Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

Yue Cao; Yue Cao

doi:10.3934/era.2020003

Electronic Research Archive

2020, Volume 28, Issue 1: 27-46. doi: 10.3934/era.2020003

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Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

Yue Cao ^{1,2
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1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematics, Georgia Institute of Technology, Atlanta 30332, USA

Received: 01 September 2019 Revised: 01 September 2019
Primary: 35B44, 35A01, 35A09; Secondary: 35G55, 76N99

In this paper, the Cauchy problem of the $ 3 $D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the $ L^\infty $ norm of the deformation tensor $ D(u) $ ($ u $: the velocity of fluids) and the $ L^6 $ norm of $ \nabla \log \rho $ ($ \rho $: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $ D(u) $ or $ \nabla \log \rho $ as the critical time approaches; equivalently, if both $ D(u) $ and $ \nabla \log \rho $ remain bounded, a regular solution persists.
- Compressible Navier-Stokes equations,
- three dimensions,
- blow-up criterion,
- regular solutions,
- degenerate viscosity,
- far field vacuum
Citation: Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities[J]. Electronic Research Archive, 2020, 28(1): 27-46. doi: 10.3934/era.2020003

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Abstract

In this paper, the Cauchy problem of the $ 3 $D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the $ L^\infty $ norm of the deformation tensor $ D(u) $ ($ u $: the velocity of fluids) and the $ L^6 $ norm of $ \nabla \log \rho $ ($ \rho $: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $ D(u) $ or $ \nabla \log \rho $ as the critical time approaches; equivalently, if both $ D(u) $ and $ \nabla \log \rho $ remain bounded, a regular solution persists.

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