A note on the entropy production of the radiative transfer equation

E. Gabetta; P. A. Markowich; A. Unterreiter

doi:10.1016/S0893-9659(99)00044-0

A note on the entropy production of the radiative transfer equation

E. Gabetta^*, P. A. Markowich, A. Unterreiter

^*Corresponding author for this work

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

4 Scopus citations

Abstract

In recent years, the entropy approach to the asymptotic (large-time) analysis of homogeneous kinetic models has led to remarkable new proofs of convex-type (e.g., logarithmic) Sobolev inequalities. The crucial point of this method lies in computing the entropy e_φ(t), the entropy production I_φ(t), and the entropy production rate I_φ(t) of the kinetic model. I_φ(t) has to be estimated in terms of I_φ(t). Then e_φ(t) is estimated in terms of I_φ(t). We apply this approach to the (explicitly solvable) homogeneous radiative transfer equation obtaining a Jensen-type inequality involving a convex function as corresponding "Sobolev inequality". All the computations are highly transparent and serve to highlight and ultimately clarify the approach.

Original language	English (US)
Pages (from-to)	111-116
Number of pages	6
Journal	Applied Mathematics Letters
Volume	12
Issue number	4
DOIs	https://doi.org/10.1016/S0893-9659(99)00044-0
State	Published - May 1999
Externally published	Yes

Keywords

Convex Sobolev inequality
Entropy
Entropy production
Radiative transfer

ASJC Scopus subject areas

Applied Mathematics

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10.1016/S0893-9659(99)00044-0

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title = "A note on the entropy production of the radiative transfer equation",

abstract = "In recent years, the entropy approach to the asymptotic (large-time) analysis of homogeneous kinetic models has led to remarkable new proofs of convex-type (e.g., logarithmic) Sobolev inequalities. The crucial point of this method lies in computing the entropy eφ(t), the entropy production Iφ(t), and the entropy production rate Iφ(t) of the kinetic model. Iφ(t) has to be estimated in terms of Iφ(t). Then eφ(t) is estimated in terms of Iφ(t). We apply this approach to the (explicitly solvable) homogeneous radiative transfer equation obtaining a Jensen-type inequality involving a convex function as corresponding {"}Sobolev inequality{"}. All the computations are highly transparent and serve to highlight and ultimately clarify the approach.",

keywords = "Convex Sobolev inequality, Entropy, Entropy production, Radiative transfer",

author = "E. Gabetta and Markowich, {P. A.} and A. Unterreiter",

year = "1999",

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doi = "10.1016/S0893-9659(99)00044-0",

language = "English (US)",

volume = "12",

pages = "111--116",

journal = "Applied Mathematics Letters",

issn = "0893-9659",

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TY - JOUR

T1 - A note on the entropy production of the radiative transfer equation

AU - Gabetta, E.

AU - Markowich, P. A.

AU - Unterreiter, A.

PY - 1999/5

Y1 - 1999/5

N2 - In recent years, the entropy approach to the asymptotic (large-time) analysis of homogeneous kinetic models has led to remarkable new proofs of convex-type (e.g., logarithmic) Sobolev inequalities. The crucial point of this method lies in computing the entropy eφ(t), the entropy production Iφ(t), and the entropy production rate Iφ(t) of the kinetic model. Iφ(t) has to be estimated in terms of Iφ(t). Then eφ(t) is estimated in terms of Iφ(t). We apply this approach to the (explicitly solvable) homogeneous radiative transfer equation obtaining a Jensen-type inequality involving a convex function as corresponding "Sobolev inequality". All the computations are highly transparent and serve to highlight and ultimately clarify the approach.

AB - In recent years, the entropy approach to the asymptotic (large-time) analysis of homogeneous kinetic models has led to remarkable new proofs of convex-type (e.g., logarithmic) Sobolev inequalities. The crucial point of this method lies in computing the entropy eφ(t), the entropy production Iφ(t), and the entropy production rate Iφ(t) of the kinetic model. Iφ(t) has to be estimated in terms of Iφ(t). Then eφ(t) is estimated in terms of Iφ(t). We apply this approach to the (explicitly solvable) homogeneous radiative transfer equation obtaining a Jensen-type inequality involving a convex function as corresponding "Sobolev inequality". All the computations are highly transparent and serve to highlight and ultimately clarify the approach.

KW - Convex Sobolev inequality

KW - Entropy

KW - Entropy production

KW - Radiative transfer

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U2 - 10.1016/S0893-9659(99)00044-0

DO - 10.1016/S0893-9659(99)00044-0

M3 - Article

AN - SCOPUS:0041857837

SN - 0893-9659

VL - 12

SP - 111

EP - 116

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

IS - 4

ER -

A note on the entropy production of the radiative transfer equation

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Keywords

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