A note on the entropy production of the radiative transfer equation

In recent years, the entropy approach to the asymptotic (large-time) analysis of homogeneous kinetic models has led to remarkable new proofs of convex-type (e.g., logarithmic) Sobolev inequalities. The crucial point of this method lies in computing the entropy e(phi)(t), the entropy production I-phi(t), and the entropy production rate (I) over dot(phi)(t) of the kinetic model. (I) over dot(phi)(t) has to be estimated in terms of I-phi(t). Then e(phi)(t) is estimated in terms of I-phi(t). We apply this approach to the (explicitly solvable) homogeneous radiative transfer equation obtaining a Jensen-type inequality involving a convex function as corresponding "Sobolev inequality". All the computations are highly transparent and serve to highlight and ultimately clarify the approach. (C) 1999 Elsevier Science Ltd. All rights reserved.