Time Dependent Kinetic Equations: Semigroup Approach

Abstract

In the previous chapter we have solved time dependent kinetic problems using integration along trajectories and perturbation techniques. In many applications, however, the phase space, the transport operator and the boundary conditions do not depend on time, in which case a semigroup approach is natural. In this chapter we shall discuss the semigroup approach in detail. Throughout we maintain the notation and terminology introduced in Chapter XI with minor modifications. In particular, the open set Λ ⊂ ℝn×ℝn specifying the position-velocity domain will now be referred to as the phase space of the system. (We recall that in the previous chapter Σ ⊂ ℝn×ℝn×(0,T) was taken as the phase space, in order to treat time in a somewhat symmetric fashion with spatial and velocity variables). Similarly, μ will be a Borel measure on Λ such that subsets of Λ with finite Lebesgue measure have finite μ-measure, υ± will be appropriate Borel measures on the parts D± of ∂Λ corresponding to the outgoing (resp. incoming) “fluxes”, J and K are bounded linear operators defined on Lp (Λ,dμ) and from Lp(D+,dυ+) into Lp(D−,dυ−), respectively, and h(x,ξ) is a nonnegative Lebesgue measurable function on Λ that is integrable on each subset of finite Lebesgue measure.