Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence
type: Item
instance of: scholarly article
title: Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence
zbMATH Open document ID: 1009.35015
author: Daniel N. Ostrov
published in: Journal of Differential Equations
publication date: 2002-09-08T00:00:00Z
review text: A unique stable solution \(u(x,t)\) to Hamilton-Jacobi equations of the form \[ u_t+ H(K(x, t), u_x)= 0,\quad x\in (-\infty,\infty),\quad t\in [0,\infty),\quad u(x,0)= g(x), \] where \(H\) is convex in \(u_x\) and has superlinear growth as \(u_x\to \pm\infty\), \(g(x)\) is Lipschitz continuous, and \(K\) is discontinuous along a finite number of curves in the \((x,t)\) plane, is constructed. The solution is determined by showing that if \(K\) is made smooth by convolving \(K\) in the \(x\) direction with the standard mollifier, then the control theory representation of the viscosity solution to the resulting Hamiltonian-Jacobi equation must converge uniformly as the mollification decreases to a Lipschitz continuous solution with an explicit control theory representation. This also defines the unique stable solution to the corresponding scalar conservation law \(u_t+ (f(K(x, t), u))_x= 0\), \(x\in (-\infty,\infty)\), \(t\in [0,\infty)\) with \(K\) discontinuous.
Mathematics Subject Classification ID: 35F25
zbMATH DE Number: 1797553
zbMATH Keywords: control theory representation
zbMATH Keywords: viscosity solution
reviewed by: Vasile Iftode
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4088
OpenAlex ID: W2069739494
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DOI: 10.1006/JDEQ.2001.4088
description: scientific article; zbMATH DE number 1797553