Geodesic
Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 223-246 
V. V. Zhikov; S. E. Pastukhova. Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 223-246. http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a1/
@article{CMFD_2017_63_2_a1,
     author = {V. V. Zhikov and S. E. Pastukhova},
     title = {Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {223--246},
     year = {2017},
     volume = {63},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a1/}
}
TY  - JOUR
AU  - V. V. Zhikov
AU  - S. E. Pastukhova
TI  - Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2017
SP  - 223
EP  - 246
VL  - 63
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a1/
LA  - ru
ID  - CMFD_2017_63_2_a1
ER  - 
%0 Journal Article
%A V. V. Zhikov
%A S. E. Pastukhova
%T Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging
%J Contemporary Mathematics. Fundamental Directions
%D 2017
%P 223-246
%V 63
%N 2
%U http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a1/
%G ru
%F CMFD_2017_63_2_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The diffusion equation is considered in an infinite $1$-periodic medium. For its fundamental solution we find approximations at large values of time $t$. Precision of approximations has pointwise and integral estimates of orders $O(t^{-\frac{d+j+1}2})$ and $O(t^{-\frac{j+1}2}),$ $j=0,1,\dots$, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in $L^p$-spaces, $1\le p\le\infty$. For the analogous equation in an $\varepsilon$-periodic medium (here $\varepsilon$ is a small parameter) we obtain approximations of the operator exponential in $L^p$-operator norms for a fixed time with precision of order $O(\varepsilon^n)$, $n=1,2,\dots$.

[1] I. A. Aleksandrova, “Spectral method in asymptotic problems of diffusion with drift”, Mat. zametki, 59:5 (1996), 768–770 (in Russian) | DOI | MR | Zbl

[2] A. Iu. Beliaev, “Waves of compression in a fluid with air bubbles”, Prikl. mat. mekh., 52:3 (1988), 444–449 (in Russian) | Zbl

[3] A. Iu. Beliaev, Averaging in Problems of the Filtration Theory, Nauka, Moscow, 2004 (in Russian)

[4] M. S. Birman, T. A. Suslina, “Periodic differential operators of second order. Threshold properties and averaging”, Algebra i analiz, 15:5 (2003), 1–108 (in Russian) | MR | Zbl

[5] E. S. Vasilevskaia, “Averaging of parabolic Cauchy problem with periodic coefficients using a corrector”, Algebra i analiz, 21:1 (2008), 3–60 (in Russian) | MR | Zbl

[6] E. S. Vasilevskaia, T. A. Suslina, “Threshold approximations of a factorized self-adjoint operator family using the first and the second correctors”, Algebra i analiz, 23:2 (2011), 102–146 (in Russian) | MR | Zbl

[7] V. V. Zhikov, “Asymptotic behavior and stabilization of solutions of a second-order parabolic equation with lower-order terms”, Tr. Mosk. Mat. ob-va, 46, 1983, 69–98 (in Russian) | MR | Zbl

[8] V. V. Zhikov, “Spectral approach to asymptotic diffusion problems”, Diff. uravn., 25:1 (1989), 44–50 (in Russian) | MR | Zbl

[9] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Averaging of Differential Operators, Nauka, Moscow, 1993 (in Russian) | MR

[10] V. V. Zhikov, S. E. Pastukhova, “On operator estimates in the theory of averaging”, Usp. mat. nauk, 71:3 (2016), 27–122 (in Russian) | DOI | MR | Zbl

[11] T. Kato, Perturbation Theory for Linear Operators, Mir, Moscow, 1972, (Russian translation) | MR

[12] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Mir, Moscow, 1983, (Russian translation) | MR

[13] V. B. Korotkov, Integral Operators, Nauka, Novosibirsk, 1983 (in Russian) | MR

[14] O. A. Ladyzhenskaia, N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1973 (in Russian) | MR

[15] S. E. Pastukhova, “Approximations of operator exponential in a periodic diffusion problem with a drift”, Mat. sb., 204:2 (2013), 133–160 (in Russian) | DOI | MR | Zbl

[16] E. V. Sevostianova, “Asymptotic expansion of solution of a second-order elliptic equation with periodic fast oscillating coefficients”, Mat. sb., 115(157):2 (1981), 204–222 (in Russian) | MR | Zbl

[17] T. A. Suslina, “On averaging of periodic parabolic systems”, Funkts. analiz i ego prilozh., 38:4 (2004), 86–90 (in Russian) | DOI | MR | Zbl

[18] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, Mir, Moscow, 1967, (Russian translation) | MR

[19] Bensousan A., Lions J. L., Papanicolaou G., Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978 | MR

[20] Ortega J. H., Zuazua E., “Large time behavior in $\mathbb R^d$ for linear parabolic equations with periodic coefficients”, Asymptot. Anal., 22:1 (2000), 51–85 | MR | Zbl

[21] Pastukhova S. E., “Approximations of the exponential of an operator with periodic coefficients”, J. Math. Sci. (N.Y.), 181:5 (2012), 668–700 | DOI | MR | Zbl

[22] Pastukhova S. E., Tikhomirov R. N., “Error estimates of homogenization in the Neumann boundary problem for an elliptic equation with multiscale coefficients”, J. Math. Sci. (N.Y.), 216:2 (2016), 325–344 | DOI | MR | Zbl

[23] Zhikov V. V., Pastukhova S. E., “Estimates of homogenization for a parabolic equation with periodic coefficients”, Russ. J. Math. Phys., 13:2 (2006), 224–237 | DOI | MR | Zbl

[24] Zhikov V. V., Pastukhova S. E., “Bloch principle for elliptic differential operators with periodic coefficients”, Russ. J. Math. Phys., 23:2 (2016), 257–277 | DOI | MR | Zbl