二维Boussinesq 系统的动力学和Liouville型定理.pdf

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1、᪥10530201330091046ᑖO175.29硕士学位论文BoussinesqḄLiouvilleᳮᵭ谢+匕曰教J导ᜧᦻᦟᣴ▾ᦪ$%&'▾学科专研究方ᦪ偏微分方程()*+,ᨴᓝ/万方数据BoussinesqḄLiouvilleᳮᵭ0123456⎅ᜧᦻᦟᣴ▾ᦪ$%&'▾'89ᦪẆ;<ᔣ>?ᑖ<@ᵭAᳮắCᣴDᓫFGᜧHᦻIJ/K()*+,ᨴᓝ/万方数据Dynamicsofthetwodim

2、ensionalBoussinesqsystemandaLiouvilletypetheoremCandidateXieYeSupervisorProfessorTangTai-ManCollegeMathematicsandComputationalScienceProgramMathematicsSpecializationPartialDifferentialEquationDegreeMasterofScienceUniversityXiangtanUniversityDateApril10,2016万方

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4、}~⊤ᡈᑏḄᡂc。QᦻḄẆ;S⌕Ḅz

5、}，ᙳᙠᦻgjP<᪗P。QᐰᑮQOPḄcᵫQ。ὅ：0AK：+ᨴHᦻ᱐ᩗ¡ᵨᣴᩗ¢QHᦻὅᐰf£᪥ᨵᐵ¦ᶇ、¡ᵨHᦻḄ©，ª᪥¦ᶇ«ᔣ¬ᨵᐵ®¯ᡈ°᪀〈JHᦻḄ³ᓺµᵯ·᱐，ᐕ¹Hᦻº»▅½▅。QᣴᩗFGᜧ¾j¿QHᦻḄᐰ®ᡈ®ᑖᑁpÀᐭᨵᐵᦪÂÃ]^ÄÅ，¾jÆᵨÇᓺ、È

6、ᓺᡈÉÊ˳ᑴÍΦÏÐÀQHᦻ。ÑHᦻᢥ᪥©ᜐᳮ。ὅ：K：+sᨴ23：/KÔÕ+5ᨴ)r/万方数据摘要QᦻÖHBoussinesq<@Ḅ×z<☢.Ù)®ᑖẆ_BoussinesqgÚ⌕ḄÛÜÝÞßàḄáᓄãä，ᒹuåæ，ç，èæ，éÛ，ᵨᙠê}?ëḄᡈìíḄ2ᦪ.

7、ᨵẆ_，ᜧ®ᑖWÖH〉ឋᑣឋ，ÖH£ḄឋÝ.ᙠ~÷ឋᦪᡈøù2ᦪ(ḄBoussinesqᐰᐝ£Ïᙠ，QᦻẆ_ìíḄឋÝ.ᐵ：Boussinesq<@；áᓄ；ᐰ£；ឋÝ；ᑢᳮ.I万方数据AbstractTwoaspectsofthetwodimensionalBoussinesqequationsarediscussedinthisthesis.Inthefirstpart,weinvestigatetheevolutionalongthetrajector

8、iesofthemajorquantitiesoftheBoussinesqsystem.Theseincludevelocity,pressure,temperature,vorticity,theforceonthefluidortheirderivatives.FormtheBoussinesqequations,wederiveordinarydifferentialequationsgoverningtheirevolutionsandthendeduceresultsfromthem.Inthesec

9、ondpart,aLiouvilletypetheoremfortheBoussinesqsystemwithoutviscousityandthermaldiffusivi-tyisproved.Aconditiononthepressurethatguaranteesthetrivialityoftheflowisgiven.TherearealotofresearchontheBoussinesqsystem.Manyofthemareonwell-posednessandregularityissuesa

10、ndrelativefewofthemdiscussthepropertiesofsolutions.Itiswellknownthatifeithertheviscousityorthermaldiffusivityisnon-zero,globalsmoothsolutionsofthetwodimensionalBoussinesqsystemexist.Thist