Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations

TL;DR

This paper proves a linear convergence rate in L^p norm for a nonlocal vanishing viscosity approximation of Hamilton-Jacobi equations, showing faster convergence than classical methods and revealing a nonlocal phenomenon.

Contribution

It introduces a nonlocal regularization using the half-Laplacian that achieves a linear convergence rate, surpassing classical second-order viscosity estimates.

Findings

Established a linear L^p convergence rate with respect to viscosity parameter

Demonstrated faster convergence than classical second-order regularization

Revealed a nonlocal phenomenon in the convergence behavior

Abstract

We establish a linear $L^p$ rate of convergence, $1<p<\infty$, with respect to the viscosity $\varepsilon$ for the vanishing viscosity process of semiconcave solutions of Hamilton-Jacobi equations by regularizing the PDE with the half-Laplacian $-\varepsilon(-\Delta)^{1/2}$. Our result reveals a nonlocal phenomenon, since it improves the known estimates obtained via the classical second order vanishing viscosity regularization $\varepsilon\Delta u$. It also highlights a faster rate of convergence than the available $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate in sup-norm obtained by the doubling of variable technique for this nonlocal approximation. The result is based on integral methods and does not use the maximum principle.