Title

Singularities in a Modified Kuramoto-Sivashinsky Equation Describing Interface Motion for Phase Transition

Document Type

Article

Department

Mathematics (HMC)

Publication Date

1995

Abstract

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(hxxhyyhxy2), 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 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.

Rights Information

© 1995 Elsevier Ltd.

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