Nonlocal Approximation Principle for Entropy Solutions of Scalar Conservation Laws

TL;DR

This paper introduces a nonlocal approximation principle for scalar conservation law entropy solutions, enabling their construction via nonlocal models and extending to sign-changing data with convergence guarantees.

Contribution

It establishes a new nonlocal approximation framework for entropy solutions, including a quantitative convergence estimate for convex fluxes and extension to sign-changing initial data.

Findings

Entropy solutions can be obtained as limits of nonlocal conservation laws.

The approximation extends to sign-changing initial data after a shift.

A quantitative convergence estimate is provided for convex fluxes.

Abstract

We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star limit of a corresponding scalar nonlocal conservation law. The flux function of the nonlocal conservation law depends on suitable spatial averages of the density. The proof is based on a reformulation on the Hamilton--Jacobi level: working with the primitives, we identify the limit via the stability properties of viscosity solutions; we then recover the entropy solution using the classical relation between Hamilton--Jacobi equations and scalar conservation laws. We further show that the approximation extends, after a suitable shift, to sign-changing initial data, and we prove a quantitative convergence estimate for convex fluxes in terms of the first moments of the nonlocal kernels. This result makes it possible to define entropy solutions for general fluxes using their nonlocal approximations, which satisfy the requirement for a finite speed of mass propagation, a key feature of hyperbolic conservation laws.