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Estimating the number of asymptotic degrees of Freedom for Nonlinear Dissipative Systems
Author(s):
Bernardo
Cockburn;
Don
A.
Jones;
Edriss
S.
Titi.
Journal:
Math. Comp.
66
(1997),
1073-1087.
MSC (1991):
Primary 35B40, 35Q30
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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
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
Email:
etiti@math.uci.edu
DOI:
10.1090/S0025-5718-97-00850-8
PII:
S 0025-5718(97)00850-8
Received by editor(s):
July 27, 1995
Received by editor(s) in revised form:
June 5, 1996
Copyright of article:
Copyright
1997,
American Mathematical Society
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