  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.
