Skip to main content
SpringerLink
Log in

Approximation of the exponential of a diffusion operator with multiscale coefficients

  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

A multiscale homogenization estimate for a parabolic diffusion equation under minimal regularity conditions is proved. This makes it possible to treat the result as an estimate in the operator norm for the difference of the operator exponentials of the initial and homogenized equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam-New York, 1978.

    MATH  Google Scholar 

  2. G. Allaire and M. Briane, “Multiscale convergence and reiterated homogenization,” Proc. Roy. Soc. Edinburgh Sect. A, 126:2 (1996), 297–342.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall, “Reiterated homogenization of monotone operators,” C. R. Acad. Sci. Paris Ser. I Math., 330:8 (2000), 675–680.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall, “Reiterated homogenization of nonlinear monotone operators,” Chinese Ann. Math. Ser. B, 22:1 (2001), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Lukkassen, G. Nguetseng, and P. Wall, “Two-scale convergence,” Int. J. Pure Appl. Math., 2:1 (2002), 35–86.

    MathSciNet  MATH  Google Scholar 

  6. M. Sh. Birman and T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization,” Algebra i Analiz, 15:5 (2003), 1–108; English transl.: St. Petersburg Math. J., 15:5 (2004), 639–714.

    MathSciNet  Google Scholar 

  7. V. V. Zhikov, “On operator estimates in homogenization theory,” Dokl. Ross. Akad. Nauk, 403:3 (2005), 305–308; English transl.: Russian Acad. Sci. Dokl. Math., 72:1 (2005), 535–538.

    MathSciNet  Google Scholar 

  8. V. V. Zhikov, “Some estimates from homogenization theory,” Dokl. Ross. Akad. Nauk, 406:5 (2006), 597–601; English transl.: Russian Acad. Sci. Dokl. Math., 73:1 (2006), 96–99.

    MathSciNet  Google Scholar 

  9. S. E. Pastukhova, “Some estimates from homogenized elasticity problems,” Dokl. Ross. Akad. Nauk, 406:5 (2006), 604–608; English transl.: Russian Acad. Sci. Dokl. Math., 73:1 (2006), 102–106.

    MathSciNet  Google Scholar 

  10. V. V. Zhikov and S. E. Pastukhova, “On operator estimates for some problems in homogenization theory,” Russ. J. Math. Phys., 12:4 (2005), 515–524.

    MathSciNet  MATH  Google Scholar 

  11. V. V. Zhikov and S. E. Pastukhova, “Estimates of homogenization for a parabolic equation with periodic coefficients,” Russ. J. Math. Phys., 13:2 (2006), 351–265.

    Article  MathSciNet  Google Scholar 

  12. V. V. Zhikov and S. V. Tikhomirova, “On operator estimates in nonsymmetric averaging problems,” Sovrem. Matem. i Ee Prilozhen., 33 (2005), 124–128; English transl.: J. Math. Sci., 144:1 (2007), 3870–3874.

    Google Scholar 

  13. G. Cardone, S. E. Pastukhova, and V. V. Zhikov, “Some estimates for non-linear homogenization,” Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 29 (2005), 101–110.

    MathSciNet  Google Scholar 

  14. V. V. Zhikov and S. E. Pastukhova, “Homogenization of degenerate elliptic equations,” Sibirsk. Mat. Zh., 49:1 (2008), 101–124; English transl.: Siberian Math. J., 49:1 (2008), 80–101.

    MathSciNet  MATH  Google Scholar 

  15. S. E. Pastukhova and S. V. Tikhomirova, “Elliptic equation with nonsymmetric matrix: Averaging of the “variational solutions”,” Mat. Zametki, 81:4 (2007), 631–635; English transl.: Math. Notes, 81:3–4 (2007), 560–565.

    Article  MathSciNet  Google Scholar 

  16. S. E. Pastukhova, “Operator estimates in nonlinear problems of reiterated homogenization,” Trudy Mat. Inst. Steklov., 261 (2008), 220–233; English transl.: Proc. Steklov Inst. Math., 261 (2008), 214–228.

    MathSciNet  Google Scholar 

  17. S. E. Pastukhova and R. N. Tikhomirov, “Operator estimates in reiterated and locally periodic homogenization,” Dokl. Ross. Akad. Nauk, 415:3 (2007), 304–309; English transl.: Russian Acad. Sci. Dokl. Math., 76:1 (2007), 548–553.

    MathSciNet  Google Scholar 

  18. S. E. Pastukhova, “Estimates in homogenization for parabolic equations with locally periodic coefficients,” Asymptot. Anal., 66:3–4 (2010), 207–228.

    MathSciNet  MATH  Google Scholar 

  19. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.

    Book  Google Scholar 

  20. T. A. Suslina, “On homogenization of periodic parabolic systems,” Funkts. Anal. Prilozhen., 38:4 (2004), 86–90; English transl.: Functional Anal. Appl., 38:4 (2004), 309–312.

    Article  MathSciNet  Google Scholar 

  21. T. A. Suslina, “Homogenization of periodic parabolic Cauchy problem,” in: Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, 201–233.

    Google Scholar 

  22. E. S. Vasilevskaya and T. A. Suslina, “Homogenization of parabolic and elliptic periodic operators in L 2(ℝd) with the first and second correctors taken into account,” Algebra i Analiz, 24:2 (2012), 1–103; English transl.: St. Petersburg Math. J., 24:2 (2013), 185–261.

    MathSciNet  Google Scholar 

  23. V. V. Zhikov, “On the spectral method in homogenization theory,” Trudy Mat. Inst. Steklov., Ross. Akad. Nauk, 250 (2005), 95–104; English transl.: Proc. Steklov Inst. Math., 250 (2005), 85–94.

    Google Scholar 

  24. S. E. Pastukhova, “On approximations of operator exponentials with periodic coefficients,” Probl. Mat. Anal., 2012, No. 63, 81–108.

    Google Scholar 

  25. S. E. Pastukhova, “Approximations of the operator exponential in a periodic diffusion problem with drift,” Mat. Sb., 204:2 (2013), 133–160; English transl.: Sb. Math., 204:2 (2013), 280–306.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow State Institute of Radio Engineering, Electronics, and Automation, Moscow, Russia

    S. E. Pastukhova

Authors
  1. S. E. Pastukhova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. E. Pastukhova.

Additional information

__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 3, pp. 34–51, 2014

Original Russian Text Copyright © by S. E. Pastukhova

This work was supported by RFBR grant no. 14-01-00192A and by the Russian Science Foundation (project no. 14-11-00398).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pastukhova, S.E. Approximation of the exponential of a diffusion operator with multiscale coefficients. Funct Anal Its Appl 48, 183–197 (2014). https://doi.org/10.1007/s10688-014-0060-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10688-014-0060-1

Key words

Search

Navigation