## Communications in Mathematical Sciences

- Commun. Math. Sci.
- Volume 8, Number 4 (2010), 1067-1078.

### Periodic homogenization of the inviscid G-equation for incompressible flows

#### Abstract

G-equations are popular front propagation models in combustion literature and
describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid
velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We
prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends
a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects,"
*Nonlinearity*, to appear. We construct approximate correctors to bypass the lack of
compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on
a local reachability property of the controlled flow trajectory as well as incompressibility of the flow.
Homogenization then follows from the comparison principle and the perturbed test function method.
The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further
assume that the flow is mean zero.

#### Article information

**Source**

Commun. Math. Sci., Volume 8, Number 4 (2010), 1067-1078.

**Dates**

First available in Project Euclid: 2 November 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.cms/1288725272

**Mathematical Reviews number (MathSciNet)**

MR2744920

**Zentralblatt MATH identifier**

1372.76085

**Subjects**

Primary: 70H20: Hamilton-Jacobi equations 76M50: Homogenization 76M45: Asymptotic methods, singular perturbations 76N20: Boundary-layer theory

**Keywords**

Non-coercive Hamilton-Jacobi equations controlled flow local reachability incompressibility periodic homogenization

#### Citation

Xin, Jack; Yu, Yifeng. Periodic homogenization of the inviscid G-equation for incompressible flows. Commun. Math. Sci. 8 (2010), no. 4, 1067--1078. https://projecteuclid.org/euclid.cms/1288725272