Homogenization of viscous and non-viscous HJ equations: a remark and an application

TL;DR

This paper extends the homogenization results of Hamilton-Jacobi equations to viscous cases, demonstrating that affine initial data imply homogenization for general data, and applies this to ergodic settings with non-convex Hamiltonians in one dimension.

Contribution

It proves the homogenization implication for viscous and non-viscous HJ equations using a variant of Evans's method, including new results for viscous equations with non-convex Hamiltonians.

Findings

Homogenization holds for viscous and non-viscous HJ equations starting from affine initial data.

The proof uses a modified Evans's perturbed test function method.

New homogenization results are established in one-dimensional ergodic settings with non-convex Hamiltonians.

Abstract

It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general uniformly continuous initial data. The argument makes use of some properties of the HJ semi-group, in particular, the finite speed of propagation. The last property is lost for viscous HJ equations. In this paper we prove the above mentioned implication in both viscous and non-viscous cases. Our proof relies on a variant of Evans's perturbed test function method. As an application, we show homogenization in the stationary ergodic setting for viscous and non-viscous HJ equations in one space dimension with non-convex Hamiltonians of specific form. The results are new in the viscous case.