The Boltzmann Equation Without Angular Cutoff in the Whole Space: Qualitative Properties of Solutions

Abstract

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoffWe refer the reader for the complete framework, definitions and bibliography, to our previous papers [9,10]. General details about Boltzmann equation for non cutoff cross sections can be found in [1,13,37]. Let us just recall herein that the Boltzmann bilinear collision operator is given by Q(g, f ) =

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

“…n! , ∀r ≥ 1 2 , ∀ε > 0, ∀n, k, l ≥ 0, v k ∂ l v e n L 2 (R) ≤ √ 2((1 − δ n,0 ) exp(εrn 1 2r ) + δ n,0 ) 2 3 2 +r e r inf(ε r , 1) k+l (k!) r (l!)…”

“…Alexandre-Morimoto-Ukai-Xu-Yang [3] highlighted the importance of regularization effects for Boltzmann equation (see also [1,4,10,11]). They studied C ∞ smoothing properties of the spatially inhomogeneous non-cutoff Boltzmann equation in [1,2,3]. In [19], Lerner-Morimoto-Starov-Xu studied the linearized Landau and Boltzmann equation and proved that the linearized non-cutoff Boltzmann operator with Maxwellian is exactly equal to a fractional power of the linearized Landau operator.…”

“…The non-integrability of the cross section is essential for the smoothing effect, see for example [9]. Alexandre-Morimoto-Ukai-Xu-Yang [3] highlighted the importance of regularization effects for Boltzmann equation (see also [1,4,10,11]). They studied C ∞ smoothing properties of the spatially inhomogeneous non-cutoff Boltzmann equation in [1,2,3].…”

“…3) with parameter ξ ∈ R, where the operator A = a w 0 (v, D v ) stands for the pseudo-differential operatora w 0 (v, D v )u = 1 2π R 2 e i(v−w)η a 0 ( v + w 2 , η)u(w)dwdηdefined by the Weyl quantization of the symbol a 0 (v, η) = c 0 (1 + η 2 + v 2 4 ) s with some constants c 0 > 0, 0 < s < 1. This operator corresponds to the principle part of the linear inhomogeneous Kac operator v∂ x + K on the Fourier side in the position variable.…”

Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

“…The method of the proof is the almost same as the one of Proposition 5.2 in [6]. For the self-containedness, we reproduce it.…”