Time-asymptotic stability of generic Riemann solutions for the system of heat-conductive ideal gas without viscosity

Abstract

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.