Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping

ABSTRACT

We study the isentropic Euler equations with time-dependent damping, given by $\frac{\mu }{(1+t{)}^{\lambda }}\rho u$. Here, λ and μ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when $0<\lambda <1,0<\mu$ in multi-dimensions $n\ge 1$. Our strategy of proving the global existence is to convert the Euler system to a time-dependent damped wave equation and use a kind of weighted energy estimate. Investigation to the asymptotic behavior of the solution is based on the detailed analysis to the fundamental solutions of the corresponding linear damped wave equation and it coincides with that of standard results if λ deduces to zero.

Keywords:

• Euler equations
• time-dependent damping
• global existence
• asymptotic behavior

Mathematical Subject Classification 2010:

• 35L70
• 35L65
• 76N15

Acknowledgments

The author wants to express his gratitude to Prof. H. Yin and Dr F. Hou in Nanjing Normal University for many helpful discussion and communication. Most of this work was done when the author was visiting Department of Mathematics in University of California, Riverside. So he also thanks Department of Mathematics in UCR for its hospitality to provide him with nice working condition.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is supported by Natural Science Foundation of Jiangsu Province [grant number SBK2018041027] and National Natural Science Foundation of China [grant number 11801268].