## Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems

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- by Bernardo Cockburn, Don A. Jones and Edriss S. Titi PDF
- Math. Comp.
**66**(1997), 1073-1087 Request permission

## Abstract:

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative 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.## References

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## Additional Information

**Bernardo Cockburn**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: cockburn@math.umn.edu
**Don A. Jones**- Affiliation: IGPP, University of California, Los Alamos National Laboratory, Mail Stop C305, Los Alamos, New Mexico 87544
- Email: dajones@kokopelli.lanl.gov
**Edriss S. Titi**- Affiliation: Department of Mathematics, and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California 92697
- MR Author ID: 172860
- Email: etiti@math.uci.edu
- Received by editor(s): July 27, 1995
- Received by editor(s) in revised form: June 5, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp.
**66**(1997), 1073-1087 - MSC (1991): Primary 35B40, 35Q30
- DOI: https://doi.org/10.1090/S0025-5718-97-00850-8
- MathSciNet review: 1415799