On the Boltzmann equation for long‐range interactions

Abstract

We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long-range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions.

On the Cauchy problem of the Boltzmann and Landau equations with soft potentials

“…Although there is extensive mathematical literature for the Boltzmann theory (see [1,4,8,9,15,21,31,27,28,35] and their references), much less is known for soft potentials γ < 0. Global smooth small-amplitude solutions of the Vlasov-Poisson-Boltzmann system near vacuum were constructed with −3 ≤ γ < −2 in [11].…”

Existence of Weak Solutions to Some Vortex Density Models

“…A similar definition was used by Alexandre and Villani in the study of the Boltzmann equation without Grad's angular assumption to the cross section [1]. There, f (t, x, v) was defined as a renormalized solution to the Boltzmann equation with a defect measure µ (t, x, v) …”

The work of Cédric Villani

“…Finally, we mention that Villani's other work related to the Boltzmann equation includes a series of papers on the influence of grazing collisions, mainly with L. Desvillettes and R. Alexandre: existence of renormalized solutions (with defect measure) for the Boltzmann equation without cutoff [2], the rigorous derivation of the Fokker-Planck-Landau equation from the Boltzmann equation in the grazing collision limit [46,3], and sharp regularity bounds associated with entropy production [1].…”