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

Yue Cao, Yachun Li, Shaojun Yu

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., $μ(ρ)=ρ^δ, δ>1$). 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 has 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 $δ\leq γ$ in Li-Pan-Zhu[21] and Huang-Wang-Zhu[13].

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

Abstract

In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in , where the viscosity depends on the density in a super-linear power law(i.e., ). 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 has 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 bound of the density, which helps to remove the restriction in Li-Pan-Zhu[21] and Huang-Wang-Zhu[13].
Paper Structure (5 sections, 10 theorems, 71 equations)

This paper contains 5 sections, 10 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The uniform $L^\infty$ bound of the density
  4. Singularity Formation
  5. Acknowledgments

Key Result

Theorem 1.1

(Local existence) Assume that If initial data $\left(\rho_0, u_0\right)$ satisfies then there exists a time $T_*>0$ and a unique regular solution $(\rho, u)(t,x)$ in $\left[0, T_*\right] \times \mathbb{R}$ to Cauchy problem 1-44 satisfying for positive constant $C^0=C^0( \gamma, \delta, \rho_0, u_0)$. In fact, $(\rho, u)$ satisfies Cauchy problem 1-44 classically in positive time $\left(0, T_*\

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1: Simon1987compact
  • Lemma 2.2
  • Lemma 2.3: Hayk2018Alge
  • Lemma 3.1
  • ...and 9 more