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March  2020, 28(1): 27-46. doi: 10.3934/era.2020003

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

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematics, Georgia Institute of Technology, Atlanta 30332, USA

* Corresponding author

Received  September 2019 Revised  September 2019 Published  March 2020

Fund Project: The first author is supported in part by China Scholarship Council 201806230126 and National Natural Science Foundation of China under grant 11571232.

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.

Citation: Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28 (1) : 27-46. doi: 10.3934/era.2020003
References:
[1]

J. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[2]

Y. ChoH. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pure. Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

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M. Ding and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pure. Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar

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E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

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G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

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Y. GengY. Li and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with vacuum, Arch. Rational. Mech. Anal., 234 (2019), 727-775.  doi: 10.1007/s00205-019-01401-9.  Google Scholar

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X. HuangJ. Li and Z. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commum. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.  Google Scholar

[8]

X. HuangJ. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure. Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

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O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI 1968.  Google Scholar

[10]

Y. LiR. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bulletin of the Brazilian Mathematical Society, 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar

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Y. LiR. Pan and S. Zhu, On classical solutions to 2D shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar

[12]

Y. LiR. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational. Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.  Google Scholar

[13]

Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, J. Dyn. Differ. Equ., 29 (2017), 549-595.  doi: 10.1007/s10884-015-9455-9.  Google Scholar

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[15]

W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Advances in Calculus of Variations, 2019. doi: 10.1515/acv-2019-0039.  Google Scholar

[16] P. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, Oxford University Press, USA, 1996.   Google Scholar
[17]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci., vol. 53, Springer, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[18]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Berlin, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[19]

G. Ponce, Remarks on a paper: Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 98 (1985), 349-353.   Google Scholar

[20]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equation, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[21] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton NJ, 1970.   Google Scholar
[22]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure. Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[23]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[24]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[25]

S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar

[26]

S. Zhu, On classical solutions of the compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 47 (2015), 2722-2753.  doi: 10.1137/14095265X.  Google Scholar

show all references

References:
[1]

J. BealeT. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  doi: 10.1007/BF01212349.  Google Scholar

[2]

Y. ChoH. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pure. Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[3]

M. Ding and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum, J. Math. Pure. Appl., 107 (2017), 288-314.  doi: 10.1016/j.matpur.2016.07.001.  Google Scholar

[4]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[5]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[6]

Y. GengY. Li and S. Zhu, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with vacuum, Arch. Rational. Mech. Anal., 234 (2019), 727-775.  doi: 10.1007/s00205-019-01401-9.  Google Scholar

[7]

X. HuangJ. Li and Z. Xin, Blowup criterion for viscous barotropic flows with vacuum states, Commum. Math. Phys., 301 (2011), 23-35.  doi: 10.1007/s00220-010-1148-y.  Google Scholar

[8]

X. HuangJ. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure. Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[9]

O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI 1968.  Google Scholar

[10]

Y. LiR. Pan and S. Zhu, Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum, Bulletin of the Brazilian Mathematical Society, 47 (2016), 507-519.  doi: 10.1007/s00574-016-0165-7.  Google Scholar

[11]

Y. LiR. Pan and S. Zhu, On classical solutions to 2D shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151-190.  doi: 10.1007/s00021-016-0276-3.  Google Scholar

[12]

Y. LiR. Pan and S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational. Mech. Anal., 234 (2019), 1281-1334.  doi: 10.1007/s00205-019-01412-6.  Google Scholar

[13]

Y. Li and S. Zhu, Existence results and blow-up criterion of compressible radiation hydrodynamic equations, J. Dyn. Differ. Equ., 29 (2017), 549-595.  doi: 10.1007/s10884-015-9455-9.  Google Scholar

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[15]

W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Advances in Calculus of Variations, 2019. doi: 10.1515/acv-2019-0039.  Google Scholar

[16] P. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, Oxford University Press, USA, 1996.   Google Scholar
[17]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci., vol. 53, Springer, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[18]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Berlin, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[19]

G. Ponce, Remarks on a paper: Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Commun. Math. Phys., 98 (1985), 349-353.   Google Scholar

[20]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equation, J. Differential Equations, 245 (2008), 1762-1774.  doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[21] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton NJ, 1970.   Google Scholar
[22]

Y. SunC. Wang and Z. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pure. Appl., 95 (2011), 36-47.  doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[23]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[24]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete and Continuous Dynamical Systems, 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[25]

S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84-119.  doi: 10.1016/j.jde.2015.01.048.  Google Scholar

[26]

S. Zhu, On classical solutions of the compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal., 47 (2015), 2722-2753.  doi: 10.1137/14095265X.  Google Scholar

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