Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations

doi:10.1007/s00220-008-0495-4

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	Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
	Li, Hai-Liang 1,3; Li, Jing2,3 ; Xin, Zhouping 1,3
	2008-07-01
发表期刊	COMMUNICATIONS IN MATHEMATICAL PHYSICS
ISSN	0010-3616
卷号	281 期号:2 页码:401-444
摘要	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
语种	英语
WOS研究方向	Physics
WOS类目	Physics, Mathematical
WOS记录号	WOS:000256471300005
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/5291
专题	数学所
通讯作者	Li, Hai-Liang
作者单位	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
推荐引用方式 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.