Time–dependent Kinetic Equations With Collision Terms Relatively Bounded With Respect to the Collision Frequency

Abstract

In this article the existence and uniqueness theory of time dependent kinetic equations is developed for collision terms dominated in the norm by the collision frequency, thus generalizing prior work by Beals and Protopopescu.

A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

“…This theory was modiÿed and extended in Reference [9] in order to deal with the existence issues in the unbounded case. However, for phase spaces, force terms, collision frequencies, collision loss operators and boundary re ection operators not depending on time, in the L 1 -setting and for a positive loss term not exceeding or even balancing the gain term, the kinetic equations studied in References [7][8][9] turn out to be applications of the abstract theory of this paper.…”

On General Transport Equations with Abstract Boundary Conditions. The Case of Divergence Free Force Field

“…H = 0 in (1.1b)). More general fields and boundary conditions (but still mostly associated with the Lebesgue measure) have been considered in a series of works [12,22,26]; unfortunately, sometimes they lack clarity. Problems with a general measure μ have been addressed very recently by the authors in [8] which should be considered as the first part of the present paper.…”

On linear kinetic equations involving unbounded cross‐sections

“…A ÿrst generation result for singular transport equations is given by the following (see also Reference [21]). …”

“…As a consequence, we are in position to prove the following generation result, without any more assumption on d and K (see also [21,Theorem A.1]). …”