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.
Mots clés : kinetic entropy, convex analysis, nonlinear approximation, moments systems, maxwellian molecules
@article{M2AN_2004__38_3_541_0,
author = {Schneider, Jacques},
title = {Entropic approximation in kinetic theory},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {541--561},
publisher = {EDP-Sciences},
volume = {38},
number = {3},
year = {2004},
doi = {10.1051/m2an:2004025},
mrnumber = {2075759},
zbl = {1084.82010},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004025/}
}
TY - JOUR AU - Schneider, Jacques TI - Entropic approximation in kinetic theory JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 541 EP - 561 VL - 38 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004025/ DO - 10.1051/m2an:2004025 LA - en ID - M2AN_2004__38_3_541_0 ER -
Schneider, Jacques. Entropic approximation in kinetic theory. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 541-561. doi : 10.1051/m2an:2004025. http://www.numdam.org/articles/10.1051/m2an:2004025/
[1] ,, and, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19 (2000) 813-830. | Zbl
[2] , On the Boltzmann equation. Arch. Rational Mech. Anal. 45 (1972) 1-34. | Zbl
[3] , and, An example of MUSCL method satisfying all the entropy inequalities. C.R. Acad Sc. Paris, Serie I 317 (1993) 619-624. | Zbl
[4] and, An entropy satisfying muscl scheme for systems of conservation laws. Numerische Math. 74 (1996) 1-34. | Zbl
[5] , I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3 (1975) 146-158. | Zbl
[6] and, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989) 321-366. | Zbl
[7] , On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407. | Zbl
[8] , Domain of definition of Levermore's five moments system. J. Stat. Phys. 93 (1998) 1143-1167. | Zbl
[9] , Maximum entropy for reduced moment problems. M3AS 10 (2000) 1001-1025. | Zbl
[10] , Some results about entropic projections, in Stochastic Analysis and Mathematical Analysis, Vol. 50, Progr. Probab., Birkhaüser, Boston, MA (2001) 59-73. | Zbl
[11] , Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021-1065. | Zbl
[12] , Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci. 10 (2000) 1121-1149. | Zbl
[13] , The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans. 47 (1965) 193-214. | Zbl
[14] and, 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. | Zbl
[15] , Fisher information bounds for Boltzmann's collision operator. J. Math. Pures Appl. 77 (1998) 821-837. | Zbl
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