Homogenization: In mathematics or physics?

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October 2016, 9(5): 1575-1590. doi: 10.3934/dcdss.2016064

Homogenization: In mathematics or physics?

Shixin Xu ^1, , Xingye Yue ^1, and Changrong Zhang ^2,

1.	Department of Mathematics, Soochow University, Suzhou 215006, China
2.	High Speed Aerodynamics Institute, China Aerodynamisc Development and Research Center, Mianyang 622661, China

Received November 2014 Revised September 2015 Published October 2016

Abstract
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In mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.

Keywords: unit transformation., Homogenization, re-scaling.

Mathematics Subject Classification: Primary: 35B27, 35B40; Secondary: 35M3.

Citation: Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064

References:

[1]	G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586.
[2]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[3]	I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277.
[4]	I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153.
[5]	I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116.
[6]	I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316.
[7]	A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999.
[9]	S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327.
[10]	K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158.
[11]	J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881.
[12]	F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3.
[13]	G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[14]	O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992.
[15]	S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822.
[16]	S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.
[17]	T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6.
[18]	L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212.
[19]	L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.
[20]	T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12.
[21]	V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601.
[22]	V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147.
[23]	V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

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References:

[1]	G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586.
[2]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[3]	I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277.
[4]	I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153.
[5]	I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116.
[6]	I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316.
[7]	A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999.
[9]	S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327.
[10]	K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158.
[11]	J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881.
[12]	F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3.
[13]	G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[14]	O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992.
[15]	S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822.
[16]	S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.
[17]	T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6.
[18]	L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212.
[19]	L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.
[20]	T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12.
[21]	V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601.
[22]	V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147.
[23]	V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.

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