Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

TL;DR

This paper studies the homogenization process for weakly coupled Hamilton--Jacobi systems with rapid switching, revealing initial layers, convergence rates, and unique properties of the effective Hamiltonian.

Contribution

It introduces a rigorous analysis of initial layers and convergence rates in coupled systems, expanding understanding beyond single Hamilton--Jacobi equations.

Findings

Solutions converge to a common limit governed by an effective equation.

Initial layers appear naturally with different initial data.

Examples show properties of the effective Hamiltonian unique to coupled systems.

Abstract

We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.