Compactness in kinetic transport equations and hypoellipticity

Diogo Arsénio; Laure Saint-Raymond

doi:10.1016/j.jfa.2011.07.020

Compactness in kinetic transport equations and hypoellipticity

Diogo Arsénio, Laure Saint-Raymond

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation. ν·∇_xf_λ=(1-Δ_x)^β2(1-Δν)^α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Arsénio and Saint-Raymond, in preparation [4]).

Original language	English (US)
Pages (from-to)	3044-3098
Number of pages	55
Journal	Journal of Functional Analysis
Volume	261
Issue number	10
DOIs	https://doi.org/10.1016/j.jfa.2011.07.020
State	Published - Nov 15 2011

Keywords

Averaging lemma
Hypoellipticity
Kinetic theory
Microlocal decomposition
Transport equation

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2011.07.020

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title = "Compactness in kinetic transport equations and hypoellipticity",

abstract = "We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1 satisfying some appropriate transport relation. ν·∇xfλ=(1-Δx )β2(1-Δν)α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Ars{\'e}nio and Saint-Raymond, in preparation [4]).",

keywords = "Averaging lemma, Hypoellipticity, Kinetic theory, Microlocal decomposition, Transport equation",

author = "Diogo Ars{\'e}nio and Laure Saint-Raymond",

note = "Funding Information: * Corresponding author. E-mail addresses: [email protected] (D. Ars{\'e}nio), [email protected] (L. Saint-Raymond). 1 D. Ars{\'e}nio would like to acknowledge the support from the foundation Sciences Math{\'e}matiques de Paris during the genesis of this work.",

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doi = "10.1016/j.jfa.2011.07.020",

language = "English (US)",

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T1 - Compactness in kinetic transport equations and hypoellipticity

AU - Arsénio, Diogo

AU - Saint-Raymond, Laure

N1 - Funding Information: * Corresponding author. E-mail addresses: [email protected] (D. Arsénio), [email protected] (L. Saint-Raymond). 1 D. Arsénio would like to acknowledge the support from the foundation Sciences Mathématiques de Paris during the genesis of this work.

PY - 2011/11/15

Y1 - 2011/11/15

N2 - We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1 satisfying some appropriate transport relation. ν·∇xfλ=(1-Δx )β2(1-Δν)α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Arsénio and Saint-Raymond, in preparation [4]).

AB - We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1 satisfying some appropriate transport relation. ν·∇xfλ=(1-Δx )β2(1-Δν)α2gλ may be inferred solely from additional integrability and compactness with respect to v. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field (Arsénio and Saint-Raymond, in preparation [4]).

KW - Averaging lemma

KW - Hypoellipticity

KW - Kinetic theory

KW - Microlocal decomposition

KW - Transport equation

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DO - 10.1016/j.jfa.2011.07.020

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JF - Journal of Functional Analysis

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Compactness in kinetic transport equations and hypoellipticity

Abstract

Keywords

ASJC Scopus subject areas

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