arXiv 2408.021542024-08-04math.AP

关于空间非均匀 Cahn-Hilliard 能量的一个注记

A note on spatially inhomogeneous Cahn-Hilliard energies

Stephan Wojtowytsch

AI总结：

本文为空间非均匀 Cahn-Hilliard 能量收敛到均匀周长泛函给出简洁新证明，并推广至随机势与空洞势，指出变化势阱需 δ≪ε^{3/2}，进而提出无需该尺度分离的修正 Modica-Mortola 能量。

摘要：

2023年，Cristoferi、Fonseca 和 Ganedi 证明：如果双势阱势中空间非均匀性的长度尺度 δ 相比相变长度尺度 ε 较小，则具有空间非均匀势的 Cahn-Hilliard 型能量会收敛到通常的（各向同性且均匀的）周长泛函。我们在关于 W 相对于相参数正则性的一个略强假设下，给出了一个简单的新证明。证明的简洁性使我们能够容易地找到其他方向上的多种推广，包括随机势以及在势阱之外可能变为零的势（“空洞”）。理论结果由数值实验加以补充。我们的主要信息是，在多种多样的设定中，任何在相之间的转变长度尺度上看起来足够均匀的事物，都不会对极限行为产生丝毫影响。显著的例外是具有空间变化势阱的势的设定，此时需要更强的尺度分离 δ≪ε^{3/2}。基于该分析，我们进一步提出了一种修正的 Modica-Mortola 型能量，它不需要这种尺度分离。理论分析由数值实验加以补充。

英文摘要：

In 2023, Cristoferi, Fonseca and Ganedi proved that Cahn-Hilliard type energies with spatially inhomogeneous potentials converge to the usual (isotropic and homogeneous) perimeter functional if the length-scale $δ$ of spatial inhomogeneity in the double-well potential is small compared to the length-scale $\varepsilon$ of phase transitions. We give a simple new proof under a slightly stronger assumption on the regularity of $W$ with respect to the phase parameter. The simplicity of the proof allows us to easily find multiple generalizations in other directions, including stochastic potentials and potentials which may become zero outside the wells ('voids'). The theoretical results are complemented by numerical experiments. Our main message is that, across a wide variety of settings, nothing that looks sufficiently homogeneous at the transition length scale between phases, affects the limiting behavior in the slightest. The notable exception is the setting of potentials with spatially varying potential wells, where a stronger scale separation $δ\ll \varepsilon^{3/2}$ is needed. Based on the analysis, we further provide a modified Modica-Mortola type energy which does not require such a scale separation. The theoretical analysis is complemented by numerical experiments.

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