Zero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock

The zero dissipation limit of the compressible heat-conducting Navier—Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ɛ satisfy κ =O(ɛ), , as ɛ → 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier—Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ɛ. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].