ¸_6.ᙠDEᡃG¹ºᙢ▊U¼ᓫ¬ḄORᵨ¾¿ḄᐝÁ)Ḅ<ᙠឋ，Â

7、ᐸ¸_6▬ÅḄᱯ9ḄA.Æ'，ᡃGÇÈXᵨᱯ9ḄÉÊËḄᱯ9ᑖ)Ḅ*+ᩭẆKx¼ᓫ¬ḄORI万方数据cdefᜧᦻᵨ.ᦻZᡃGV¬ḄORᵨÌÍÎÏḄ¦ Ðᓄ.ᐵÒÓ：ᑖᦪ▤ឤ，Õ⚪，'⚣Ö×cd，ᨬq⊦012，Green4ᦪ+，⌲612，!，ᱯ9ᑖ)，ª«¬，¼ᓫ¬，Goursat⚪，OØឋᩩÚ.-II-万方数据InitialValueProblemsforTwoTypesofEvolutionEquationsABSTRACTInthisthesis,westudytwotypeso

8、fnonlinearevolutionequations:thefractalparabolicconservationlawsandthepressure-gradientsystem.Firstly,wetaketheCauchyproblemofthefractalparabolicconservationlawsasanexampletoconsiderhowthesolutionsbehaveintheL2spaceandthehomogeneousSobolevspaces.Fortheclassicalparab

9、olicconservationlaws,theGreen'sfunctionmethodisappliedtotheinitialvalueproblem.Secondly,weconsiderthepressure-gradientsystemwhichisasimplifiedmodelofEulerequation.WestudyaspecialRiemannproblem.Thethesisisarrangedasfollows:InChapter1,wereviewthephysicalbackgroundofth

10、efractalconservationlawsandthepressuregradientsystem.Westatethehistoryofstudyforthesystems.Wealsointroducetheproblemswewillstudyandthemain