论文标题
分数Allen-CaHn方程中的亚稳态速度
Metastable Speeds in the Fractional Allen-Cahn Equation
论文作者
论文摘要
我们从数值上研究具有均匀的neumann边界条件的间隔，具有光谱分数拉普拉斯$（-δ）^{α/2} $的一维allen-cahn方程。特别是，我们对接近和灭绝的尖锐接口的速度感兴趣。在经典的拉普拉斯式的情况下，已知此过程呈指数速度。在这里，我们调查了界面的宽度和速度如何改变，如果我们改变了分数拉普拉斯式的指数$α$。对于现实线上的相关模型，我们根据$α$和缩放参数$ \ varepsilon $得出了接口速度和接口速度的渐近公式。我们通过有限元的方法使用数值方法，该方法基于将分数拉普拉斯式扩展到上半平面的圆柱，并计算界面速度，$α\ in（0.2,2] $的$α\ $α\的界面宽度（0.2,2] $）。一个比较表明，对于界面的速度和时间较大的速度使得较大的速度速度为近似速度，以使得良好的速度差异。
We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-Δ)^{α/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $α$ of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of $α$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for $α\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.
