Nonlocal conservation laws. I. A new class of monotonicity-preserving models

TL;DR

This paper introduces a new class of nonlocal conservation laws with finite horizon interactions, establishing well-posedness, monotonicity, and convergence to local laws as the horizon shrinks, along with a compatible numerical scheme.

Contribution

It presents a novel nonlocal conservation law model with proven well-posedness, monotonicity properties, and a convergence framework to local laws, including a compatible numerical discretization.

Findings

Model inherits maximum principle and monotonicity

Proven global-in-time well-posedness for broad initial data

Numerical solutions converge to local conservation law solutions

Abstract

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 continuum 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 for these models which covers a broad class of initial data. Moreover, in the limit when the horizon parameter approaches zero, we are able to prove that our nonlocal model reduces to the conventional class 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-product both the existence theory for the nonlocal model and the convergence property relating the nonlocal regime and the asymptotic local regime.