Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Abstract

Abstract Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modified Kuramoto-Sivashinsky equation, h t + ∇ 2 h + ∇ 4 h = (1 − λ )|∇ h | 2 ± λ (∇ 2 h ) 2 + δλ ( h xx h yy − h xy 2 ), 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 > 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 |h xx | l ∞ and indicate that the singularities are all locally described by a unique self-similar profile in h xx . We discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics.