# American Institute of Mathematical Sciences

June  2018, 11(3): 547-596. doi: 10.3934/krm.2018024

## High order approximation for the Boltzmann equation without angular cutoff

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Mathematics, Yunnan Normal University, Kunming 650500, China

* Corresponding author: Yulong Zhou

Received  March 2017 Revised  September 2017 Published  March 2018

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

Citation: Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic and Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024
##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463.  doi: 10.1215/21562261-1625154. [3] L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal., 45 (1972), 1-16. [4] Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions I: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8. [5] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923. [6] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part i: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512. [7] N. Fournier and D. Godinho, Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff, Comm. Math. Phys., 316 (2012), 307-344.  doi: 10.1007/s00220-012-1578-9. [8] N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 157-199.  doi: 10.24033/asens.2318. [9] L. He, Asymptotic analysis of the spatially homogeneous Boltzmann equation: grazing collisions limit, J. Stat. Phys., 155 (2014), 151-210.  doi: 10.1007/s10955-014-0932-z. [10] L. He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys., 312 (2012), 447-476.  doi: 10.1007/s00220-012-1481-4. [11] L. He, Sharp bounds for Boltzmann and Landau collision operators, preprint, arXiv: 1604.06981. [12] L. -B. He and J. -C. Jiang, On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules, preprint, arXiv: 1710.00315. [13] Z. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489.  doi: 10.3934/krm.2008.1.453. [14] 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. [15] S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3. [16] S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8. [17] 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. [18] C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212. [19] L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.

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##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff, Kyoto J. Math., 52 (2012), 433-463.  doi: 10.1215/21562261-1625154. [3] L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal., 45 (1972), 1-16. [4] Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions I: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8. [5] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transp. Theory Stat. Phys., 21 (1992), 259-276.  doi: 10.1080/00411459208203923. [6] L. Desvillettes and C. Villani, On the spatially homogeneous landau equation for hard potentials part i: Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512. [7] N. Fournier and D. Godinho, Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff, Comm. Math. Phys., 316 (2012), 307-344.  doi: 10.1007/s00220-012-1578-9. [8] N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 157-199.  doi: 10.24033/asens.2318. [9] L. He, Asymptotic analysis of the spatially homogeneous Boltzmann equation: grazing collisions limit, J. Stat. Phys., 155 (2014), 151-210.  doi: 10.1007/s10955-014-0932-z. [10] L. He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys., 312 (2012), 447-476.  doi: 10.1007/s00220-012-1481-4. [11] L. He, Sharp bounds for Boltzmann and Landau collision operators, preprint, arXiv: 1604.06981. [12] L. -B. He and J. -C. Jiang, On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules, preprint, arXiv: 1710.00315. [13] Z. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinet. Relat. Models, 1 (2008), 453-489.  doi: 10.3934/krm.2008.1.453. [14] 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. [15] S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3. [16] S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8. [17] 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. [18] C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212. [19] L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.
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