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

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

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.

Citation: Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic & 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. Google Scholar

[2]

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

[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.  Google Scholar

[4]

N. Ben AbdallahA. 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.  Google Scholar

[5]

D. A. BensonR. SchumerM. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

L. CesbronA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[16]

M. JaraT. 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.  Google Scholar

[17]

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

[18]

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

E. D. NezzaG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

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. Google Scholar

[2]

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

[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.  Google Scholar

[4]

N. Ben AbdallahA. 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.  Google Scholar

[5]

D. A. BensonR. SchumerM. 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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

L. CesbronA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[16]

M. JaraT. 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.  Google Scholar

[17]

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

[18]

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

E. D. NezzaG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[1]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019

[2]

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

[3]

Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic & Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873

[4]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[5]

Yuanwei Qi. Anomalous exponents and RG for nonlinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 738-745. doi: 10.3934/proc.2005.2005.738

[6]

Arnaud Debussche, Julien Vovelle. Diffusion limit for a stochastic kinetic problem. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2305-2326. doi: 10.3934/cpaa.2012.11.2305

[7]

Martin Frank, Weiran Sun. Fractional diffusion limits of non-classical transport equations. Kinetic & Related Models, 2018, 11 (6) : 1503-1526. doi: 10.3934/krm.2018059

[8]

Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic & Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045

[9]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[10]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[11]

Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173

[12]

Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3195-3210. doi: 10.3934/dcdsb.2018340

[13]

Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020029

[14]

María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157

[15]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028

[16]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114

[17]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[18]

Hongjie Dong, Dong Li. On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437

[19]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257

[20]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (18)
  • HTML views (9)
  • Cited by (1)

Other articles
by authors

[Back to Top]