### Abstract

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex (L^{1} type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue H̄ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s_{T}. An important problem in turbulent combustion theory is to study properties of s_{T}, in particular how s_{T} depends on the flow amplitude A. In this paper, we study the behavior of as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that H̄ (A,d) ≦ C(d) for all d > 0,where C(d) is a constant depending on d, but independent of A. Compared with H̄(A,0) = O(A/log A), A ≫ 1, of the inviscid G-equation (d = 0), presence of diffusion dramatically slows down front propagation. For shear flow, where λ (d) is strictly decreasing in d, and has zero derivative at d = 0. The linear growth law is also valid for s_{T} of the curvature dependent G-equation in shear flows.

Original language | English |
---|---|

Pages (from-to) | 461-492 |

Number of pages | 32 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 202 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 Nov 1 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

*Archive for Rational Mechanics and Analysis*,

*202*(2), 461-492. https://doi.org/10.1007/s00205-011-0418-y