Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum

Abstract

In this paper, we study the one-dimensional motion of viscous gas near a vacuum, with the gas connecting to a vacuum state with a jump in density. The interface behavior, the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficient μ(ρ) = ρ^α for any 0 < α < 1; this includes the time-weighted boundedness from below and above. The smoothness of the solution is discussed. Moreover, we construct a class of self-similar classical solutions which exhibit some interesting properties, such as optimal estimates. The present paper extends the results in [Luo T, Xin Z P, Yang T. SIAM J Math Anal, 2000, 31(6): 1175–1191] to the jump boundary conditions case with density-dependent viscosity.