Regularity of the moments of the solution of a Transport Equation

François Golse; Pierre Louis Lions; Benoît Perthame; Rémi Sentis

doi:10.1016/0022-1236(88)90051-1

Regularity of the moments of the solution of a Transport Equation

François Golse
, Pierre Louis Lions
, Benoît Perthame
, Rémi Sentis

Université Paris Dauphine
Centre d'Études de Limeil Valenton
Centre d'Etudes de Limeil-Valenton
Centre de Mathématiques Appliquées
PSL research University & IPSL

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

355 Citations (Scopus)

Abstract

Let u = u(x, v) satisfy the Transport Equation u+v·∂ _xu=f, x∈R^N, v∈R^Nwhere f belongs to some space of type L^p(dx ⊗ dμ(v)) (where μ is a positive bounded measure on R^N). We study the resulting regularity of the moment ∝ u(x, v) dμ(v) (in terms of fractional Sobolev spaces, for example). Counter-examples are given in order to test the optimality of our results.

Original language	English
Pages (from-to)	110-125
Number of pages	16
Journal	Journal of Functional Analysis
Volume	76
Issue number	1
DOIs	https://doi.org/10.1016/0022-1236(88)90051-1
Publication status	Published - 1 Jan 1988

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10.1016/0022-1236(88)90051-1

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title = "Regularity of the moments of the solution of a Transport Equation",

abstract = "Let u = u(x, v) satisfy the Transport Equation u+v·∂ xu=f, x∈RN, v∈RNwhere f belongs to some space of type Lp(dx ⊗ dμ(v)) (where μ is a positive bounded measure on RN). We study the resulting regularity of the moment ∝ u(x, v) dμ(v) (in terms of fractional Sobolev spaces, for example). Counter-examples are given in order to test the optimality of our results.",

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AU - Sentis, Rémi

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N2 - Let u = u(x, v) satisfy the Transport Equation u+v·∂ xu=f, x∈RN, v∈RNwhere f belongs to some space of type Lp(dx ⊗ dμ(v)) (where μ is a positive bounded measure on RN). We study the resulting regularity of the moment ∝ u(x, v) dμ(v) (in terms of fractional Sobolev spaces, for example). Counter-examples are given in order to test the optimality of our results.

AB - Let u = u(x, v) satisfy the Transport Equation u+v·∂ xu=f, x∈RN, v∈RNwhere f belongs to some space of type Lp(dx ⊗ dμ(v)) (where μ is a positive bounded measure on RN). We study the resulting regularity of the moment ∝ u(x, v) dμ(v) (in terms of fractional Sobolev spaces, for example). Counter-examples are given in order to test the optimality of our results.

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Regularity of the moments of the solution of a Transport Equation

Abstract

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