Entropy based moment closure for kinetic equations: Riemann problem and invariant regions
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- defined up
- solution does
- ?u ?
- r?1
- system
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- corresponding homogeneous
- based moment closure
- corresponding eigenvectors
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English
Entropy-based moment closure for kinetic equations:
Riemann problem and invariant regions
Jean-Fran¸coisCoulombeland ThierryGoudon
CNRS & Universit´ Lille 1, Laboratoire Paul Painlev´, UMR CNRS 8524
Cit´ scientifique, 59655 VILLENEUVE D’ASCQ Cedex, France
and Team SIMPAF, INRIA Futurs
E-mails:jfcoulom@math.univ-lille1.fr, thierry.goudon@math.univ-lille1.fr
November8,2005
Abstract
We study a nonlinear hyperbolic system of balance laws that arises from an
entropybased moment closure of a kinetic equation.We show that the corresponding homogeneous
Riemann problem can be solved without smallness assumption, and we exhibit invariant
regions.
AMS subject classification:82C40, 35L60 35L67
1
Introduction
This paper is devoted to the analysis of the following PDEs system
∂tρ+∂xJ= 0,
εJ(1)
2
ε ∂tJ+∂xρ ψ=−J ,
ρ
where the unknown are the densityρ, and the currentJ, whileεis a positive scaling parameter.
The functionψthat appears in (1) is defined in the following way:
ψ: (−1,+1)−→]0,+∞[
′′
F(2)
2′ −1−1
u7−→u+G G(u) =G(u),
F
where we have let
′
sinh(β) 1F(β)
∀β∈R,F(β) :=,G(β) := coth(β)−=.(3)
β βF(β)
∞ −1
We note thatGis aCdiffeomorphism fromRonto (−1,1), so the use of the inverseGis
legitimate. Forfuture purposes, it is convenient to remark that
Z
+1
βv
F(β) =e dµ(v),
−1
where, here and below,dµstands for the normalized Lebesgue measure on (−1,+1). Itis also
worth noting thatF, andψare even functions, whileGis an odd function.We will show below
thatψis strictly convex.The following relations will be often used throughout the paper:
1
′ ′
F(0) = 1,G(0) = 0, ψ(0) =G(0) =, ψ(0) = 0.
3
1
