Homogenization of a spectral equation with drift in linear transport

Bal, Guillaume

ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 613-627.

This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.

[1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 9 (1992) 1482-1518. | MR | Zbl

[2] G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. | Numdam | MR | Zbl

[3] G. Bal, Couplage d'équations et homogénéisation en transport neutronique, Thèse de Doctorat de l'Université Paris 6 (1997).

[4] G. Bal, Boundary layer analysis in the homogenization of neutron transport equations in a cubic domain. Asymptot. Anal. 20 (1999) 213-239. | MR | Zbl

[5] G. Bal, First-order Corrector for the Homogenization of the Criticality Eigenvalue Problem in the Even Parity Formulation of the Neutron Transport. SIAM J. Math. Anal. 30 (1999) 1208-1240. | MR | Zbl

[6] G. Bal, Diffusion Approximation of Radiative Transfer Equations in a Channel. Transport Theory Statist. Phys. (to appear). | MR | Zbl

[7] P. Benoist, Théorie du coefficient de diffusion des neutrons dans un réseau comportant des cavités, Note CEA-R 2278 (1964).

[8] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland (1978). | MR | Zbl

[9] , Boundary Layers and Homogenization of Transport Processes. RIMS, Kyoto Univ. (1979).

[10] J. Bergh and L. Löfström, Interpolation spaces. Springer, New York (1976). | Zbl

[11] J. Bussac and P. Reuss, Traité de neutronique. Hermann, Paris (1978).

[12] Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 807-812. | MR | Zbl

[13] , Homogenization of a Neutronic Critical Diffusion Problem with Drift. Proc. Roy Soc. Edinburgh Sect. A (accepted). | Zbl

[14] F. Chatelin, Spectral approximation of linear operators. Academic Press, Comp. Sci. Appl. Math. (1983). | MR | Zbl

[15] R. Dautray and J.L. Lions, Mathematical analysis and numerical methods for Science and Technology, Vol. 6. Springer Verlag, Berlin (1993). | MR | Zbl

[16] V. Deniz, The theory of neutron leakage in reactor lattices, in Handbook of nuclear reactor calculations, Vol. II, edited by Y. Ronen (1968) 409-508.

[17] J. Garnier, Homogenization in a periodic and time dependent potential. SIAM J. Appl. Math. 57 (1997) 95-111. | MR | Zbl

[18] F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. | MR | Zbl

[19] T. Kato, Perturbation theory for linear operators. Springer Verlag, Berlin (1976). | MR | Zbl

[20] M.L. Kleptsyna and A.L. Piatnitski, On large deviation asymptotics for homgenized SDE with a small diffusion. Probab. Theory Appl. (submitted).

[21] S. Kozlov, Reductibility of quasiperiodic differential operators and averaging. Trans. Moscow Math. Soc. 2 (1984) 101-136. | MR | Zbl

[22] E.W. Larsen, Neutron transport and diffusion in inhomogeneous media. I. J. Math. Phys. 16 (1975) 1421-1427. | MR

[23] , Neutron transport and diffusion in inhomogeneous media. II. Nuclear Sci. Engrg. 60 (1976) 357-368.

[24] E.W. Larsen and J.B. Keller, Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15 (1974) 75-81. | MR

[25] E.W. Larsen and M. Williams, Neutron Drift in Heterogeneous Media. Nuclear Sci. Engrg. 65 (1978) 290-302.

[26] M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. World Scientific, Singapore (1997). | Zbl

[27] J. Planchard, Méthodes mathématiques en neutronique, Collection de la Direction des Études et Recherches d'EDF. Eyrolles (1995).

[28] L. Ryzhik, G. Papanicolaou and J.B. Keller, Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327-370. | MR | Zbl

[29] R. Sentis, Study of the corrector of the eigenvalue of a transport operator. SIAM J. Math. Anal. 16 (1985) 151-166. | MR | Zbl

[30] M. Struwe, Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin (1990). | MR | Zbl