  Fluid dynamic limit to compressible Euler equations from compressible
Navier-Stokes equations and Boltzmann equation has been an active topic with
limited success so far. In this paper, we consider the case when the solution
of the Euler equations is a Riemann solution consisting two rarefaction waves
and a contact discontinuity and prove this limit for both Navier-Stokes
equations and the Boltzmann equation when the viscosity, heat conductivity
coefficients and the Knudsen number tend to zero respectively. In addition, the
uniform convergence rates in terms of the above physical parameters are also
obtained. It is noted that this is the first rigorous proof of this limit for a
Riemann solution with superposition of three waves even though the fluid
dynamic limit for a single wave has been proved.
