Nonlocal Transport Equations---Existence and Uniqueness of Solutions and Relation to the Corresponding Conservation Laws

Abstract

In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term “nonlocal” refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions for $TV\cap L^{\infty}$ initial datum and provide stability estimates. Moreover, we investigate the convergence of the nonlocal transport equation to the corresponding local conservation law when the nonlocal reach tends to zero. For quadratic flux functions (including Burgers' equation and the Lighthill--Whitham--Richards traffic flow model), we establish convergence to a weak solution of the local conservation law for “symmetric” nonlocal terms. For specific quasi-convex and quasi-concave initial datum we even obtain convergence to the local entropy solution. We demonstrate that for “nonsymmetric” nonlocal approximations the solution cannot converge to the entropy solution or even a weak solution. We conclude with additional numerical examples showing that convergence appears to hold for more general initial datum.