Communications in Mathematical Sciences

Volume 9 (2011)

Number 2

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

Pages: 561 – 582



James Nolen (Department of Mathematics, Duke University, Durham, North Carolina)

Alexei Novikov (Department of Mathematics, Pennsylvania State University, University Park)


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.


front propagation, homogenization, random media

2010 Mathematics Subject Classification

35B40, 35F21, 35R60

Published 21 December 2010