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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 |
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.
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] |
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. |
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