Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock data

This paper is concerned with the vanishing dissipation limiting problem of one-dimensional non-isentropic Navier-Stokes equations with shock data. The limiting problem was solved in 1989 by Hoff-Liu in [13] for isentropic gas with single shock, but was left open for non-isentropic case. In this paper, we solve the non-isentropic case, i.e., we first establish the global existence of solutions to the non-isentropic Navier-Stokes equations with initial discontinuous shock data, and then show these solutions converge in $L^{\infty}$ norm to a single shock wave of the corresponding Euler equations away from the shock curve in any finite time interval, as both the viscosity and heat-conductivity tend to zero. Different from [13] in which an integrated system was essentially used, motivated by [21,22], we introduce a time-dependent shift $\mathbf{X}^\varepsilon(t)$ to the viscous shock so that a weighted Poincar\'{e} inequality can be applied to overcome the difficulty generated from the ``bad" sign of the derivative of viscous shock velocity, and the anti-derivative technique is not needed. We also obtain an intrinsic property of non-isentropic viscous shock, see Lemma 2.2 below. With the help of Lemma 2.2, we can derive the desired uniform a priori estimates of solutions, which can be shown to converge in $L^{\infty}$ norm to a single inviscid shock in any given finite time interval away from the shock, as the vanishing dissipation limit. Moreover, the shift $\mathbf{X}^\varepsilon(t)$ tends to zero in any finite time as viscosity tends to zero. The proof consists of a scaling argument, $L^2$-contraction technique with time-dependent shift to the shock, and relative entropy method.