GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM

doi:10.1007/s10473-025-0316-2

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 1045-1062.doi: 10.1007/s10473-025-0316-2

GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM

Yue ZHANG¹, Minyan YU^1,†, Houzhi TANG²

1. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;
2. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China

收稿日期:2023-11-23 修回日期:2024-05-28 出版日期:2025-05-25 发布日期:2025-09-30

GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM

Yue ZHANG¹, Minyan YU^1,†, Houzhi TANG²

1. School of Mathematical Sciences, Capital Normal University, Beijing 100048, China;
2. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China

Received:2023-11-23 Revised:2024-05-28 Online:2025-05-25 Published:2025-09-30
Contact: ^†Minyan YU, E-mail: ymathmy@126.com
About author:Yue ZHANG, E-mail: yuezhangmath@126.com; Houzhi TANG, E-mail: houzhitang@ahnu.edu.cn
Supported by:
National Natural Science Foundation of China (11931010, 12226326, 12226327), and the Key Research Project of Academy for Multidisciplinary Studies, Capital Normal University. The third author was supported by the Anhui Provincial Natural Science Foundation (2408085QA031).

摘要/Abstract

摘要： We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system, which consists of the pressureless Navier-Stokes equations for $(n,w)$ coupled with the isentropic compressible Navier-Stokes equations for $(\rho,u)$ through a drag force term $n(w-u)$. We prove the global existence of strong solutions to the coupled system when the initial data are small perturbations of the constant equilibrium state. However, due to the pressureless structure, one can only deal with the density $n$ of the pressureless flow through the transport equation and it is crucial to obtain the exact time-decay rates for the corresponding velocity $w$ of the pressureless flow. To this end, we make use of the spectral analysis, low-high frequency decomposition and time-weighted energy method to deduce the large time behavior of $(w,\rho,u)$ and consequently establish the Lyapunov stability of the density $n$ in Sobolev space.

关键词: pressureless Navier-Stokes/Navier-Stokes system, Cauchy problem, strong solution, global existence

Abstract: We consider the Cauchy problem for the three-dimensional pressureless Navier-Stokes/Navier-Stokes system, which consists of the pressureless Navier-Stokes equations for $(n,w)$ coupled with the isentropic compressible Navier-Stokes equations for $(\rho,u)$ through a drag force term $n(w-u)$. We prove the global existence of strong solutions to the coupled system when the initial data are small perturbations of the constant equilibrium state. However, due to the pressureless structure, one can only deal with the density $n$ of the pressureless flow through the transport equation and it is crucial to obtain the exact time-decay rates for the corresponding velocity $w$ of the pressureless flow. To this end, we make use of the spectral analysis, low-high frequency decomposition and time-weighted energy method to deduce the large time behavior of $(w,\rho,u)$ and consequently establish the Lyapunov stability of the density $n$ in Sobolev space.

Key words: pressureless Navier-Stokes/Navier-Stokes system, Cauchy problem, strong solution, global existence

中图分类号:

76N10

Yue ZHANG, Minyan YU, Houzhi TANG. GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM[J]. 数学物理学报(英文版), 2025, 45(3): 1045-1062.

Yue ZHANG, Minyan YU, Houzhi TANG. GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM[J]. Acta mathematica scientia,Series B, 2025, 45(3): 1045-1062.

参考文献

[1] Bresch D, Huang X D, Li J. Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system. Comm Math Phys, 2012, 309: 737-755
[2] Choi Y P. Global classical solutions and large-time behavior of the two-phase fluid model. SIAM J Math Anal, 2016, 48: 3090-3122
[3] Choi Y P, Kwon B. The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations. J Differential Equations, 2016, 261: 654-711
[4] Dong J W, Yuen M W. Some special self-similar solutions for a model of inviscid liquid-gas two-phase flow. Acta Math Sci, 2021, 41B(1): 114-126
[5] Duan X L, Hao F F, Li H L, Zhao S. Nonlinear stability of planar stationary solutions to the viscous two-phase flow model in $\Bbb R^3_+$. J Differential Equations, 2023, 367: 749-782
[6] Evje S, Wang W J, Wen H Y. Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model. Arch Ration Mech Anal, 2016, 221: 1285-1316
[7] Guo S S, Wu G C, Zhang Y H. Global existence and large time behaviour for the pressureless Euler-Navier-Stokes system in $\mathbb{R}^3$. Proc Roy Soc Edinburgh Sect A, 2024, 154: 328-352
[8] Hao F F, Li H L, Shang L Y, Zhao S. Recent progress on outflow/inflow problem for viscous multi-phase flow. Commun Appl Math Comput, 2023, 5: 987-1014
[9] Jung J. Global-in-time dynamics of the two-phase fluid model in a bounded domain. Nonlinear Anal, 2022, 223: Paper No 113044
[10] Li H L, Shou L Y. Global existence of weak solutions to the drift-flux system for general pressure laws. Sci China Math, 2023, 66: 251-284
[11] Li H L, Shou L Y. Global existence and optimal time-decay rates of the compressible Navier-Stokes-Euler system. SIAM J Math Anal, 2023, 55: 1810-1846
[12] Li H L, Wang T, Wang Y. Wave phenomena to the three-dimensional fluid-particle model. Arch Ration Mech Anal, 2022, 243: 1019-1089
[13] Li H L, Zhao S. Existence and nonlinear stability of stationary solutions to the full two-phase flow model in a half line. Appl Math Lett, 2021, 116: Paper No 107039
[14] Li H L, Zhao S, Zuo H W. Existence and nonlinear stability of steady-states to outflow problem for the full two-phase flow. J Differential Equations, 2022, 309: 350-385
[15] Matsumura A, Nishida T. The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc Japan Acad Ser A Math Sci, 1979, 55: 337-342
[16] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67-104
[17] Novotný A, Pokorný M. Weak solutions for some compressible multicomponent fluid models. Arch Ration Mech Anal, 2020, 235: 355-403
[18] Tang H Z, Zhang Y. Large time behavior of solutions to a two phase fluid model in $\Bbb{R}^3$. J Math Anal Appl, 2021, 503: Paper No 125296
[19] Wen H Y, Yao L, Zhu C J. Review on mathematical analysis of some two-phase flow models. Acta Math Sci, 2018, 38B: 1617-1636
[20] Wu G C, Zhang Y H, Zou L. Optimal large-time behavior of the two-phase fluid model in the whole space. SIAM J Math Anal, 2020, 52: 5748-5774
[21] Wu Y K, Zhang Y, Tang H Z. Optimal decay rate of solutions to the two-phase flow model. Math Methods Appl Sci, 2023, 46: 2538-2568

GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM

GLOBAL STRONG SOLUTION OF THE PRESSURELESS NAVIER-STOKES/NAVIER-STOKES SYSTEM

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