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

Abstract

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. Due to the 3D setting the approach for the two-dimensional case could not be applied directly. Instead, the analysis of the 3D case is carried out in the original non-scaled variables, and consequently the dissipation terms are more singular. Novel ideas and techniques are developed to establish the uniform estimates. More accurate 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.