Analytical solutions to the pressureless Navier-Stokes equations with density-dependent viscosity coefficients

In this paper, we construct a class of spherically symmetric and self-similar analytical solutions to the pressureless Navier-Stokes equations with density-dependent viscosity coefficients satisfying h(rho) = rho(0), g(rho) = (theta - 1)rho(0) for all theta > 0. Under the continuous density free boundary conditions imposed on the free surface, we investigate the large-time behavior of the solutions according to various theta > 1 and 0 < theta < 1. When the time grows up, such solutions exhibit interesting information: Case (i) If the free surface initially moves inward, then the free surface infinitely approaches to the symmetry center and the fluid density blows up at the symmetry center, or the free surface tends to an equilibrium state; Case (ii) If the free surface initially moves outward, then the free surface infinitely expands outward and the fluid density decays and tends to zero almost everywhere away from the symmetry center, or the free surface tends to an equilibrium state. We also study the large-time behavior of the solutions for theta = 1 without any boundary conditions.