Convergence Rate Toward Planar Stationary Waves for Compressible Viscous Fluid in Multidimensional Half Space

Abstract

The present paper concerns a large-time behavior of motion of isentropic and compressible viscous fluid in a two- or three-dimensional half space. We obtain a convergence rate of a solution toward a planar stationary wave for an outflow problem, where fluid flows out through a boundary. For a supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate provided that an initial perturbation decays, with respect to a normal direction, with the algebraic or the exponential rate, respectively. The algebraic convergence rate is also obtained for a transonic flow. Owing to a degenerate property of the transonic flow, the convergence rate is worse than that for the supersonic flow. The proofs are based on deriving a priori estimates of the perturbation from the stationary wave by using a time and space weighted energy method.