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

Pedro Aceves-Sánchez; Christian Schmeiser

doi:10.3934/krm.2017021

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2017, Volume 10, Issue 3: 541-551. Doi: 10.3934/krm.2017021

This issue Previous Article Next Article Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime

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
* Corresponding author

Received: July 2016

Revised: October 2016

Published: September 2017

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Abstract

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.

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Mathematics Subject Classification: 76P05, 35B40, 26A33.

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References

	P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, preprint, arXiv: 1607.00855.
	P. Aceves-Sanchez and A. Mellet, Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous scaling, preprint, arXiv: 1606.01023.
	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.
	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.
	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.
	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.
	L. Cesbron, Anomalous diffusion limit of kinetic equations on spatially bounded domains, work in progress.
	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.
	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.
	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.
	P. Degond , T. Goudon and F. Poupaud , Diffusion limit for nonhomogeneous and non-micro-reversible processes, Indiana Univ. Math. J., 49 (2000) , 1175-1198.
	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.
	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.
	M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Vol. 19, Walter de Gruyter, 2011. doi: 10.1515/9783110889741.
	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.
	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.
	M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, preprint, arXiv: 1507.07356.
	E. Larsen and J. Keller , Asymptotic solution of neutron transport processes for small free paths, J. Math. Phys., 15 (1974) , 75-81.
	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.
	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.
	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.
	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.
	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.
	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.
	E. Wigner, Nuclear Reactor Theory, AMS, 1961.