A note on the entropy production of the radiative transfer equation

Abstract

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 ϕ ( t), the entropy production I ϕ ( t), and the entropy production rate I ϕ ( t) of the kinetic model. I ϕ ( t) has to be estimated in terms of I ϕ ( t). Then e ϕ ( t) is estimated in terms of I ϕ ( 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.