$L^2$-estimates of error in homogenization of parabolic equations with correctors taken into account

S. E. Pastukhova

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 1, pp. 134-151.

Résumé

We consider second-order parabolic equations with bounded measurable $\varepsilon$-periodic coefficients. To solve the Cauchy problem in the layer $\mathbb{R}^d\times(0,T)$ with the nonhomogeneous equation, we obtain approximations in the norm $\|\cdot\|_{L^2(\mathbb{R}^d\times(0,T))}$ with remainder of order $\varepsilon^2$ as $\varepsilon\to 0.$

Détail

Mots-clés : parabolic equations
Keywords: homogenization of solutions, homogenization error, corrector.

Bibliographie

[1] Bakhvalov N. S., “Osrednenie differentsialnykh uravnenii s chastnymi proizvodnymi s bystro ostsilliruyuschimi koeffitsientami”, Dokl. AN SSSR, 221:3 (1975), 516–519 | MR | Zbl

[2] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, Nauka, M., 1984

[3] Birman M. Sh., Suslina T. A., “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva usredneniya”, Algebra i analiz, 15:3 (2003), 1–108

[4] Birman M. Sh., Suslina T. A., “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104

[5] Vasilevskaya E. S., “Usrednenie parabolicheskoi zadachi Koshi s periodicheskimi koeffitsientami pri uchete korrektora”, Algebra i analiz, 21:1 (2009), 3–60 | MR

[6] Vasilevskaya E. S., Suslina T. A., “Usrednenie parabolicheskikh i ellipticheskikh periodicheskikh operatorov v $L_2(\mathbb{R}^d)$ pri uchete pervogo i vtorogo korrektorov”, Algebra i analiz, 24:2 (2012), 1–103 | MR

[7] Zhikov V. V., “Spektralnyi podkhod k asimptoticheskim zadacham diffuzii”, Diff. uravn., 25:1 (1989), 44–50 | MR | Zbl

[8] Zhikov V. V., “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

[9] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993

[10] Zhikov V. V., Pastukhova S. E., “Ob operatornykh otsenkakh v teorii usredneniya”, Usp. mat. nauk, 71:3 (2016), 3–98 | MR

[11] Meshkova Yu. M., Suslina T. A., “Usrednenie pervoi nachalno-kraevoi zadachi dlya parabolicheskikh sistem: operatornye otsenki pogreshnosti”, Algebra i analiz, 29:9 (2017), 99–158 | MR

[12] Miloslova A. A., Suslina T. A., “Usrednenie parabolicheskikh uravnenii vysokogo poryadka s periodicheskimi koeffitsientami”, Sovr. mat. Fundam. napravl., 67, no. 1 (2021), 130–191 | MR

[13] Pastukhova S. E., “Approksimatsiya eksponenty operatora diffuzii s mnogomasshtabnymi koeffitsientami”, Funkts. analiz i ego prilozh., 48:3 (2014), 34–51 | DOI | MR | Zbl

[14] Pastukhova S. E., “$L^2$-approksimatsii rezolventy ellipticheskogo operatora v perforirovannom prostranstve”, Sovr. mat. Fundam. napravl., 66, no. 2 (2020), 314–334

[15] Pastukhova S. E., “Uluchshennye $L^2$-approksimatsii rezolventy v usrednenii operatorov chetvertogo poryadka”, Algebra i analiz, 34:4 (2022), 74–106

[16] Pastukhova S. E., “Ob uluchshennykh approksimatsiyakh rezolventy v usrednenii operatorov vtorogo poryadka s periodicheskimi koeffitsientami”, Funkts. analiz i ego prilozh., 56:4 (2022), 93–104 | DOI

[17] Pastukhova S. E., Tikhomirov R. N., “Otsenki lokalno periodicheskogo i povtornogo usredneniya: parabolicheskie uravneniya”, Dokl. RAN, 428:2 (2009), 166–170 | Zbl

[18] Senik N. N., “Ob usrednenii nesamosopryazhennykh lokalno periodicheskikh ellipticheskikh operatorov”, Funkts. analiz i ego prilozh., 51, no. 2 (2017), 92–96 | DOI | Zbl

