Traveling wave solutions of the heat flow of director fields
type: Item
instance of: scholarly article
title: Traveling wave solutions of the heat flow of director fields
zbMATH Open document ID: 1176.35093
author: Michiel Bertsch
author: Ivano Primi
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2009-02-26T00:00:00Z
full work available at URL: https://eudml.org/doc/78733
review text: The authors consider the heat flow of harmonic maps from infinite cylinder \(\{(x_1,x_2,x_3)\in\mathbb R^3: x_1^2+x_2^2<1\}\) to the unit sphere \(\mathbb S^2: u_t=\Delta u+|\nabla u|^2u\), to which by cylindrical coordinates there corresponds a scalar angle function \(h(r,x_3,t)\). In order to study nonuniqueness of axially symmetric solutions of the heat flow, they look for traveling wave solutions \(h(r,x_3,t)=h(r,z), z=x_3-ct\) which satisfy the following singular elliptic equation \[ h_{rr}+h_{zz}+\frac{h_r}{r}+ch_z-\frac{\sin(2h)}{2r^2}=0, \quad r\in{(0,1)},\;z\in{\mathbb{R}};\qquad h(1,z)=g(z), \] where \(c>0\) is the wave speed and \(g\) is a given function satisfying some special conditions. By the variation methods and the concept of relaxed energy, they prove in two cases the existence of traveling wave solutions with point singularity of topological degree 0 or 1.
reviewed by: Wuqing Ning
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A21
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5519838
zbMATH Keywords: harmonic maps
zbMATH Keywords: topological degree
zbMATH Keywords: relaxed energy
zbMATH Keywords: point singularity
MaRDI profile type: Publication
OpenAlex ID: W2011621184
cites work: Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps
cites work: Nonuniqueness for the heat flow of harmonic maps on the disk
cites work: Traveling wave solutions of harmonic heat flow
cites work: Nonuniqueness of the traveling wave speed for harmonic heat flow
cites work: On the dynamics of deformable ferromagnets. I: Global weak solutions for soft ferromagnets at rest
cites work: Q3972645
cites work: Finite-time blow-up of the heat flow of harmonic maps from surfaces
cites work: Nonuniqueness for the heat flow of harmonic maps
cites work: Harmonic map heat flow for axially symmetric data
cites work: Concentrated boundary data and axially symmetric harmonic maps
cites work: Harmonic maps of manifolds with boundary
cites work: Axially symmetric harmonic maps minimizing a relaxed energy
cites work: Q3330158
cites work: Rearrangements and convexity of level sets in PDE
cites work: Nonuniqueness of the heat flow of director fields
cites work: Q3961188
cites work: Reverse bubbling of currents and harmonic heat flows with prescribed singular set
cites work: Q4531669
cites work: Flow alignment in nematic liquid crystals in flows with cylindrical symmetry
DOI: 10.1016/J.ANIHPC.2006.03.008
description: scientific article; zbMATH DE number 5519838