Generalized solutions of the radiative transfer equations in a singular case
This paper deals with a pair of nonlinear PDEs which may be regarded as a perturbation of the well-known transport equation. These equations are classical in astrophysics and represent the evolution of a stellar atmosphere in the absence of hydrodynamic motion and heat conduction. The local thermodynamic equilibrium is assumed for the matter. The purpose of this paper is to study the Cauchy problem. The main text of this paper is organized as follows. The assumptions and main results are stated in Section I, where some heuristic derivations for the theorems are also given. Section II was devoted to the existence proof and the maximum principle. This proof relies mainly on a nonlinear version of Hille-Yosida theorem. In Section III, the generalized solution of the radiative transfer equations is proved to be classical when the initial energy is uniformly positive. The uniqueness of the semigroup solving these equations is deduced from this proof. A bounded variation regularity theorem of the generalized solutions in the degenerate case is proved in Section IV, when the temperature is allowed to vanish. Finally the convergence of some splitting algorithms associated to the radiative transfer equations was proved in Section V.
scientific article; zbMATH DE number 3994156
scientific article