New estimations of the nonlinear Boltzmann operator and their application to existence theorems

Abstract

Abstract New estimations of the nonlinear Boltzmann collision operator J(F) are given, which include all known estimations of J(F) in sup-norm. These new estimations are used to prove the existence of a mild solution to the nonlinear Boltzmann equation both in R3 and in the case of the flow past a body in R3 . The solution is global in time if the initial data decay fast enough at infinity and satisfy a smallness condition. Furthermore, the existence theorem admits solutions that are unbounded both in the space variable and the velocity variable. Examples are given in which it is shown that the smallness condition can be realized either through the smallness of supnorm of the initial datum or through the smallness of the support of the initial datum with respect to x or v variables. The last condition, physically speaking, corresponds to an initial datum that represents a small cloud of gas, or an arbitrary cloud of gas whose particle velocities are sufficiently close to some vo ∊ R3.