Singularities in a Modified 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 modified Kuramoto-Sivashinsky equation, ht + ∇2h + ∇4h = (1 − λ)|∇h|2 ± λ(∇2h)2 + δλ(hxxhyy − hxy2), describes near planar interfaces which are marginally long-wave unstable. We study the question of finite-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 (λ = 0). For all 1 ≥ λ \u3e 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite-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 finite-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 and indicate that the singularities are all locally described by a unique self-similar profile in hxx. We discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics