A linear condition determining local or global existence for nonlinear problems

John Neuberger; John Neuberger; James Swift

Open Mathematics (2013)

  • Volume: 11, Issue: 8, page 1361-1374
  • ISSN: 2391-5455

Abstract

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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].

How to cite

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John Neuberger, John Neuberger, and James Swift. "A linear condition determining local or global existence for nonlinear problems." Open Mathematics 11.8 (2013): 1361-1374. <http://eudml.org/doc/269565>.

@article{JohnNeuberger2013,
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].},
author = {John Neuberger, John Neuberger, James Swift},
journal = {Open Mathematics},
keywords = {Nonlinear semigroups; Lie generators; Local-global existence; Polish space; local semigroup; Lie generator; eigenfunction; steepest descent method; local-global existence},
language = {eng},
number = {8},
pages = {1361-1374},
title = {A linear condition determining local or global existence for nonlinear problems},
url = {http://eudml.org/doc/269565},
volume = {11},
year = {2013},
}

TY - JOUR
AU - John Neuberger
AU - John Neuberger
AU - James Swift
TI - A linear condition determining local or global existence for nonlinear problems
JO - Open Mathematics
PY - 2013
VL - 11
IS - 8
SP - 1361
EP - 1374
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].
LA - eng
KW - Nonlinear semigroups; Lie generators; Local-global existence; Polish space; local semigroup; Lie generator; eigenfunction; steepest descent method; local-global existence
UR - http://eudml.org/doc/269565
ER -

References

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  1. [1] Dorroh J.R., Neuberger J.W., A theory of strongly continuous semigroups in terms of Lie generators, J. Funct. Anal., 1996, 136(1), 114–126 http://dx.doi.org/10.1006/jfan.1996.0023 
  2. [2] 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, available at http://www.claymath.org/millennium/Navier-Stokes_Equations/navierstokes.pdf Zbl1194.35002
  3. [3] Lie S., Differentialgleichungen, AMS Chelsea Publishing, Providence, 1967 
  4. [4] Neuberger J.W., Lie Generators for Local Semigroups, Contemp. Math., 513, American Mathematical Society, Providence, 2010 Zbl1203.22005
  5. [5] Neuberger J.W., Sobolev Gradients and Differential Equations, 2nd ed., Lecture Notes Math., 1670, Springer, Berlin, 2010 http://dx.doi.org/10.1007/978-3-642-04041-2 Zbl1203.35004
  6. [6] 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 

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