Global stability of rarefaction wave for the outflow problem governed by the radiative Euler equations

This paper is devoted to the study of the initial-boundary value problem for the radiative full Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena. It turns out that the pattern of the asymptotic states is not unique and depends on the data both on the boundary and at the far field. In this paper, we focus our attention on the outflow problem when the flow velocity on the boundary is negative, and give a rigorous proof of the asymptotic stability of the rarefaction wave without restrictions on the smallness of the wave strength. Unlike our previous work on the inflow problem for the radiative Euler equations in [L. Fan, L. Ruan and W. Xiang, Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019), 595–625], lack of boundary conditions on the density and velocity prevents us from applying the integration by part to derive the energy estimates directly. Thus, the outflow problem is more challenging in mathematical analysis than the inflow problem studied in [L. Fan, L. Ruan and W. Xiang, Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019),595–625]. New weighted energy estimates are introduced and the trace of the density and velocity on the boundary are handled by some subtle analysis. The weight is chosen based on the new observation on the key decay properties of the smooth rarefaction wave. Our investigations on the inflow and outflow problem provide a good understanding on the radiative effect and boundary effect in the setting of rarefaction wave.