Stability of the Hamilton–Jacobi corrector under perturbations

Establish the stability of the corrector associated with the Hamilton–Jacobi equation with a flux limiter under perturbations of the microscopic velocity law $W$, in order to determine whether the corrector approach can yield continuity of the flux-limiter map.

Background

The paper studies perturbed Follow-the-Leader models whose macroscopic limits are scalar conservation laws with discontinuous flux and an associated flux limiter. One of the principal results is continuity of the map that assigns a flux limiter $f_W$ to a microscopic perturbation $W$.

The authors note that existing approaches based on a corrector for a Hamilton–Jacobi equation do not readily provide this continuity result because the dependence of the corrector on $W$ is unresolved. Establishing such stability would clarify whether the Hamilton–Jacobi homogenization framework can independently support continuity of the flux-limiter map.