Blow-up of Smooth Solutions to the Compressible Fluid Models of Korteweg Type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin＇s result [Xin, Z.： Blow-up of smooth solutions to the compressible Navier-Stokes equations with compac...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2012-03, Vol.28 (3), p.645-652
Main Authors: Zhang, Ying Hui, Tan, Zhong
Format: Article
Language:English
Subjects:
Blow-up
Capillarity
Cauchy problems
Compressible fluids
Decay
Density
Entropy
Fluid dynamics
KdV方程
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Navier-Stokes equations
Studies
Velocity
Viscosity
光滑解
可压缩Navier-Stokes方程
可压缩流体
流体模型
爆破
类型
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Summary:We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin＇s result [Xin, Z.： Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229-240 （1998）] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-012-9042-5