Homogenization of the G-equation with incompressible random drift in two dimensions

TL;DR

This paper investigates the homogenization of the G-equation with random incompressible drift in two dimensions, establishing conditions under which solutions behave predictably in turbulent fluid models.

Contribution

It provides new sufficient conditions for homogenization in the two-dimensional divergence-free case with random velocity fields.

Findings

Homogenization holds with probability one under certain conditions.

Conditions are verified for divergence-free flows with specific stream function growth.

Results apply to modeling flame propagation in turbulent fluids.

Abstract

We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about the growth of the random stream function.