Cockburn, Bernardo; Jones, Don A.; Titi, Edriss S. Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems. (English) Zbl 0866.35091 Math. Comput. 66, No. 219, 1073-1087 (1997). Summary: 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. Cited in 43 Documents MSC: 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs Keywords:long-time behavior; Navier-Stokes equations; dissipative evolution equations; finite-dimensional space of interpolant polynomials; dynamics of the evolution equation; dynamics of the projection PDFBibTeX XMLCite \textit{B. Cockburn} et al., Math. Comput. 66, No. 219, 1073--1087 (1997; Zbl 0866.35091) Full Text: DOI References: [1] A. V. Babin and M. I. Vishik, Attractors of evolution partial differential equations and estimates of their dimension, Uspekhi Mat. Nauk 38 (1983), no. 4(232), 133 – 187 (Russian). · Zbl 0541.35038 [2] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058 [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] Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. · Zbl 0687.35071 [5] P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, vol. 70, Springer-Verlag, New York, 1989. · Zbl 0683.58002 [6] P. Constantin, C. Foias, and R. Temam, On the large time Galerkin approximation of the Navier-Stokes equations, SIAM J. Numer. Anal. 21 (1984), no. 4, 615 – 634. · Zbl 0551.76021 [7] P. Constantin, C. Foias, and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Phys. D 30 (1988), no. 3, 284 – 296. · Zbl 0658.58030 [8] Ciprian Foias and Igor Kukavica, Determining nodes for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations 7 (1995), no. 2, 365 – 373. · Zbl 0835.35067 [9] C. Foias, O. P. Manley, R. Temam, and Y. M. Trève, Asymptotic analysis of the Navier-Stokes equations, Phys. D 9 (1983), no. 1-2, 157 – 188. · Zbl 0584.35007 [10] C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. (9) 67 (1988), no. 3, 197 – 226. · Zbl 0694.35028 [11] C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1 – 34 (French). · Zbl 0176.54103 [12] Ciprian Foias, George R. Sell, and Roger Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309 – 353. · Zbl 0643.58004 [13] Ciprian Foias, George R. Sell, and Edriss S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dynam. Differential Equations 1 (1989), no. 2, 199 – 244. · Zbl 0692.35053 [14] 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 · Zbl 0555.76030 [15] Ciprian Foias and Roger Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comp. 43 (1984), no. 167, 117 – 133. · Zbl 0563.35058 [16] Ciprian Foias and Edriss S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity 4 (1991), no. 1, 135 – 153. · Zbl 0714.34078 [17] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. · Zbl 0413.65081 [18] Jack K. Hale, Xiao-Biao Lin, and Geneviève Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988), no. 181, 89 – 123. · Zbl 0666.35013 [19] John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal. 23 (1986), no. 4, 750 – 777. · Zbl 0611.76036 [20] D.A. JONES, A.M. STUART, E.S. TITI, Persistence of invariant sets for dissipative evolution equations, (submitted). · Zbl 0909.35061 [21] Don A. Jones and Edriss S. Titi, On the number of determining nodes for the 2D Navier-Stokes equations, J. Math. Anal. Appl. 168 (1992), no. 1, 72 – 88. · Zbl 0773.35050 [22] Don A. Jones and Edriss S. Titi, Determining finite volume elements for the 2D Navier-Stokes equations, Phys. D 60 (1992), no. 1-4, 165 – 174. Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). · Zbl 0778.35084 [23] Don A. Jones and Edriss S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993), no. 3, 875 – 887. · Zbl 0796.35128 [24] 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 [25] R.H. KRAICHNAN, Interial ranges in two-dimensional turbulence, Phys. Fluids, 10, (1967), 1417-1423. [26] Igor Kukavica, On the number of determining nodes for the Ginzburg-Landau equation, Nonlinearity 5 (1992), no. 5, 997 – 1006. · Zbl 0758.34014 [27] L. LANDAU, E. LIFSCHITZ, Fluid Mechanics, Addison-Wesley, New-York, 1953. [28] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). · Zbl 0189.40603 [29] Vincent Xiaosong Liu, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Comm. Math. Phys. 158 (1993), no. 2, 327 – 339. · Zbl 0790.35085 [30] John Mallet-Paret and George R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc. 1 (1988), no. 4, 805 – 866. · Zbl 0674.35049 [31] Roger Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. · Zbl 0522.35002 [32] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. · Zbl 0662.35001 [33] Edriss Saleh Titi, On a criterion for locating stable stationary solutions to the Navier-Stokes equations, Nonlinear Anal. 11 (1987), no. 9, 1085 – 1102. · Zbl 0642.76031 [34] Edriss Saleh Titi, Un critère pour l’approximation des solutions périodiques des équations de Navier-Stokes, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 1, 41 – 43 (French, with English summary). · Zbl 0725.35010 [35] R. Wait and A. R. Mitchell, Finite element analysis and applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. · Zbl 0577.65093 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.