ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

doi:10.1007/s10473-025-0406-1

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1343-1354.doi: 10.1007/s10473-025-0406-1

ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

Yue CAO¹, Yachun LI², Shaojun YU^3,*

1. School of Mathematics, East China University of Science and Technology, Shanghai 200237, China;
2. School of Mathematical Sciences, MOE-LSC, and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China;
3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

收稿日期:2024-03-27 修回日期:2024-06-06 出版日期:2025-10-10 发布日期:2025-10-10

ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

Yue CAO¹, Yachun LI², Shaojun YU^3,*

1. School of Mathematics, East China University of Science and Technology, Shanghai 200237, China;
2. School of Mathematical Sciences, MOE-LSC, and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China;
3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2024-03-27 Revised:2024-06-06 Online:2025-10-10 Published:2025-10-10
Contact: ^*Shaojun YU, E-mail: edwardsmith123@sjtu.edu.cn
About author:Yue CAO, E-mail: cao yue12@ecust.edu.cn; Yachun LI, E-mail: ycli@sjtu.edu.cn
Supported by:
National Natural Science Foundation of China (12371221, 12161141004, 11831011), the Fundamental Research Funds for the Central Universities and Shanghai Frontiers Science Center of Modern Analysis.

摘要/Abstract

摘要： In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law (i.e., $\mu(\rho)=\rho^\delta, \delta>1$). We first obtain the local existence of the regular solution, then show that the regular solution will blow up in finite time if initial data have an isolated mass group, no matter how small and smooth the initial data are. It is worth mentioning that based on the transport structure of some intrinsic variables, we obtain the $L^\infty$ bound of the density, which helps to remove the restriction $\delta\leq \gamma$ in Li-Pan-Zhu [21] and Huang-Wang-Zhu [13].

关键词: compressible Navier-Stokes system, degenerate viscosity, vacuum, singular formation

Abstract: In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law (i.e., $\mu(\rho)=\rho^\delta, \delta>1$). We first obtain the local existence of the regular solution, then show that the regular solution will blow up in finite time if initial data have an isolated mass group, no matter how small and smooth the initial data are. It is worth mentioning that based on the transport structure of some intrinsic variables, we obtain the $L^\infty$ bound of the density, which helps to remove the restriction $\delta\leq \gamma$ in Li-Pan-Zhu [21] and Huang-Wang-Zhu [13].

Key words: compressible Navier-Stokes system, degenerate viscosity, vacuum, singular formation

Yue CAO, Yachun LI, Shaojun YU. ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM[J]. 数学物理学报(英文版), 2025, 45(4): 1343-1354.

Yue CAO, Yachun LI, Shaojun YU. ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1343-1354.

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ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

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