Estimating the number of asymptotic degrees of freedom for nonlinear dissipatlve systems

Bernardo Cockburn, Don A. Jones, Edriss S. Titi

Research output: Contribution to journalArticle

53 Scopus citations

Abstract

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipat ve evolution equations on the finite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is fine enough. We also provide an explicit estimate on the size of the mesh. Moreover, we show that if the evolution equation has an inertial manifold, then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite-dimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation.

Original languageEnglish (US)
Pages (from-to)1073-1087
Number of pages15
JournalMathematics of Computation
Volume66
Issue number219
DOIs
StatePublished - Jul 1997
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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