Convergence rates in homogenization of higher-order parabolic systems

Weisheng Niu; Yao Xu

doi:10.3934/dcds.2018183

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2018, Volume 38, Issue 8: 4203-4229. Doi: 10.3934/dcds.2018183

This issue Previous Article Bifurcation of limit cycles for a family of perturbed Kukles differential systems Next Article Second order regularity for degenerate nonlinear elliptic equations

Convergence rates in homogenization of higher-order parabolic systems

Weisheng Niu^1, and
Yao Xu^2, ,

1.
School of Mathematical Science, Anhui University, Hefei 230601, China
2.
Department of Mathematics, Nanjing University, Nanjing 210093, China

^* Corresponding author: Yao Xu
^* Corresponding author: Yao Xu

Received: January 2018

Published: May 2018

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Abstract

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\varepsilon )$ convergence rate in the space $L^2(0, T; H^{m-1}(\Omega ))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by [25] is used here.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35K52.

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[1]	S. N. Armstrong, A. Bordas and J. C. Mourrat, Quantitative stochastic homogenization and regularity theory of parabolic equations, preprint, arXiv: 1705.07672.
[2]	M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847. doi: 10.1002/cpa.3160400607.
[3]	A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5, North-Holland Publishing Company Amsterdam, 1978.
[4]	S. Byun and Y. Jang, Calderón-Zygmund estimate for homogenization of parabolic systems, Discrete Contin. Dyn. Syst., 36 (2016), 6689-6714. doi: 10.3934/dcds.2016091.
[5]	H. Dong and H. Zhang, Conormal problem of higher-order parabolic systems with time irregular coefficients, Trans. Amer. Math. Soc., 368 (2016), 7413-7460. doi: 10.1090/tran/6605.
[6]	J. Geng and Z. Shen, Uniform regularity estimates in parabolic homogenization, Indiana Univ. Math. J., 64 (2015), 697-733. doi: 10.1512/iumj.2015.64.5503.
[7]	J. Geng and Z. Shen, Convergence rates in parabolic homogenization with time-dependent periodic coefficients, J. Funct. Anal., 272 (2017), 2092-2113. doi: 10.1016/j.jfa.2016.10.005.
[8]	G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286.
[9]	S. Gu, Convergence rates in homogenization of Stokes systems, J. Differential Equations, 260 (2016), 5796-5815. doi: 10.1016/j.jde.2015.12.017.
[10]	V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.
[11]	C. E. Kenig, F. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal., 203 (2012), 1009-1036. doi: 10.1007/s00205-011-0469-0.
[12]	C. E. Kenig, F. Lin and Z. Shen, Homogenization of elliptic systems with {N}eumann boundary conditions, J. Amer. Math. Soc., 26 (2013), 901-937. doi: 10.1090/S0894-0347-2013-00769-9.
[13]	C. E. Kenig, F. Lin and Z. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure Appl. Math., 67 (2014), 1219-1262. doi: 10.1002/cpa.21482.
[14]	A. A. Kukushkin and T. A. Suslina, Homogenization of high-order elliptic operators with periodic coefficients, Algebra i Analiz, 28 (2016), 89-149. doi: 10.1090/spmj/1439.
[15]	Y. M. Meshkova and T. A. Suslina, Homogenization of solutions of initial boundary value problems for parabolic systems, Funct. Anal. Appl., 49 (2015), 72-76. doi: 10.1007/s10688-015-0087-y.
[16]	Y. M. Meshkova and T. A. Suslina, Homogenization of initial boundary value problems for parabolic systems with periodic coefficients, Appl. Anal., 95 (2016), 1736-1775. doi: 10.1080/00036811.2015.1068300.
[17]	W. Niu, Z. Shen and Y. Xu, Convergence rates and interior estimates in homogenization of higher order elliptic systems, J. Funct. Anal., 274 (2018), 2356-2398. doi: 10.1016/j.jfa.2018.01.012.
[18]	W. Niu and Y. Xu, Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems, preprint, arXiv: 1709.04097.
[19]	S. E. Pastukhova, Estimates in homogenization of higher-order elliptic operators, Appl. Anal., 95 (2016), 1449-1466. doi: 10.1080/00036811.2016.1151495.
[20]	S. E. Pastukhova, Operator error estimates for homogenization of fourth order elliptic equations, St. Petersburg Math. J., 28 (2017), 273-289. doi: 10.1090/spmj/1450.
[21]	Z. Shen, Boundary estimates in elliptic homogenization, Anal. PDE, 10 (2017), 653-694. doi: 10.2140/apde.2017.10.653.
[22]	Z. Shen, Lectures on Periodic Homogenization of Elliptic Systems, preprint, arXiv: 1710.11257.
[23]	Z. Shen and J. Zhuge, Convergence rates in periodic homogenization of systems of elasticity, Proc. Amer. Math. Soc., 145 (2017), 1187-1202. doi: 10.1090/proc/13289.
[24]	T. A. Suslina, Homogenization of the parabolic Cauchy problem in the Sobolev class $H^1(\mathbb{R}^d)$, Funct. Anal. Appl., 44 (2010), 318-322. doi: 10.1007/s10688-010-0043-9.
[25]	T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: $L^2$-operator error estimates, Mathematika, 59 (2013), 463-476. doi: 10.1112/S0025579312001131.
[26]	T. A. Suslina, Homogenization of the Neumann problem for elliptic systems with periodic coefficients, SIAM J. Math. Anal., 45 (2013), 3453-3493. doi: 10.1137/120901921.
[27]	T. A. Suslina, Homogenization of the dirichlet problem for higher-order elliptic equations with periodic coefficients, Algebra i Analiz, 29 (2017), 139-192. doi: 10.1090/spmj/1496.
[28]	T. A. Suslina, Homogenization of the neumann problem for higher order elliptic equations with periodic coefficients, Complex Variables and Elliptic Equations, 0 (2017), 1-31. doi: 10.1080/17476933.2017.1365845.
[29]	Q. Xu and S. Zhou, Quantitative estimates in homogenization of parabolic systems of elasticity in lipschitz cylinders, preprint, arXiv: 1705.01479.
[30]	Y. Xu and W. Niu, Convergence rates in almost-periodic homogenization of higher-order elliptic systems, preprint, arXiv: 1712.01744.
[31]	V. V. Zhikov and S. E. Pastukhova, Estimates of homogenization for a parabolic equation with periodic coefficients, Russ. J. Math. Phys., 13 (2006), 224-237. doi: 10.1134/S1061920806020087.