Configurational nonlocal Hamilton-Jacobi framework for interface propagation

Abstract

We introduce a class of nonlocal Hamilton–Jacobi equations in which the classical gradient is replaced by a finite-horizon interaction operator, thereby embedding an intrinsic length scale directly into first-order nonlinear evolution. In contrast with integro-differential formulations where nonlocality appears as an additive correction, the proposed framework modifies the underlying differential structure of the equation, leading to propagation laws governed by neighborhood-dependent configurational interactions. A rigorous mechanical interpretation is developed by identifying the nonlocal operator as a finite-horizon configurational driving field. The resulting evolution law is derived from a rate-dependent kinetic relation for interface motion and shown to be consistent with a thermodynamic dissipation principle. This provides a sharp-interface, interaction-driven alternative to gradient-based regularization mechanisms such as phase-field models. From a mathematical standpoint, we establish consistency with the classical Hamilton–Jacobi equation in the local limit and prove well-posedness in a Banach-space setting. A canonical quadratic model is analyzed, and asymptotic expansions reveal higher-order corrections induced by nonlocality. Numerical experiments on front propagation demonstrate that the interaction horizon induces scale-dependent dynamics and regularization without introducing higher-order diffusion. The proposed framework defines a new class of nonlocal first-order propagation laws that retain the hyperbolic character of Hamilton–Jacobi equations while incorporating mesoscale interaction effects relevant to continuum mechanics.