Maximal gain of regularity in velocity averaging Lemmas

Diogo Arsénio, Nader Masmoudi

Research output: Contribution to journalArticle

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· ∇xf = g belong to Lx r Lv r', where 2n/(n+1) < r ≤ 2 and n ≥ 1 is the dimension, then the velocity averages belong to Hx 1/2. We further explore the setting where the densities belong to Lx 4/3 Lv 2 and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to Wx n/(4(n-1),4/3 in this case, in any dimension n ≥ 2, which strongly indicates that velocity averages should almost belong to Wx 1/2,2n/(n+1) whenever the densities belong to Lx 2n/(n+1) Lv 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.

Original languageEnglish (US)
Pages (from-to)333-388
Number of pages56
JournalAnalysis and PDE
Volume12
Issue number2
DOIs
StatePublished - Jan 1 2018

Fingerprint

Averaging Lemmas
Regularity
Kinetic theory
Kinetic Theory
Vega
Dispersive Equations
Harmonic analysis
Harmonic Analysis
Wave equations
Transport Equation
Multiplier
Smoothing
Wave equation
Boundedness
Restriction
Derivatives
Derivative
Operator

Keywords

  • Kinetic theory
  • Kinetic transport equation
  • Velocity averaging lemmas

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

Maximal gain of regularity in velocity averaging Lemmas. / Arsénio, Diogo; Masmoudi, Nader.

In: Analysis and PDE, Vol. 12, No. 2, 01.01.2018, p. 333-388.

Research output: Contribution to journalArticle

Arsénio, Diogo ; Masmoudi, Nader. / Maximal gain of regularity in velocity averaging Lemmas. In: Analysis and PDE. 2018 ; Vol. 12, No. 2. pp. 333-388.
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