Stability of Stationary Solutions to the Inflow Problem for Full Compressible Navier--Stokes Equations with a Large Initial Perturbation

Abstract

The large-time behavior of stationary solutions to the inflow problem of full compressible Navier--Stokes equations in the half line is investigated. We first give some necessary and sufficient conditions for the existence of the stationary solution with the aid of center manifold theory. We also prove that if $L^2$-norm of initial perturbation is small in accordance with any given $L^2$-norm of its derivative, then the stationary solutions with small strength for the general gas including ideal polytropic gas are asymptotically stable. Last, we show that the subsonic stationary solution with small strength for the ideal polytropic gas is asymptotically stable under large perturbations in both the $H^1$-norm and the $L^\infty$-norm without the condition that the adiabatic exponent $\gamma$ be close to 1, provided that the strength of the stationary wave is suitably small.