On Mittag-Leffler Moments for the Boltzmann Equation for Hard Potentials Without Cutoff

We study generation and propagation properties of Mittag-Leffler moments for solutions of the spatially homogeneous Boltzmann equation for scattering collision kernels corresponding to hard potentials without angular Grad's cutoff assumption, i.e. the angular part of the scattering kernel is non-integrable with prescribed singularity rate. These kind of moments are infinite sums of renormalized polynomial moments associated to the probability density that solves the Cauchy problem under consideration. Such moments renormalization generates a fractional power series. In particular, Mittag-Leffler moments can be viewed as a generalization of fractional exponential moments. The summability of such fractional power series is proved, in both cases, by analyzing the convergence of partial sums sequences. More specifically, we show the propagation of exponential moments with orders that depend on the angular singularity rates given by the scattering cross section. The proof uses a subtle combination of angular averaging and angular singularity cancellation, that generates a negative contribution of highest order while controlling all positive terms. We ultimately obtain that the partial sums satisfy an ordinary differential inequality, whose solutions are uniformly bounded in time and number of terms. These techniques apply to both generation and propagation of Mittag-Leffler moments, with some variations depending on the case.