A linear condition determining local or global existence for nonlinear problems

• Autorzy: John Neuberger , John Neuberger , James Swift
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

Nonlinear semigroups; Lie generators; Local-global existence

Kategorie tematyczne

• 35A01: Existence problems: global existence, local existence, non-existence

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 8
• Strony: 1361-1374

Daty

wydano

2013-08-01

online

2013-05-22

Twórcy

autor

John Neuberger

• Northern Arizona University

autor

John Neuberger

• University of North Texas

autor

James Swift

• Northern Arizona University

Bibliografia

• [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] 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
• [3] Lie S., Differentialgleichungen, AMS Chelsea Publishing, Providence, 1967
• [4] Neuberger J.W., Lie Generators for Local Semigroups, Contemp. Math., 513, American Mathematical Society, Providence, 2010
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-013-0249-1