Remarks on analytical solutions to compressible Navier-Stokes equations with free boundaries

In this paper, we consider the free boundary problem of the radially symmetric compressible Navier-Stokes equations with viscosity coefficients of the form mu(rho) = rho (theta ), lambda(rho) = (theta - 1)rho (theta )in R-N. Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, "Analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients and free boundaries," J. Differ. Equ., vol. 253, no. 1, pp. 1-19, 2012) for N = 3, theta = gamma > 1 and improve the spreading rate of the free boundary, where gamma is the adiabatic exponent. Moreover, we construct an analytical solution for theta = 2/3, N = 3 and gamma > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When theta = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, gamma = 2 and N = 3, gamma = 5/3, respectively.