Formation of singularities of Hamilton-Jacobi equation. II
type: Item
instance of: scholarly article
title: Formation of singularities of Hamilton-Jacobi equation. II
zbMATH Open document ID: 0655.35009
author: Mikio Tsuji
published in: Journal of Mathematics of Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: [For part 1 see Proc. Jap. Acad., Ser. A 59, 55-58 (1983; Zbl 0531.35015).] Consider the Cauchy problem for Hamilton-Jacobi equation in two space dimensions: \[ (1)\quad \partial u/\partial t+f(\partial u/\partial x)=0\quad in\quad \{t>0,\quad x\in R^ 2\};\quad (2)\quad u(0,x)=\phi (x)\in {\mathcal S}(R^ 2). \] We assume that f is \(C^{\infty}\) and uniformly convex. It is well known that, even for smooth initial data, the Cauchy problem (1) and (2) does not admit a smooth solution for all t. Therefore we consider a generalized solution. The existence of global generalized solution for (1) and (2) is already established [for example see \textit{S. N. Kruzkov}, Math. USSR Sb. 10, 217-243 (1970); translation from Mat. Sb., Nov. Ser. 81(123), 228-255 (1970; Zbl 0202.112)]. For detailed bibliography, refer to \textit{Stanley H. Benton} [``The Hamilton-Jacobi equations. A global approach'' (1977; Zbl 0418.49001)]. This paper is concerned with the singularities of global generalized solutions.
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 4069398
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: uniformly convex
zbMATH Keywords: smooth initial data
zbMATH Keywords: generalized solution
zbMATH Keywords: existence
zbMATH Keywords: global generalized solution
zbMATH Keywords: singularities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520923
OpenAlex ID: W1546841982
DOI: 10.1215/KJM/1250520923
description: scientific article; zbMATH DE number 4069398