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After revisiting the existence and uniqueness theory of solutions to the homogeneous Boltzmann equation whose transition probabilities (or collision kernels) (Alonso and Gamba, 2022; Mischler and Wennberg, 1999) are given by Maxwell type and hard intramolecular potentials, under just integrability condition for the angular scattering kernel, we present in this manuscript several new results. We start by showing the Lebesgue and Sobolev propagation of the exponential tails for such solutions. Previous results required stronger angular scattering kernel integrability conditions (Alonso and Gamba, 2008; Gamba et al., 2009). We point out that one of the novel tools for obtaining these results includes pointwise (i.e. strong) commutators between fractional derivatives and the collision operator. The paper includes the analysis for the critical case of Maxwell interactions corresponding to propagation of tails rather than generation. In addition, we show new estimates giving 𝐿𝑝 -integrability generation of exponential tails in the case of hard potential interactions in the range 𝑝 ∈ [1, ∞], exponentially-fast convergence rate to thermodynamical equilibrium (under rather general physical initial data), and regularization in the sense of exponential attenuation of singularities. In many ways, this work is an improvement and an extension of several classical works in the area (Alonso and Gamba, 2007; Alonso and Gamba, 2008; Arkeryd, 1982; Bobylev and Gamba, 2017; Gamba et al., 2009; Mouhot and Villani, 2004; Wennberg, 1993). We, both, use known techniques and introduce new and flexible ideas that achieve the proofs in a rather elementary manner. 

In this paper, we discuss a situation, which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local differential operators (short-range interactions) by non-local ones (long-range interactions), we arrive at a new Vlasov-type kinetic model in the mean field limit under the molecular chaos assumption reminiscent of models for collective behavior in which anharmonic oscillators replace individual particles.