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Entropic approximation in kinetic theory

Published online by Cambridge University Press:  15 June 2004

Jacques Schneider
Jacques Schneider*
Affiliation:
Laboratoire Modélisation Numérique et Couplages, Université de Toulon et du Var, 83162 La Valette Cedex, France. [email protected].
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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.

Keywords

Kinetic entropyconvex analysisnonlinear approximationmomentssystemsMaxwellian molecules.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 3 , May 2004 , pp. 541 - 561
Copyright
© EDP Sciences, SMAI, 2004

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References

Andries, P., Le Tallec, P., Perlat, J.P. and Perthame, B., The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19 (2000) 813–830. CrossRef
Arkeryd, L., 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 I 317 (1993) 619–624.
Coquel, F. and LeFloch, P., An entropy satisfying muscl scheme for systems of conservation laws. Numerische Math. 74 (1996) 1–34. CrossRef
Csiszár, I., I-divergence geometry of probability distributions and minimization problems Sanov property. Ann. Probab. 3 (1975) 146–158.
DiPerna, R. and Lions, P.-L., On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. 130 (1989) 321–366. CrossRef
Grad, H., On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331–407. CrossRef
Junk, M., Domain of definition of Levermore's five moments system. J. Stat. Phys. 93 (1998) 1143-1167. CrossRef
Junk, M., Maximum entropy for reduced moment problems. M3AS 10 (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.
Levermore, C.D., Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021–1065. CrossRef
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.