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.
qd2125@columbia.edu
² Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA.
lipton@math.lsu.edu
³ Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
mengesha@utk.edu

Received: 17 January 2014
Revised: 20 February 2015
Accepted: 14 September 2015

Abstract

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.