Convergence to nonlinear diffusion waves for solutions of M1 model

TL;DR

This paper studies the long-term behavior of solutions to the M1 radiative transfer model, proving convergence to nonlinear diffusion waves under general conditions with optimal rates using energy estimates and Green functions.

Contribution

It introduces a more general system related to the M1 model, proving global existence and convergence to nonlinear diffusion waves with sharper results and weaker initial data conditions.

Findings

Solutions converge to nonlinear diffusion waves

Optimal convergence rates are established

Results hold under weaker initial data assumptions

Abstract

In this paper, we are concerned with the asymptotic behavior of solutions of M1 model proposed in the radiative transfer fields. Starting from this model, combined with the compressible Euler equation with damping, we introduce a more general system. We rigorously prove that the solutions to the Cauchy problem of this system globally exist and time-asymptotically converge to the shifted nonlinear diffusion waves whose profile is self-similar solution to the corresponding parabolic equation governed by the classical Darcy's law. Moreover, the optimal convergence rates are also obtained. Compared with previous results obtained by Nishihara, Wang and Yang in [29], we have a weaker and more general condition on the initial data, and the conclusions are more sharper. The approach adopted in the paper is the technical time-weighted energy estimates with the Green function method together.