Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain
Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain
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DOI：
10.1007/bf03167468
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发表时间：
2004-06
影响因子：
0.9
通讯作者：
Y. Morita
中科院分区：
文献类型：
--
作者：
Y. Morita
本文讨论了三维薄区域中具有磁效应的Ginzburg-Landau方程，其中薄区域的厚度由一个小的正参数控制，变厚度由光滑函数的图形表示。考虑薄域的平台由一族具有窄通道的不相交子域组成的情况。这给出了一个弱连接的例子，称为超导中的S-c-S结。我们证明了，如果约化方程，在极限薄度为零，有一个非退化的稳定的解决方案，在每个子域，如果体积的通道是足够小，那么存在一个稳定的解决方案，以原方程在薄域。利用同样的论证，我们还可以证明薄区域的变量曲面在尺度上具有深威尔斯井的情况下非平凡稳定解的存在性。
We are dealing with the Ginzburg-Landau equation with magnetic effect in a 3-dimensional thin domain, where the thinness of the domain is controlled by a small positive parameter, epsilon, and the variable thickness is represented by the graph of a smooth function. Consider the case that the platform of the thin domain consists of a family of disjoint subdomains with narrow channels. This gives an example of a weak link, called an S-c-S junction in superconductivity. We prove that if the reduced equation, obtained in the limit as the thinness vanishes, has a nondegenerate stable solution in each subdomain and if the volume of the channels is sufficiently small, then there exists a stable solution to the original equation in the thin domain. Using the same argument, we can also prove the existence of non-trivial stable solutions for the case that the variable surface of a thin domain has deep wells in the scale by epsilon.