Decay rates to viscous contact wave for the one-dimensional compressible Navier–Stokes equations with radiation

Abstract

In this paper, we consider the stability and decay rate of viscous contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with radiation under the zero mass condition. we consider the case that the constitutive relations for the pressure p, the specific internal energy e, the specific volume v and the absolute temperature θ are given by p=Rθ/v+aθ4/3, e=Cvθ+avθ4, with R>0, Cv>0, and a>0 being the perfect gas constant, the specific heat and the radiation constant, respectively.Our main idea is to use the smallness of the strength of the contact wave to control the possible growth of its solutions induced by the nonlinearity of the system and interactions between the solutions and the contact wave. The key point in our analysis is to obtain the uniform convergence rate of a contact wave pattern (1+t)−58ln12(2+t) by using a new inequality (see Lemma 2.3) and a weighted characteristic energy estimate.