August  2018, 38(8): 4203-4229. doi: 10.3934/dcds.2018183

Convergence rates in homogenization of higher-order parabolic systems

1. 

School of Mathematical Science, Anhui University, Hefei 230601, China

2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Yao Xu

Received  January 2018 Published  May 2018

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.

Citation: Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
References:
[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. BensoussanJ. 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. JikovS. 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. KenigF. 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. KenigF. 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. KenigF. 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. NiuZ. 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.

show all references

References:
[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. BensoussanJ. 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. JikovS. 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. KenigF. 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. KenigF. 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. KenigF. 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. NiuZ. 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.

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