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

TL;DR

This paper proves short-time well-posedness of a coordinate-free flame front model for any instability parameter and shows its solutions approximate the Kuramoto--Sivashinsky equation near a critical threshold.

Contribution

It establishes the well-posedness of the model and links its behavior to the well-studied Kuramoto--Sivashinsky equation near a critical parameter value.

Findings

Proves short-time well-posedness for all positive alpha.

Demonstrates solutions approximate KS equation near alpha ≈ 1.

Provides insights into flame front stability and dynamics.

Abstract

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.