Stability of Planar Stationary Solutions to the Compressible Navier-Stokes Equation on the Half Space

Abstract

Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space $${\bf R}_+^n (n\geq 2)$$ under outflow boundary condition is investigated. It is shown that the planar stationary solution is stable with respect to small perturbations in $$H^s\left({\bf R}_{+}^{n}\right)$$ with s≥ [n/2]+1 and the perturbations decay in L ∞ norm as t →∞, provided that the magnitude of the stationary solution is sufficiently small. The stability result is proved by the energy method. In the proof an energy functional based on the total energy of the system plays an important role.