The Annals of Applied Probability

Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials

Liping Xu

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Abstract

We prove a strong/weak stability estimate for the 3D homogeneous Boltzmann equation with moderately soft potentials [$\gamma\in(-1,0)$] using the Wasserstein distance with quadratic cost. This in particular implies the uniqueness in the class of all weak solutions, assuming only that the initial condition has a finite entropy and a finite moment of sufficiently high order. We also consider the Nanbu $N$-stochastic particle system, which approximates the weak solution. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 2 (2018), 1136-1189.

Dates
Received: May 2016
Revised: February 2017
First available in Project Euclid: 11 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1523433633

Digital Object Identifier
doi:10.1214/17-AAP1327

Mathematical Reviews number (MathSciNet)
MR3784497

Zentralblatt MATH identifier
06897952

Subjects
Primary: 82C40: Kinetic theory of gases 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Kinetic theory Boltzmann equation stochastic particle systems propagation of chaos Wasserstein distance

Citation

Xu, Liping. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. Ann. Appl. Probab. 28 (2018), no. 2, 1136--1189. doi:10.1214/17-AAP1327. https://projecteuclid.org/euclid.aoap/1523433633


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References

  • [1] Aldous, D. (1978). Stopping times and tightness. Ann. Probab. 6 335–340.
  • [2] Alexandre, R., Desvillettes, L., Villani, C. and Wennberg, B. (2000). Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 327–355.
  • [3] Bhatt, A. G. and Karandikar, R. L. (1993). Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21 2246–2268.
  • [4] Cortez, R. and Fontbona, J. (2015). Quantitative uniform propagation of chaos for Maxwell molecules. Arvix preprint. Available at arXiv:1512.09308.
  • [5] Desvillettes, L. and Mouhot, C. (2009). Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch. Ration. Mech. Anal. 193 227–253.
  • [6] Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 109–153.
  • [7] Fontbona, J., Guérin, H. and Méléard, S. (2009). Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Probab. Theory Related Fields 143 329–351.
  • [8] Fournier, N. (2015). Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 860–897.
  • [9] Fournier, N. and Guérin, H. (2008). On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131 749–781.
  • [10] Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields 162 707–738.
  • [11] Fournier, N. and Hauray, M. (2016). Propagation of chaos for the Landau equation with moderately soft potentials. Ann. Probab. 44 3581–3660.
  • [12] Fournier, N., Hauray, M. and Mischler, S. (2014). Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc. (JEMS) 16 1423–1466.
  • [13] Fournier, N. and Méléard, S. (2002). A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comp. 71 583–604. DOI:10.1090/S0025-5718-01-01339-4.
  • [14] Fournier, N. and Mischler, S. (2016). Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules. Ann. Probab. 44 589–627.
  • [15] Fournier, N. and Mouhot, C. (2009). On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Comm. Math. Phys. 289 803–824.
  • [16] Godinho, D. and Quiñinao, C. (2015). Propagation of chaos for a subcritical Keller–Segel model. Ann. Inst. Henri Poincaré Probab. Stat. 51 965–992.
  • [17] Graham, C. and Méléard, S. (1997). Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 115–132.
  • [18] Grünbaum, F. A. (1971). Propagation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 42 323–345.
  • [19] Hauray, M. and Jabin, P.-E. (2015). Particle approximation of Vlasov equations with singular forces: Propagation of chaos. Ann. Sci. Éc. Norm. Supér. (4) 48 891–940.
  • [20] Horowitz, J. and Karandikar, R. L. (1990). Martingale problems associated with the Boltzmann equation. In Seminar on Stochastic Processes, 1989. Progress in Probability 18 75–122. Birkhäuser, Boston, MA.
  • [21] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [22] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [23] Kac, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, Vol. III 171–197. Univ. California Press, Berkeley and Los Angeles.
  • [24] Lu, X. and Mouhot, C. (2012). On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates. J. Differential Equations 252 3305–3363.
  • [25] McKean, H. P. Jr. (1967). An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Combin. Theory 2 358–382.
  • [26] Mischler, S. and Mouhot, C. (2013). Kac’s program in kinetic theory. Invent. Math. 193 1–147.
  • [27] Mischler, S. and Wennberg, B. (1999). On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 467–501.
  • [28] Nanbu, K. (1983). Interrelations between various direct simulation methods for solving the Boltzmann equation. J. Phys. Soc. Jpn. 52 3382–3388.
  • [29] Rousset, M. (2014). A N-uniform quantitative Tanaka’s theorem for the conservative Kac’s N-particle system with Maxwell molecules. Arxiv preprint. Available at arXiv:1407.1965.
  • [30] Sznitman, A.-S. (1984). Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66 559–592.
  • [31] Tanaka, H. (1978/79). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 67–105.
  • [32] Toscani, G. and Villani, C. (1999). Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94 619–637.
  • [33] Villani, C. (1998). On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 273–307.
  • [34] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [35] Xu, L. (2016). The multifractal nature of Boltzmann processes. Stochastic Process. Appl. 126 2181–2210.