On Large-Time Behavior of Solutions to the Compressible Navier-Stokes Equations in the Half Space in R 3

Abstract

The Navier-Stokes equation for compressible viscous fluid is considered on the half space in R3 under the zero-Dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum. Time decay properties in L2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in L2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in \(\). Some lower bounds are derived for solutions to the linearized problem, one of which indicates a nonlinear phenomenon not appearing in the case of the Cauchy problem on the whole space.