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Rate of convergence for homogenization of nonlinear weakly coupled Hamilton-Jacobi systems

Hiroyoshi Mitake, Panrui Ni

TL;DR

The paper addresses periodic homogenization for nonlinear weakly coupled Hamilton-Jacobi systems with convex Hamiltonians and nonlinear coupling, introducing an effective Hamiltonian via cell problems. It develops a fixed-point iteration framework to control the multiscale error and proves a sharp convergence rate $|\mathbf{u}^\varepsilon-\bar{\mathbf{u}}|\le C\sqrt{\varepsilon}$ for times away from initial, along with a small-time bound, using quantitative homogenization estimates. The results extend to a stationary discounted problem under a strict monotonicity condition $\lambda>\Theta$, yielding the same $O(\sqrt{\varepsilon})$ rate. The methods combine iteration techniques with recent quantitative homogenization theory for Hamilton-Jacobi equations, establishing the first rate-of-convergence result for nonlinear, non-monotone, weakly coupled HJ systems and broadening applicability beyond single-equation settings.

Abstract

Here, we study the periodic homogenization problem of nonlinear weakly coupled systems of Hamilton-Jacobi equations in the convex setting. We establish a rate of convergence $O(\sqrt{\varepsilon})$ which is sharp.

Rate of convergence for homogenization of nonlinear weakly coupled Hamilton-Jacobi systems

TL;DR

The paper addresses periodic homogenization for nonlinear weakly coupled Hamilton-Jacobi systems with convex Hamiltonians and nonlinear coupling, introducing an effective Hamiltonian via cell problems. It develops a fixed-point iteration framework to control the multiscale error and proves a sharp convergence rate for times away from initial, along with a small-time bound, using quantitative homogenization estimates. The results extend to a stationary discounted problem under a strict monotonicity condition , yielding the same rate. The methods combine iteration techniques with recent quantitative homogenization theory for Hamilton-Jacobi equations, establishing the first rate-of-convergence result for nonlinear, non-monotone, weakly coupled HJ systems and broadening applicability beyond single-equation settings.

Abstract

Here, we study the periodic homogenization problem of nonlinear weakly coupled systems of Hamilton-Jacobi equations in the convex setting. We establish a rate of convergence which is sharp.

Paper Structure

This paper contains 4 sections, 15 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

For each $\varepsilon>0$, there is a unique solution of both E and E0, which is denoted by $\mathbf{u}^\varepsilon$ and $\bar{\mathbf{u}}$, respectively. There is a constant $C>0$ depending only on $n$, $\mathbf{H}$, $\|\varphi_i\|_{W^{1,\infty}}$ and $T$ such that for all $t\in[\sqrt{\varepsilon}, and for all $t\in(0,\sqrt{\varepsilon})$ we have

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1
  • Example 1.1
  • Theorem 2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 21 more