On the Boltzmann equation with the symmetric stable Lévy process

Abstract

As for the spatially homogeneous Boltzmann equation of Maxwellian moleculeswith the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can beviewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Lévy processand the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establisha global existence theorem with maximum growth estimate, uniqueness and stability of solutions.