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

We constructed a class of analytical, rotational and self-similar solutions to the isentropic compressible Euler equations with time-dependent damping mu/(1+t)(lambda)rho u (mu >= 0, lambda is an element of R) and vacuum free boundary in cylindrical coordinates. These solutions indicate the fact that the sound speed is C-1/2-Holder continuous across the boundary (i.e., the physical vacuum) and is determined by the free boundary a(t), which is the solution of a second order nonlinear ordinary differential equation with parameters mu and lambda (see (1.12)). The global existence and detailed spreading rate of the free boundary a(t) are presented, while the stability for the free boundary problem is still unknown. Precisely, we show that for lambda > 1 the free boundary 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 -> +infinity (see Remark 1.6). However, if lambda <= 1, the free boundary will grow sub-linearly in time. In particular, if -1 <= lambda <= 1/(2 gamma-1), where gamma is the adiabatic exponent of polytropic gases, the free boundary will grow more slowly as lambda becomes smaller, and possesses a finite upper bound when lambda < -1.