Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media∗,∗∗,∗∗∗
1 Department of Applied Physics and Applied Mathematics,
Columbia University, New York, 10027, USA.
2 Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA.
3 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
Revised: 20 February 2015
Accepted: 14 September 2015
The method of two scale convergence is implemented to study the homogenization of time-dependent nonlocal continuum models of heterogeneous media. Two integro-differential models are considered: the nonlocal convection-diffusion equation and the state-based peridynamic model in nonlocal continuum mechanics. The asymptotic analysis delivers both homogenized dynamics as well as strong approximations expressed in terms of a suitable corrector theory. The method provides a natural analog to that for the time-dependent local PDE models with highly oscillatory coefficients with the distinction that the driving operators considered in this work are bounded.
Mathematics Subject Classification: 74Q05 / 74E05 / 74H10 / 45F99 / 45P05
Key words: Multiscale analysis / peridynamics / nonlocal equations / Navier equation / homogenization / heterogeneous materials / two-scale convergence
Qiang Du’s research is supported in part by NSF grant DMS-1318586, DMS-1312809 and AFOSR MURI center for Material Failure Prediction through peridynamics.
Robert Lipton’s research is supported in part by NSF grant DMS-1211066 and NSF EPSCOR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents.
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