Asymptotic Behavior of Solutions of the Compressible Navier-Stokes Equations on the Half Space

Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space R ⁿ₊(n≥2) is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and L^p estimates for solutions of the linearized problem are obtained for 2≤p≤∞. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one that behaves like diffusion waves and the other one purely diffusively. There, however, are some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary Stokes problem in some L^p spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained. Large-time behavior of solutions of the nonlinear problem is then investigated in L^p spaces for 2≤p≤∞ by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.