Higher Order Convergence Rates in Theory of Homogenization III: viscous Hamilton-Jacobi Equations

TL;DR

This paper establishes higher order convergence rates in the periodic homogenization of viscous Hamilton-Jacobi equations, highlighting the role of initial data geometry and nonlinear effects absorption in advanced approximations.

Contribution

It introduces a framework for higher order convergence analysis in viscous Hamilton-Jacobi homogenization, emphasizing the influence of initial data shape and nonlinear effect absorption.

Findings

Higher order convergence rates are achieved in homogenization.

Nonlinear effects are absorbed as external sources in higher order approximations.

The shape of initial data must be carefully chosen for optimal convergence.

Abstract

In this paper, we establish the higher order convergence rates in periodic homogenization of viscous Hamilton-Jacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates.