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Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

Published online by Cambridge University Press:  15 September 2004

Hélène Guérin
Hélène Guérin*
Affiliation:
Université Paris 10, UFR SEGMI, Modal'X, 200 avenue de la République, 92000 Nanterre, France; hguerin@ccr.jussieu.fr.
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Abstract


Using probabilistic tools, this work states a pointwise convergence offunction solutions of the 2-dimensional Boltzmann equation to the functionsolution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results ofFournier (2000) on the Malliavin calculus for the Boltzmannequation. Moreover, using the particle system introduced by Guérin andMéléard (2003), some simulations of the solution of the Landau equation will be given. This result isoriginal and has not been obtained for the moment by analytical methods.

Keywords

Boltzmann equation without cutoff for a Maxwell gasLandau equation for a Maxwell gasnonlinear stochastic differential equationsMalliavin calculus.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 36 - 55
Copyright
© EDP Sciences, SMAI, 2004

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References

R. Alexandre et C. Villani, On the Landau approximation in plasma physics (in preparation).
Arsenev, A.A. and Buryak, O.E., On the connection between a solution of the Boltzmann equation and a solution of the Landau–Fokker–Planck equation. Math. USSR Sbornik 69 (1991) 465-478. CrossRef
K. Bichteler, J.B. Gravelreaux and J. Jacod, Malliavin calculus for processes with jumps, Theory and Application of stochastic Processes. Gordon and Breach, New York (1987).
K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts, existence d'une densité pour le cas unidimensionel, in Séminaire de probabilités XVII. Springer, Berlin, Lecture Notes in Math. 986 (1983) 132-157. CrossRef
Boltzmann, L., Weitere studien über das wärme gleichgenicht unfer gasmoläkuler. Sitzungsber. Akad. Wiss. 66 (1872) 275-370. Translation: Further Studies on the thermal equilibrium of gas molecules, S.G. Brush Ed., Pergamon, Oxford, Kinetic Theory 2 (1966) 88-174.
L. Boltzmann, Lectures on gas theory. Reprinted by Dover Publications (1995).
Degon, P. and Lucquin–Desreux, B., The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. Appl. Sci. 2 (1992) 167-182. CrossRef
Desvillettes, L., On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Theory Statist. Phys. 21 (1992) 259-276. CrossRef
Desvillettes, L., Graham, C. and Méléard, S., Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 (1999) 115-135. CrossRef
Fournier, N., Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach. Ann. Appl. Probab. 10 (2000) 434-462.
Fournier, N. and Méléard, S., A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput. 70 (2002) 583-604.
Goudon, T., Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau–Fokker–Planck : influence des collisions rasantes. C. R. Acad. Sci. Paris 324 (1997) 265-270. CrossRef
Graham, C. and Méléard, S., Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations. Comm. Math. Phys. 205 (1999) 551-569. CrossRef
Guérin, H., Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab. 13 (2003) 515-539. CrossRef
Guérin, H., Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. Stochastic Process. Appl. 101 (2002) 303-325. CrossRef
Guérin, H. and Méléard, S., Convergence from Boltzmann to Landau processes with soft potential and particle approximation. J. Statist. Phys. 111 (2003) 931-966. CrossRef
J. Horowitz and R.L. Karandikar, Martingale problem associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989, E. Cinlar, K.L. Chung and R.K. Getoor Eds., Birkhäuser, Boston (1990).
J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer (1987).
E.M. Lifchitz and L.P. Pitaevskii, Physical kinetics – Course in theorical physics. Pergamon, Oxford 10 (1981).
D. Nualart, The Malliavin calculus and related topics. Springer-Verlag (1995).
Tanaka, H., Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Geb. 46 (1978) 67-105. CrossRef
Villani, C., On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Meth. Mod. Appl. Sci. 8 (1998) 957-983. CrossRef
Villani, C., On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 (1998) 273-307. CrossRef