Unconditional global stability of solutions to a parabolic–elliptic coupled system: Cauchy problem

We derive a parabolic–elliptic coupled system as a new simplified model for the hydrodynamics of radiating gas with viscosity and thermal conductivity. This model captures the interplay between viscous dissipation and nonlocal radiative effect. Our main contributions are the establishment of unconditional global stability of solutions to the Cauchy problem for constant states, rarefaction waves, and planar rarefaction waves under different initial conditions, and the derivation of precise decay rates for the asymptotic behaviors of solutions. Unlike previous studies, our results do not require any smallness assumptions on the initial perturbation or wave strength. This can be viewed as the first result on the unconditional global stability for fluid dynamics equations. Our analysis relies on a combination of energy estimates, maximum principle and Fourier analysis, with the dissipation term playing a crucial role in controlling nonlinear effect.