Long-time homogenization and asymptotic ballistic transport of classical waves

Benoit, Antoine; Gloria, Antoine

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 703-759.

Considérons un opérateur elliptique sous forme divergence à coefficients symétriques non constants. Si ces coefficients sont périodiques, la théorie de Floquet-Bloch permet de diagonaliser l'opérateur elliptique, ce qui est crucial pour l'étude des propriétés spectrales de l'opérateur et le point de départ usuel pour l'étude des propriétés d'homogénéisation en temps long de l'opérateur des ondes associé. Quand les coefficients ne sont pas périodiques (disons quasi-périodiques, presque périodiques, ou aléatoires stationnaires ergodiques), la théorie de Floquet-Bloch ne s'applique plus et les propriétés spectrales ainsi que le comportement en temps long de l'opérateur des ondes associé ne sont pas claires a priori. Aux basses fréquences, nous pouvons cependant considérer un développement de Taylor formel des ondes de Bloch (que celles-ci existent ou non) en se basant sur des correcteurs introduits en homogénéisation elliptique. Ces ondes de Taylor-Bloch diagonalisent l'opérateur elliptique à un terme d'erreur près (un “défaut propre”), que nous exprimons à l'aide d'une nouvelle famille de correcteurs étendus. Nous utilisons cette formulation des défauts propres pour quantifier les propriétés de transport et d'homogénéisation en temps long pour l'équation des ondes associée en termes de croissance spatiale des correcteurs étendus. D'une part, cela quantifie la validité de l'homogénéisation en temps long (à la fois pour l'opérateur homogénéisé standard et pour des versions d'ordre supérieur). D'autre part, cela nous permet d'établir le transport balistique asymptotique des ondes classiques aux basses énergies pour des opérateurs presque périodiques et aléatoires.

Consider an elliptic operator in divergence form with symmetric coefficients. If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization. When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associated operator are unclear. At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization. The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an “eigendefect”), which we express with the help of a new family of extended correctors. We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times for the wave equation in terms of the spatial growth of these extended correctors. On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions). On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.

Allaire, G.; Briane, M.; Vanninathan, M. A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA J., Volume 73 (2016), pp. 237-259 (ISSN: 2254-3902) | DOI | MR | Zbl

Allaire, G.; Conca, C. Analyse asymptotique spectrale de l'équation des ondes. Complétude du spectre de Bloch, C. R. Acad. Sci. Paris Sér. I Math., Volume 321 (1995), pp. 557-562 (ISSN: 0764-4442) | MR | Zbl

Allaire, G.; Conca, C. Analyse asymptotique spectrale de l'équation des ondes. Homogénéisation par ondes de Bloch, C. R. Acad. Sci. Paris Sér. I Math., Volume 321 (1995), pp. 293-298 (ISSN: 0764-4442) | MR | Zbl

Armstrong, S.; Gloria, A.; Kuusi, T. Bounded correctors in almost periodic homogenization, Arch. Ration. Mech. Anal., Volume 222 (2016), pp. 393-426 (ISSN: 0003-9527) | DOI | MR | Zbl

Abdulle, A.; Grote, M. J.; Stohrer, C. Finite element heterogeneous multiscale method for the wave equation: long-time effects, Multiscale Model. Simul., Volume 12 (2014), pp. 1230-1257 (ISSN: 1540-3459) | DOI | MR | Zbl

Armstrong, S.; Kuusi, T.; Mourrat, J.-C. The additive structure of elliptic homogenization, Invent. math., Volume 208 (2017), pp. 999-1154 (ISSN: 0020-9910) | DOI | MR | Zbl

Abdulle, A.; Pouchon, T. A priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time, SIAM J. Numer. Anal., Volume 54 (2016), pp. 1507-1534 (ISSN: 0036-1429) | DOI | MR | Zbl

Allaire, G.; Palombaro, M.; Rauch, J. Diffractive behavior of the wave equation in periodic media: weak convergence analysis, Ann. Mat. Pura Appl., Volume 188 (2009), pp. 561-589 (ISSN: 0373-3114) | DOI | MR | Zbl

Allaire, G.; Palombaro, M.; Rauch, J. Diffractive geometric optics for Bloch wave packets, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 373-426 (ISSN: 0003-9527) | DOI | MR | Zbl

Armstrong, S. N.; Smart, C. K. Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016), pp. 423-481 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Aizenman, M.; Warzel, S., Graduate Studies in Math., 168, Amer. Math. Soc., 2015, 326 pages (ISBN: 978-1-4704-1913-4) | MR

Bella, P.; Fehrman, B.; Fischer, J.; Otto, F. Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors, SIAM J. Math. Anal., Volume 49 (2017), pp. 4658-4703 (ISSN: 0036-1410) | DOI | MR | Zbl

Brahim-Otsmane, S.; Francfort, G. A.; Murat, F. Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl., Volume 71 (1992), pp. 197-231 (ISSN: 0021-7824) | MR | Zbl

