Self-similar solutions and large time behavior to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary and rotation

Abstract

This paper is concerned with a class of self-similar analytical solutions to the 2D isentropic compressible Navier–Stokes equations with vacuum free boundary for polytropic gases. It is shown that the free boundary will grow linearly in time (see (1.11)), moreover, both angular velocity uϕ and its derivatives, and the derivative of radial velocity ur will tend to zero as t→+∞, while radial velocity itself is bounded. In particular, it is interesting to see that when the adiabatic exponent γ>2 and for a suitably large time, the rotation plays a dominant role in accelerating the growth of the free boundary compared with the effect of pressure.