Determination of the solutions of the Navier-Stokes equations by a set of nodal values

Foias, Ciprian; Temam, Roger

doi:10.1090/S0025-5718-1984-0744927-9

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
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

Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires

Nonhomogeneous Boundary Value Problems and Applications

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