On a fluid model of neutron star

In a recent paper, C. Monrozeau et al. [27] analyze the influence of neutron superfluidity on the cooling time of inner crust matter in neutron stars, in the case of a rapid cooling of the core of the star. The model used to describe the evolution of temperature [23] in the star follows Lattimer et al. [23]. It supposes a linear dependence of the specific heat as a function of temperature and assumes that a mechanical equilibrium is reached, so the problem reduces to the study of large time asymptotics for a Fast Diffusion Equation satisfied by the temperature. In a more general setting, it is interesting to consider the complete problem where temperature is coupled to density and velocity fluctuations through a thermo-mechanical system. The simplest description of such a model is achieved [21] through the compressible Navier-Stokes system. Concerning the fully 3D compressible case with heat conductivity the basic references are the works of Lions [26], Feireisl [11, 12] and Bresch-Desjardins [3], in which global existence of a weak solution is proved. Concerning asymptotics we can mention results done by Feireisl and his collaborator on the problem of the long-time behavior of solutions to the complete system with a time dependent driving force [13]. However spherical symmetry is considered in the major part of astrophysical literature [4, 5, 14, 21] as a quite reliable approximation (at least when rotation and magnetic aspects are neglected) and in this quasi-monodimensional situation, global existence and uniqueness of a classical solution and its large-time behavior have been obtained in some spherically symmetric cases (see [19, 16, 15]). Our purpose is then to prove well-posedness and large time asymptotics for this model (the compressible Navier-Stokes system for a spherical symmetric flow with specific temperaturedependent specific heat and thermal conductivity) leading to a simple estimate of a “cooling time”, in the spirit of Lattimer et al. [23]. The general formulation of the system reads [15]  ρt + (ρv)r + 2ρv