# Entropic approximation in kinetic theory

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 38, Issue: 3, page 541-561
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topSchneider, Jacques. "Entropic approximation in kinetic theory." ESAIM: Mathematical Modelling and Numerical Analysis 38.3 (2010): 541-561. <http://eudml.org/doc/194227>.

@article{Schneider2010,

abstract = {
Approximation theory in the context of probability density
function turns out to go beyond the classical idea of orthogonal
projection. Special tools have to be designed so as to respect the
nonnegativity of the approximate function. We develop here and
justify from the theoretical point of view an approximation
procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an
entropy minimization principle under moment constraints. We prove
in particular a global existence theorem for such an approximation
and derive as a by-product a necessary and sufficient condition
for the so-called problem of moment realizability.
Applications of the above result are given in kinetic theory:
first in the context of Levermore's approach and second to design
generalized BGK models for Maxwellian molecules.
},

author = {Schneider, Jacques},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Kinetic entropy; convex analysis; nonlinear approximation; moments
systems; Maxwellian molecules.; Entropic approximation; moment closure; kinetic theory},

language = {eng},

month = {3},

number = {3},

pages = {541-561},

publisher = {EDP Sciences},

title = {Entropic approximation in kinetic theory},

url = {http://eudml.org/doc/194227},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Schneider, Jacques

TI - Entropic approximation in kinetic theory

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 3

SP - 541

EP - 561

AB -
Approximation theory in the context of probability density
function turns out to go beyond the classical idea of orthogonal
projection. Special tools have to be designed so as to respect the
nonnegativity of the approximate function. We develop here and
justify from the theoretical point of view an approximation
procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an
entropy minimization principle under moment constraints. We prove
in particular a global existence theorem for such an approximation
and derive as a by-product a necessary and sufficient condition
for the so-called problem of moment realizability.
Applications of the above result are given in kinetic theory:
first in the context of Levermore's approach and second to design
generalized BGK models for Maxwellian molecules.

LA - eng

KW - Kinetic entropy; convex analysis; nonlinear approximation; moments
systems; Maxwellian molecules.; Entropic approximation; moment closure; kinetic theory

UR - http://eudml.org/doc/194227

ER -

## References

top- P. Andries, P. Le Tallec, J.P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids19 (2000) 813–830.
- L. Arkeryd, On the Boltzmann equation. Arch. Rational Mech. Anal.45 (1972) 1–34.
- F. Bouchut, C. Bourdarias and B. Perthame, An example of MUSCL method satisfying all the entropy inequalities. C.R. Acad Sc. Paris, Serie I317 (1993) 619–624.
- F. Coquel and P. LeFloch, An entropy satisfying muscl scheme for systems of conservation laws. Numerische Math.74 (1996) 1–34.
- I. Csiszár, I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab.3 (1975) 146–158.
- R. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math.130 (1989) 321–366.
- H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math.2 (1949) 331–407.
- M. Junk, Domain of definition of Levermore's five moments system. J. Stat. Phys.93 (1998) 1143-1167.
- M. Junk, Maximum entropy for reduced moment problems. M3AS10 (2000) 1001–1025.
- C. Léonard, Some results about entropic projections, in Stochastic Analysis and Mathematical Analysis, Vol. 50, Progr. Probab., Birkhaüser, Boston, MA (2001) 59–73.
- C.D. Levermore, Moment closure hierarchies for kinetic theories. J. Stat. Phys.83 (1996) 1021–1065.
- L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci. 10 (2000) 1121–1149.
- A.J. Povzner, The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans. 47 (1965) 193–214.
- F. Rogier and J. Schneider, A Direct Method for Solving the Boltzmann Equation. Proc. Colloque Euromech n0287 Discrete Models in Fluid Dynamics, Transport Theory Statist. Phys. 23 (1994) 1–3.
- C. Villani, Fisher information bounds for Boltzmann's collision operator. J. Math. Pures Appl. 77 (1998) 821–837.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.