  We study the vanishing dissipation limit of the three-dimensional (3D)
compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler
equations. Our results are twofold. First, we prove that the 3D compressible
Navier-Stokes-Fourier equations admit a family of smooth solutions that
converge to the planar rarefaction wave solution of the 3D compressible Euler
equations with arbitrary strength. Second, we obtain a uniform convergence rate
in terms of the viscosity and heat-conductivity coefficients. For this
multi-dimensional problem, we first need to introduce the hyperbolic wave to
recover the physical dissipations of the inviscid rarefaction wave profile as
in our previous work [29] on the two-dimensional (2D) case. However, due to the
3D setting that makes the analysis significantly more challenging than the 2D
problem, the hyperbolic scaled variables for the space and time could not be
used to normalize the dissipation coefficients as in the 2D case. Instead, the
analysis of the 3D case is carried out in the original non-scaled variables,
and consequently the dissipation terms are more singular compared with the 2D
scaled case. Novel ideas and techniques are developed to establish the uniform
estimates. In particular, more accurate {\it a priori} assumptions with respect
to the dissipation coefficients are crucially needed for the stability
analysis, and some new observations on the cancellations of the physical
structures for the flux terms are essentially used to justify the 3D limit.
Moreover, we find that the decay rate with respect to the dissipation
coefficients is determined by the nonlinear flux terms in the original
variables for the 3D limit in this paper, but fully determined by the error
terms in the scaled variables for the 2D case in [29].
