Maximal gain of regularity in velocity averaging lemmas

Arsénio, Diogo; Masmoudi, Nader

doi:10.2140/apde.2019.12.333

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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

Abstract

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 $v \cdot \nabla_{x} f = g$ belong to $L_{x}^{r} L_{v}^{r^{'}}$ , where $2 n ∕ (n + 1) < r \leq 2$ and $n \geq 1$ is the dimension, then the velocity averages belong to $H_{x}^{1 ∕ 2}$ .

We further explore the setting where the densities belong to $L_{x}^{4 ∕ 3} L_{v}^{2}$ and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to $W_{x}^{n ∕ (4 (n - 1)), 4 ∕ 3}$ in this case, in any dimension $n \geq 2$ , which strongly indicates that velocity averages should almost belong to $W_{x}^{1 ∕ 2, 2 n ∕ (n + 1)}$ whenever the densities belong to $L_{x}^{2 n ∕ (n + 1)} L_{v}^{2}$ .

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.

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Keywords

velocity averaging lemmas, kinetic theory, kinetic transport equation

Mathematical Subject Classification 2010

Primary: 35B65

Secondary: 42B37, 82C40

Milestones

Received: 16 January 2017

Revised: 4 December 2017

Accepted: 14 March 2018

Published: 14 August 2018

Authors

Diogo Arsénio
Institut de Mathématiques de Jussieu–Paris Rive Gauche Université Paris Diderot Paris France	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

Vol. 12, No. 2, 2019