KMS Of Academy of mathematics and systems sciences, CAS
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 Publication | COMMUNICATIONS IN MATHEMATICAL PHYSICS |
ISSN | 0010-3616 |
Volume | 281Issue:2Pages:401-444 |
Abstract | The 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. |
DOI | 10.1007/s00220-008-0495-4 |
Language | 英语 |
WOS Research Area | Physics |
WOS Subject | Physics, Mathematical |
WOS ID | WOS:000256471300005 |
Publisher | SPRINGER |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/5291 |
Collection | 中国科学院数学与系统科学研究院 |
Corresponding Author | Li, Hai-Liang |
Affiliation | 1.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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