Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

Meshkova, Yu.; Suslina, T.

doi:10.1090/spmj/1521

Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates
HTML articles powered by AMS MathViewer

by Yu. M. Meshkova and T. A. Suslina
Translated by: T. A. Suslina

St. Petersburg Math. J. 29 (2018), 935-978

DOI: https://doi.org/10.1090/spmj/1521

Published electronically: September 4, 2018

PDF | Request permission

Abstract:

Let $\mathcal {O}\subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal {O};\mathbb {C}^n)$, a selfadjoint matrix second order elliptic differential operator $B_{D,\varepsilon }$, $0<\varepsilon \leq 1$, is considered with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator $B_{D,\varepsilon }$ is positive definite; its coefficients are periodic and depend on $\mathbf {x}/\varepsilon$. The behavior of the operator exponential $e^{-B_{D,\varepsilon }t}$, $t>0$, is studied as $\varepsilon \rightarrow 0$. Approximations for the exponential $e^{-B_{D,\varepsilon }t}$ are obtained in the operator norm on $L_2(\mathcal {O};\mathbb {C}^n)$ and in the norm of operators acting from $L_2(\mathcal {O};\mathbb {C}^n)$ to the Sobolev space $H^1(\mathcal {O};\mathbb {C}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

References

Grégoire Allaire, Yves Capdeboscq, Andrey Piatnitski, Vincent Siess, and M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Ration. Mech. Anal. 174 (2004), no. 2, 179–220. MR 2098106, DOI 10.1007/s00205-004-0332-7
N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788, DOI 10.1007/978-94-009-2247-1
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
Michael Birman and Tatyana Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71–107. MR 1882692
M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1
M. Sh. Birman and T. A. Suslina, Averaging of periodic elliptic differential operators taking a corrector into account, Algebra i Analiz 17 (2005), no. 6, 1–104 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897–973. MR 2202045, DOI 10.1090/S1061-0022-06-00935-6
M. Sh. Birman and T. A. Suslina, Averaging of periodic differential operators taking a corrector into account. Approximation of solutions in the Sobolev class $H^2(\Bbb R^d)$, Algebra i Analiz 18 (2006), no. 6, 1–130 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 6, 857–955. MR 2307356, DOI 10.1090/S1061-0022-07-00977-6
D. I. Borisov, Asymptotics of solutions of elliptic systems with rapidly oscillating coefficients, Algebra i Analiz 20 (2008), no. 2, 19–42 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 2, 175–191. MR 2423995, DOI 10.1090/S1061-0022-09-01043-7
Hi Jun Choe, Ki-Bok Kong, and Chang-Ock Lee, Convergence in $L^p$ space for the homogenization problems of elliptic and parabolic equations in the plane, J. Math. Anal. Appl. 287 (2003), no. 2, 321–336. MR 2022721, DOI 10.1016/S0022-247X(02)00464-X
Jun Geng and Zhongwei Shen, Convergence rates in parabolic homogenization with time-dependent periodic coefficients, J. Funct. Anal. 272 (2017), no. 5, 2092–2113. MR 3596717, DOI 10.1016/j.jfa.2016.10.005
Georges Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal. 40 (2004), no. 3-4, 269–286. MR 2107633
Georges Griso, Interior error estimate for periodic homogenization, Anal. Appl. (Singap.) 4 (2006), no. 1, 61–79. MR 2199793, DOI 10.1142/S021953050600070X
V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546, DOI 10.1007/978-3-642-84659-5
V. V. Zhikov, Asymptotic behavior and stabilization of solutions of a second-order parabolic equation with lowest terms, Trudy Moskov. Mat. Obshch. 46 (1983), 69–98 (Russian). MR 737901
V. V. Zhikov, On operator estimates in homogenization theory, Dokl. Akad. Nauk 403 (2005), no. 3, 305–308 (Russian). MR 2164541
V. V. Zhikov and S. E. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys. 12 (2005), no. 4, 515–524. MR 2201316
V. V. Zhikov and S. E. Pastukhova, Estimates of homogenization for a parabolic equation with periodic coefficients, Russ. J. Math. Phys. 13 (2006), no. 2, 224–237. MR 2262826, DOI 10.1134/S1061920806020087
V. V. Zhikov and S. E. Pastukhova, On operator estimates in homogenization theory, Uspekhi Mat. Nauk 71 (2016), no. 3(429), 27–122 (Russian, with Russian summary); English transl., Russian Math. Surveys 71 (2016), no. 3, 417–511. MR 3535364, DOI 10.4213/rm9710
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Carlos E. Kenig, Fanghua Lin, and Zhongwei Shen, Convergence rates in $L^2$ for elliptic homogenization problems, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 1009–1036. MR 2928140, DOI 10.1007/s00205-011-0469-0
