The existence and nonlinear stability of stationary solutions to outflow problem for a reduced gravity two and a half layer model

Abstract

The main concern of this paper is to study large-time behavior of solutions for an outflow problem to the reduced gravity two and a half layer model in a half space. We establish the existence of stationary solutions and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small. Moreover, we obtain the convergence rate of the solution toward the stationary solution in some weighted Sobolev spaces only for the supersonic case. The proof is based on the basic energy method. A key point for the proof of convergence rates is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity satisfy some additional conditions.