Well-posedness of classical solutions to the 3D isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum

Peng Lu; Shaojun Yu; Peng Lu; Shaojun Yu

doi:10.3934/cam.2025031

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 779-809. doi: 10.3934/cam.2025031

Previous Article Next Article

Research article Special Issues

Well-posedness of classical solutions to the 3D isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum

Peng Lu ^{1
,
,},
Shaojun Yu ²

1.
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, P.R. China
2.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China

Received: 08 May 2025 Revised: 25 August 2025 Accepted: 27 August 2025 Published: 01 September 2025
35Q35, 35A09, 76N10

We consider the isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum in a three-dimensional torus. The local well-posedness of classical solution is established by introducing a "quasi-symmetric hyperbolic"–"degenerate elliptic" coupled structure to control the behavior of the velocity of the fluid near the vacuum and give some uniform estimates. In particular, the initial data allows vacuum in an open set and we do not need any initial compatibility conditions.
- density-dependent viscosities,
- compressible Navier-Stokes-Poisson equations,
- vacuum,
- local well-posedness,
- classical solutions
Citation: Peng Lu, Shaojun Yu. Well-posedness of classical solutions to the 3D isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum[J]. Communications in Analysis and Mechanics, 2025, 17(3): 779-809. doi: 10.3934/cam.2025031

Related Papers:

Abstract

We consider the isentropic compressible Navier-Stokes-Poisson equations with degenerate viscosities and vacuum in a three-dimensional torus. The local well-posedness of classical solution is established by introducing a "quasi-symmetric hyperbolic"–"degenerate elliptic" coupled structure to control the behavior of the velocity of the fluid near the vacuum and give some uniform estimates. In particular, the initial data allows vacuum in an open set and we do not need any initial compatibility conditions.

