Non-local Hamilton-Jacobi equations arising in dislocation dynamics
type: Item
instance of: scholarly article
title: Non-local Hamilton-Jacobi equations arising in dislocation dynamics
zbMATH Open document ID: 1207.35097
DOI: 10.4171/ZAA/1411
author: Hitoshi Ishii
author: Yutaka Matsumura
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2010-08-16T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: The authors consider the following nonlocal Hamilton-Jacobi equation arising in dislocation dynamics: \[ u_t=[c(x,t)+M_p[u(\cdot,t)](x)]|p+Du(x,t)|,\tag{1} \] where \(u:\mathbb R^N\times [0,\infty)\to\mathbb R\), \(p\in\mathbb R^N\), \(Du=(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_N})\), \[ M_p[\phi](x)=\int_{\mathbb R^N}J(z)[E(\varphi(x+z)-\varphi(x)+p\cdot z)-p\cdot z]\,dz. \] Here, \(J\) is a measurable function on \(\mathbb R^N\), and \(E(r)\) denotes the greatest integer less than or equal to \(r\in\mathbb R\). Firstly, they establish some stability properties of viscosity solutions, comparison theorems between subsolutions and supsolutions. By using the above mentioned results, an existence and unique theorem for the initial value problem for (1) is proved. Furthermore, for \(N=1\), they show the well-posedness for the initial value problem for (1) by weakening the conditions.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35F21
Mathematics Subject Classification ID: 35D40
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 5770970
zbMATH Keywords: comparison theorem
zbMATH Keywords: subsolutions and supsolutions
reviewed by: Peixuan Weng
MaRDI profile type: Publication
cites work: Small-time solvability of a non-local Hamilton-Jacobi equation describing dislocation dynamics
cites work: Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels
cites work: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations
OpenAlex ID: W2036641120
description: scientific article; zbMATH DE number 5770970