Recent progress in velocity averaging

A classical result in kinetic theory establishes that if f(x, v) and v ·∇xf(x, v) both belong to L2 (Rx × Rv ), then ∫ K fdv ∈ H 1 2 (Rx), for any compact set K ⊂ Rv . Such regularity statements are known as velocity averaging lemmas and have important implications in the analysis of kinetic equations. It was asked in [2] whether other settings of velocity averaging could produce a similar maximal gain of regularity of half a derivative. This question, motivated by an earlier work of Pierre-Emmanuel Jabin and Luis Vega [17] on the subject, turns out to be surprisingly rich and difficult, and it is, for the moment, far from being fully understood. In this article, after recalling some classical results in the field, we survey the recent developments from [2], where new settings of velocity averaging lemmas were investigated. We also formulate a few conjectures, mainly derived from a dimensional analysis and by analogy with known results, thus delimiting the possibilities for other new settings of velocity averaging. Velocity averaging lemmas concern the regularity theory of solutions to the kinetic transport equation (∂t + v · ∇x) f(t, x, v) = g(t, x, v), (0.1) where (t, x, v) ∈ R× R × R, or its stationary counterpart v · ∇xf(x, v) = g(x, v), (0.2) where (x, v) ∈ R × R, with n ≥ 1. Variants of the above equations are also relevant. Indeed, different spatial and velocity domains, as well as non-linear velocity fields (consider the relativistic case), are sometimes studied. Nevertheless, for the sake of simplicity, we will mainly focus on the Euclidean stationary setting (0.2), which, we believe, captures the essential features of kinetic transport (at least as far as velocity averaging is concerned). We will nevertheless make brief references to the non-stationary case (0.1) in Section 2 below without expanding on the subject. After describing a modern viewpoint on classical velocity averaging in Sections 1, 2 and 3, we intend to survey, in the remainder of this text, recent theorems and conjectures on the subject. Most results presented here are taken directly from [2] and we will systematically sketch proofs of the asserted results. Nevertheless, we refer the reader to [2] and the references therein for further details and complete justifications. 1. Classical velocity averaging, the Hilbertian case The classical Hilbertian case of velocity averaging is contained in the following result. It is the starting point of the regularity theory of kinetic transport equations and has been established first in [12]. Note, however, that such regularity results had already been suggested in weaker forms in [1, 13].