Planar Stationary Solution to Outflow Problem for Compressible Heat Conducting Gas in $\mathbb{R}^3_+$: Stability and Convergence Rate

Source abstract

Abstract. We are concerned with the stability and convergence rate of planar stationary solution to the compressible heat conducting gas in half space [Formula: see text] under outflow condition. (1) It is shown that a corresponding planar stationary solution is time asymptotically stable in three cases (i.e., supersonic, subsonic, and transonic), respectively, provided the initial perturbation in a certain Sobolev space and the boundary strength are sufficiently small. (2) Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Precisely, we obtain an algebraic decay rate provided that an initial perturbation decays in a tangential direction with the algebraic rate for a supersonic flow. The corresponding algebraic convergence rate is also obtained for a transonic flow, which is worse than that for the supersonic flow due to a degenerate property of the transonic flow. Each proof is given by deriving a priori estimates of the perturbation from the stationary wave by using a time and space weighted energy method.