[19] Suslina T. A., “Ob usrednenii periodicheskikh parabolicheskikh sistem”, Funkts. analiz i ego prilozh., 38, no. 4 (2004), 86–90 | DOI | Zbl

[20] Bensoussan A., Lions J. L., Papanicolaou G., Asymptotic analysis for periodic structures, North Holland Publishing Co., Amsterdam—New York, 1978 | MR | Zbl

[21] Geng J., Shen Z., “Convergence rates in parabolic homogenization with time-dependent periodic coefficients”, J. Funct. Anal., 272 (2017), 2092–2113 | DOI | MR | Zbl

[22] Geng J., Shen Z., “Homogenization of parabolic equations with non-self-similar scales”, Arch. Ration. Mech. Anal., 236:8 (2020), 145–188 | DOI | MR | Zbl

[23] Meshkova Y., “Note on quantitative homogenization results for parabolic systems in $\mathbb{R}^d$”, J. Evol. Equ., 21 (2021), 763–769 | DOI | MR | Zbl

[24] Meshkova Yu. M., Suslina T. A., “Homogenization of initial boundary value problem for parabolic systems with periodic coefficients”, Appl. Anal., 95:8 (2016), 1736–1775 | DOI | MR | Zbl

[25] Niu W., Xu Y., “Convergence rates in homogenization of higher-order parabolic systems”, Discrete Contin. Dyn. Syst., 38:8 (2018), 4203–4229 | DOI | MR | Zbl

[26] Niu W., Yuan Y., “Convergence rate in homogenization of elliptic systems with singular perturbations”, J. Math. Phys., 60 (2019), 111509 | DOI | MR | Zbl

[27] Pastukhova S. E., “Estimates in homogenization of parabolic equations with locally periodic coefficients”, Asymptot. Anal., 66:3-4 (2010), 207–228 | MR | Zbl

[28] Pastukhova S. E., “Operator estimates in homogenization of elliptic systems of equations”, J. Math. Sci. (N. Y.), 226:4 (2017), 445–461 | DOI | MR | Zbl

[29] Pastukhova S. E., “$L^2$-estimates for homogenization of elliptic operators”, J. Math. Sci. (N. Y.), 244:4 (2020), 671–685 | DOI | MR | Zbl

[30] Pastukhova S. E., “Homogenization estimates for singularly perturbed operators”, J. Math. Sci. (N. Y.), 251:5 (2020), 724–747 | DOI | MR | Zbl

[31] Pastukhova S. E., “On resolvent approximations of elliptic differential operators with locally periodic coefficients”, Lobachevskii J. Math., 41:5 (2020), 818–838 | DOI | MR | Zbl

[32] Pastukhova S. E., “On resolvent approximations of elliptic differential operators with periodic coefficients”, Appl. Anal., 101:13 (2022), 4453–4474 | DOI | MR | Zbl

[33] Pastukhova S. E., “$L^2$-estimates for homogenization of diffusion operators with unbounded nonsymmetric matrices”, J. Math. Sci. (N. Y.), 268:4 (2022), 473–492 | DOI | MR | Zbl

[34] Senik N., “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898 | DOI | MR | Zbl

[35] Senik N., “Homogenization for locally periodic elliptic operators”, J. Math. Anal. Appl., 505:2 (2022), 125581 | DOI | MR | Zbl

[36] Suslina T. A., “Homogenization of a periodic parabolic Cauchy problem”, Nonlinear equations and spectral theory, Am. Math. Soc., Providence, 2007, 201–233 | MR | Zbl

[37] Suslina T. A., “Homogenization of a periodic parabolic Cauchy problem in the Sobolev space”, Math. Model. Nat. Phenom., 5:4 (2010), 390–447 | DOI | MR | Zbl

[38] Zhikov V. V., Pastukhova S. E., “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[39] Zhikov V. V., Pastukhova S. E., “Estimates of homogenization for a parabolic equation with periodic coefficients”, Russ. J. Math. Phys., 12:2 (2006), 224–237 | DOI | MR