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We study the vanishing dissipation limit of the three-dimensional (3D)compressible Navier-Stokes-Fourier equations to the corresponding 3D full Eulerequations. Our results are twofold. First, we prove that the 3D compressibleNavier-Stokes-Fourier equations admit a family of smooth solutions thatconverge to the planar rarefaction wave solution of the 3D compressible Eulerequations with arbitrary strength. Second, we obtain a uniform convergence ratein terms of the viscosity and heat-conductivity coefficients. For thismulti-dimensional problem, we first need to introduce the hyperbolic wave torecover the physical dissipations of the inviscid rarefaction wave profile asin our previous work [29] on the two-dimensional (2D) case. However, due to the3D setting that makes the analysis significantly more challenging than the 2Dproblem, the hyperbolic scaled variables for the space and time could not beused to normalize the dissipation coefficients as in the 2D case. Instead, theanalysis of the 3D case is carried out in the original non-scaled variables,and consequently the dissipation terms are more singular compared with the 2Dscaled case. Novel ideas and techniques are developed to establish the uniformestimates. In particular, more accurate {\it a priori} assumptions with respectto the dissipation coefficients are crucially needed for the stabilityanalysis, and some new observations on the cancellations of the physicalstructures for the flux terms are essentially used to justify the 3D limit.Moreover, we find that the decay rate with respect to the dissipationcoefficients is determined by the nonlinear flux terms in the originalvariables for the 3D limit in this paper, but fully determined by the errorterms in the scaled variables for the 2D case in [29].

Vanishing dissipation limit to the planar rarefaction wave for the three-dimensional compressible Navier-Stokes-Fourier equations