Inverse problems, non-roundness and flat pieces of the effective burning velocity from an inviscid quadratic Hamilton-Jacobi model

TL;DR

This paper investigates the geometric properties of the effective burning velocity in a combustion model, revealing flat regions under certain flow conditions and discussing implications for flame front behavior and inverse problems.

Contribution

It identifies conditions under which the level set of the effective burning velocity has flat pieces in a 2D setting, advancing understanding of flame front geometry in turbulent flows.

Findings

Flat pieces occur in the level set when flow is weak or very strong in 2D.

The exact location of at least one flat piece is determined.

Implications for inverse problems and flame front analysis are discussed.

Abstract

The main goal of this paper is to understand finer properties of the effective burning velocity from a combustion model introduced by Majda and Souganidis [19]. Motivated by results in [4] and applications in turbulent combustion, we show that when the dimension is two and the flow of the ambient fluid is either weak or very strong, the level set of the effective burning velocity has flat pieces. Due to the lack of an applicable Hopf-type rigidity result, we need to identify the exact location of at least one flat piece. Implications on the effective flame front and other related inverse type problems are also discussed.