Surface tension of contrast structures
type: Item
instance of: scholarly article
title: Surface tension of contrast structures
zbMATH Open document ID: 0962.35104
author: A. A. Bykov
author: O. V. Kozyreva
author: V. Yu. Popov
author: Vl. Val. Voevodin
author: D. Sokoloff
published in: Doklady Mathematics
publication date: 2001-01-28T00:00:00Z
review text: The article is devoted to studying the nonlinear evolution of physical fields of various natures where specific configurations called contrast structures appear. A two-dimensional medium is considered where a physical field \(u(x,y,t)\) participates in transfer at a velocity \(\mathbf V\) and in diffusion with a small diffusion coefficients \(\mu\). In addition, the medium is characterized by a growth parameter \(\gamma\) whose dependence on the field strength is nonlinear: \[ \frac{\partial u}{\partial t}+V_x\frac{\partial u}{\partial x}+V_y\frac{\partial u}{\partial y}=\mu\Biggl(\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial y^2}\Biggr)+\gamma(u)u, \] where \(\gamma > 0\) if \(|u|< U\), \(\gamma = 0\) if \(|u|= U\), and \(\gamma < 0\) if \(|u|> U\). To be specific, it is assumed that \(\gamma(u) = \gamma_0[1-(\frac{u}{U})^2]\). The equation which is considered inside a bounded domain \(D\) with a boundary \(\Sigma\), is augmented by the initial condition \(u|_{t=0} = u_0(x,y)\) and a boundary condition \(u|_{\Sigma} = 0\). The aim of the article is to be specify a factor determining the character of the evolution and lifetime of an unstable nonstationary contrast structure. The authors study the evolution of the initial field specified in the form of one or several circular or elliptical spots that possibly include spots of opposite sign. The authors show that the domain of rapid field variation in a contrast structure has a surface tension which determines the character of evolution in the examples considered, and suggests a formula for evaluating the surface tension. The methods of asymptotic analysis is used; to determine constraints on various parameters, the authors also employ computer experiments by CRAY Y-MP C90 supercomputer.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1557595
zbMATH Keywords: circular or elliptical spots
zbMATH Keywords: transition layer
zbMATH Keywords: lifetime
zbMATH Keywords: numerical experiment
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1557595