Analytical solutions to the 3-D compressible Navier-Stokes equations with free boundaries

Abstract

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity $h(\rho)=\mu\geq0$ , and the bulk viscosity $g(\rho)=\rho^{\beta}$ ( $\beta>0$ ). By constructing a class of radial symmetric and self-similar analytical solutions in $\mathbb {R}^{N}$ ( $N\geq2$ ) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity field.