Determination of the solutions of the Navier-Stokes equations by a set of nodal values
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- by Ciprian Foias and Roger Temam PDF
- Math. Comp. 43 (1984), 117-133 Request permission
Abstract:
We consider the Navier-Stokes equations of a viscous incompressible fluid, and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions. The results presented here are exact results and not approximate ones: we show, in several cases, that the solutions are entirely determined by their values on a discrete set, provided this set contains enough points and these points are sufficiently densely distributed (in a sense described in the article). Two typical results are the following ones: two stationary solutions coincide if they coincide on a set sufficiently dense but finite; similarly if the large time behavior of the solutions to the Navier-Stokes equations is known on an appropriate discrete set, then the large time behavior of the solution itself is totally determined.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Jim Douglas Jr. and Todd Dupont, Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on $C^{1}$-piecewise-polynomial spaces. MR 0483559
- 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. MR 732571, DOI 10.1016/0167-2789(83)90297-X
- 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). MR 223716
- C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 339–368. MR 544257
- C. Foias and R. Temam, Asymptotic numerical analysis for the Navier-Stokes equations, Nonlinear dynamics and turbulence, Interaction Mech. Math. Ser., Pitman, Boston, MA, 1983, pp. 139–155. MR 755529
- Colette Guillopé, Comportement à l’infini des solutions des équations de Navier-Stokes et propriété des ensembles fonctionnels invariants (ou attracteurs), Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, ix, 1–37 (French, with English summary). MR 688020 J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationnary Navier-Stokes problem. Part I: Regularity of solutions and second-order spatial discretization." (To appear.)
- G. Iooss, Bifurcation of a $T$-periodic flow towards an $nT$-periodic flow and their non-linear stabilities, Arch. Mech. (Arch. Mech. Stos.) 26 (1974), 795–804 (English, with Russian and Polish summaries). MR 390549
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401 J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, Heidelberg, New York, 1972.
- Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
- 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. MR 764933
- P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for $2$D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 1–27. MR 768102, DOI 10.1002/cpa.3160380102
- Peter Constantin, Ciprian Foias, Oscar Manley, and Roger Temam, Connexion entre la théorie mathématique des équations de Navier-Stokes et la théorie conventionnelle de la turbulence, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 11, 599–602 (French, with English summary). MR 735689
- P. Constantin, C. Foias, and R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1985), no. 314, vii+67. MR 776345, DOI 10.1090/memo/0314
- R. Temam, Infinite-dimensional dynamical systems in fluid mechanics, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 431–445. MR 843630, DOI 10.1063/1.865959
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp. 43 (1984), 117-133
- MSC: Primary 35Q10; Secondary 76D05
- DOI: https://doi.org/10.1090/S0025-5718-1984-0744927-9
- MathSciNet review: 744927