Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)2775-2788
Journal / PublicationPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume366
Issue number1876
Publication statusPublished - 13 Aug 2008
Externally publishedYes

Abstract

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier-Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/t or some positive power of 1/t). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally. Moreover, computation of the solution of the IE over an interval [0,p0] provides sharper control of its p →∞ behaviour. Preliminary numerical computations give encouraging results. © 2008 The Royal Society.

Research Area(s)

  • Borel summation, Smooth solution, Three-dimensional Navier-Stokes

Citation Format(s)

Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation. / Costin, Ovidiu; Luo, Guo; Tanveer, Saleh.
In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 366, No. 1876, 13.08.2008, p. 2775-2788.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review