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

Abstract

This paper is concerned with affine solutions to the isentropic compressible Navier–Stokes equations with physical vacuum free boundary. Motivated by the result for Euler equations by Sideris (Arch Ration Mech Anal 225:141–176, 2017), we established the existence theories of affine solutions for the Navier–Stokes equations in ${ℝ}^N(N\ge 2)$ space under the homogeneity assumption that the pressure and the nonlinear viscosity parameters as functions of the density have a common degree of homogeneity. We derived an $N\times N$ second-order system of nonlinear ODEs of the deformation gradient $A\left(t\right)$ and provided an asymptotic analysis of the corresponding matrix system. The results show that both the diameter and volume of viscous fluids expand to infinity as time goes to infinity, and the algebraic rate of expansion is not bigger than that of inviscid fluids (Euler equations). In particular, the results contain the spherically 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 the Navier–Stokes equations with constant viscosity and the Euler equations.