A linear condition determining local or global existence for nonlinear problems

John M Neuberger, John W. Neuberger, James W Swift

Research output: Contribution to journalArticle

Abstract

Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57-67].

Original languageEnglish (US)
Pages (from-to)1361-1374
Number of pages14
JournalCentral European Journal of Mathematics
Volume11
Issue number8
DOIs
StatePublished - Aug 2013

Fingerprint

Local Existence
Global Existence
Nonlinear Problem
Semigroup
Systems of Partial Differential Equations
Autonomous Systems
System of Ordinary Differential Equations
Smoothness
Navier-Stokes Equations
Existence and Uniqueness
Nonlinear Systems
Sufficient
Testing
Necessary

Keywords

  • Lie generators
  • Local-global existence
  • Nonlinear semigroups

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A linear condition determining local or global existence for nonlinear problems. / Neuberger, John M; Neuberger, John W.; Swift, James W.

In: Central European Journal of Mathematics, Vol. 11, No. 8, 08.2013, p. 1361-1374.

Research output: Contribution to journalArticle

@article{5aa37565715e44229596f6f7a74d1ef1,
title = "A linear condition determining local or global existence for nonlinear problems",
abstract = "Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57-67].",
keywords = "Lie generators, Local-global existence, Nonlinear semigroups",
author = "Neuberger, {John M} and Neuberger, {John W.} and Swift, {James W}",
year = "2013",
month = "8",
doi = "10.2478/s11533-013-0249-1",
language = "English (US)",
volume = "11",
pages = "1361--1374",
journal = "Open Mathematics",
issn = "1895-1074",
publisher = "Walter de Gruyter GmbH & Co. KG",
number = "8",

}

TY - JOUR

T1 - A linear condition determining local or global existence for nonlinear problems

AU - Neuberger, John M

AU - Neuberger, John W.

AU - Swift, James W

PY - 2013/8

Y1 - 2013/8

N2 - Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57-67].

AB - Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This work is the second part of a program, the first part being [Neuberger J.W., How to distinguish local semigroups from global semigroups, Discrete Contin. Dyn. Syst. (in press), available at http://arxiv.org/abs/1109.2184]. Future work points to a distant goal for problems as in [Fefferman C.L., Existence and Smoothness of the Navier-Stokes Equation, In: The Millennium Prize Problems, Clay Mathematics Institute, Cambridge/American Mathematical Society, Providence, 2006, 57-67].

KW - Lie generators

KW - Local-global existence

KW - Nonlinear semigroups

UR - http://www.scopus.com/inward/record.url?scp=84878028911&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878028911&partnerID=8YFLogxK

U2 - 10.2478/s11533-013-0249-1

DO - 10.2478/s11533-013-0249-1

M3 - Article

VL - 11

SP - 1361

EP - 1374

JO - Open Mathematics

JF - Open Mathematics

SN - 1895-1074

IS - 8

ER -