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<p>In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in ℝ, where the viscosity depends on the density in a super-linear power law (i.e., <i>μ</i>(<i>ρ</i>) = <i>ρ</i><sup><i>δ</i></sup>, <i>δ</i> &gt; 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 <i>L</i><sup>∞</sup> bound of the density, which helps to remove the restriction <i>δ</i> ≤ <i>γ</i> in Li-Pan-Zhu [21] and Huang-Wang-Zhu [13].</p>

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On blow-up to the one-dimensional Navier-Stokes equations with degenerate viscosity and vacuum

  • 期刊论文

  • 发表日期：

    2025年4月8日

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  • Yue Cao,
  • Yachun Li,
  • Shaojun Yu

摘要

In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in ℝ, where the viscosity depends on the density in a super-linear power law (i.e., μ(ρ) = ρδ, δ > 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∞ bound of the density, which helps to remove the restriction δ ≤ γ in Li-Pan-Zhu [21] and Huang-Wang-Zhu [13].