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Navier and Stokes meet Poincaré and Dulac. (English) Zbl 1457.35024

This nice survey discuss asymptotic behavior of solutions of the Navier-Stokes system with potential forces either with space periodic conditions or in bounded domains, beginning with lower bounds on solutions and the limit of the ratio of the enstrophy over the energy, and culminating in the normal form of those equations in the sense generalizing Poincaré and Dulac theory of normal forms of (ordinary) differential equations. On the top of the announcements (and sometimes sketches or ideas of the proofs) of results, their clear geometric interpretations (invariant manifolds etc.) and historical miscellanea are given. Moreover, open problems are listed, and an extensive set of references is provided.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
37G05 Normal forms for dynamical systems
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35C20 Asymptotic expansions of solutions to PDEs
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