Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations

Zhigang Wang; Zhigang Wang

doi:10.3934/cam.2025007

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 145-158. doi: 10.3934/cam.2025007

Previous Article Next Article

Research article Special Issues

Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations

Zhigang Wang ^{1
,
,}

School of Mathematics and Statistics, Fuyang Normal University, Anhui 236037, P.R.China

Received: 02 July 2024 Revised: 20 January 2025 Accepted: 10 February 2025 Published: 18 February 2025
Primary: 35A01, 35B40; Secondary: 35B65, 35A09

In this paper, we consider the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. When the shear and bulk viscosity coefficients are both given as a constant multiple of the mass density's power ($ \rho^\delta $ with $ \delta > 1 $), we show that the $ L^{\infty} $ norms of $ \nabla u $, $ \nabla\rho^{\frac{\gamma-1}{2}} $ and $ \nabla\rho^{\frac{\delta-1}{2}} $ control the possible breakdown of classical solutions with far-field vacuum; this criterion is analogous to Serrin's blowup criterion for the compressible Navier–Stokes equations.
- Compressible Navier–Stokes equations,
- degenerate viscosity,
- far-field vacuum,
- classical solutions,
- Serrin-type blowup Criterion
Citation: Zhigang Wang. Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2025, 17(1): 145-158. doi: 10.3934/cam.2025007

Related Papers:

Abstract

In this paper, we consider the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. When the shear and bulk viscosity coefficients are both given as a constant multiple of the mass density's power ($ \rho^\delta $ with $ \delta > 1 $), we show that the $ L^{\infty} $ norms of $ \nabla u $, $ \nabla\rho^{\frac{\gamma-1}{2}} $ and $ \nabla\rho^{\frac{\delta-1}{2}} $ control the possible breakdown of classical solutions with far-field vacuum; this criterion is analogous to Serrin's blowup criterion for the compressible Navier–Stokes equations.

