Abstract:

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 ε satisfy κ=O(ε), κ/ε ≥ c ＞ 0, as ε → 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 ε. The proof is given by a
combination of the energy estimates and the matched asymptotic analysis introduced in [3].

Key words: Zero dissipation limit, Navier--Stokes equations, shock waves