Infinite energy solutions to the homogeneous Boltzmann equation

Abstract

The goal of this work is to present an approach to the homogeneous Boltzmann equation for Maxwellian molecules with a physical collision kernel that allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the Fourier transform. In that space, the second moment of a measure is not assumed to be finite, so infinite energy solutions are not a priori excluded from our considerations. Moreover, we study the large-time asymptotics of solutions and, in a particular case, we give an elementary proof of the asymptotic stability of self-similar solutions obtained by A. V. Bobylev and C.

A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation

“…This significantly improves the previous result by Cannone and Karch (2010) [2] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution is justified as for the finite energy solution except for a single Dirac mass initial datum.…”

“…By Proposition 2.2, we get (1.14), that is, M α ⊂ F(P α (R d )). Since it follows from Lemma 3.15 of [2] that F(P α (R d )) ⊂ K α , we have M α ⊂ K α . More precisely, there exist C 1 , C 2 > 0 such that for ϕ ∈ M α , F = F −1 (ϕ), we have…”

“…´ v α f 0 dv, it follows from Lemma 3.15 of [2] that {f 0,m } belongs to a bounded set of P α , equivalently, {f 0,m } belongs to a bounded set of K α , that is, where D(f )(t) is defined by…”

“…Thanks to the new characterization of P α by its exact Fourier image M α , we can improve results, given in [2,5,7], concerning the existence and the smoothing effect of measure valued solutions to the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecule type cross section without angular cutoff that will be stated as follows.…”

Convergence to self-similar solutions for the homogeneous Boltzmann equation

“…for positive constants s ∈ (0, 1) and b 0 > 0. As in [7,8,9,11,16], the Cauchy problem (1.1) and (1.2) is considered in the set of probability measures on R 3 . For presentation, we first introduce some function spaces defined in the previous literatures.…”