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.
@article{M2AN_2004__38_3_541_0,
author = {Schneider, Jacques},
title = {Entropic approximation in kinetic theory},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {38},
year = {2004},
pages = {541-561},
doi = {10.1051/m2an:2004025},
mrnumber = {2075759},
zbl = {1084.82010},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2004__38_3_541_0}
}
Schneider, Jacques. Entropic approximation in kinetic theory. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 541-561. doi : 10.1051/m2an:2004025. http://gdmltest.u-ga.fr/item/M2AN_2004__38_3_541_0/
[1] ,, and, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19 (2000) 813-830. | Zbl 0967.76082
[2] , On the Boltzmann equation. Arch. Rational Mech. Anal. 45 (1972) 1-34. | Zbl 0245.76059
[3] , and, An example of MUSCL method satisfying all the entropy inequalities. C.R. Acad Sc. Paris, Serie I 317 (1993) 619-624. | Zbl 0779.65061
[4] and, An entropy satisfying muscl scheme for systems of conservation laws. Numerische Math. 74 (1996) 1-34. | Zbl 0860.65076
[5] , I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3 (1975) 146-158. | Zbl 0318.60013
[6] and, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989) 321-366. | Zbl 0698.45010
[7] , On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407. | Zbl 0037.13104
[8] , Domain of definition of Levermore's five moments system. J. Stat. Phys. 93 (1998) 1143-1167. | Zbl 0952.82024
[9] , Maximum entropy for reduced moment problems. M3AS 10 (2000) 1001-1025. | Zbl 1012.44005
[10] , Some results about entropic projections, in Stochastic Analysis and Mathematical Analysis, Vol. 50, Progr. Probab., Birkhaüser, Boston, MA (2001) 59-73. | Zbl 1020.60014
[11] , Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021-1065. | Zbl 1081.82619
[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 1078.82526
[13] , The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans. 47 (1965) 193-214. | Zbl 0188.21204
[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 0811.76050
[15] , Fisher information bounds for Boltzmann's collision operator. J. Math. Pures Appl. 77 (1998) 821-837. | Zbl 0918.60093