References

[1]	P. A. Markowich, C. A. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer, 1990. https://doi.org/10.1007/978-3-7091-6961-2
[2]	S. Kawashima, Systems of a Hyperbolic-Parabolic composite type, with applications to the equations of Magnetohydrodynamics, Ph.D. thesis, Kyoto University, 1983.
[3]	A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67–104. https://doi.org/10.1215/kjm/1250522322 doi: 10.1215/kjm/1250522322
[4]	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
[5]	Y. Cho, H. Kim, On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91–129. https://doi.org/10.1007/s00229-006-0637-y doi: 10.1007/s00229-006-0637-y
[6]	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, Comm. Pure Appl. Math., 65 (2012), 549–585. https://doi.org/10.1002/cpa.21382 doi: 10.1002/cpa.21382
[7]	Q. Jiu, M. Li, Y. Ye, Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data, J. Differential Equations, 257 (2014), 311–350. https://doi.org/10.1016/j.jde.2014.03.014 doi: 10.1016/j.jde.2014.03.014
[8]	H. Gong, J. Li, X.-G. Liu, X. Zhang, Local well-posedness of isentropic compressible Navier–Stokes equations with vacuum, Commun. Math. Sci., 18 (2020), 1891–1909. https://doi.org/10.4310/CMS.2020.v18.n7.a4 doi: 10.4310/CMS.2020.v18.n7.a4
[9]	X. Huang, On local strong and classical solutions to the three-dimensional barotropic compressible Navier–Stokes equations with vacuum, Sci. China Math., 64 (2021), 1771–1788. https://doi.org/10.1007/s11425-019-9755-3 doi: 10.1007/s11425-019-9755-3
[10]	D. Bresch, B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211–223. https://doi.org/10.1007/s00220-003-0859-8 doi: 10.1007/s00220-003-0859-8
[11]	D. Bresch, A. Vasseur, C. Yu, Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density-dependent viscosities, J. Eur. Math. Soc., 24 (2022), 1791–1837. https://doi.org/10.4171/jems/1143 doi: 10.4171/jems/1143
[12]	Q. Jiu, Z. Xin, The cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1 (2008), 313–330. https://doi.org/10.3934/krm.2008.1.313 doi: 10.3934/krm.2008.1.313
[13]	J. Li, Z. Xin, Global existence of weak solutions to the barotropic compressible Navier–Stokes flows with degenerate viscosities, preprint, arXiv: 1504.06826.
[14]	A. Mellet, A. Vasseur, On the barotropic compressible Navier–Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431–452. https://doi.org/10.1080/03605300600857079 doi: 10.1080/03605300600857079
[15]	A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, Invent. Math., 206 (2016), 935–974. https://doi.org/10.1007/s00222-016-0666-4 doi: 10.1007/s00222-016-0666-4
[16]	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
[17]	Y. Li, R. Pan, S. Zhu, On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum, Arch. Ration. Mech. Anal., 234 (2019), 1281–1334. https://doi.org/10.1007/s00205-019-01412-6 doi: 10.1007/s00205-019-01412-6
[18]	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. Ration. Mech. Anal., 234 (2019), 727–775. https://doi.org/10.1007/s00205-019-01401-9 doi: 10.1007/s00205-019-01401-9
[19]	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
[20]	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/10.1016/j.matpur.2021.05.004 doi: 10.1016/j.matpur.2021.05.004
[21]	Y. Cao, H. Li, S. Zhu, Global regular solutions for one-dimensional degenerate compressible Navier–Stokes equations with large data and far field vacuum, SIAM J. Math. Anal., 54 (2022), 4658–4694. https://doi.org/10.1137/21M1464609 doi: 10.1137/21M1464609
[22]	P. Germain, P. Lefloch, Finite energy method for compressible fluids: the Navier–Stokes–Korteweg model, Comm. Pure Appl. Math., 69 (2016), 3–61. https://doi.org/10.1002/cpa.21622 doi: 10.1002/cpa.21622
[23]	Z. Guo, H.-L. Li, Z. Xin, Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier–Stokes equations, Comm. Math. Phys., 309 (2012), 371–412. https://doi.org/10.1007/s00220-011-1334-6 doi: 10.1007/s00220-011-1334-6
[24]	P.-L. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, volume 2, Oxford University Press, New York, 1998.
[25]	T. Yang, H. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163–184. https://doi.org/10.1006/jdeq.2001.4140 doi: 10.1006/jdeq.2001.4140
[26]	D. Donatelli, Local and global existence for the coupled Navier–Stokes–Poisson problem, Quart. Appl. Math., 61 (2003), 345–361. https://doi.org/10.1090/qam/1976375 doi: 10.1090/qam/1976375
[27]	H.-L. Li, A. Matsumura, G. Zhang, Optimal decay rate of the compressible Navier–Stokes–-Poisson system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681–713. https://doi.org/10.1007/s00205-009-0255-4 doi: 10.1007/s00205-009-0255-4
[28]	X. Zheng, Global well-posedness for the compressible Navier–Stokes–Poisson system in the ${L}^p$ framework, Nonlinear Anal., 75 (2012), 4156–4175. https://doi.org/10.1016/j.na.2012.03.006 doi: 10.1016/j.na.2012.03.006
[29]	Z. Tan, Y. Wang, Y. Wang, Stability of steady states of the Navier–-Stokes–-Poisson equations with non-flat doping profile, SIAM J. Math. Anal., 47 (2015), 179–209. https://doi.org/10.1137/130950069 doi: 10.1137/130950069
[30]	S. Liu, X. Xu, J. Zhang, Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier–Stokes–Poisson equations subject to large and non-flat doping profile, J. Differential Equations, 269 (2020), 8468–8508. https://doi.org/10.1016/j.jde.2020.06.006 doi: 10.1016/j.jde.2020.06.006
[31]	H. Liu, T. Luo, H. Zhong, Global solutions to compressible Navier–Stokes–Poisson and Euler–Poisson equations of plasma on exterior domains, J. Differential Equations, 269 (2020), 9936–10001. https://doi.org/10.1016/j.jde.2020.07.005 doi: 10.1016/j.jde.2020.07.005
[32]	Y. Chen, B. Huang, X. Shi, Global well-posedness of classical solutions to the compressible Navier–Stokes–Poisson equations with slip boundary conditions in 3D bounded domains, J. Math. Fluid Mech., 26 (2024), 41. https://doi.org/10.1007/s00021-024-00875-2 doi: 10.1007/s00021-024-00875-2
[33]	B. Ducomet, Š. Nečasová, A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities, Z. Angew. Math. Phys., 61 (2010), 191–211. https://doi.org/10.1007/s00033-009-0035-x doi: 10.1007/s00033-009-0035-x
[34]	B. Ducomet, Š. Nečasová, A. Vasseur, On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas, J. Math. Fluid Mech., 13 (2011), 191–211. https://doi.org/10.1007/s00021-009-0010-5 doi: 10.1007/s00021-009-0010-5
[35]	A. Zlotnik, On global in time properties of the symmetric compressible barotropic Navier–-Stokes–Poisson flows in a vacuum, Int. Math. Ser., 7 (2008), 329–376. https://doi.org/10.1007/978-0-387-75219-8_7 doi: 10.1007/978-0-387-75219-8_7
[36]	Y. Ye, C. Dou, Global weak solutions to 3D compressible Navier–Stokes–Poisson equations with density-dependent viscosity, J. Math. Anal. Appl., 455 (2017), 180–211. https://doi.org/10.1016/j.jmaa.2017.05.044 doi: 10.1016/j.jmaa.2017.05.044
[37]	H. Yu, Global existence and decay of strong solution to the 3D compressible Navier–Stokes–-Poisson equations with large data, J. Differential Equations, 444 (2025), 113672. https://doi.org/10.1016/j.jde.2025.113672 doi: 10.1016/j.jde.2025.113672
[38]	O. Ladyzenskaja, N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968.
[39]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, 1986. https://doi.org/10.1007/978-1-4612-1116-7
[40]	J. Simon, Compact sets in ${L}^p(0, {T}; {B})$, Ann. Mat. Pura. Appl., 146 (1987), 65–96. https://doi.org/10.1007/BF01762360 doi: 10.1007/BF01762360
[41]	J. Boldrini, M. Rojas-Medar, E. Fernández-Cara, Semi-Galerkin approximation and regular solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pure Appl., 82 (2003), 1499–1525. https://doi.org/10.1016/j.matpur.2003.09.005 doi: 10.1016/j.matpur.2003.09.005
[42]	C. Evans, Partial Differential Equations, American Mathematical Society, 2010. https://doi.org/10.1090/gsm/019

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)