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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 show global well-posedness near vacuum for the binary–ternary Boltzmann equation. The binary–ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel–Shinbrot iteration and related work to approximate the solution of the non-linear equation by monotone sequences of super- solutions and subsolutions. This analysis required establishing new convolution-type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, and consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions. 

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.