A new approach to velocity averaging lemmas in Besov spaces

Abstract

We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results, which display, in some cases, a gain of one full derivative. Moreover, the study of dispersion allows to treat the case of L r x L p v integrability with r ≤ p. We also establish results on the control of concentrations in the degenerate L 1x,v case, which is fundamental in the study of the hydrodynamic limit of the Boltzmann equation.

Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

“…For readers we would also point out another two kinds of applications of the Besov space to the Boltzmann equation. In fact, in Arsénio-Masmoudi [3], a new approach to velocity averaging lemmas in some Besov spaces is developed basing on the dispersive property of the kinetic transport equation, and in Sohinger-Strain [23], the optimal time decay rates in the whole space are investigated in the framework of [13] under the additional assumption that initial data belongs to a negative power Besov space B s 2,∞ for some s < 0 with respect to x variable. In what follows let us a little detailedly recall some related works as far as the choice of different function spaces for well-posedness of the Boltzmann equation near Maxwellians is concerned.…”

The Boltzmann equation, Besov spaces, and optimal time decay rates inRxn

“…We note that the method of [20] does not obtain the optimal L 1 decay rates. Related recent results, concerning Besov spaces and the Boltzmann equation, which appeared after this work was complete can be found in [2] and [7].…”

“…To overcome these types of difficulties, it is of great interest to prove decay rates in an L 2 (R n x ) based space which is larger than L 1 (R n x ). In this paper we accomplish this task for the non-cutoff Boltzmann equation in the homogeneous Besov-Lipschitz spaceḂ −̺,∞ 2 ⊃ L p (R n x ) where for p ∈ [1,2] we use ̺ = n p − n 2 . We remark that these spaces can be thought of as a physical choice since it is possible to obtain the L 1 (R n x ) embedding.…”

On propagation of higher space regularity for nonlinear Vlasov equations

“…There exist many versions of these, involving several different assumptions on the functional spaces, on the number of derivatives in v or in x in the source etc., see e.g. [21,52,32,13,42] and [41,4,3] for more recent advances. The closest (to Theorem (6.1)) avatar of averaging lemmas is arguably the following result of Perthame and Souganidis [52].…”