Optimal Decay Rate to the Viscous Contact Discontinuity for the System of Heat-Conductive Ideal Gas Without Viscosity

Abstract

This paper is concerned with the time-decay rates of the viscous contact wave in 1D heat-conductive ideal gas without viscosity. For the case without the zero mass condition, although the heat-conductive ideal gas without viscosity studied in this paper lacks dissipation in terms of velocity, we can still prove the same optimal decay rate as the compressible Navier–Stokes equation. Precisely, we can obtain the optimal decay rate \((1+t)^{-1/2}\) for the perturbation in \(L^{\infty }\)-norm. The crucial point of the analysis is as follows. We refine the estimates for both anti-derivatives and the original perturbations in the first, and then, we use an innovative transformation to ensure the structural conditions continue to hold for the system of derivatives. With this approach, we obtain the optimal decay rate.