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Investigation of the Foias–Saut Normalization in the Finite-Dimensional Case

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Abstract

We study the normalization obtained by C. Foias and J. C. Saut for the Navier–Stokes equations in the classical case of analytic ordinary differential equations in the neighborhood of a stationary point. We show that in this general (finite-dimensional) case, this normalization coincides with the distinguished normalization in the sense of A. D. Brjuno. Moreover, their approach leads to a spectral formula for the (inverse of the) distinguished normalizing map. In the particular case of an asymptotically stable, by linearization, stationary point, the Foias–Saut device gives the inverse of the already known link between the Lyapunov exponential expansion and normalization.

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REFERENCES

  1. Arnold, V. I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York.

    Google Scholar 

  2. Bibikov, Y. N. (1979). Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lect. Notes Math. 702. Springer, Berlin, Heidelberg, New York.

    Google Scholar 

  3. Brjuno, A. D. (1972). Analytical form of differential equations. Trans. Moscow Math. Soc. 25–26.

  4. Foias, C., and Saut, J. C. (1987). Linearization and normal form of the Navier-Stokes equations with potential forces. Ann. Inst H. Poincaré 4, 1–47.

    Google Scholar 

  5. Foias, C., and Saut, J. C. (1991). Asymptotic integration of Navier-Stokes equations with potential forces. Ind. Univ. Math. J. 40, 305–320.

    Google Scholar 

  6. Hartman, P. (1964). Ordinary Differential Equations, Wiley, New York.

    Google Scholar 

  7. Lefschetz, S. (1977). Differential Equations: Geometric Theory, Dover, New York.

    Google Scholar 

  8. Sell, G. (1985). Smooth linearization near a fixed point. Am. J. Math. 107, 1035–1091.

    Google Scholar 

  9. Sternberg, S. (1957). Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824.

    Google Scholar 

  10. Sternberg, S. (1961). Infinite Lie groups and the formal aspects of dynamical systems. J. Math. Mech. 10, 451–471.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania

    Gheorghe Minea

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  1. Gheorghe Minea
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Minea, G. Investigation of the Foias–Saut Normalization in the Finite-Dimensional Case. Journal of Dynamics and Differential Equations 10, 189–207 (1998). https://doi.org/10.1023/A:1022696614020

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  • DOI: https://doi.org/10.1023/A:1022696614020

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