Vanishing viscosity limit of the radiation hydrodynamic equations with far field vacuum

Abstract

We prove the vanishing viscosity limit of the Navier–Stokes–Boltzmann equations (see (1.3)) to the Euler–Boltzmann equations (see (1.9)) for a three-dimensional compressible isentropic flow in radiation hydrodynamics. It is shown that under some reasonable assumptions for the radiation coefficients, there exists a unique regular solution of Navier–Stokes–Boltzmann equations with degenerate viscosities, arbitrarily large initial data and far field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention to the fact that, via introducing two different symmetric structures and applying some techniques dealing with the complexity caused by the strong coupling between fluid and radiation field, we can also give some uniform estimates of (I,ργ−12,u) in H3 and of ∇ρ/ρ in D1, which provide the convergence of the regular solution of the viscous radiation flow to that of the inviscid radiation flow (see Li–Zhu [17]) in L∞([0,T];Hs′) space for any s′∈[2,3) with a rate of ϵ2(1−s′3).