# American Institute of Mathematical Sciences

June  2019, 12(3): 483-505. doi: 10.3934/krm.2019020

## A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff

 Sorbonne Université, CNRS, LPSM, UMR 8001, F-75005 Paris, France

Received  March 2017 Revised  February 2018 Published  February 2019

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\infty)\times {\mathbb{R}}^3$ random variable $(M_t,V_t)$ such that $\mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express $f_t$ in terms of $f_0$, we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [18] and of its interpretation by McKean [10,11].

Citation: Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020
##### References:
 [1] E. A. Carlen, M. C. Carvalho and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion, Comm. Pure Appl. Math., 53 (2000), 370-397.  doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0. [2] E. A. Carlen, M. C. Carvalho and E. Gabetta, On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation, J. Funct. Anal., 220 (2005), 362-387.  doi: 10.1016/j.jfa.2004.06.011. [3] E. A. Carlen and F. Salvarani, On the optimal choice of coefficients in a truncated Wild sum and approximate solutions for the Kac equation, J. Statist. Phys., 109 (2002), 261-277.  doi: 10.1023/A:1019943813176. [4] C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [5] E. Dolera, E. Gabetta and E. Regazzini, Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem, Ann. Appl. Probab., 19 (2009), 186-209.  doi: 10.1214/08-AAP538. [6] N. Fournier and J. S. Giet, Exact simulation of nonlinear coagulation processes, Monte Carlo Methods Appl., 10 (2004), 95-106.  doi: 10.1515/156939604777303253. [7] N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff, Math. Comp., 71 (2002), 583-604.  doi: 10.1090/S0025-5718-01-01339-4. [8] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. Ⅲ, University of California Press, 1956,171–197. [9] X. Lu and C. Mouhot, On measure solutions of the boltzmann equation part Ⅰ: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021. [10] H. P. McKean, Speed of approach to equilibrium for Kacs caricature of a Maxwellian gas, Arch. Ration. Mech. Anal., 21 (1966), 343-367.  doi: 10.1007/BF00264463. [11] H. P. McKean, An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382.  doi: 10.1016/S0021-9800(67)80035-8. [12] S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0. [13] L. Pareschi and G. Russo, Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput., 23 (2001), 1253-1273.  doi: 10.1137/S1064827500375916. [14] A. J. Povzner, About the Boltzmann equation in kinetic gas theory, Mat. Sborn, 58 (1962), 65-86. [15] H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. und Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689. [16] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0. [17] B. Wennberg, An example of nonuniqueness for solutions to the homogeneous Boltzmann equation, J. Statist. Phys., 95 (1999), 469-477.  doi: 10.1023/A:1004546031908. [18] E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc., 47 (1951), 602-609.  doi: 10.1017/S0305004100026992.

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##### References:
 [1] E. A. Carlen, M. C. Carvalho and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion, Comm. Pure Appl. Math., 53 (2000), 370-397.  doi: 10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0. [2] E. A. Carlen, M. C. Carvalho and E. Gabetta, On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation, J. Funct. Anal., 220 (2005), 362-387.  doi: 10.1016/j.jfa.2004.06.011. [3] E. A. Carlen and F. Salvarani, On the optimal choice of coefficients in a truncated Wild sum and approximate solutions for the Kac equation, J. Statist. Phys., 109 (2002), 261-277.  doi: 10.1023/A:1019943813176. [4] C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. [5] E. Dolera, E. Gabetta and E. Regazzini, Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem, Ann. Appl. Probab., 19 (2009), 186-209.  doi: 10.1214/08-AAP538. [6] N. Fournier and J. S. Giet, Exact simulation of nonlinear coagulation processes, Monte Carlo Methods Appl., 10 (2004), 95-106.  doi: 10.1515/156939604777303253. [7] N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff, Math. Comp., 71 (2002), 583-604.  doi: 10.1090/S0025-5718-01-01339-4. [8] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. Ⅲ, University of California Press, 1956,171–197. [9] X. Lu and C. Mouhot, On measure solutions of the boltzmann equation part Ⅰ: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021. [10] H. P. McKean, Speed of approach to equilibrium for Kacs caricature of a Maxwellian gas, Arch. Ration. Mech. Anal., 21 (1966), 343-367.  doi: 10.1007/BF00264463. [11] H. P. McKean, An exponential formula for solving Boltmann's equation for a Maxwellian gas, J. Combinatorial Theory, 2 (1967), 358-382.  doi: 10.1016/S0021-9800(67)80035-8. [12] S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 467-501.  doi: 10.1016/S0294-1449(99)80025-0. [13] L. Pareschi and G. Russo, Time relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput., 23 (2001), 1253-1273.  doi: 10.1137/S1064827500375916. [14] A. J. Povzner, About the Boltzmann equation in kinetic gas theory, Mat. Sborn, 58 (1962), 65-86. [15] H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. und Verw. Gebiete, 46 (1978/79), 67-105.  doi: 10.1007/BF00535689. [16] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0. [17] B. Wennberg, An example of nonuniqueness for solutions to the homogeneous Boltzmann equation, J. Statist. Phys., 95 (1999), 469-477.  doi: 10.1023/A:1004546031908. [18] E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Cambridge Philos. Soc., 47 (1951), 602-609.  doi: 10.1017/S0305004100026992.
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