Homogenization of Solutions of Initial Boundary Value Problems for Parabolic Systems

Yu. M. Meshkova   ; T. A. Suslina

Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 88-93

Résumé

Let ${\mathcal O}\subset {\mathbb R}^d$ be a bounded $C^{1,1}$ domain. In $L_2({\mathcal O};{\mathbb C}^n)$ we consider strongly elliptic operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by the differential expression $b({\mathbf D})^*g({\mathbf x}/\varepsilon)b({\mathbf D})$, $\varepsilon>0$, with Dirichlet and Neumann boundary conditions, respectively. Here $g({\mathbf x})$ is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and $b({\mathbf D})$ is a first-order differential operator. We find approximations of the operators $\exp(-A_{D,\varepsilon} t)$ and $\exp(-A_{N,\varepsilon} t)$ for fixed $t>0$ and small $\varepsilon$ in the $L_2 \to L_2$ and $L_2 \to H^1$ operator norms with error estimates depending on $\varepsilon$ and $t$. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.

Détail

Keywords: homogenization of periodic differential operators, parabolic systems, initial boundary value problems, effective operator, corrector, operator error estimates.

Bibliographie

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Math. and Appl., 5, North-Holland, Amsterdam–New York, 1978 | MR

[2] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[3] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Nauka, M., 1993 | MR | Zbl

[4] T. A. Suslina, Funkts. analiz i ego pril., 38:4 (2004), 86–90 | DOI | MR | Zbl

[5] T. A. Suslina, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. (2), 220, Amer. Math. Soc., Providence, RI, 2007, 201–233 | MR | Zbl

[6] V. V. Zhikov, S. E. Pastukhova, Russ. J. Math. Phys., 13:2 (2006), 224–237 | DOI | MR | Zbl

[7] T. A. Suslina, Math. Model. Nat. Phenom., 5:4 (2010), 390–447 | DOI | MR | Zbl

[8] E. S. Vasilevskaya, Algebra i analiz, 21:1 (2009), 3–60 | MR

[9] E. S. Vasilevskaya, T. A. Suslina, Algebra i analiz, 24:2 (2012), 1–103 | MR

[10] Yu. M. Meshkova, Algebra i analiz, 25:6 (2013), 125–177 | MR

[11] M. A. Pakhnin, T. A. Suslina, Algebra i analiz, 24:6 (2012), 139–177 | MR

[12] T. A. Suslina, Mathematika, 59:2 (2013), 463–476 | DOI | MR | Zbl

[13] T. A. Suslina, SIAM J. Math. Anal., 45:6 (2013), 3453–3493 | DOI | MR | Zbl

[14] T. A. Suslina, Funkts. analiz i ego pril., 48:4 (2014), 88–94 | DOI | Zbl

[15] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Math., Springer-Verlag, Heidelberg–Dordrecht–London–New York, 2012 | MR

[16] I. M. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR