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We consider the zero heat conductivity limit to a contact discontinuity for the mono-dimensional full compressible Navier-Stokes-Fourier system. The method is based on the relative entropy method, and do not assume any smallness conditions on the discontinuity, nor on the $BV$ norm of the initial data. It is proved that for any viscosity $\nu\geq0$, the solution of the compressible Navier-Stokes-Fourier system (with well prepared initial value) converges, when the heat conductivity $\kappa$ tends to zero, to the contact discontinuity solution to the corresponding Euler system. We obtain the decay rate $\kappa^{\frac12}$. It implies that the heat conductivity dominates the dissipation in the regime of the limit to a contact discontinuity. This is the first result, based on the relative entropy, of an asymptotic limit to a discontinuous solutions for a system.

The Inviscid Limit to a Contact Discontinuity for the Compressible Navier--Stokes--Fourier System Using the Relative Entropy Method