Paper Info
Title
Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems
Abstract
We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equa- tions on the nite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is ne enough. We also provide an explicit estimate on the size of the mesh. More- over, 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 nite-dimensional space spanned by the ap- proximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation. Physical arguments indicate that the asymptotic behavior of the solutions of certain dissipative evolution equations can be described by only a nite number of degrees of freedom. Such equations include, but are not limited to, systems of reaction-diusion equations as well as systems that arise in fluid mechanics. See for example (24), (26). This assertion was rst rigorously proven by (10), in the case of the two-dimensional Navier-Stokes equations (NSE). Specically, they show that if the dierence between the rst N Fourier modes of any two solutions of the Navier-Stokes equations tends to zero, asymptotically in time as time goes to innity, for N suciently large, then the dierence between the two solutions tends to zero, in an appropriate norm, as time goes to innity. An explicit estimate on N, in terms of the Reynolds number, was rst given by Foias, Manley, Temam and Treve (8) and later improved by Jones and Titi (22). After this rigorous pioneering work of Foias and Prodi (10) several authors proved similar results for other degrees of freedom than the components of the Fourier modes. Foias and Temam (14) used the values of the solutions at nodes in the spatial domain as degrees of freedom. The work (15) and later (21) used local averages of the solutions on nite volumes as the degrees of freedom. Explicit estimates for these kinds of degrees of freedom were presented in (22). In this paper, we further extend the above results and show that for a large class of dissipative evolution equations, including the Navier-Stokes equations, there is a larger collection of determining sets of degrees of freedom (see Denition 1.1 below), than the ones mentioned above. The existence of such a collection was asserted in (13). Part of the results of this paper have been announced in (3).
Year
DOI
Venue
1997
10.1090/S0025-5718-97-00850-8
Math. Comput.
Keywords
Field
DocType
asymptotic degree,nonlinear dissipative system,reynolds number,dissipative system,exact solution,degree of freedom,fluid mechanics
Exact solutions in general relativity,Nonlinear system,Polynomial interpolation,Polynomial,Mathematical analysis,Dissipative system,Degrees of freedom (mechanics),Partial differential equation,Mathematics,Navier–Stokes equations
Journal
Volume
Issue
ISSN
66
219
0025-5718
Citations 
PageRank 
References 
7
2.16
0
Authors
3
Name
Order
Citations
PageRank
Bernardo Cockburn12796434.40
Don Jones2165.62
Edriss S. Titi38728.28