Generation and propagation of Lebesgue norms for the homogeneous Boltzmann equation without angular cutoff

We consider kinetic systems comprised of a large number of interacting particles and discuss one specific approach to modelling such an object from a physical point of view. The subsequent kinetic model obtained through this process is a nonlinear integro-partial differential equation; namely, the Boltzmann equation. We focus, in particular, on the spatially homogeneous Boltzmann equation for soft potentials (HBESP) and without angular cutoff and use techniques adapted from the hard potential theory together with classical ideas to study the instantaneous appearance (generation) and propagation of Lᵖ -norms. By considering solutions over a fixed interval of time and with sufficient assumptions regarding their L¹ and L² moments, we are able to demonstrate that Lᵖ -norms of solutions to the non-cutoff (HBESP) are both generated and propagated in time.