Concentration phenomena for neutronic multigroup diffusion in random environments

Armstrong, Scott N.; Souganidis, Panagiotis E.

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 419-439.

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton–Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.

[1] M.A. Akcoglu, U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math. 323 (1981), 53-67 | EuDML | Zbl

[2] G. Allaire, G. Bal, Homogénéisation dʼune équation spectrale du transport neutronique, C. R. Acad. Sci. Paris Sér. I Math. 325 no. 9 (1997), 1043-1048

[3] G. Allaire, G. Bal, Homogenization of the criticality spectral equation in neutron transport, M2AN Math. Model. Numer. Anal. 33 no. 4 (1999), 721-746 | EuDML | Numdam | Zbl

[4] G. Allaire, Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg. 187 no. 1–2 (2000), 91-117 | Zbl

[5] G. Allaire, Y. Capdeboscq, A. Piatnitski, Homogenization and localization with an interface, Indiana Univ. Math. J. 52 no. 6 (2003), 1413-1446 | Zbl

[6] G. Allaire, Y. Capdeboscq, A. Piatnitski, V. Siess, M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Ration. Mech. Anal. 174 no. 2 (2004), 179-220 | Zbl

[7] G. Allaire, F. Malige, Analyse asymptotique spectrale dʼun problème de diffusion neutronique, C. R. Acad. Sci. Paris Sér. I Math. 324 no. 8 (1997), 939-944 | Zbl

[8] G. Allaire, R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM Control Optim. Calc. Var. 13 no. 4 (2007), 735-749 | EuDML | Numdam | Zbl

[9] G. Allaire, I. Pankratova, A. Piatnitski, Homogenization and concentration for a diffusion equation with large convection in a bounded domain, J. Funct. Anal. 262 (2012), 300-330 | Zbl

[10] G. Allaire, A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. Partial Differential Equations 27 no. 3–4 (2002), 705-725 | Zbl

[11] L. Ambrosio, H. Frid, Multiscale Young measures in almost periodic homogenization and applications, Arch. Ration. Mech. Anal. 192 no. 1 (2009), 37-85 | Zbl

[12] S.N. Armstrong, P.E. Souganidis, Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments, J. Math. Pures Appl. 97 (2012), 460-504 | Zbl

[13] M.E. Becker, Multiparameter groups of measure-preserving transformations: A simple proof of Wienerʼs ergodic theorem, Ann. Probab. 9 no. 3 (1981), 504-509 | Zbl

[14] J. Busca, B. Sirakov, Harnack type estimates for nonlinear elliptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 5 (2004), 543-590 | EuDML | Zbl

[15] Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Sér. I Math. 327 no. 9 (1998), 807-812 | Zbl

[16] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. Roy. Soc. Edinburgh Sect. A 132 no. 3 (2002), 567-594 | Zbl

[17] M.G. Crandall, H. Ishii, P.-L. Lions, Userʼs guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 no. 1 (1992), 1-67

[18] V. Deniz, The theory of neutron leakage in reactor lattices, Y. Ronen (ed.), Handbook of Nuclear Reactor Calculations, vol. II, Chemical Rubber Company, Boca Raton (1986), 409-508

[19] L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111 no. 3–4 (1989), 359-375 | Zbl

[20] H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac. 38 no. 1 (1995), 101-120 | Zbl

[21] H. Ishii, Almost periodic homogenization of Hamilton–Jacobi equations, International Conference on Differential Equations, vols. 1, 2, Berlin, 1999, World Sci. Publ., River Edge, NJ (2000), 600-605 | Zbl

[22] H. Ishii, S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations 16 no. 6–7 (1991), 1095-1128 | Zbl

[23] E. Kosygina, F. Rezakhanlou, S.R.S. Varadhan, Stochastic homogenization of Hamilton–Jacobi–Bellman equations, Comm. Pure Appl. Math. 59 no. 10 (2006), 1489-1521 | Zbl

[24] S.M. Kozlov, The averaging method and walks in inhomogeneous environments, Uspekhi Mat. Nauk 40 no. 2(242) (1985), 61-120 | Zbl

[25] N.V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and Its Applications (Soviet Series) vol. 7, D. Reidel Publishing Co., Dordrecht (1987) | Zbl

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

[27] E.W. Larsen, Neutron transport and diffusion in inhomogeneous media, Nucl. Sci. Eng. 60 (1976), 357-368

[28] E.W. Larsen, M. Williams, Neutron drift in heterogeneous media, Nucl. Sci. Eng. 65 (1978), 290-302

[29] P.-L. Lions, P.E. Souganidis, Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math. 56 no. 10 (2003), 1501-1524 | Zbl

[30] P.-L. Lions, P.E. Souganidis, Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations 30 no. 1–3 (2005), 335-375 | Zbl

[31] P.-L. Lions, P.E. Souganidis, Stochastic homogenization of Hamilton–Jacobi and “viscous”-Hamilton–Jacobi equations with convex nonlinearities—Revisited, Commun. Math. Sci. 8 no. 2 (2010), 627-637 | Zbl

[32] E. Mitidieri, G. Sweers, Weakly coupled elliptic systems and positivity, Math. Nachr. 173 (1995), 259-286 | Zbl

[33] B. Perthame, P.E. Souganidis, Asymmetric potentials and motor effect: A homogenization approach, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2055-2071 | EuDML | Numdam | Zbl

[34] A.L. Piatnitski, Asymptotic behaviour of the ground state of singularly perturbed elliptic equations, Comm. Math. Phys. 197 no. 3 (1998), 527-551 | Zbl

[35] A.L. Pyatnitskiĭ, A.S. Shamaev, On the asymptotic behavior of the eigenvalues and eigenfunctions of a nonselfadjoint operator in ${ℝ}^n$, Tr. Semin. im. I.G. Petrovskogo 23 (2003), 287-308 | Zbl

[36] G. Sweers, Strong positivity in $C\left(\overline{\Omega }\right)$ for elliptic systems, Math. Z. 209 no. 2 (1992), 251-271 | EuDML

[37] A.M. Weinberg, E.P. Wigner, The Physical Theory of Neutron Chain Reactors, The University of Chicago Press (1958)

[38] A. Wilkinson, personal communication.