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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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

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