Thermostatted kinetic theory in measure spaces: Well-posedness

This paper is devoted to the generalization of the thermostatted kinetic theory within the framework of probability measures. Specifically well-posedness of the Cauchy problem related to a thermostatted kinetic equation for measure-valued functions is established. The external force applied to the system is assumed to be Lipschitz, in contrast to previous work where external forces are generally constant. Existence is obtained by employing an Euler-like approximation scheme which is shown to converge assuming the initial condition has moment of order greater than 2. Uniqueness is proved assuming the gain operator is Lipschitz w.r.t a (new) Monge–Kantorovich–Wasserstein distance $W_{2-}$, intermediate between the classical $W_2$ and $W_r$, $r<2$, distances. The assumptions on the gain operator are quite general covering $n$-ary interaction, and apply in particular to the Kac equation.