TY - JOUR
T1 - Maximal gain of regularity in velocity averaging Lemmas
AU - Arsénio, Diogo
AU - Masmoudi, Nader
N1 - Funding Information:
We would like to thank Pierre-Emmanuel Jabin for sharing his insight on the subject with us. Masmoudi was partially supported by NSF grant no. DMS-1716466. MSC2010: primary 35B65; secondary 42B37, 82C40. Keywords: velocity averaging lemmas, kinetic theory, kinetic transport equation.
Publisher Copyright:
© 2019 Mathematical Sciences Publishers.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Kinetic theory
KW - Kinetic transport equation
KW - Velocity averaging lemmas
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U2 - 10.2140/apde.2019.12.333
DO - 10.2140/apde.2019.12.333
M3 - Article
AN - SCOPUS:85057983260
SN - 2157-5045
VL - 12
SP - 333
EP - 388
JO - Analysis and PDE
JF - Analysis and PDE
IS - 2
ER -