Abstract: We investigate the vanishing viscosity limit of a parabolic-elliptic coupled system arising in radiation hydrodynamics. The limit is the solution to a hyperbolic-elliptic coupled system. We study the problem on both the whole line $\mathbb{R}$ and the half line $\mathbb{R}_{+}$. Two types of conditions are considered: (1) the initial data are sufficiently close to a given initial state with small wave strength, and (2) the initial data are monotonically increasing. Under these conditions, we establish uniform convergence rates for both Cauchy problems and initial-boundary value problems. Specifically, we demonstrate that the solutions to the parabolic-elliptic system converge to those of the hyperbolic-elliptic system as the viscosity coefficient $\varepsilon$ approaches zero, with convergence rates of $O(\varepsilon^\frac{3}{4})$ for $u$ and $O(\varepsilon)$ for $q$ in $L^\infty$-norm. Additionally, we prove the global well-posedness of the parabolic-elliptic coupled system by the maximum principle and energy method. Our results extend previous work by providing explicit convergence rates and addressing both Cauchy problems and initial-boundary value problems under various conditions.

Key words: vanishing viscosity limit, parabolic-elliptic coupled system, hyperbolic-elliptic coupled system