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Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbb R^2\). (English) Zbl 1042.37058
The authors study the long time behavior of solutions of the Navier-Stokes equations and the related vorticity equation on unbounded domains, with an approach based on ideas from the theory of dynamical systems. They prove that there exist finite-dimensional invariant manifolds in the phase space of these equations, and that all solutions in a neighborhood of the origin approach one of these manifolds with a rate that can be explicitly computed. Computing the asymptotics of solutions up to that order is reduced to determining the asymptotics of the system of ordinary differential equations that result by restriction of the original equations to the invariant manifolds. By a different method the analogous result for a bounded domain was obtained by C. Foiaş and J. C. Saut [Indiana Univ. Math. J. 33, 459–477 (1984; Zbl 0565.35087)].

MSC:
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B40 Asymptotic behavior of solutions to PDEs
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