Singularities in a Modiied Kuramoto-sivashinsky Equation Describing Interface Motion for Phase Transition

Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modiied Kuramoto-Sivashinsky equation, h t + r 2 h + r 4 h = (1 ?)jrhj 2 (r 2 h) 2 + (h xx h yy ? h 2 xy); describes near planar interfaces which are marginally long-wave unstable. We study the question of nite-time singularity formation in this equation in one and two space dimensions on a periodic domain. Such singularity formation does not occur in the Kuramoto-Sivashinsky equation (= 0). For all 1 > 0 we provide suucient conditions on the initial data and size of the domain to guarantee a nite-time blow up in which a second derivative of h becomes unbounded. Using a bifurcation theory analysis, we show a parallel between the stability of steady periodic solutions and the question of nite-time blow up in one dimension. Finally, we consider the local structure of the blow up in the one-dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self-similar grid. The simulations resolve the singularity to over 25 decades in jh xx j L 1 and indicate that the singularities are all locally described by a unique self-similar proole in h xx. We discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics.