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This manuscript focus on an extensive survey with new techniques on theproblem of solving the Boltzmann flow by bringing a unified approach to theCauchy problem to homogeneous kinetic equations with Boltzmann-like collisionoperators under integrability assumption of the scattering profile in theparticle-particle interaction mechanism. The work focuses on the relevant hardpotential case where the solution properties are studied with a modern take.While many of the discussed results can be found the literature spread overseveral papers along the years, we bring a complete program that includes a newapproach to the existence and uniqueness theorem securing the well-posednesstheory, to moments estimates, and integrability propagation for the homogeneousBoltzmann flow. In particular, a detailed calculation of classical polynomialmoments upper bounds as function of the coerciveness is described, whichcharacterized the rate of exponential moments obtained by summability of thepolynomial ones. In addition, a proof of uniform propagation of $L^\infty$regularity under general integrable scattering kernels is performed. Along theway, constants appearing in estimates are carefully calculated, improving mostof previous exiting results in the literature. For the non expert reader wealso include a general discussion of the basic elements of the Boltzmann modeland important key results for the understanding of the mathematical discussionof the equation and include an extensive set of references that enrich andmotivate further discussions on Boltzmann flows for broader gas modelingconfiguration such as gas mixtures systems, polyatomic gases, multilinearcollisional forms such as, ternary or quartic, to those derived from symmetrybraking quantum mean field theories or weak turbulence models from spectralenergy waves in classical fluid.

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