Global well-posedness of a binary–ternary Boltzmann equation

Abstract

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 supersolutions 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.

Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

“…Trying out techniques which work for nonlinear dispersive PDEs on the Boltzmann equation is not without precursors. For years, there have been many nice developments [1,2,4,5,42,43,45,58,60,66,68,70] which have hinted at or used space-time estimates like the Chermin-Lerner spaces or the harmonic analysis related to nonlinear dispersive PDEs, and many of them have reached global (strong and mild) solutions if the datum is close enough to the Maxwellians or satisifes some conditions. In the same period of time, the theory of nonlinear dispersive PDEs has matured into a stage on which the working function spaces have been cleanly unified, the well-posedness and ill-posedness (See, for example, the now well-known work [38, 39, 49-51, 56, 57] and also the survey [69] and the references within.)…”

“…In §3, we prove the gain term estimates in Theorem 1.5 by appealing to a Hölder type estimate in [3] together with a Littlewood-Paley decomposition. 4 For completeness, we apply in the short section §4, the loss term estimate in Theorem 1.4 and the gain term estimate in Theorem 1.5 to prove the well-posedness part of Theorem 1.3. We prove the illposedness part of Theorem 1.3 in §5.…”

Rigorous derivation of a binary-ternary Boltzmann equation for a dense gas of hard spheres

“…(1. 19) We should mention that in [3], global well-posedness near vacuum has been shown for (1.16) for potentials ranging from moderately soft to hard in spaces of functions bounded by Maxwellian. In fact in…”

“…Let N ∈ N, with N ≥ 3, and 0 < ǫ2 < ǫ3 < 1. The natural phase space 3 to capture both binary and ternary interactions is: Let us describe the evolution in time of such a system. Consider an initial configuration ZN ∈ DN,ǫ 2 ,ǫ 3 .…”

“…IOAKEIM AMPATZOGLOU AND NATA ŠA PAVLOVI Ć [3], it is seen that the ternary collisional operator allows consideration of softer potentials that the binary operator. In other words the ternary correction to the Boltzmann equation does not behave worse than the classical Boltzmann equation.…”