High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials

Zheng-an Yao; Yu-Long Zhou

doi:10.3934/krm.2020015

Article Contents

2020, Volume 13, Issue 3: 435-478. Doi: 10.3934/krm.2020015

This issue Previous Article Next Article A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials

Zheng-an Yao and
Yu-Long Zhou^,

School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, P. R. China

^* Corresponding author: Yu-Long Zhou
^* Corresponding author: Yu-Long Zhou

Received: May 2019

Revised: November 2019

Published: March 2020

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

The Boltzmann equation has an angular singularity inherent in the long range interaction between molecules. Angular singularity produces many difficulties in both theoretical and numerical study of Boltzmann equation. As a result, many rely on angular cutoff models to approximate the Boltzmann equation. Cutoff models have no angular singularity and can be solved by existent numerical methods. However, as the singularity goes stronger, the proportion of grazing collision becomes larger, which renders ineffectiveness of the cutoff Boltzmann equation. Based on the theoretical result that the limit of grazing collision is Landau operator, we propose to add a suitably scaled Landau operator to the cutoff equation to form a new approximate equation. This new approximate equation was studied in [27] in the case of hard potentials and its approximation accuracy is proved to be one order higher than that of angular cutoff models, which is a significant improvement in numerical computing. In this work, under moderately soft potentials, we establish the well-posedness theory of the new approximate equation, prove regularity propagation of its solution, check the high order accuracy. The new approximate equation can be solved by existing numerical methods, and this work may provide a theoretical foundation and a new direction to high order numerical methods for solving the Boltzmann equation.

Keywords:

Mathematics Subject Classification: Primary: 35Q20, 35R11; Secondary: 75P05.

Citation:

Full Text(HTML)

Related Papers

Cited by

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]	R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012.
[4]	R. J. Alonso, I. M. Gamba and S. H. Tharkabhushanam, Convergence and error estimates for the Lagrangian-based conservative spectral method for Boltzmann equations, SIAM J. Numer. Anal., 56 (2018), 3534-3579. doi: 10.1137/18M1173332.
[5]	R. J. Alonso and W. Sun, The radiative transfer equation in the forward-peaked regime, Comm. Math. Phys., 338 (2015), 1233-1286. doi: 10.1007/s00220-015-2395-8.
[6]	L. Arkeryd, On the Boltzmann equation, Arch. Ration. Mech. Anal., 45 (1972), 1-16. doi: 10.1007/BF00253392.
[7]	G. A. Bird, Molecular Gas Dynamics, Clarendon Press, Oxford, 1994.
[8]	E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 1 (2010), 85-122. doi: 10.3934/krm.2010.3.85.
[9]	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.
[10]	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.
[11]	L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Rat. Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x.
[12]	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.
[13]	L. Fainsilber, P. Kurlberg and B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal., 37 (2006), 1903-1922. doi: 10.1137/040618916.
[14]	F. Filbet and L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, J. Comput. Phys., 179 (2002), 1-26. doi: 10.1006/jcph.2002.7010.
[15]	N. Fournier, Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff, J. Stat. Phys., 125 (2006), 927-946. doi: 10.1007/s10955-006-9208-6.
[16]	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.
[17]	N. Fournier and C. Mouhot, On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Commun. Math. Phys., 289 (2009), 803-824. doi: 10.1007/s00220-009-0807-3.
[18]	N. Fournier and H. Guerin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys., 131 (2008), 749-781. doi: 10.1007/s10955-008-9511-5.
[19]	N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Ec. Norm. Super., 50 (2017), 157-199.
[20]	I. M. Gamba and J. R. Haack, A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit, J. Comput. Phys., 270 (2014), 40-57. doi: 10.1016/j.jcp.2014.03.035.
[21]	I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036. doi: 10.1016/j.jcp.2008.09.033.
[22]	I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation, J. Comput. Phys., 28 (2010), 430-460. doi: 10.4208/jcm.1003-m0011.
[23]	Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9.
[24]	L.-B. 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.
[25]	L.-B. 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.
[26]	L.-B. He, Sharp bounds for Boltzmann and Landau collision operators, Ann. Sci. Ec. Norm. Super., 51 (2018), 1253-1341. doi: 10.24033/asens.2375.
[27]	L.-B. He and Y.-L. Zhou, High order approximation for the Boltzmann equation without angular cutoff, Kinet. Relat. Models, 11 (2018), 547-596. doi: 10.3934/krm.2018024.
[28]	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.
[29]	M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, Berkeley and Los Angeles, 1956, University of California Press, 171–197.
[30]	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.
[31]	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.
[32]	S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147. doi: 10.1007/s00222-012-0422-3.
[33]	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.
[34]	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.
[35]	C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212. doi: 10.1007/s00205-004-0316-7.
[36]	K. Nanbu, Direct simulation scheme derived from the boltzmann equation I. monocomponent gases, J. Phys. Soc. Japan, 52 (1983), 2042-2049.
[37]	A. Palczewski, J. Schneider and A. V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal., 34 (1997), 1865-1883. doi: 10.1137/S0036142995289007.
[38]	A. Palczewski and J. Schneider, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Stat. Phys., 91 (1998), 307-326. doi: 10.1023/A:1023000406921.
[39]	V. A. Panferov and A. G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci., 25 (2002), 571-593. doi: 10.1002/mma.303.
[40]	L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Trans. Theo. Stat. Phys., 25 (1996), 369-382. doi: 10.1080/00411459608220707.
[41]	L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal., 37 (2000), 1217-1245. doi: 10.1137/S0036142998343300.
[42]	L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Trans. Theo. Stat. Phys., 29 (2000), 431-447. doi: 10.1080/00411450008205883.
[43]	L. Pareschi, G. Russo and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys., 165 (2000), 216-236. doi: 10.1006/jcph.2000.6612.
[44]	S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253. doi: 10.1006/jcph.1996.0057.
[45]	S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer, Berlin, 2005.
[46]	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.
[47]	Y.-L. Zhou, A refined estimate of the grazing limit from Boltzmann to Landau operator in Coulomb potential, Appl. Math. Lett., 100 (2020), 106039, 8pp. doi: 10.1016/j.aml.2019.106039.