Convergence rates in the periodic homogenization of vanishing-viscous Hamilton-Jacobi equations

Abstract

We study convergence rates when periodic homogenization and vanishing viscosity occur at the same scale in Hamilton--Jacobi equations whose momentum Hessians are positive definite at every point. For bounded Lipschitz initial data and a class of smooth Hamiltonians, we establish an optimal rate $O(\varepsilon|\log\varepsilon|)$ on fixed time intervals. The second-order term plays a key role: the elliptic cell problem provides higher derivative bounds for the effective Hamiltonian $\bar{H}$ and its Legendre dual $\bar{L}$. The proof is purely PDE, based on corrector expansions and viscosity comparison, while its guiding idea comes from the control interpretation of the Hopf--Lax formula. For a fixed target $(x,t)$, a minimizing origin selects a characteristic velocity and its dual momentum, these guide the construction of smooth comparison profiles, whose gradients supply the momentum argument of the first-order corrector. Thus we never need to differentiate the effective solution, even at points where it is nonsmooth.