# American Institute of Mathematical Sciences

2018, 11(3): 597-613. doi: 10.3934/krm.2018025

## Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

 1 Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia 2 Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab., 209 South 33rd Street, Philadelphia, PA 19104, USA

* Corresponding author: Milana Pavić-Čolić

Received  April 2017 Revised  September 2017 Published  March 2018

We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its 1-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

Citation: Milana Pavić-Čolić, Maja Tasković. Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules. Kinetic & Related Models, 2018, 11 (3) : 597-613. doi: 10.3934/krm.2018025
##### References:
 [1] R. J. Alonso, J. A. Cañizo, I. M. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169. doi: 10.1080/03605302.2012.715707. [2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403. [3] A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas, Teoret. Mat. Fiz., 60 (1984), 280-310. [4] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Soviet Sci. Rev. Sect. C. Math. Phys., Vol. 7, Harwood Academic Publ., Chur, 7 (1988), 111-233. [5] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214. doi: 10.1007/BF02732431. [6] A. V. Bobylev and I. M. Gamba, Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules, Kinet. Relat. Models, 10 (2017), 573-585. doi: 10.3934/krm.2017023. [7] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys., 116 (2004), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea. [8] L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404. doi: 10.1007/BF00375586. [9] L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440. doi: 10.1007/BF02101556. [10] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9. [11] 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. [12] C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672. doi: 10.1007/s00220-005-1455-x. [13] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869, arXiv: 1512.06769. doi: 10.1137/17M1117926. [14] 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. doi: 10.1007/s002050050106. [15] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

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##### References:
 [1] R. J. Alonso, J. A. Cañizo, I. M. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Comm. Partial Differential Equations, 38 (2013), 155-169. doi: 10.1080/03605302.2012.715707. [2] L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Arch. Rational Mech. Anal., 77 (1981), 11-21. doi: 10.1007/BF00280403. [3] A. V. Bobylev, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas, Teoret. Mat. Fiz., 60 (1984), 280-310. [4] A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, in Soviet Sci. Rev. Sect. C. Math. Phys., Vol. 7, Harwood Academic Publ., Chur, 7 (1988), 111-233. [5] A. V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys., 88 (1997), 1183-1214. doi: 10.1007/BF02732431. [6] A. V. Bobylev and I. M. Gamba, Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules, Kinet. Relat. Models, 10 (2017), 573-585. doi: 10.3934/krm.2017023. [7] A. V. Bobylev, I. M. Gamba and V. A. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys., 116 (2004), 1651-1682. doi: 10.1023/B:JOSS.0000041751.11664.ea. [8] L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Arch. Rational Mech. Anal., 123 (1993), 387-404. doi: 10.1007/BF00375586. [9] L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys., 168 (1995), 417-440. doi: 10.1007/BF02101556. [10] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282. doi: 10.1007/s00205-009-0250-9. [11] 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. [12] C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672. doi: 10.1007/s00220-005-1455-x. [13] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869, arXiv: 1512.06769. doi: 10.1137/17M1117926. [14] 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. doi: 10.1007/s002050050106. [15] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.
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