Zero dissipation limit with two interacting shocks of the 1D non–isentropic Navier–Stokes equations

We investigate the zero dissipation limit problem of the one dimensional compressible non–isentropic Navier–Stokes equations with Riemann initial data in the case of the composite wave of two shock waves. It is shown that the unique solution of the Navier–Stokes equations exists for all time, and converge to the Riemann solution of the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks, as both the viscosity and heat-conductivity tend to zero. In contrast to previous related works, where either shock waves are absent or the effects of initial layers are ignored, this gives the first mathematical justification of this limit for the compressible non–isentropic Navier–Stokes equations in the presence of both shocks and initial layers. Our method of proof consists of a scaling argument, the construction of the approximate solution and delicate energy estimates.