The Kuramoto–Sivashinsky equation

The paper formulates the “stripes” conjecture—define v as a Gaussian smoothing of u_x and declare stripes to be the connected components of {v<0}; numerics suggest that after trajectories approach the inertial manifold, stripes only undergo births and merges (no deaths or splits), and the authors explicitly state this as a conjecture and note that proving it “seems quite challenging,” providing no proof or theorem-level argument. This matches the model’s conclusion that the statement was likely open as of the cutoff. The paper’s narrative supports openness (and even calls out related open problems like stripe density), so the correct combined assessment is that the problem remained open at the time.