Homogenization and enhancement of the $G-$equation in random environments

Details Homogenization and enhancement of the $G-$equation in random environments Homogenization and enhancement of the $G-$equation in random environments

2011-10-08

arxiv.org/abs/1110.1760v1

Mathematics

Optimization and Control

Scientific paper

We study the homogenization of a $G$-equation which is advected by a divergence free stationary vector field in a general ergodic random environment. We prove that the averaged equation is an anisotropic deterministic G-equation and we give necessary and sufficient conditions in order to have enhancement. Since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions. In addition, as we show, the associated minimal (first passage) time function does not satisfy, in general, the uniform integrability condition which is necessary to apply the sub-additive ergodic theorem. We overcome these obstacles by (i) establishing a new reachability (controllability) estimate for the minimal function and (ii) constructing, for each direction and almost surely, a random sequence which has both a long time averaged limit (due to the sub-additive ergodic theorem) and stays (in the same sense) asymptotically close to the minimal time.

Affiliated with

Also associated with

No associations

Rating

Homogenization and enhancement of the $G-$equation in random environments does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Homogenization and enhancement of the $G-$equation in random environments, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Homogenization and enhancement of the $G-$equation in random environments will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-497831