Global existence and asymptotic behavior of affine solutions to Navier-Stokes equations in RN with degenerate viscosity and free boundary

This paper is concerned with affine solutions to the isentropic compressibleNavier-Stokes equations with physical vacuum free boundary. Motivated bythe result for Euler equations by Sideris (Arch Ration Mech Anal 225:141-176,2017), we established the existence theories of affine solutions for theNavier-Stokes equations in R-N(N >= 2)space under the homogeneityassumption that the pressure and the nonlinear viscosity parameters as func-tions of the density have a common degree of homogeneity. We derived anNxNsecond-order system of nonlinear ODEs of the deformation gradientA(t)andprovided an asymptotic analysis of the corresponding matrix system. The resultsshow that both the diameter and volume of viscous fluids expand to infinity astime goes to infinity, and the algebraic rate of expansion is not bigger than thatof inviscid fluids (Euler equations). In particular, the results contain the spher-ically symmetric case, in which the free boundary will grow linearly in time,exactly as that in inviscid fluids. Moreover, these results can be applied to theNavier-Stokes equations with constant viscosity and the Euler equations.