The Boltzmann equation for hard potentials with integrable angular transition: Coerciveness, exponential tails rates, and Lebesgue integrability

This manuscript focus on an extensive survey with new techniques on the problem of solving the Boltzmann flow by bringing a unified approach to the Cauchy problem to homogeneous kinetic equations with Boltzmann-like collision operators under integrability assumption of the scattering profile in the particle-particle interaction mechanism. The work focuses on the relevant hard potential case where the solution properties are studied with a modern take. While many of the discussed results can be found the literature spread over several papers along the years, we bring a complete program that includes a new approach to the existence and uniqueness theorem securing the well-posedness theory, to moments estimates, and integrability propagation for the homogeneous Boltzmann flow. In particular, a detailed calculation of classical polynomial moments upper bounds as function of the coerciveness is described, which characterized the rate of exponential moments obtained by summability of the polynomial ones. In addition, a proof of uniform propagation of $L^\infty$ regularity under general integrable scattering kernels is performed. Along the way, constants appearing in estimates are carefully calculated, improving most of previous exiting results in the literature. For the non expert reader we also include a general discussion of the basic elements of the Boltzmann model and important key results for the understanding of the mathematical discussion of the equation and include an extensive set of references that enrich and motivate further discussions on Boltzmann flows for broader gas modeling configuration such as gas mixtures systems, polyatomic gases, multilinear collisional forms such as, ternary or quartic, to those derived from symmetry braking quantum mean field theories or weak turbulence models from spectral energy waves in classical fluid.