Existence And Uniqueness Results For A Class Of Nonlocal Conservation Laws By Means Of A Lax-Hopf-Type Solution Formula

We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given "local" conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax-Hopf-type solution formula for the corresponding Hamilton-Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax-Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax-Oleinik-type formula for the nonlocal conservation law.