Blowup Criterion for the Compressible Flows with Vacuum States

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precise...

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Bibliographic Details
Published in:arXiv.org 2010-04
Main Authors: Huang, Xiangdi, Li, Jing, Zhouping Xin
Format: Article
Language:English
Subjects:
Boundary layer
Compressibility
Criteria
Deformation
Euler-Lagrange equation
Fluid dynamics
Fluid flow
Mathematical analysis
Navier-Stokes equations
Regularity
Tensors
Velocity gradient
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Summary:We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method can be generalized to the full Compressible Navier-Stokes system which improve the previous results. In addition, initial vacuum states are allowed in our cases.
ISSN:2331-8422
DOI:10.48550/arxiv.1004.5469