Well-posedness and asymptotics of a coordinate-free model of flame fronts

We investigate a coordinate-free model of flame fronts introduced by Frankel and Sivashinsky; this model has a parameter $\alpha$ which relates to how unstable the front might be. We first prove short-time well-posedness of the coordinate-free model, for any value of $\alpha>0.$ We then argue that near the threshold $\alpha \approx 1,$ the solution stays arbitrarily close to the solution of the weakly nonlinear Kuramoto--Sivashinsky (KS) equation, as long as the initial values are close.