Smoothing properties of a linearization of the three-wave collision operator in the bosonic Boltzmann–Nordheim equation

Abstract

We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann–Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three-wave collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol ln⁡(1+p2). Our main result in this context is that for initial data in a certain Banach space of functions satisfying a Hölder type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.