Singularity formation for compressible Euler equations with time-dependent damping
Ying Sui, Huimin Yu
Abstract
<p style='text-indent:20px;'>In this paper, we consider the compressible Euler equations with time-dependent damping <inline-formula><tex-math id="M1">\begin{document}$ \frac{{\alpha}}{(1+t)^\lambda}u $\end{document}</tex-math></inline-formula> in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case <inline-formula><tex-math id="M2">\begin{document}$ \lambda\neq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lambda = 1 $\end{document}</tex-math></inline-formula> respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for <inline-formula><tex-math id="M4">\begin{document}$ 1<\gamma<3 $\end{document}</tex-math></inline-formula> we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.