Analysis of the nonlocal wave propagation problem with volume constraints

International audienceIn the current paper, we investigate a nonlocal hyperbolic problem with volume constraints. The main motivation of this work is to apply the nonlocal vector calculus, introduced and developed by DU et al. [3] to such problem. Moreover, based on some density arguments, some a priori estimates and using the Galerkin approach, we prove existence and uniqueness of a weak solution to the nonlocal wave equation. 1. Introduction. The study of nonlocal problems has gained great attention over the last two decades. Nonlocal models involve integral equations and fractional derivatives allowing nonlocal interactions, that is to say, the interaction may occur even when the closures of two domains have an empty intersection. Such models are effective in modeling material singularities and are widely considered in a variety of applications, including image analyses [6]-[10], phase transition [4][11], machine learning [12] and obstacle problem [5]... In a major advance in 2013, Du et al. [3] introduced nonlocal vector calculus as a nonlocal framework to understand and analyze nonlocal problems. It defines non-local fluxes , nonlocal analogues of the gradient, divergence, and curl operators, and presentes some basic nonlocal integral theorems that mimic the classical integral theorems of the vector calculus for differential operators, the authors have also provided connection between the nonlocal operators and their usual differential counterparts in a distributional sense then in a weak sense by introducing nonlocal weighted operators. The present paper was motivated by [2], where the authors threw light on the analogy between nonlocal and local diffusion problems with a convincing explanation of the usefulness, in the nonlocal case, of volume constraints which represent the nonlocal analogue of the boundary conditions of the classical theory. Our purpos