Search

We introduce a new class of nonlocal nonlinear conservation laws in one space dimension that allow for nonlocal interactions over a finite horizon. The proposed model, which we refer to as the nonlocal pair-interaction model, inherits at the co ntinuum level the unwinding feature of finite difference schemes for local hyperbolic conservation laws, so that the maximum principle and certain monotonicity properties hold and, consequently, the entropy inequalities are naturally satisfied. We establish a global-in-time well-posedness theory fo r these models which covers a broad class of initial data. Moreover, in the limit when the horizon par ameter approaches zero, we are able to prove that our nonlocal model reduces to the conventional c lass of local hyperbolic conservation laws. Furthermore, we propose a numerical discretization method adapted to our nonlocal model, which relies on a monotone numerical flux and a uniform mesh, and we establish that these numerical solutions converge to a solution, providing as by-products both the existence theory for the nonlocal model and the convergence property relating the nonlocal regime and the asymptotic local regime. [ABSTRACT FROM AUTHOR]