Chen, T. Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3, J. Stat. Phys., Volume 120 (2005), pp. 279-337 (ISSN: 0022-4715) | DOI | MR | Zbl

Conca, C.; Orive, R.; Vanninathan, M. On Burnett coefficients in periodic media, J. Math. Phys., Volume 47 (2006) (ISSN: 0022-2488) | DOI | MR | Zbl

Conca, C.; Vanninathan, M. Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., Volume 57 (1997), pp. 1639-1659 (ISSN: 0036-1399) | DOI | MR | Zbl

Duerinckx, M.; Gloria, A. Multiscale functional inequalities in probability: Concentration properties (preprint arXiv:1711.03148 ) | MR

Duerinckx, M.; Gloria, A. Multiscale functional inequalities in probability: Constructive approach (preprint arXiv:1711.03152 ) | MR

Duerinckx, M.; Gloria, A.; Shirley, C. Approximate normal form via Floquet-Bloch theory: Nehoroser stability for linear waves in quasiperiodic media (preprint arXiv:1809.07106 ) | MR

Dohnal, T.; Lamacz, A.; Schweizer, B. Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simul., Volume 12 (2014), pp. 488-513 (ISSN: 1540-3459) | DOI | MR | Zbl

Dohnal, T.; Lamacz, A.; Schweizer, B. Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptot. Anal., Volume 93 (2015), pp. 21-49 (ISSN: 0921-7134) | DOI | MR | Zbl

Figotin, A.; Klein, A. Localization of classical waves. I. Acoustic waves, Comm. Math. Phys., Volume 180 (1996), pp. 439-482 http://projecteuclid.org/euclid.cmp/1104287356 (ISSN: 0010-3616) | DOI | MR | Zbl

Francfort, G. A.; Murat, F. Oscillations and energy densities in the wave equation, Comm. Partial Differential Equations, Volume 17 (1992), pp. 1785-1865 (ISSN: 0360-5302) | DOI | MR | Zbl

Gloria, A.; Habibi, Z. Reduction in the resonance error in numerical homogenization II: Correctors and extrapolation, Found. Comput. Math., Volume 16 (2016), pp. 217-296 (ISSN: 1615-3375) | DOI | MR | Zbl

Gloria, A.; Neukamm, S.; Otto, F. A regularity theory for random elliptic operators (preprint arXiv:1409.2678 ) | MR

Gloria, A.; Neukamm, S.; Otto, F. Quantitative stochastic homogenization for correlated fields (preprint arXiv:1409.2678 )

Gloria, A.; Neukamm, S.; Otto, F. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. math., Volume 199 (2015), pp. 455-515 (ISSN: 0020-9910) | DOI | MR | Zbl

Gloria, A.; Otto, F. The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations (preprint arXiv:1510.08290 )

Gloria, A.; Otto, F. An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., Volume 39 (2011), pp. 779-856 (ISSN: 0091-1798) | DOI | MR | Zbl

Gloria, A.; Otto, F. An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., Volume 22 (2012), pp. 1-28 (ISSN: 1050-5164) | DOI | MR | Zbl

Gloria, A.; Otto, F. Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc. (JEMS), Volume 19 (2017), pp. 3489-3548 (ISSN: 1435-9855) | DOI | MR | Zbl

Gu, Y. High order correctors and two-scale expansions in stochastic homogenization, Probab. Theory Related Fields, Volume 169 (2017), pp. 1221-1259 (ISSN: 0178-8051) | DOI | MR | Zbl

Kozlov, S. M. Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), Volume 107(149) (1978), p. 199-217, 317 (ISSN: 0368-8666) | MR | Zbl

Lamacz, A. Dispersive effective models for waves in heterogeneous media, Math. Models Methods Appl. Sci., Volume 21 (2011), pp. 1871-1899 (ISSN: 0218-2025) | DOI | MR | Zbl

Naddaf, A.; Spencer, T. Estimates on the variance of some homogenization problems (preprint https://pdfs.semanticscholar.org/83de/4ab157b8ff3f7a14b2fd709bd0df1004ed20.pdf )

Papanicolaou, G. Mathematical problems in geophysical wave propagation, Doc. Math. (Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)) (1998), pp. 403-427 (ISSN: 1431-0635) | MR | Zbl

Ryzhik, L.; Papanicolaou, G.; Keller, J. B. Transport equations for elastic and other waves in random media, Wave Motion, Volume 24 (1996), pp. 327-370 (ISSN: 0165-2125) | DOI | MR | Zbl

Santosa, F.; Symes, W. W. A dispersive effective medium for wave propagation in periodic composites, SIAM J. Appl. Math., Volume 51 (1991), pp. 984-1005 (ISSN: 0036-1399) | DOI | MR | Zbl

Stollmann, P., Progress in Mathematical Physics, 20, Birkhäuser, 2001, 166 pages (ISBN: 0-8176-4210-2) | DOI | MR | Zbl