  In this article, some elementary observations are made regarding the behavior
of solutions to the two-dimensional curl-free Burgers equation which suggest
the distinguished role played by the scalar divergence field in determining the
dynamics of the solution. These observations inspire a new divergence-based
regularity condition for the two-dimensional Kuramoto-Sivashinsky equation
(KSE) that provides conceptual clarity to the nature of the potential blow-up
mechanism for this system. The relation of this regularity criterion to the
Ladyzhenskaya-Prodi-Serrin-type criterion for the KSE is also established, thus
providing the basis for the development of an alternative framework of
regularity criterion for this equation based solely on the low-mode behavior of
its solutions. The article concludes by applying these ideas to identify a
conceptually simple modification of KSE that yields globally regular solutions,
as well as providing a straightforward verification of this regularity
criterion to establish global regularity of solutions to the 2D
Burgers-Sivashinsky equation. The proofs are direct, elementary, and concise.
