Viscous limits of the compressible Navier–Stokes equations to piecewise smooth solutions with two interacting out-going shocks

Abstract

We study the viscous 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 system is piecewise smooth with a 1-shock and a 3-shock starting from the same point, there exists a family of smooth solutions to the compressible Navier–Stokes equations, which converges to the inviscid solution away from the shock discontinuities at a rate of εη, ∀η ∈ (0, 1) as the viscosity ε tends to zero, provided that the heat-conducting coefficient κ = O(ε).