Analytical solutions and asymptotic behaviors to the vacuum free boundary problem for 2D Navier-Stokes equations with degenerate viscosity

In this paper, we constructed a new class of analytical solutions to the isentropic compressible Navier-Stokes equations with vacuum free boundary in polar coordinates. These rotational solutions captured the physical vacuum phenomenon that the sound speed was $C^{1/2}$-Hölder continuous across the boundary, and they provided some new information on our understanding of ocean vortices and reference examples for simulations of computing flows. It was shown that both radial and angular velocity components and their derivatives will tend to zero as $t\to +\mathrm{\infty }$ and the free boundary will grow linearly in time, which happens to be consistent with the linear growth properties of inviscid fluids. The large time behavior of the free boundary $r=a(t)$ was completely determined by a second order nonlinear ordinary differential equation (ODE) with parameters of rotational strength $\xi$, adiabatic exponent $\gamma$, and viscosity coefficients. We tracked the profile and large time behavior of $a(t)$ by exploring the intrinsic structure of the ODE and the contradiction argument, instead of introducing some physical quantities, such as the total mass, the momentum weight and the total energy, etc., which are usually used in the previous literature. In particular, these results can be applied to the 2D Navier-Stokes equations with constant viscosity and the Euler equations.