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August  2018, 11(4): 1011-1036. doi: 10.3934/krm.2018039

Linear Boltzmann equation and fractional diffusion

1. 

Laboratoire J.-L. Lions, BP 187, 75252 Paris Cedex 05, France

2. 

CMLS, École polytechnique, 91128 Palaiseau Cedex, France

3. 

DPMMS, University of Cambridge, Wilberforce Road, CB3 0WA Cambridge, United Kingdom

* Corresponding author: François Golse

Received  August 2017 Revised  February 2018 Published  April 2018

Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma\to+∞$ and $ 1-\alpha \sim C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of $\sqrt{-\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions ("heavy tails") or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].

Citation: Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039
References:
[1]

P. Aceves-Sánchez and C. Schmeiser, Fractional diffusion limit of a linear kinetic equation in a bounded domain, Kinetic Relat. Mod., 10 (2017), 541-551.  doi: 10.3934/krm.2017021.  Google Scholar

[2]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations, existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460.  doi: 10.1016/0022-1236(88)90096-1.  Google Scholar

[3]

C. BardosE. BernardF. Golse and R. Sentis, The diffusion approximation for the linear Boltzmann equation with vanishing scattering coefficient, Commun. Math. Sci., 13 (2015), 641-671.  doi: 10.4310/CMS.2015.v13.n3.a3.  Google Scholar

[4]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[5]

N. Ben AbdallahA. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models and Methods Appl. Sci., 21 (2011), 2249-2262.  doi: 10.1142/S0218202511005738.  Google Scholar

[6]

A. BensoussanJ.-L. Lions and G.C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.  Google Scholar

[7]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New-York, Dordrecht, Heidelberg, London, 2011.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C.R. Acad. Sci. Paris Sér. I, 299 (1984), 831-834.   Google Scholar

[10]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 1960.  Google Scholar

[11]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6. Evolution problems, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.  Google Scholar

[12]

L. Desvillettes and F. Golse, A remark concerning the Chapman-Enskog asymptotics, in Advances in Kinetic Theory and Computing (ed. B. Perthame), Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, 1994,191-203.  Google Scholar

[13]

U. Frisch and H. Frisch, Non LTE Transfer. Asymptotic Expansion for Small $\epsilon$, Mon. Not. R. Astr. Not., 181 (1977), 273-280.   Google Scholar

[14]

F. Golse, Fluid dynamic limits of the kinetic theory of gases, in From Particle Systems to Partial Differential Equations (eds. C. Bernardin and Patrícia Gonçalves), Springer Proc. in Math. and Statist. 75, Springer Verlag, Berlin, Heidelberg, 2014, 3-91. doi: 10.1007/978-3-642-54271-8_1.  Google Scholar

[15]

D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.  doi: 10.1007/BF01456676.  Google Scholar

[16]

A. M. Il'in and R. Z. Has'minskii (Khasminskii), On the equations of Brownian motion (Russian), Teor. Verojatnost. i Primenen., 9 (1964), 466-491.   Google Scholar

[17]

M. Kwasnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[18]

E. W. Larsen and J. B. Keller, Asymptotics solutions of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.  doi: 10.1063/1.1666510.  Google Scholar

[19]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[20]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[21]

G. C. Papanicolaou, Asymptotic analysis of transport processes, Bull. Amer. Math. Soc., 81 (1975), 330-392.  doi: 10.1090/S0002-9904-1975-13744-X.  Google Scholar

[22]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1963.  Google Scholar

[23]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer-Verlag, Berlin, Heidelberg, 2007.  Google Scholar

[24]

A. Weinberg and E. Wigner, The Physical Theory of Neutron Chain Reactors, The University of Chicago Press, 1958.  Google Scholar

show all references

References:
[1]

P. Aceves-Sánchez and C. Schmeiser, Fractional diffusion limit of a linear kinetic equation in a bounded domain, Kinetic Relat. Mod., 10 (2017), 541-551.  doi: 10.3934/krm.2017021.  Google Scholar

[2]

C. BardosF. GolseB. Perthame and R. Sentis, The nonaccretive radiative transfer equations, existence of solutions and Rosseland approximation, J. Funct. Anal., 77 (1988), 434-460.  doi: 10.1016/0022-1236(88)90096-1.  Google Scholar

[3]

C. BardosE. BernardF. Golse and R. Sentis, The diffusion approximation for the linear Boltzmann equation with vanishing scattering coefficient, Commun. Math. Sci., 13 (2015), 641-671.  doi: 10.4310/CMS.2015.v13.n3.a3.  Google Scholar

[4]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.  Google Scholar

[5]

N. Ben AbdallahA. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models and Methods Appl. Sci., 21 (2011), 2249-2262.  doi: 10.1142/S0218202511005738.  Google Scholar

[6]

A. BensoussanJ.-L. Lions and G.C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci., 15 (1979), 53-157.  doi: 10.2977/prims/1195188427.  Google Scholar

[7]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer New-York, Dordrecht, Heidelberg, London, 2011.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eq., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C.R. Acad. Sci. Paris Sér. I, 299 (1984), 831-834.   Google Scholar

[10]

S. Chandrasekhar, Radiative Transfer, Dover Publications, Inc., New York, 1960.  Google Scholar

[11]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6. Evolution problems, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.  Google Scholar

[12]

L. Desvillettes and F. Golse, A remark concerning the Chapman-Enskog asymptotics, in Advances in Kinetic Theory and Computing (ed. B. Perthame), Ser. Adv. Math. Appl. Sci., 22, World Sci. Publ., River Edge, NJ, 1994,191-203.  Google Scholar

[13]

U. Frisch and H. Frisch, Non LTE Transfer. Asymptotic Expansion for Small $\epsilon$, Mon. Not. R. Astr. Not., 181 (1977), 273-280.   Google Scholar

[14]

F. Golse, Fluid dynamic limits of the kinetic theory of gases, in From Particle Systems to Partial Differential Equations (eds. C. Bernardin and Patrícia Gonçalves), Springer Proc. in Math. and Statist. 75, Springer Verlag, Berlin, Heidelberg, 2014, 3-91. doi: 10.1007/978-3-642-54271-8_1.  Google Scholar

[15]

D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann., 72 (1912), 562-577.  doi: 10.1007/BF01456676.  Google Scholar

[16]

A. M. Il'in and R. Z. Has'minskii (Khasminskii), On the equations of Brownian motion (Russian), Teor. Verojatnost. i Primenen., 9 (1964), 466-491.   Google Scholar

[17]

M. Kwasnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[18]

E. W. Larsen and J. B. Keller, Asymptotics solutions of neutron transport problems for small mean free paths, J. Math. Phys., 15 (1974), 75-81.  doi: 10.1063/1.1666510.  Google Scholar

[19]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[20]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[21]

G. C. Papanicolaou, Asymptotic analysis of transport processes, Bull. Amer. Math. Soc., 81 (1975), 330-392.  doi: 10.1090/S0002-9904-1975-13744-X.  Google Scholar

[22]

G. C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1963.  Google Scholar

[23]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer-Verlag, Berlin, Heidelberg, 2007.  Google Scholar

[24]

A. Weinberg and E. Wigner, The Physical Theory of Neutron Chain Reactors, The University of Chicago Press, 1958.  Google Scholar

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