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Estimating the number of asymptotic degrees of Freedom for Nonlinear Dissipative Systems

Authors: Bernardo Cockburn, Don A. Jones and Edriss S. Titi
Journal: Math. Comp. 66 (1997), 1073-1087
MSC (1991): Primary 35B40, 35Q30
MathSciNet review: 1415799
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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.

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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Don A. Jones
Affiliation: IGPP, University of California, Los Alamos National Laboratory, Mail Stop C305, Los Alamos, New Mexico 87544

Edriss S. Titi
Affiliation: Department of Mathematics, and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California 92697

Received by editor(s): July 27, 1995
Received by editor(s) in revised form: June 5, 1996
Article copyright: © Copyright 1997 American Mathematical Society