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On large-time behavior of solutions to the compressible Navier-Stokes equations in the half space in $$\mathbb R^3$$. (English) Zbl 1016.35055
Summary: The Navier-Stokes equation for a compressible viscous fluid is considered on the half space in $$\mathbb R^3$$ 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 $$L^2$$ norms for solutions of the linearized problem are investigated to obtain the rate of convergence in $$L^2$$ norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in $$H^3\cap L^1$$. 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.

##### MSC:
 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B40 Asymptotic behavior of solutions to PDEs
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