Guérin, Hélène Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilistic interpretation. (English) Zbl 1033.60088 ESAIM, Probab. Stat. 8, 36-55 (2004). Summary: Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of N. Fournier [J. Stat. Phys. 99, 725–749 (2000; Zbl 0959.82020)] on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by the author and S. Méléard [ibid. 111, 931–966 (2003; Zbl 1031.82035)], some simulations of the solution of the Landau equation will be given. This result is original and has not been obtained for the moment by analytical methods. Cited in 1 Document MSC: 60J75 Jump processes (MSC2010) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 82C40 Kinetic theory of gases in time-dependent statistical mechanics Keywords:Boltzmann equation without cutoff for a Maxwell gas; Landau equation for a Maxwell gas; nonlinear stochastic differential equations; Malliavin calculus Citations:Zbl 0959.82020; Zbl 1031.82035 PDFBibTeX XMLCite \textit{H. Guérin}, ESAIM, Probab. Stat. 8, 36--55 (2004; Zbl 1033.60088) Full Text: DOI Numdam EuDML References: [1] R. Alexandre et C. Villani , On the Landau approximation in plasma physics (in preparation). Numdam | Zbl 1044.83007 · Zbl 1044.83007 [2] A.A. Arsenev and O.E. Buryak , 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 . Zbl 0724.35090 · Zbl 0724.35090 [3] 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 ). MR 1008471 | Zbl 0706.60057 · Zbl 0706.60057 [4] 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 . Numdam | Zbl 0525.60067 · Zbl 0525.60067 [5] L. Boltzmann , 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 . JFM 04.0566.01 · JFM 04.0566.01 [6] L. Boltzmann , Lectures on gas theory . Reprinted by Dover Publications ( 1995 ). [7] P. Degon and B. Lucquin-Desreux , The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case . Math. Mod. Meth. Appl. Sci. 2 ( 1992 ) 167 - 182 . Zbl 0755.35091 · Zbl 0755.35091 [8] L. Desvillettes , On asymptotics of the Boltzmann equation when the collisions become grazing . Transp. Theory Statist. Phys. 21 ( 1992 ) 259 - 276 . MR 1165528 | Zbl 0769.76059 · Zbl 0769.76059 [9] L. Desvillettes , C. Graham and S. Méléard , Probabilistic interpretation and numerical approximation of a Kac equation without cutoff . Stochastic Process. Appl. 84 ( 1999 ) 115 - 135 . MR 1720101 | Zbl 1009.76081 · Zbl 1009.76081 [10] N. Fournier , Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach . Ann. Appl. Probab. 10 ( 2000 ) 434 - 462 . Article | MR 1768239 | Zbl 1056.60052 · Zbl 1056.60052 [11] N. Fournier and S. Méléard , A stochastic particle numerical method for 3D Boltzmann equations without cutoff . Math. Comput. 70 ( 2002 ) 583 - 604 . MR 1885616 | Zbl 0990.60085 · Zbl 0990.60085 [12] T. Goudon , 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 . Zbl 0882.76079 · Zbl 0882.76079 [13] C. Graham and S. Méléard , 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 . MR 1711273 | Zbl 0953.60057 · Zbl 0953.60057 [14] H. Guérin , Solving Landau equation for some soft potentials through a probabilistic approach . Ann. Appl. Probab. 13 ( 2003 ) 515 - 539 . Article | MR 1970275 | Zbl 1070.82027 · Zbl 1070.82027 [15] H. Guérin , Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach . Stochastic Process. Appl. 101 ( 2002 ) 303 - 325 . MR 1931271 | Zbl 1075.60058 · Zbl 1075.60058 [16] H. Guérin and S. Méléard , Convergence from Boltzmann to Landau processes with soft potential and particle approximation . J. Statist. Phys. 111 ( 2003 ) 931 - 966 . MR 1972130 | Zbl 1031.82035 · Zbl 1031.82035 [17] 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 ). MR 1042343 | Zbl 0696.60095 · Zbl 0696.60095 [18] J. Jacod and A.N. Shiryaev , Limit theorems for stochastic processes . Springer ( 1987 ). MR 959133 | Zbl 0635.60021 · Zbl 0635.60021 [19] E.M. Lifchitz and L.P. Pitaevskii , Physical kinetics - Course in theorical physics . Pergamon, Oxford 10 ( 1981 ). [20] D. Nualart , The Malliavin calculus and related topics . Springer-Verlag ( 1995 ). MR 1344217 | Zbl 0837.60050 · Zbl 0837.60050 [21] H. Tanaka , Probabilistic treatment of the Boltzmann equation of Maxwellian molecules . Z. Wahrsch. Verw. Geb. 46 ( 1978 ) 67 - 105 . MR 512334 | Zbl 0389.60079 · Zbl 0389.60079 [22] C. Villani , On the spatially homogeneous Landau equation for Maxwellian molecules . Math. Meth. Mod. Appl. Sci. 8 ( 1998 ) 957 - 983 . MR 1646502 | Zbl 0957.82029 · Zbl 0957.82029 [23] C. Villani , On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations . Arch. Rational Mech. Anal. 143 ( 1998 ) 273 - 307 . MR 1650006 | Zbl 0912.45011 · Zbl 0912.45011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.