Taiwanese Journal of Mathematics

ON THE HOMOGENIZATION OF SECOND ORDER DIFFERENTIAL EQUATIONS

Jiann-Sheng Jiang, Kung-Hwang Kuo, and Chi-Kun Lin

Full-text: Open access

Abstract

We discuss the homogenization process of second order differential equations involving highly oscillating coefficients in the time and space variables. It generate memory or nonlocal effect. For initial value problems, the memory kernels are described by Volterra integral equations; and for boundary value problems, they are characterized by Fredholm integral equations. When the equation is translation (in time or in space) invariant, the memory or nonlocal kernel can be represented explicitly in terms of the Young's measure.

Article information

Source
Taiwanese J. Math., Volume 9, Number 2 (2005), 215-236.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407797

Digital Object Identifier
doi:10.11650/twjm/1500407797

Mathematical Reviews number (MathSciNet)
MR2142574

Zentralblatt MATH identifier
1077.35020

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35B35: Stability

Keywords
homogenization weak limit green's function Volterra and Fredholm integral equations Young's measure kinetic formulation dunford-Taylor integral eigenfunction expansion

Citation

Jiang, Jiann-Sheng; Kuo, Kung-Hwang; Lin, Chi-Kun. ON THE HOMOGENIZATION OF SECOND ORDER DIFFERENTIAL EQUATIONS. Taiwanese J. Math. 9 (2005), no. 2, 215--236. doi:10.11650/twjm/1500407797. https://projecteuclid.org/euclid.twjm/1500407797


Export citation

References

  • [1.] G. Allaire, Homogenization and Applications to Material Science, Lecture note at Newton Institute, Cambridge, 1999.
  • [2.] Y. Amirat, K. Hamdache and A. Ziani, Homogénéisation d`équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux, Ann. Inst. Henri Poincaré, Analyse non linéaire, 6, (1989), 397-417.
  • [3.] Y. Amirat, K. Hamdache and A. Ziani, Kinetic formulation for a transport equation with memory, Commun. in Partial Differential Equations, 16, (1991), 1287-1311.
  • [4.] Y. Amirat, K. Hamdache and A. Ziani, Some results on homogenization of convection diffusion equations, Arch. Rational Mech. Anal., 114, (1991), 155-178.
  • [5.] Y. Amirat, K. Hamdache and A. Ziani, On homogenization of ordinary differential equations and linear transport equations, Calculus of Variations, Homogenization and Continuum Mechanics, (G. Bouchitté, G. Buttazza and P. Suquet ed.) pp. 29-50, World Scientific, Singapore, 1994.
  • [6.] F. Bornemann, Homogenization in time of Singularly Perturbed Mechanical Systems, Lecture Note in Mathematics, 1687, Springer-Verlag, Berlin and New York, 1998.
  • [7.] N. Antonić, Memory effects in homogenization linear second-order equations, Arch. Rational Mech. Anal., 125, (1993), 1-24.
  • [8.] E. De Giorgi, Some remarks on $\Gamma$-convergence and least squares method, In Composite Media and Homogenization Theory, (ed. Dall Maso and Dell'Antonio), Birkhäuser, Boston, (135-142) 1991.
  • [9.] J.-S. Jiang and C.-K. Lin, Nonlocal effects induced by homogenization of a second order differential equation, In: Proceedings of the IMC'94, (Y. Fong, W.-J. Huang, Y.-J. Lee and N.-C. Wong ed.) pp. 131-148, World Scientific, Singapore, 1996.
  • [10.] J.-S. Jiang and C.-K. Lin, Homogenization of the Dirac-like system, Mathematical Models and Methods in Applied Sciences, 11 (2001), 433-458.
  • [11.] J.-S. Jiang, K.-H. Kuo and C.-K. Lin, Homogenization of second order equation with spatial dependent coefficient, Discrete and Continuous Dynamical System, 12, No.2 (2005) 303-313.
  • [12.] J.-S. Jiang, K.-H. Kuo and C.-K. Lin, Homogenization and memory effect of a $3\times 3$ systems, Journal of Mathematics of Kyoto University, (under revised) (2005).
  • [13.] D. L. Koch and J. F. Brady, A nonlocal description of advection-diffusion with application to dispersion in porous media, J. Fluid Mech.,180, (1987), 387-403.
  • [14.] D. L. Koch, R. G. Cox, H. Brenner and J. F. Brady, The effect of order on dispersion in porous media, J. Fluid Mech., 200, (173-188) 1989.
  • [15.] R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II, Nonequilibrium statistical mchanics, Springer-Verlag, Berlin and New York, 1991
  • [16.] M. L. Mascarenhas, A linear homogenization problem with time dependent coefficient, Trans. Amer. Math. Soc., 281, (1984), 179-195.
  • [17.] M. L. Mascarenhas, Memory effect and $\Gamma$-convergence, Proceedings of the Royal Society of Edinburgh, 123A, (1993), 311-322.
  • [18.] A. Mikelić, Homogenization Theory and Applications to Filtration Through Porous Media, in Filtration in Porous Media and Industrial Application, (ed. M. S. Espedal, A. Fasano and A. Mikelić) pp. 127-214, Lecture Note in Mathematics 1734, Springer-Verlag, Berlin and New York, 1998.
  • [19.] M. Renardy and W. J. Hrusa, Mathematical Problems in Viscoelasticity, Longman, New York, 1987.
  • [20.] E. Sanchez-Palencia, Non-homogeneous media and vibration theory, Lecture Note in Physics, Vol. 127, Springer-Verlag, Berlin and New York, 1980.
  • [21.] L. Tartar, Compensated compactness and applications to partial differential equations, In: Research notes in mathematics, nonlinear analysis and mechanics, (R. J. Knops ed.) pp. 136-211, Heriot-Watt Symposium, Vol. 4, Pitman Press, New York, 1979.
  • [22.] L. Tartar, Remark on homogenization, In: Homogenization and effective moduli of materials and media, (J. L. Ericksen et al. ed.) pp. 228-246, Springer-Verlag, Berlin and New York, 1986.
  • [23.] L. Tartar, Nonlocal effects induced by homogenization, In: Partial Differential Equations and the Calculus of Variations (Essays in Honor of E. De Giorgi) Vol. II, (F. Colombini et al. ed.) pp. 925-938, Birkhäuser, Boston, 1989.
  • [24.] L. Tartar, Memory effects and homogenization, Arch. Rational Mech. Anal., 111, (1990), 121-133.
  • [25.] L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proceedings of the Royal Society of Edinburgh, 115A, (1990), 193-230.
  • [26.] L. Tartar, On mathematical tools for studying partial differential equations of continuum physica: H-measure and Young's measure, In: Developments in Partial Differential Equations and Applications to Mathematical Physics, (G. Buttazzo et al. ed.) pp. 201-217, Plenum Press, New York, 1992.
  • [27.] L. Tartar, Homogenization and Hyperbolicity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25, (1997), 785-805.
  • [28.] L. Tartar, An introduction to the homogenization method in optimal design, in Filtration in Porous Media and Industrial Application, (ed. A. Cellina and A. Ornelas) pp. 47-156, Lecture Note in Mathematics 1740, Springer-Verlag, Berlin and New York, 1998
  • [29.] V. V. Zhikov, Estimates for the averaged matrix and the averaged tensor, Russian Math. Surveys, 46, (1991), 65-136.