References

[1]	S. Chapman, T. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, 1990. https://doi.org/10.2307/3609795
[2]	T. Li, T. Qin, Physics and Partial Differential Equations, Vol. Ⅱ., Beijing: Higher Education Press, 2014.
[3]	G. Galdi, An Introduction to the Mathmatical Theory of the Navier–Stokes Equations, New York: Springer, 1994. https://doi.org/10.1007/978-0-387-09620-9
[4]	S. Kawashima, Systems of A Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University in Kyoto, 1983. https://doi.org/10.14989/doctor.k3193
[5]	J. Nash, Le probleme de Cauchy pour les équations différentielles dún fluide général, Bull. Soc. Math. France, 90 (1962), 487–491. https://doi.org/10.24033/bsmf.1586 doi: 10.24033/bsmf.1586
[6]	Y. Cho, H. Choe, H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243–275. https://doi.org/10.1016/j.matpur.2003.11.004 doi: 10.1016/j.matpur.2003.11.004
[7]	X. Huang, J. Li, 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. https://doi.org/10.1002/cpa.21382 doi: 10.1002/cpa.21382
[8]	Y. Cho, B. Jin, Blow-up of viscous heat-conducting compressible flows, J. Math. Anal. Appl., 320 (2006), 819–826. https://doi.org/10.1016/j.jmaa.2005.08.005 doi: 10.1016/j.jmaa.2005.08.005
[9]	Y. Cho, H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manu. Math., 120 (2006), 91–129. https://doi.org/10.1007/s00229-006-0637-y doi: 10.1007/s00229-006-0637-y
[10]	X. Huang, J. Li, Z. Xin, Blow-up criterion for viscous baratropic flows with vacuum states, Commun. Math. Phys., 301 (2010) 23–35. https://doi.org/10.1007/s00220-010-1148-y doi: 10.1007/s00220-010-1148-y
[11]	H. Li, Y. Wang, Z. Xin, Non-existence of classical solutions with finite energy to the Cauchy problem of the compressible Navier–Stokes equations, Arch. Rational. Mech. Anal., 232 (2019), 557–590. https://doi.org/10.1007/s00205-018-1328-z doi: 10.1007/s00205-018-1328-z
[12]	O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier–Stokes equations, J. Differential Equations, 245 (2010), 1762–1774. https://doi.org/10.1016/j.jde.2008.07.007 doi: 10.1016/j.jde.2008.07.007
[13]	Y. Sun, C. Wang, Z. Zhang, A Beale–Kato–Majda blow-up criterion to the compressible Navier–Stokes equation, J. Math. Pures Appl., 95 (2011), 36–47. https://doi.org/10.1016/j.matpur.2010.08.001 doi: 10.1016/j.matpur.2010.08.001
[14]	H. Wen, C. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier–Stokes equations with vacuum, Adv. Math., 248 (2013), 534–572. https://doi.org/10.1016/j.aim.2013.07.018 doi: 10.1016/j.aim.2013.07.018
[15]	Z. Xin, W. Yan, On blow-up of classical solutions to the compressible Navier–Stokes equations, Commun. Math. Phys., 321 (2013), 529–541. https://doi.org/10.1007/s00220-012-1610-0 doi: 10.1007/s00220-012-1610-0
[16]	R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Commun. Partial Differ. Equ., 32 (2007), 1373–1397. https://doi.org/10.1080/03605300600910399 doi: 10.1080/03605300600910399
[17]	Q. Jiu, Y. Wang, Z. Xin, Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities, J. Differential Equations, 259 (2015), 2981–3003. https://doi.org/10.1016/j.jde.2015.04.007 doi: 10.1016/j.jde.2015.04.007
[18]	T. Liu, Z. Xin, T. Yang, Vacuum states for compressible flow, Discrete Contin. Dyn. Syst., 4 (1998), 1–32. https://doi.org/10.3934/dcds.1998.4.1 doi: 10.3934/dcds.1998.4.1
[19]	L. Sundbye, Global existence for Dirichlet problem for the viscous shallow water equation, J. Math. Anal. Appl., 202 (1996), 236–258. https://doi.org/10.1006/jmaa.1996.0315 doi: 10.1006/jmaa.1996.0315
[20]	L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mt. J. Math., 28 (1998), 1135–1152. https://doi.org/10.1216/rmjm/1181071760 doi: 10.1216/rmjm/1181071760
[21]	T. Yang, C. Zhu, Compressible Navier–Stokes equations with degnerate viscosity coefficient and vacuum, Commun. Math. Phys., 230 (2002), 329–363. https://doi.org/10.1007/s00220-002-0703-6 doi: 10.1007/s00220-002-0703-6
[22]	T. Zhang, D. Fang, Compressible flows with a density-dependent viscosity coefficient, SIAM. J. Math. Anal., 41 (2010), 2453–2488. https://doi.org/10.1137/090758878 doi: 10.1137/090758878
[23]	Y. Geng, Y. Li, 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. https://doi.org/10.1007/s00205-019-01401-9 doi: 10.1007/s00205-019-01401-9
[24]	Y. Li, R. Pan, S. Zhu, On classical solutions to 2D Shallow water equations with degenerate viscosities, J. Math. Fluid Mech., 19 (2017), 151–190. https://doi.org/10.1007/s00021-016-0276-3 doi: 10.1007/s00021-016-0276-3
[25]	Y. Li, R. Pan, S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Rational. Mech. Anal., 234 (2019), 1281–1334. https://doi.org/10.1007/s00205-019-01412-6 doi: 10.1007/s00205-019-01412-6
[26]	Z. Xin, S. Zhu, Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum, Adv. Math., 393 (2021), 108072. https://doi.org/10.1016/j.aim.2021.108072 doi: 10.1016/j.aim.2021.108072
[27]	Z. Xin, S. Zhu, Well-posedness of three-dimensional isentropic compressible Navier–Stokes equations with degenerate viscosities and far field vacuum, J. Math. Pures Appl., 152 (2021), 94–144. https://doi.org/10.48550/arXiv.1811.04744 doi: 10.48550/arXiv.1811.04744
[28]	S. Zhu, On the breakdown of regular solutions with finite energy for 3D degenerate compressible Navier–Stokes equations, J. Math. Fluid Mech., 23 (2021), 52. https://doi.org/10.48550/arXiv.1911.07965 doi: 10.48550/arXiv.1911.07965
[29]	S. Zhu, Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum, J. Differential Equations, 259 (2015), 84–119. https://doi.org/10.1016/j.jde.2015.01.048 doi: 10.1016/j.jde.2015.01.048
[30]	O. Ladyzenskaja, N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. https://doi.org/10.1090/mmono/023
[31]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York: Spinger-Verlag, 1986. https://doi.org/10.1007/978-1-4612-1116-7

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)