David Ambrose, PhD, Drexel University

Abstract: The Kuramoto-Sivashinsky equation is a model for the motion of flame fronts. In one spatial dimension much is understood, including that solutions exist for all time, and small solutions remain small for all time. Analagous results in two dimensions are much more limited; most results in 2D assume that the domain is "thin," or approximately one-dimensional. We will give an overview of the results in one dimension and anisotropic results in two dimensions. We will then show some global existence theorems for small data in two dimensions without making use of any anisotropy. This includes joint work with Anna Mazzucato.