Different approaches to the nonlinear dynamics of premixed flames exist in the literature: equations based on developments in a gas expansion parameter, weak nonlinearity approximation, and potential model equation in a coordinate-free form. However, the relation between these different equations is often unclear. Starting here with the low-vorticity approximation proposed recently by one of the authors, we are able to recover from this formulation the dynamic equations usually obtained at the lowest orders in gas expansion for planar on average flames, as well as obtain a new second-order coordinate-free equation extending the potential flow model known as the Frankel equation. It is also common to modify gas expansion theories into phenomenological equations, which agree quantitatively better with numerical simulations. We discuss here the restrictions imposed by the gas expansion development results on this process.