Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0025-5718-97-00850-8
MathSciNet review: 1415799
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. V. BABIN AND M. I. VISHIK, Attractors of partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk, 38, (1983), 133-187 (in Russian); Russian Math. Surveys, 38, 151-213 (in English). MR 84k:58133
  • 2. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, 1978. MR 58:25001
  • 3. B. COCKBURN, D.A. JONES, E.S. TITI, Degrés de liberté déterminants pour équations nonlinéaires dissipatives, C.R. Acad. Sci.-Paris, Sér. I, 321, (1995), 563-568.
  • 4. P. CONSTANTIN, C. FOIAS, Navier-Stokes Equations, University of Chicago Press, 1988. MR 90b:35190
  • 5. P. CONSTANTIN, C. FOIAS, B. NICOLAENKO, R. TEMAM, Integral Manifolds and Inertial Manifolds for Dissipative Partial differential Equations, Applied Mathematics Sciences, 70, Springer-Verlag, 1989. MR 90a:35026
  • [5.5] P. CONSTANTIN, C. FOIAS, R. TEMAM, On the large time Galerkin approximation of the Navier-Stokes equations, SIAM J. Numer. Anal. 21 (1984), 615-634. MR 85i:65145
  • 6. P. CONSTANTIN, C. FOIAS, R. TEMAM, On the dimension of the attractors in two-dimensional turbulence, Physica, D30, (1988), 284-296. MR 89j:76056
  • 7. C. FOIAS, I. KUKAVICA, Determining nodes for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eq., 7 (1995), 365-373. MR 96b:35190
  • 8. C. FOIAS, O.P. MANLEY, R. TEMAM, Y. TREVE, Asymptotic analysis of the Navier-Stokes equations, Physica, D9, (1983), 157-188. MR 85e:35097
  • 9. C. FOIAS, B. NICOLAENKO, G.R. SELL, R. TEMAM, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, (1988), 197-226. MR 90e:35137
  • 10. C. FOIAS, G. PRODI, Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension two, Rend. Sem. Mat. Univ. Padova, 39, (1967), 1-34. MR 36:6764
  • 11. C. FOIAS, G. SELL, R. TEMAM, Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eq., 73, (1988), 309-353. MR 89e:58020
  • 12. C. FOIAS, G. SELL, E.S. TITI, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynam. and Diff. Eq., 1, No. 2, (1989), 199-243. MR 90k:35031
  • 13. C. FOIAS, R. TEMAM, Asymptotic numerical analysis for the Navier-Stokes equations, Nonlinear Dynamics and Turbulence, Edit. by Barenblatt, Iooss, Joseph, Boston: Pitman Advanced Pub. Prog., 1983. CMP 16:17
  • 14. C. FOIAS, R. TEMAM, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comput., 43, (1984), 117-133. MR 85f:35165
  • 15. C. FOIAS, E.S. TITI, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4, (1991), 135-153. MR 92a:65241
  • 16. V. GIRAULT, P.A. RAVIART, Finite Element Approximations of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer Verlag, 1979. MR 83b:65122
  • 17. J.K. HALE, X.-B. LIN, G. RAUGEL, Upper Semicontinuity of Attractors for Approximations of Semigroups and Partial Differential Equations. Math. Comp., 50, (1988), 89-123. MR 89a:47093
  • 18. J.G. HEYWOOD, R. RANNACHER, Finite element approximation of the nonstationary Navier-Stokes problems. Part II Stability of solutions and error estimates uniform in time, SIAM J. Num. Anal., 23, (1986), 750-777. MR 88b:65132
  • 19. D.A. JONES, A.M. STUART, E.S. TITI, Persistence of invariant sets for dissipative evolution equations, (submitted).
  • 20. D.A. JONES, E.S. TITI, On the number of determining nodes for the 2D Navier-Stokes equations, J. Math. Anal. Appl., 168, (1992), 72-88. MR 93f:35179
  • 21. D.A. JONES, E.S. TITI, Determining finite volume elements for the $2D$ Navier-Stokes equations, Physica, D60, (1992), 165-174. MR 93j:35133
  • 22. D.A. JONES, E.S. TITI, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Math. J., 42, (1993), 875-887. MR 94k:35249
  • 23. D.A. JONES, E.S. TITI, $C^1$ Approximations of inertial manifolds for dissipative nonlinear equations, J. Diff. Eq., 127, (1996), 54-86. CMP 96:12
  • 24. R.H. KRAICHNAN, Interial ranges in two-dimensional turbulence, Phys. Fluids, 10, (1967), 1417-1423.
  • 25. I. KUKAVICA, On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity, 5, (1992), 997-1006. MR 93h:35108
  • 26. L. LANDAU, E. LIFSCHITZ, Fluid Mechanics, Addison-Wesley, New-York, 1953.
  • 27. J.L. LIONS, Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. MR 41:4326
  • 28. V.X. LIU, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Commun. Math. Phys., 158, (1993), 327-339. MR 94k:35251
  • 29. J. MALLET-PARET, G. SELL, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1, (1988), 805-866. MR 90h:58056
  • 30. R. TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983. MR 86f:35152
  • 31. R. TEMAM, Infinite Dimensional Dynamical Systems in Mechanics and Physics , Springer Verlag, New York, 1988. MR 89m:58056
  • 32. E.S. TITI, On a criterion for locating stable stationary solutions to the Navier-Stokes equations, Nonlinear Anal. TMA, 11, (1987), 1085-1102. MR 89f:35173
  • 33. E.S. TITI, Un critère pour l'approximation des solutions périodiques des équations de Navier-Stokes, C.R. Acad. Sci. Paris, 312, Série I. No. 1, (1991), 41-43. MR 92e:35130
  • 34. R. WAIT, A.R. MITCHELL, Finite Element Analysis and Applications, John Wiley & Sons, 1985. MR 87i:65192

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 35B40, 35Q30

Retrieve articles in all journals with MSC (1991): 35B40, 35Q30


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: https://doi.org/10.1090/S0025-5718-97-00850-8
Received by editor(s): July 27, 1995
Received by editor(s) in revised form: June 5, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society