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Investigation of the Foias-Saut normalization in the finite-dimensional case. (English) Zbl 0970.34045

Summary: The author studies the normalization obtained by C. Foias and J. C. Saut [Ann. Inst. Henri PoincarĂ©, Anal. Non Lineaire 4, 1-47 (1987; Zbl 0635.35075) and Indiana Univ. Math. J. 40, No. 1, 305-320 (1991; Zbl 0739.35066)] for the Navier-Stokes equations in the classical case of analytic ordinary differential equations in the neighborhood of a stationary point. They show that in this general (finite-dimensional) case, this normalization coincides with the distinguished normalization in the sense of A. D. Brjuno [Trans. Moscow Math. Soc. 25(1971), 131-288 (1973; Zbl 0272.34018)]. 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.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
35Q30 Navier-Stokes equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
76D05 Navier-Stokes equations for incompressible viscous fluids
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