The mathematical properties of the radiative transfer equation

Abstract

The kinetic equation describing the interaction of matter with electromagnetic waves is the radiative transfer equation. This thesis has as aim the development of a comprehensive mathematical theory for this equation. In particular, it collects several results about different problems which study the behavior of the temperature distribution in a body where the heat is transferred mainly by radiation. This work is a cumulative thesis which collects five articles written by the author with several collaborators during her PhD studies. <br/>

Chapter 1 gives a detailed introduction of the radiative transfer equation describing its derivation, its main mathematical features and the available mathematical literature. <br/>

Chapter 2 deals with the existence theory of the stationary radiative transfer equation in the case in which the absorption and the scattering coefficients depend on the temperature and it is based on the article in Appendix A. <br/>

Chapter 3 is based on the article in Appendix B and it treats the diffusion approximation of the radiative transfer equation, which is studied via matched asymptotic expansions, introducing the concepts of equilibrium and non-equilibrium diffusion approximations. <br/>

Chapter 4 studies rigorously the diffusion approximation of the stationary radiative transfer equation when the radiation interacts with matter only through emission and absorption. This chapter is based on the article in Appendix C. <br/>

Chapter 5 develops the well-posedness theory of a free boundary problem for the melting of ice in the case in which the heat is transferred by conduction in the liquid and by conduction and radiation in the solid. This chapter is based on the article in Appendix D. <br/>

Chapter 6 is based on the article in Appendix E and it continues the study of the free boundary problem introduced in Chapter 5 and it deals with the existence theory of traveling wave solutions. <br/>

Finally, Chapter 7 summarizes the main results of this thesis and presents several future research directions. <br/>

The articles upon which this thesis is based can be found in the appendices.