Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian

Abstract

The purpose of this paper is to show how the combination of the well-known results for convergence to equilibrium and conditional regularity, in addition to a short-time existence result, lead to a quick proof of the existence of global smooth solutions for the non cutoff Boltzmann equation when the initial data is close to equilibrium. We include a short-time existence result for polynomially-weighted L ∞ initial data. From this, we deduce that if the initial data is sufficiently close to a Maxwellian in this norm, then a smooth solution exists globally in time.

Cutoff Boltzmann equation with polynomial perturbation near Maxwellian

“…x,v well-posedness result near Maxwellian is proved for the Landau equation in [37] and for the noncutoff Boltzmann equation in [8,45]. The solution with large amplitude initial data is first proved in [16] under the assumption of small entropy, we also refer to [21] for the Boltzmann equation with large amplitude initial data in bounded domains and [52] for the relativistic Boltzmann equation with large amplitude initial data.…”

Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials

“…For the perturbation setting with polynomial decay, the classical solutions were constructed by Alonso-Morimoto-Sun-Yang [10]; see also the independent works of He-Jiang [36] and Hérau-Tonon-Tristani [41]. The unique existence of weak solutions in the L 2 ∩ L ∞ setting was initiated by Alonso-Morimoto-Sun-Yang [9], and we mention the recent Silvestre-Snelson's work [57]. The well-posedness for Landau equations can be found in [15,17,18,31,35] and references therein Finally we mention two techniques used frequently when investigate the regularity property of kinetic equations, one referring to De Giorgi-Nash-Moser theory with the help of the averaging lemma and another to Hörmander's hypoelliptic theory.…”

Regularity estimates and open problems in kinetic equations

“…There are a number of papers dealing with solutions near equilibrium, for practically the full range of parameters γ and s. See [43,2,3,36,55,12,10,97,91]. From the proofs in any of these papers, in theory one should be able to extract a proof of short-time existence.…”

“…While global well posedness results near equilibrium are interesting in themselves, they say hardly anything about the solutions for arbitrary initial data. Moreover, any appropriate short-time existence result combined with the convergence to equilibrium [28], and Theorem 3.1, implies the existence of global solutions when the initial data is sufficiently close to a nonzero Maxwellian (see [91]). It is also possible to study the global well posedness of the equations when the initial data in near zero (see [76] and [26]).…”