Vol. 12, No. 2, 2019

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Maximal gain of regularity in velocity averaging lemmas

Diogo Arsénio and Nader Masmoudi

Vol. 12 (2019), No. 2, 333–388
DOI: 10.2140/apde.2019.12.333

We investigate new settings of velocity averaging lemmas in kinetic theory where a maximal gain of half a derivative is obtained. Specifically, we show that if the densities f and g in the transport equation v xf = g belong to LxrLvr , where 2n(n + 1) < r 2 and n 1 is the dimension, then the velocity averages belong to Hx12.

We further explore the setting where the densities belong to Lx43Lv2 and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to Wxn(4(n1)),43 in this case, in any dimension n 2, which strongly indicates that velocity averages should almost belong to Wx12,2n(n+1) whenever the densities belong to Lx2n(n+1)Lv2.

These results and their proofs bear a strong resemblance to the famous and notoriously difficult problems of boundedness of Bochner–Riesz multipliers and Fourier restriction operators, and to smoothing conjectures for Schrödinger and wave equations, which suggests interesting links between kinetic theory, dispersive equations and harmonic analysis.

velocity averaging lemmas, kinetic theory, kinetic transport equation
Mathematical Subject Classification 2010
Primary: 35B65
Secondary: 42B37, 82C40
Received: 16 January 2017
Revised: 4 December 2017
Accepted: 14 March 2018
Published: 14 August 2018
Diogo Arsénio
Institut de Mathématiques de Jussieu–Paris Rive Gauche
Université Paris Diderot
New York University Abu Dhabi
Abu Dhabi
United Arab Emirates
Nader Masmoudi
Courant Institute of Mathematical Sciences
New York University
New York, NY
United States
New York University Abu Dhabi
Abu Dhabi
United Arab Emirates