Optimal rate of convergence in the vanishing viscosity for uniformly convex Hamilton-Jacobi equations

TL;DR

This paper establishes an optimal convergence rate of O(ε log ε) for vanishing viscosity solutions of uniformly convex Hamilton-Jacobi equations, improving upon the previously believed rate of O(√ε).

Contribution

It proves the optimal convergence rate of O(ε log ε) for these equations and introduces new techniques combining regularisation and entropy estimates.

Findings

Convergence rate of O(ε log ε) is optimal.

The previous rate of O(√ε) can be improved.

Method combines sup-convolution regularisation with entropy estimates.

Abstract

The purpose of this note is to provide an optimal rate of convergence in the vanishing viscosity regime for first-order Hamilton-Jacobi equations with uniformly convex Hamiltonian. We prove that for a globally Lipschitz-continuous and semiconcave terminal condition the rate is of order O($\epsilon$log$\epsilon$), and we provide an example to show that this rate cannot be sharpened. This improves on the previously known rate of convergence O($\sqrt$$\epsilon$), which was widely believed to be optimal. Our proof combines techniques involving regularisation by sup-convolution with entropy estimates for the flow of a suitable version of the adjoint linearized equation. The key technical point is an integrated estimate of the Laplacian of the solution against this flow. Moreover, we exploit the semiconcavity generated by the equation to handle less regular data in the quadratic case.