Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (4): 1343-1354.doi: 10.1007/s10473-025-0406-1

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

Yue CAO1, Yachun LI2, Shaojun YU3,*   

  1. 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.

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

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