### Electronic Research Archive

2020, Issue 1: 27-46. doi: 10.3934/era.2020003
Special Issues

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

• 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.

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

### Related Papers:

• 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.

 [1] Remarks on the breakdown of smooth solutions for the $3$-D Euler equations. Commun. Math. Phys. (1984) 94: 61-66. [2] Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pure. Appl. (2004) 83: 243-275. [3] Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with far field vacuum. J. Math. Pure. Appl. (2017) 107: 288-314. [4] On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. (2001) 3: 358-392. [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 [6] Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow with vacuum. Arch. Rational. Mech. Anal. (2019) 234: 727-775. [7] Blowup criterion for viscous barotropic flows with vacuum states. Commum. Math. Phys. (2011) 301: 23-35. [8] Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun. Pure. Appl. Math. (2012) 65: 549-585. [9] O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI 1968. [10] Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum. Bulletin of the Brazilian Mathematical Society (2016) 47: 507-519. [11] On classical solutions to 2D shallow water equations with degenerate viscosities. J. Math. Fluid Mech. (2017) 19: 151-190. [12] On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum. Arch. Rational. Mech. Anal. (2019) 234: 1281-1334. [13] Existence results and blow-up criterion of compressible radiation hydrodynamic equations. J. Dyn. Differ. Equ. (2017) 29: 549-595. [14] Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. [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 [16] (1996) Mathematical Topics in Fluid Mechanics: Compressible Models.Oxford University Press. [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 [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 [19] Remarks on a paper: Remarks on the breakdown of smooth solutions for the $3$-D Euler equations. Commun. Math. Phys. (1985) 98: 349-353. [20] Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equation. J. Differential Equations (2008) 245: 1762-1774. [21] (1970) Singular Integrals and Differentiability Properties of Functions.Princeton Univ. Press. [22] A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations. J. Math. Pure. Appl. (2011) 95: 36-47. [23] Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. (1998) 51: 229-240. [24] The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete and Continuous Dynamical Systems (2017) 37: 5631-5649. [25] Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum. J. Differential Equations (2015) 259: 84-119. [26] On classical solutions of the compressible magnetohydrodynamic equations with vacuum. SIAM J. Math. Anal. (2015) 47: 2722-2753.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142