On large-time behavior of solutions to the compressible Navier-Stokes equations in the half space in R³

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 H³ ∩ L¹. 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.