S. M. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch. 46 (1983), 99–123 (Russian). MR 737902
V. A. Kondrat′ev and S. D. Èĭdel′man, Boundary-surface conditions in the theory of elliptic boundary value problems, Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 812–815 (Russian). MR 543538
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
O. A. Ladyženskaja and N. N. Ural′ceva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0211073
V. G. Maz′ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Monographs and Studies in Mathematics, vol. 23, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 785568
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Yu. M. Meshkova, Homogenization of the Cauchy problem for parabolic systems with periodic coefficients, Algebra i Analiz 25 (2013), no. 6, 125–177 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 25 (2014), no. 6, 981–1019. MR 3234842, DOI 10.1090/s1061-0022-2014-01326-x
Yu. M. Meshkova and T. A. Suslina, Homogenization of initial boundary value problems for parabolic systems with periodic coefficients, Appl. Anal. 95 (2016), no. 8, 1736–1775. MR 3505416, DOI 10.1080/00036811.2015.1068300
Yu. M. Meshkova and T. A. Suslina, Two-parametric error estimates in homogenization of second-order elliptic systems in $\Bbb R^d$, Appl. Anal. 95 (2016), no. 7, 1413–1448. MR 3499668, DOI 10.1080/00036811.2015.1122758
—, Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates, arXiv:1702.00550v4 (2017).
Yu. M. Meshkova and T. A. Suslina, Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients, Funktsional. Anal. i Prilozhen. 51 (2017), no. 3, 87–93 (Russian); English transl., Funct. Anal. Appl. 51 (2017), no. 3, 230–235. MR 3685305, DOI 10.1007/s10688-017-0187-y
Shari Moskow and Michael Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1263–1299. MR 1489436, DOI 10.1017/S0308210500027050
M. A. Pakhnin and T. A. Suslina, Operator error estimates for the homogenization of the elliptic Dirichlet problem in a bounded domain, Algebra i Analiz 24 (2012), no. 6, 139–177 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 6, 949–976. MR 3097556, DOI 10.1090/S1061-0022-2013-01274-X
Vyacheslav S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2) 60 (1999), no. 1, 237–257. MR 1721827, DOI 10.1112/S0024610799007723
Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
T. A. Suslina, On the averaging of periodic parabolic systems, Funktsional. Anal. i Prilozhen. 38 (2004), no. 4, 86–90 (Russian); English transl., Funct. Anal. Appl. 38 (2004), no. 4, 309–312. MR 2117512, DOI 10.1007/s10688-005-0010-z
T. A. Suslina, Homogenization of a periodic parabolic Cauchy problem, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 201–233. MR 2343612, DOI 10.1090/trans2/220/09
T. Suslina, Homogenization of a periodic parabolic Cauchy problem in the Sobolev space $H^1(\Bbb R^d)$, Math. Model. Nat. Phenom. 5 (2010), no. 4, 390–447. MR 2662463, DOI 10.1051/mmnp/20105416
T. A. Suslina, Homogenization in the Sobolev class $H^1(\Bbb R^d)$ for second-order periodic elliptic operators with the inclusion of first-order terms, Algebra i Analiz 22 (2010), no. 1, 108–222 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 1, 81–162. MR 2641084, DOI 10.1090/S1061-0022-2010-01135-X
T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: $L_2$-operator error estimates, Mathematika 59 (2013), no. 2, 463–476. MR 3081781, DOI 10.1112/S0025579312001131
Tatiana Suslina, Homogenization of the Neumann problem for elliptic systems with periodic coefficients, SIAM J. Math. Anal. 45 (2013), no. 6, 3453–3493. MR 3131481, DOI 10.1137/120901921
T. A. Suslina, Homogenization of elliptic operators with periodic coefficients that depend on the spectral parameter, Algebra i Analiz 27 (2015), no. 4, 87–166 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 27 (2016), no. 4, 651–708. MR 3580194, DOI 10.1090/spmj/1412
Qiang Xu, Uniform regularity estimates in homogenization theory of elliptic system with lower order terms, J. Math. Anal. Appl. 438 (2016), no. 2, 1066–1107. MR 3466080, DOI 10.1016/j.jmaa.2016.02.011
Qiang Xu, Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem, J. Differential Equations 261 (2016), no. 8, 4368–4423. MR 3537832, DOI 10.1016/j.jde.2016.06.027
Qiang Xu, Convergence rates for general elliptic homogenization problems in Lipschitz domains, SIAM J. Math. Anal. 48 (2016), no. 6, 3742–3788. MR 3566906, DOI 10.1137/15M1053335
Q. Xu and Sh. Zhou, Quantitative estimates in homogenization of parabolic systems of elasticity in Lipschitz cylinders, arXiv:1705.01479 (2017).