The geometry of the phase diffusion equation
type: Item
instance of: scholarly article
title: The geometry of the phase diffusion equation
zbMATH Open document ID: 0981.76087
author: Alan C. Newell
author: Nicholas M. Ercolani
author: R. A. Indik
author: T. Passot
published in: Journal of Nonlinear Science
publication date: 2000-05-07T00:00:00Z
review text: The authors consider the Cross-Newell phase diffusion equation \(\tau(|\vec k|) \Theta_T= -\nabla\cdot (B(|\vec k|) \cdot\vec k)\), \(\vec k=\nabla \Theta\) and its regularization, which describes natural patterns and defects far from onsel in large aspect ratio systems with rotational symmetry. They construct explicit solutions of the nonregularized equation and examine its weak solutions. This idea is confirmed by studying a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of free energy, and it is shown that these minimizers can be remarkably well approximated by a second-order ``self-dual'' equation. Using the results of \textit{W. Jin} and \textit{R. V. Kohn} [J. Nonlinear Sci. 10, No. 3, 355-390, doi:10.1007/s003329910014 (2000; Zbl 0973.49009)], the authors demonstrate that the energy of asymptotic minimizers converges to self-dual solutions in a viscosity limit.
Mathematics Subject Classification ID: 76R50
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1439880
zbMATH Keywords: Legendre transform
zbMATH Keywords: asymptotic minimizer of free energy
zbMATH Keywords: second-order self-dual equation
zbMATH Keywords: Cross-Newell phase diffusion equation
zbMATH Keywords: large aspect ratio systems
zbMATH Keywords: rotational symmetry
zbMATH Keywords: nonregularized equation
zbMATH Keywords: weak solution
zbMATH Keywords: fourth-order regularized equation
zbMATH Keywords: self-dual solution
zbMATH Keywords: viscosity limit
reviewed by: Elena Gavrilova
MaRDI profile type: Publication
full work available at URL: s003329910010
OpenAlex ID: W1974108099
DOI: 10.1007/S003329910010
description: scientific article; zbMATH DE number 1439880