Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries

Abstract

In this paper, we study a class of analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients, which describe compressible fluids moving into outer vacuum. For suitable viscous polytropic fluids, we construct a class of radial symmetric and self-similar analytical solutions in RN(N⩾2) with both continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum. Such solutions exhibit interesting new information such as the formation of vacuum at the center of the symmetry as time tends to infinity and explicit regularities and large time decay estimates of the velocity field.