2017/01/01 by Antoine Benoît, Benoit, Antoine, Antoine Gloria +1 · 1 citation
Computer Science · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1701.08600
openalex publication_date 2017/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an elliptic operator in divergence form with symmetric\ncoefficients.If the diffusion coefficients are periodic, the Bloch theorem\nallows one to diagonalize the elliptic operator, which is key to the spectral\nproperties of the elliptic operator and the usual starting point for the study\nof its long-time homogenization.When the coefficients are not periodic (say,\nquasi-periodic, almost periodic, or random with decaying correlations at\ninfinity), the Bloch theorem does not hold and both the spectral properties and\nthe long-time behavior of the associatedoperator are unclear.At low\nfrequencies, we may however consider a formal Taylor expansion of Bloch waves\n(whether they exist or not) based on correctors in elliptic homogenization.The\nassociated Taylor-Bloch waves diagonalize the elliptic operator up to an error\nterm (an "eigendefect"), which we express with the help of a new family of\nextended correctors.We use the Taylor-Bloch waves with eigendefects to quantify\nthe transport properties and homogenization error over large timesfor the wave\nequation in terms of the spatial growth of these extended correctors.On the one\nhand, this quantifies the validity of homogenization over large times (both for\nthe standard homogenized equation and higher-order versions).On the other hand,\nthis allows us to prove asymptotic ballistic transport of classical waves at\nlow energies for almost periodic and random operators.\n