Ginzburg-Landau equation and motion by mean curvature. II: Development of the initial interface
type: Item
instance of: scholarly article
title: Ginzburg-Landau equation and motion by mean curvature. II: Development of the initial interface
zbMATH Open document ID: 0935.35061
DOI: 10.1007/BF02921629
author: H. Mete Soner
published in: The Journal of Geometric Analysis
publication date: 2000-04-24T00:00:00Z
review text: [For part I, see ibid., 437-475 (1997; Zbl 0935.35062).] The paper deals with the study of the short time behavior of solutions to the Ginzburg-Landau equation \(u_t^\varepsilon - \Delta u^\varepsilon + 2/\varepsilon^2 u^\varepsilon (u^\varepsilon -1) (u^\varepsilon +1)=0\) in \({\mathbb R}^d\times [0,+\infty)\), with the initial condition \(u^\varepsilon (x,0)=u_0^\varepsilon (x)\). The main result of the work establishes that under appropriate assumptions on \(u_0^\varepsilon\), solutions converge to \(\pm 1\) in short time and, they behave like the one-dimensional traveling wave along the interface. As a consequence of this result, the author deduces that the energy remains uniformly bounded with respect to \(\varepsilon\). The main results are proved after deducing two key estimates: (i) a detailed description of \(u^\varepsilon\) near the initial interface, which is a sharper version of a previous result of DeMottoni and Schatzman, and (ii) a gradient estimate (Theorem 5.1 in the paper).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1326049
zbMATH Keywords: short time behavior
zbMATH Keywords: maximum principle
zbMATH Keywords: traveling waves
zbMATH Keywords: asymptotic behavior of solutions
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
cites work: Front Propagation and Phase Field Theory
cites work: Generation and propagation of interfaces for reaction-diffusion equations
cites work: Q3134551
cites work: Phase transitions and generalized motion by mean curvature
cites work: The approach of solutions of nonlinear diffusion equations to travelling front solutions
cites work: Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature
cites work: Development of interfaces in ℝN
cites work: Fast Reaction, Slow Diffusion, and Curve Shortening
cites work: Motion of a set by the curvature of its boundary
full work available at URL: https://doi.org/10.1007/bf02921629
OpenAlex ID: W1972038087
description: scientific article; zbMATH DE number 1326049