Asymptotic behavior, as \(t\to +\infty\) of solutions of Navier-Stokes equations and nonlinear spectral manifolds. (English) Zbl 0565.35087

The paper is concerned with the decay in time of the solutions of Navier- Stokes equations with zero volumic forces. Here A denotes the Stokes operator and S(t) the nonlinear semigroup delivered by the Navier-Stokes equations in the open set \({\mathcal R}\) in \(V=D(A^{1/2})\) of initial data for regular solutions on [0,\(\infty).\)
The main result states that for each \(u_ 0\) in \({\mathcal R}\) there exist an eigenvalue \(\Lambda =\Lambda (u_ 0)\) of A and a non-zero eigenvector \(U_{\Lambda}\) associated to it such that \(e^{\nu \Lambda t}S(t)u_ 0\to U_{\Lambda}\) for \(t\to \infty\) in V (like in the case of the (linear) Stokes equations, where to each \(u_ 0\) in V corresponds such an eigenvalue \(\Lambda^{lin}(u_ 0)\), the smallest for which there is a non-zero Fourier coefficient in the decomposition of \(u_ 0)\). Moreover, the terms of an asymptotic expansion of \(S(t)u_ 0\) as \(t\to \infty\), modulo \(o(e^{-2\nu \Lambda t})\), are given. Next one shows that the invariant ”spectral” sets \(M_{\lambda}=\{u_ 0\in {\mathcal R}| \Lambda (u_ 0)\geq \lambda \},\) defined for each eigenvalue \(\lambda\) of A, are smooth analytic manifolds around the origin, the tangent space at 0 being the space of finite codimension \(M_{\lambda}^{lin}\), similarly defined for the Stokes equations.
Reviewer: G.Minea


35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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