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Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
Li, Hai-Liang1,3; Li, Jing2,3; Xin, Zhouping1,3
2008-07-01
Source PublicationCOMMUNICATIONS IN MATHEMATICAL PHYSICS
ISSN0010-3616
Volume281Issue:2Pages:401-444
AbstractThe Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.
DOI10.1007/s00220-008-0495-4
Language英语
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000256471300005
PublisherSPRINGER
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/5291
Collection中国科学院数学与系统科学研究院
Corresponding AuthorLi, Hai-Liang
Affiliation1.Capital Normal Univ, Dept Math, Beijing, Peoples R China
2.Acad Sinica, AMSS, Inst Appl Math, Beijing 100080, Peoples R China
3.Chinese Univ Hong Kong, Inst Math Sci, Shatin, Hong Kong, Peoples R China
Recommended Citation
GB/T 7714
Li, Hai-Liang,Li, Jing,Xin, Zhouping. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS,2008,281(2):401-444.
APA Li, Hai-Liang,Li, Jing,&Xin, Zhouping.(2008).Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations.COMMUNICATIONS IN MATHEMATICAL PHYSICS,281(2),401-444.
MLA Li, Hai-Liang,et al."Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations".COMMUNICATIONS IN MATHEMATICAL PHYSICS 281.2(2008):401-444.
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