Finite-time singularity formation for C¹ solutions to the compressible Euler equations with time-dependent damping

Abstract

In this paper, the initial-boundary value problem of the multidimensional compressible Euler equations with time-dependent damping in radial symmetry is considered. It is shown that finite-time singularity will be developed for the $C^1$ solutions of the compressible Euler equations with time-dependent damping coefficients $\mu /\left((1+t{)}^{\lambda }\right),\mu >0,\lambda \ge 0$ if the initial value of a newly introduced functional, $F\left(t\right)={\int }_1^{\mathrm{\infty }}\rho (t,r)u(t,r)[r^N-\alpha (t\left)\right]\mathrm{d}r$ with a time-dependent parameter $\alpha \left(t\right)$ is sufficiently large. The blowup conditions imply that the initial kinetic energy of the fluid must not be less than a given constant.

Keywords:

• Blowup
• time-dependent damping
• Euler equations
• initial-boundary value problem
• differential inequality

2010 Mathematics Subject Classifications:

• 35B44
• 35L67
• 35Q31
• 35B30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research paper is supported by the Dean's Research Funds (IRS7 2018 04305 and 04171) of the Faculty of Liberal Arts and Social Sciences of the Education University of Hong Kong and the Small Scale Grant 2018/19 of the Department of Mathematics and Information Technology of the Education University of Hong Kong.