Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Abstract

We constructed a class of analytical, rotationaland self-similar solutions to the isentropic compressible Euler equations with time-dependent damping $\frac{\mu}{\left( 1+t\right) ^{\lambda}}%\rho\mathbf{u}$ ($\mu\geq0$, $\lambda\in\mathbb{R}$) and vacuum free boundary in cylindrical coordinates. These solutions capture the precise physical vacuum behavior that the sound speed is $C^{1/2}$-H\"older continuous across the boundary and are determined by the freeboundary $a(t)$, which is the solution of a second order nonlinear ordinarydifferential equation with parameters $\mu$ and $\lambda$ (see (1.12)) $). Furthermore, global well-posedness and asymptotic behavior of the free boundary $a(t)$ are presented in this paper. Precisely, we show that for $\lambda>1$ the freeboundary will grow linearly in time, and radial velocity $u^{r}$ and axial velocity $u^{z}$ are bounded, while the angular velocity $u^{\phi}$ and the radial derivatives of velocity components all tend to zero as $t\rightarrow+\infty$ (see Remark 1.4). However, if $\lambda\leq1$, the free boundary will grow sub-linearly in time. In particular, if $-1\leq \lambda\leq 1/(2\gamma-1)$, where $\gamma$ is the adiabatic exponent of polytropicgases, the free boundary will grow more slowly as $\lambda$ becomes smaller, and possesses a finite upper bound when $\lambda<-1$. Mathematics Subject Classifications (2020): 35R35, 76N10