Phase transition of an anisotropic Ginzburg–Landau equation

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-10 DOI:10.1007/s00526-024-02779-5

Yuning Liu

引用次数: 0

Abstract

We study the effective geometric motions of an anisotropic Ginzburg–Landau equation with a small parameter \(\varepsilon >0\) which characterizes the width of the transition layer. For well-prepared initial datum, we show that as \(\varepsilon \) tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field \({\textbf{u}}(x,t)\) which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that \({\textbf{u}}\) must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen–Frank model in liquid crystals.

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各向异性金兹堡-朗道方程的相变

我们研究了各向异性金兹堡-朗道方程的有效几何运动，该方程有一个小参数\(\varepsilon >0\)，它描述了过渡层的宽度。对于准备充分的初始基准，我们证明当 \(\varepsilon \)趋向于零时，解将形成一个尖锐的界面极限，该极限在平均曲率流下演化。解的体极限对应于矢量场 \({\textbf{u}}(x,t)\)，该矢量场在界面一侧为单位长度，而在另一侧为零。证明结合了调制能量法和弱收敛法。特别是，通过（边界）炸毁论证，我们证明了 \({\textbf{u}}\) 必须与尖锐界面相切。此外，它还求解了液晶中奥森-弗兰克模型的几何演化方程。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.