On a functional equation arising in the kinetic theory of gases

Arkeryd, Leif ; Cercignani, Carlo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 2, pp. 139-149

Résumé

The problem of finding the summational collision invariants for the Boltzmann equation leads to a functional equation related to the Cauchy equation. The solution of this equation is known under different assumptions on its unknown \( \psi \). Most proofs assume that the equation is pointwise satisfied, while the result needed in kinetic theory concerns the solutions of the equation when the latter is satisfied almost everywhere. The only results of this kind appear to be due to the authors of the present paper. Here the problem is tackled with the aim of giving a simple proof that the most general solution of the problem is not different from the standard one when the equation is satisfied almost everywhere in \( \mathbb{R}^{3} \times \mathbb{R}^{3} \times \mathbb{S}^{2} \) and \( \psi \) is assumed to be measurable and finite a.e.