### 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· ∇_{x}f = g belong to L_{x}
^{r} L_{v}
^{r'}, where 2n/(n+1) < r ≤ 2 and n ≥ 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 ≥ 2, which strongly indicates that velocity averages should almost belong to W_{x}
^{1/2,2n/(n+1)} whenever the densities belong to L_{x}
^{2n/(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.

Original language | English (US) |
---|---|

Pages (from-to) | 333-388 |

Number of pages | 56 |

Journal | Analysis and PDE |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2018 |

### Fingerprint

### Keywords

- Kinetic theory
- Kinetic transport equation
- Velocity averaging lemmas

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

*Analysis and PDE*,

*12*(2), 333-388. https://doi.org/10.2140/apde.2019.12.333

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

Research output: Contribution to journal › Article

*Analysis and PDE*, vol. 12, no. 2, pp. 333-388. https://doi.org/10.2140/apde.2019.12.333

}

TY - JOUR

T1 - Maximal gain of regularity in velocity averaging Lemmas

AU - Arsénio, Diogo

AU - Masmoudi, Nader

PY - 2018/1/1

Y1 - 2018/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=85057983260&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057983260&partnerID=8YFLogxK

U2 - 10.2140/apde.2019.12.333

DO - 10.2140/apde.2019.12.333

M3 - Article

VL - 12

SP - 333

EP - 388

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 2

ER -