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September  2017, 10(3): 541-551. doi: 10.3934/krm.2017021

Fractional diffusion limit of a linear kinetic equation in a bounded domain

Pedro Aceves-Sánchez , and  Christian Schmeiser

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

* Corresponding author

Received  July 2016 Revised  October 2016 Published  December 2016

A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.

Keywords: Kinetic transport equations, linear Boltzmann operator, anomalous diffusion limit, fractional diffusion, asymptotic analysis.
Mathematics Subject Classification: 76P05, 35B40, 26A33.
Citation: Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic and Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021
References:
[1]

P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, preprint, arXiv: 1607.00855.

[2]

P. Aceves-Sanchez and A. Mellet, Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous scaling, preprint, arXiv: 1606.01023.

[3]

P. Aceves-Sanchez and C. Schmeiser, Fractional-diffusion-advection limit of a kinetic model, SIAM J. Math. Anal., 48 (2016), 2806-2818.  doi: 10.1137/15M1045387.

[4]

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

[5]

D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, Fractional dispersion, lévy motion, and the made tracer tests, Transport in Porous Media, 42 (2001), 211-240.  doi: 10.1023/A:1006733002131.

[6]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[7]

L. Cesbron, Anomalous diffusion limit of kinetic equations on spatially bounded domains, work in progress.

[8]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous transport of particles in plasma physics, Appl. Math. Lett., 25 (2012), 2344-2348.  doi: 10.1016/j.aml.2012.06.029.

[9]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.  doi: 10.1007/s002080050232.

[10]

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

[11]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. 

[12]

D. del Castillo-Negrete, B. Carreras and V. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach, Physical Review Letters, 94 (2005), Article 065003. 

[13]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.

[14]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Vol. 19, Walter de Gruyter, 2011. doi: 10.1515/9783110889741.

[15]

G. J. Habetler and B. J. Matkowsky, Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation, J. Mathematical Phys., 16 (1975), 846-854. 

[16]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.

[17]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, preprint, arXiv: 1507.07356.

[18]

E. Larsen and J. Keller, Asymptotic solution of neutron transport processes for small free paths, J. Math. Phys., 15 (1974), 75-81. 

[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. Mellet, S. 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]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[22]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.

[23]

R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Related Fields, 125 (2003), 578-592.  doi: 10.1007/s00440-002-0251-1.

[24]

J.-L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.

[25]

E. Wigner, Nuclear Reactor Theory, AMS, 1961.

show all references

References:
[1]

P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, preprint, arXiv: 1607.00855.

[2]

P. Aceves-Sanchez and A. Mellet, Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous scaling, preprint, arXiv: 1606.01023.

[3]

P. Aceves-Sanchez and C. Schmeiser, Fractional-diffusion-advection limit of a kinetic model, SIAM J. Math. Anal., 48 (2016), 2806-2818.  doi: 10.1137/15M1045387.

[4]

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

[5]

D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, Fractional dispersion, lévy motion, and the made tracer tests, Transport in Porous Media, 42 (2001), 211-240.  doi: 10.1023/A:1006733002131.

[6]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198.  doi: 10.1007/s00220-006-0178-y.

[7]

L. Cesbron, Anomalous diffusion limit of kinetic equations on spatially bounded domains, work in progress.

[8]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous transport of particles in plasma physics, Appl. Math. Lett., 25 (2012), 2344-2348.  doi: 10.1016/j.aml.2012.06.029.

[9]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.  doi: 10.1007/s002080050232.

[10]

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

[11]

P. Degond, T. Goudon and F. Poupaud, Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000), 1175-1198. 

[12]

D. del Castillo-Negrete, B. Carreras and V. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach, Physical Review Letters, 94 (2005), Article 065003. 

[13]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.

[14]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Vol. 19, Walter de Gruyter, 2011. doi: 10.1515/9783110889741.

[15]

G. J. Habetler and B. J. Matkowsky, Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation, J. Mathematical Phys., 16 (1975), 846-854. 

[16]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.

[17]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, preprint, arXiv: 1507.07356.

[18]

E. Larsen and J. Keller, Asymptotic solution of neutron transport processes for small free paths, J. Math. Phys., 15 (1974), 75-81. 

[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. Mellet, S. 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]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[22]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.  doi: 10.1017/S0308210512001783.

[23]

R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Related Fields, 125 (2003), 578-592.  doi: 10.1007/s00440-002-0251-1.

[24]

J.-L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.

[25]

E. Wigner, Nuclear Reactor Theory, AMS, 1961.

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