Higher order extensions of the Boltzmann equation
This dissertation investigates extensions of the Boltzmann equation to higher order interactions and consists of two parts, which are submitted separately for publication, see [6, 4]. In the first part of the dissertation, we present a rigorous derivation of a novel kinetic equation, which we call ternary Boltzmann equation, describing the limiting behavior of a classical system of particles with three particle instantaneous interactions. Derivation of such an equation required development of new conceptual and geometrical ideas to treat interactions among three particles and their evolution in time. We also show that a symmetrized version of the ternary Boltzmann equation has the same conservation laws and entropy production properties as the classical binary operator. The superposition of this ternary equation with the classical Boltzmann equation, which we call the binary-ternary Boltzmann equation, could be understood as a step towards modeling a dense gas in non-equilibrium, since both binary and ternary interactions between particles are taken into account.
In the second part of the dissertation, we show global well-posedness near vacuum for the binary-ternary Boltzmann equation for monoatomic gases with a wide range of hard and soft potentials. Well-posedness of the ternary equation for these potentials follows as a special case. This is the first global well-posedness result for the binary-ternary Boltzmann equation and for the ternary Boltzmann equation. To prove global well-posedness, we implement a Kaniel-Shinbrot iteration and related works to the ternary correction of the Boltzmann equation to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions which converge, for small initial data, to the global in time solution of the binary-ternary equation. 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 binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution.