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Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half-space. (English) Zbl 1098.76062
The authors consider the asymptotic behavior of solutions to compressible Navier-Stokes equations in the half-space \({\mathbb R}^{n}_{+} \, (n \geq 2)\) around a given constant equilibrium. They derive a solution formula for the linearized problem, and obtain \(L^{p}\) estimates for solutions of the linearized problem for \(2 \leq p \leq \infty\). 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 diffuse. This is well-known from the Cauchy problem. But, on the other side, there are some aspects different from the Cauchy problem, namely in considering spatial derivatives.
Furthermore, the authors show that the solution of the linearized problem approaches for large times the solution for nonstationary Stokes problem in some \(L^{p}\) spaces. As a result, a solution formula for the nonstationary Stokes problem is derived. Applying the results of the linearized analysis and the weighted energy method, large-time behavior of solutions of the nonlinear problem is then studied in \(L^{p}\) spaces for \(2 \leq p \leq \infty\). These results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
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