In this paper, we study the zero dissipation limit problem for the Navier-Stokes equations of one-dimensional compressible viscous heat-conducting fluids. We prove that if the solution of the inviscid Euler equations is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from shock discontinuities at a rate of $\epsilon^1$ as the viscosity $\epsilon$ tend to zero, provided that the heat-conducting coefficient $k = 0($\epsilon$).
"Viscous Limits to Piecewise Smooth Solutions for the Navier-Stokes Equations of One-dimensional Compressible Viscous Heat-conducting Fluids." Methods Appl. Anal. 16 (1) 1 - 32, March 2009.