Journal
COMMUNICATIONS IN MATHEMATICAL PHYSICS
Volume 301, Issue 1, Pages 23-35Publisher
SPRINGER
DOI: 10.1007/s00220-010-1148-y
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Funding
- Zheng Ge Ru Foundation
- Hong Kong RGC [CUHK4028/04P, CUHK4040/06P, CUHK4042/08P]
- RGC [CA05/06.SC01]
- SRF for ROCS, SEM, NNSFC [1060105910971215]
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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 (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic 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 (Ponce in Commun Math Phys 98:349-353, 1985). In addition, initial vacuum states are allowed in our cases.
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