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

MSC:

 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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