Recovering coercivity for the G-equation in general random media

TL;DR

This paper demonstrates that coercivity for the G-equation in random media can be recovered after a finite waiting time, under minimal assumptions on the velocity field, with explicit characterization of this waiting time.

Contribution

It introduces a method to recover coercivity for the G-equation in general random media after a finite waiting time, weakening previous assumptions on the velocity field.

Findings

Coercivity is recovered almost surely after finite waiting time.

Explicit characterization of the waiting time in terms of velocity field means.

Examples illustrating the theory are provided.

Abstract

The G-equation is a popular model for premixed turbulent combustion. Mathematically it has attracted a lot of interest in part because it is a simple example of a Hamilton-Jacobi equation which is only coercive `on average'. This paper shows that, after an almost surely finite waiting time, coercivity is recovered for the G-equation in a small mean, incompressible, space-time stationary ergodic velocity field. The argument follows ideas from recent work of Burago, Ivanov and Novikov, while significantly weakening the assumption on the velocity field. The waiting time is explicitly characterized in terms of the space-time means of the velocity field and so mixing estimates on the waiting time can easily be derived. Examples are provided.