Coarsening Fronts
Coarsening Fronts
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前沿粗化
DOI：
10.1007/s00205-006-0422-9
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发表时间：
2006
影响因子：
2.5
通讯作者：
A. Scheel
中科院分区：
文献类型：
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作者：
A. Scheel
我们用非线性调制前沿的传播来描述 Allen-Cahn 方程中粗化过程的空间扩展。艾伦-卡恩方程的不稳定周期模式被前沿侵入，以振荡方式传播，并留下均匀、稳定的平衡。在振荡传播的一个周期内，两层周期性图案被湮灭。伽辽金近似和不适定空间动力学的康利指数用于显示所有参数值的调制前沿的存在。在小振幅模式或大波速的限制下，我们建立了调制前沿的唯一性和渐近稳定性。我们表明，最小传播速度可以用二分法来表征，这取决于拉锋的存在。这里的主要工具是无限维不适定动力学的埃文斯函数类型构造以及基于 Sturm-Liouville 理论的复杂色散关系分析。
We characterize the spatial spreading of the coarsening process in the Allen–Cahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the Allen–Cahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and the Conley index for ill-posed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy which depends on the existence of pulled fronts. The main tools here are an Evans function type construction for the infinite-dimensional ill-posed dynamics and an analysis of the complex dispersion relation based on Sturm–Liouville theory.